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Investing in Great Libyan Jamahiriya
Thursday 5th - Friday 6th July 2001 * Royal Garden Hotel, London
Under the Patronage of The General People's Committee For Economy and Trade
Visit http://www.ibcenergy.com/libya/?source=eq176e1c2
Dear Colleague,
I'm delighted to attach a copy of the delegation confirmed by the Libyan government and the updated programme agenda. As you will see, we have confirmation of a very high level of governmental participation including:
* H.E. Dr Abdusalam Jewer, Secretary for General People's Committee for Economy and Trade and Secretary, Management Committee of Libyan Foreign Investment Board
* H.E. Hassuna Eshawish, Undersecretary for Information & Culture, General People's Committee for Foreign Liaison and International Co-operation
* H.E. Dr Khalid Faraj Al-Zantuti, Undersecretary for the Technical Co-operation, General People's Committee for African Unity
* H.E. Mohamed Abu Alqassim Azwai, Ambassador, Libyan People's Bureau, London
We have also received a letter of support from H.E. Mohamed Abu Alqassim Azwai, Ambassador, Libyan People's Bureau, London. Visit our website at http://www.ibcenergy.com/libya/?source=eq176e1c2 to view the letter.
Also, please note we are running a pre-conference workshop on Wednesday 4th July entitled Structuring and Financing Business Transactions when Investing in Libya in association with Pitmans and British Arab Commercial Bank.
View the workshop and conference programme updates on the attached document or visit http://www.ibcenergy.com/libya/?source=eq176e1c2.
For registration or further information, contact Patricia Harris on +44 (0) 1932 893857.
Sincerely,
Charlotte Hunt
Senior Conference Producer
IBC Global Conferences
charlotte.hunt@informa.com
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Abstract
Significant epidemiological and clinical evidence has emerged that suggests Alzheimer’s disease (AD) can be added to the list of chronic illnesses that are primarily caused by modern diets and lifestyles at odds with human physiology. High intakes of refined carbohydrates insufficient physical activity, suboptimal sleep quantity and quality, and other factors that may contribute to insulin resistance combine to create a perfect storm of glycation and oxidative stress in the brain. Specific neurons lose the ability to metabolise and harness energy from glucose, ultimately resulting in neuronal degeneration and death. Simultaneously, chronic peripheral hyperinsulinaemia prevents ketogenesis, thus depriving struggling neurons of a highly efficient alternative fuel substrate. The intimate association between type 2 diabetes and AD suggests that they have common underlying causes, namely insulin resistance and perturbed glucose metabolism. Preclinical evidence of AD is detectable decades before over symptoms appear, indicating that AD progresses over time, with observable signs manifesting only after the brain’s compensatory mechanisms have failed and widespread neuronal atrophy begins to interfere with cognition and performance of daily life tasks. That dietary and environmental triggers play pivotal roles in causing AD suggests that nutrition and lifestyle based interventions may hold the key to ameliorating or preventing this debilitating condition for which conventional pharmaceutical treatments are largely ineffective. Results from small scale clinical studies indicate that dietary and lifestyle strategies may be effective for reversing dementia and cognitive impairment. Increased research efforts should be dedicated towards this promising avenue in the future. |
292 So.2d 773 (1974)
Mary Boslaugh EATON
v.
GENERAL ACCIDENT GROUP and L. M. Berry & Co.
No. 6153.
Court of Appeal of Louisiana, Fourth Circuit.
April 9, 1974.
*774 Pierre F. Gaudin, Gretna, for plaintiff-appellant.
Dufour, Levy, Marx, Lucas & Osborne, Michael Osborne, New Orleans, for defendants-appellees.
Before SAMUEL REDMANN and LEMMON, JJ.
SAMUEL, Judge.
Plaintiff filed this suit against General Accident Group and its insured, L. M. Berry & Company, for compensation benefits at the rate of $49 per week from June 2, 1971 for the duration of her disability. She also seeks penalties and attorney's fees based on the defendants' alleged arbitrary and capricious refusal to pay compensation benefits. Defendants answered, admitting plaintiff's employment, denying her disability, and affirmatively alleging compensation was not paid because she failed to prove an accident and refused to be examined by a physician of defendants' choice.
After a trial on the merits, judgment was rendered in favor of plaintiff and against both defendants, in solido, awarding her compensation benefits at the rate of $49 per week from July 1, 1971 until February 18, 1972. However, the judgment denied and dismissed plaintiff's claim for penalties and attorney's fees. Plaintiff has appealed. In this court she contends she is entitled to benefits for total and permanent disability and to penalties and attorney's fees. Defendants have answered the appeal seeking dismissal of the suit against them.
The employer, L. M. Berry & Company, handles yellow page listings and advertisements for South Central Bell Telephone & Telegraph Company. Plaintiff's work required that she sit in her employer's office and make telephone solicitations of yellow page advertisements. She used several telephone directories placed in a book case next to her desk and she could reach these directories without leaving her chair. Typing was not a part of her job. Her supervisor estimated that 99% of her work consisted of telephone solicitations and paper work. On rare occasions she personally called on a customer. In making such calls she used her car and carried one or two telephone directories, and possibly a briefcase. Outside solicitation was not a requirement of her employment, and had little effect, if any, on her income.
On June 2, 1971 plaintiff fell at work in the office and appeared to be badly shaken by the fall. She apparently struck her knee and then landed on her buttocks, causing a jolting injury to her back. She continued to work that day and worked regularly until June 9. She was absent from work June 9 through June 11 and also was absent on June 16 and 17. Her employer's records indicate, and she did report, these five days of absence resulted from an attack of influenza. From June 18 through September 10, 1971 plaintiff was carried on her employer's records as being on leave of absence. She never returned to work and no compensation payments were ever made to her.
The first physician consulted by plaintiff was Dr. Charles S. Wyckoff, an osteopath. On June 8, 1971, she complained to him of pain in the back, legs and knee. The only positive findings by Dr. Wyckoff were spasm of the musculature of her back and traumatic fibrositis, which he attributed to her fall. She again saw him on June 9, and his ultimate conclusion was traumatic fibrositis caused by the accident. However, Dr. Wyckoff speculated that this condition should have cleared in four to five weeks. We note that plaintiff did return to work after seeing Dr. Wyckoff.
Later plaintiff telephoned Dr. Wyckoff with regard to her condition. Upon *775 learning she had 101 of fever he recommended she see her family physician. Following his advice she saw that physician, Dr. Swan Ward, on June 15, 1971. Dr. Ward's primary concern was plaintiff's respiratory problems, but he also noted complaints of pain by her in the lumbar-sacral area. His examination in this area was basically negative, with no evidence of bruises or muscle spasm. He saw her on June 18 and June 22. On the latter date he found her asymptomatic and discharged her from further treatment.
Plaintiff then drove herself by automobile to her sister's home in Florida. On July 1, 1971 she was examined by Dr. James H. Pollock, an orthopedic surgeon, in Boynton Beach, Florida. Dr. Pollock diagnosed acute neck and back strain and prescribed medicine and physical exercises to alleviate her condition. On July 8 she had improved, but continued to complain of neck soreness when she drove her automobile. On July 15 she complained to Dr. Pollock of a recurrence of her neck and shoulder pain while she was swimming. Dr. Pollock was of the opinion plaintiff was making slow but progressive improvement and concluded her injuries would interfere with her working activities at that time. He estimated her disability as 5% of her body as a whole, which should gradually resolve.
Plaintiff next was seen by Dr. Blaise Salatich, in New Orleans. Her first visit to him was on September 7, 1971. At that time he found palpable muscle spasm of the neck and back indicating a lumbosacral and cervical injury. He advised ultrasonic treatments three times a week. From September 7, 1971 to February 19, 1973 plaintiff saw Dr. Salatich and/or received treatments 129 times. Dr. Salatich's opinion was that she is totally and permanently disabled from returning to her usual occupation as a telephone sales person for the defendant employer.
On February 18, 1972, Dr. Irvin Cahen examined plaintiff at the request of the defendants. Dr. Cahen found the plaintiff to be completely asymptomatic and was of the firm opinion she was able to perform the usual duties of her occupation as of that date. Plaintiff was also examined on November 27, 1972 by Dr. H. R. Soboloff, again at the request of the defendants. Dr. Soboloff's findings were essentially the same as those of Dr. Cahen. Dr. Soboloff found no objective symptoms and concluded plaintiff could return to her employment.
The issue of the extent of the plaintiff's disability is easily resolved. Prior to July 1, 1971 she was seen only twice by a local osteopath and several times thereafter by her family physician. Both of these physicians concluded she did sustain some injury from her fall, but the diagnosis of each was that her disability would be of a limited duration. On July 1, 1971 plaintiff was first seen by Dr. Pollock, an orthopedic surgeon, whom she continued to visit until August 30. Dr. Pollock was of the opinion, and there is no contradictory evidence in the record, that she was unable to return to work. A week later, she began treatment by Dr. Salatich, a physician locally recognized as a specialist in the field of orthopedic surgery. Dr. Salatich's testimony is uncontradicted for the period beginning September 7, 1971 and ending February 18, 1972 when the plaintiff was examined by Dr. Cahen, a Board certified orthopedic surgeon. Dr. Cahen's negative findings were substantiated by Dr. Soboloff's examination of November 27, 1972.
It is obvious that, on the basis of the record before him the trial court had no alternative but to award compensation from the time plaintiff first saw an orthopedic specialist until the time of Dr. Cahen's examination and negative findings on February 18, 1972. During that period there is uncontradicted medical evidence showing that, because of injuries received in the fall, she could not return to her duties with the defendant employer. However, after that date the trial court chose *776 to give more weight to the findings of Dr. Cahen than to the findings of Dr. Salatich.
Plaintiff contends the testimony of the treating physician, Dr. Salatich, is entitled to greater weight than that of the evaluating physician, Dr. Cahen, whose findings were later substantiated by Dr. Soboloff but who examined plaintiff only once. Defendants contend the latter two physicians are more reliable because of the alleged fact that the accuracy and credibility of Dr. Salatich's medical testimony frequently has been the subject of criticism by the courts.[1]
While the treating physician ordinarily is in a better position to determine the presence or extent of residual disability, this is only one consideration to be taken into account for the purpose of resolving conflicting medical opinions. Certainly, it alone is not conclusive. After a reading of the record in its entirety, we cannot conclude the trial judge committed error in deciding to reject the testimony of Dr. Salatich and accept the testimony of Dr. Cahen.
Our decision is not based solely upon the credibility of conflicting medical testimony. Plaintiff herself admitted that at the time of trial she was driving her car, visiting friends, and going to the grocery store at least twice a week. She also admitted she carried her groceries without assistance from her car into her home. Physical activity involved in driving her car and carrying groceries requires as much as, if not more than, the same amount of stress as plaintiff's position with the defendant firm. In view of plaintiff's own testimony and in view of the credible medical testimony in the record, we affirm the trial court's award of compensation benefits.
The other issue for our consideration is plaintiff's contention that she should have been awarded penalties and attorney's fees pursuant to LSA-R.S. 22:658.[2] On October 8, 1971, an attorney for plaintiff made formal demand upon the defendant for workmen's compensation benefits. On that date he addressed a letter to the claims manager of the defendant insurer to which he attached a copy of Dr. Salatich's medical report. This report stated without equivocation that plaintiff was then disabled. Again on October 29, 1971 plaintiff's attorney made formal demand on the claims manager to institute compensation payments. To this letter was attached medical authorizations executed by plaintiff, apparently for the purpose of allowing the defendant insurer to obtain information verifying plaintiff's physical condition.
Neither defendant made any compensation payment. This suit was filed on January 9, 1972 and only after suit was filed did the defendants request that the plaintiff be examined. It was at this time that Dr. Cahen performed his examination of February 18, 1972, discussed above.
The only defense offered by defendants to the claim for penalties and attorney's fees is the argued lack of credibility of Dr. Salatich and therefore of the report which was attached to the letter from plaintiff's attorney on October 8, 1971. Some mention is also made of the employer's records indicating that plaintiff was absent from *777 work on June 9, 10, 11, 16 and 17, 1971 because of an attack of influenza. However, the plaintiff's fall on June 2, 1971 was seen by her supervisor and by at least one other coworker, and was duly noted.
It cannot be questioned that the letter from plaintiff's attorney forwarding Dr. Salatich's medical report presented the defendants with their only medical opinion regarding plaintiff's condition, and this opinion supported her claim. In Bushnell v. Southern Farm Bureau Casualty Ins. Co.,[3] a similar question was presented. The court decided that where an injured employee makes demand for workmen's compensation benefits and the only medical evidence available supports her claim, the party upon whom demand is made cannot justify failure to pay because of a mere suspicion that the claim was not valid. In the Bushnell case plaintiff provided the defendants with a medical report, but the defendants chose to ignore the demand because their suspicion was aroused by the timing of the demand, since it was not received until after the claimant had been fired. The court reasoned that suspicion alone was not sufficient to withhold benefits since the defendants had no medical basis for their refusal and since the only medical evidence available at that time was favorable to the employee's claim.
While the courts have not afforded great weight to the testimony of Dr. Salatich on some occasions, he is regarded as an expert in the field of orthopedics and the courts have accepted, and do accept, his testimony as an expert in that specialty.
The statute under which penalties and attorney's fees may be imposed against an insurer is LSA-R.S. 22:658, which reads:
"All insurers issuing any type of contract other than those specified in R.S. 22:656 and 22:657 shall pay the amount of any claim due any insured including any employee under Chapter 10 of Title 23 of the Revised Statutes of 1950 within sixty days after receipt of satisfactory proofs of loss from the insured, employee or any party in interest. Failure to make such payment within sixty days after receipt of such proofs and demand therefor, when such failure is found to be arbitrary, capricious, or without probable cause, shall subject the insurer to a penalty, in addition to the amount of the loss, of 12% damages on the total amount of the loss, payable to the insured, or to any of said employees, together with all reasonable attorney's fees for the prosecution and collection of such loss, or in the event a partial payment or tender has been made, 12% of the difference between the amount paid or tendered and the amount found to be due and all reasonable attorney's fees for the prosecution and collection of such amount. Provided, that all losses on policies covering automobiles, trucks, motor propelled vehicles and other property against fire and theft, the amount of the penalty in each of the above cases shall be 25% and all reasonable attorney's fees." LSA-R.S. 22:658.
With regard to this state, the Louisiana Supreme Court made the following statement in Tullier v. Ocean Accident and Guarantee Corp.:[4]
"Under the basic statute, it is clear that penalties arise only from an arbitrary and capricious refusal by the insurer to pay a just claim after sixty days notice of the loss. The statute does not authorize the assessment of penalties merely because the insurer is the unsuccessful litigant."
While defendants are correct in their citation of Tullier to show that merely losing a workmen's compensation case does not carry with it an award of penalties and attorney's fees, LSA-R.S. 22:658 clearly provides that failure to make payment within 60 days after proper demand will *778 result in penalties provided the failure to pay is found to be arbitrary, capricious, or without probable cause.
In the case before us, the defendant insurer was furnished with demand and a medical report from a physician who has been recognized as an expert in the field of orthopedics. The report was favorable to the claimant. Without any contradictory medical evidence, it is not within the realm of the defendants to determine for themselves Dr. Salatich's credibility or the weight which is to be given to his opinion. Faced with the claim and the supporting positive report, it was the duty of the defendant insurer to take proper steps to investigate the claim in order to verify or negate its validity. This could have been easily done by virtue of the medical authorizations which the plaintiff freely and voluntarily furnished.
Penalty provisions such as that relied upon by the plaintiff have as their basic purpose the discouragement of employers and insurers from assuming attitudes of indifference to the difficulties incurred by injured employees.[5] In the present case, the defendants did nothing in response to the plaintiff's claim from its receipt immediately after October 8, 1971 until after plaintiff filed suit on January 9, 1972. Under all of these circumstances, failure to make any compensation payments within sixty days after receipt of the claim and medical report was arbitrary, capricious, and without probable cause under the statutory provisions. Consequently, an award in accordance with the statute will be made against the defendant insurer.
Revised Statute 22:658 itself provides that when penalties are assessed against an insurer, the amount of said penalties shall be 12% of the total amount of the loss, payable to the employee, together with reasonable attorney's fees for the prosecution and collection of the loss. The trial court award, which we affirm, was for 33 weeks at $49 per week, or a total of $1,617. Twelve percent of that sum is $194.04, in this case the penalty provided for by the statute.
Regarding attorney's fees, the record in the case is quite extensive and much testimony was adduced in the trial court in addition to the taking of the testimony of three physicians by deposition. After weighing the considerations available to us, including the amount recovered, our conclusion is that $700 would be a reasonable attorney's fee. Accordingly, we will assess the penalty and the attorney's fee, a total of $894.04, against the defendant insurer.
For the reasons assigned, the judgment appealed from is amended so as to add the following paragraph:
It is further ordered, adjudged and decreed that there be additional judgment in favor of the plaintiff, Mary Boslaugh Eaton, and against the defendant insurer, General Accident Group, in the full sum of $894.04.
As thus amended, and in all other respects, the judgment appealed from is affirmed; costs in the trial court to be paid by both defendants; costs in this court to be paid by the defendant insurer, General Accident Group.
Amended and affirmed.
NOTES
[1] Defendants cite: Barrere v. Commercial Union Insurance Group, La.App., 195 So.2d 461; Snell v. Intercoastal Airways, Inc., La. App., 165 So.2d 878; Herbert v. American General Insurance Company, La.App., 150 So. 2d 627; Baturo v. Employers' Liability Assurance Corp., La.App., 149 So.2d 627; Norman v. Standard Supply & Hardware Co., Inc., La.App., 126 So.2d 776; Easterling v. J. A. Jones Construction Company, La.App., 115 So.2d 888; Smith v. W. Horace Williams Company, La.App., 84 So.2d 223; Vogts v. Schwegmann, La.App., 56 So.2d 177; Noble Drilling Corporation v. Donovan, D.C., 266 F.Supp. 917.
[2] We note that penalties and attorney's fees can be imposed only against General Accident, the insurer. They cannot be imposed against the employer where, as here, the employer's liability for workmen's compensation claims is covered by insurance. LSA-R.S. 23:1201.2; Butler v. Peter Kiewit & Sons Company, La.App., 281 So.2d 467.
[3] La.App., 271 So.2d 267.
[4] 243 La. 921, 927, 928, 148 So.2d 601, 603, 604.
[5] See, for example, Poindexter v. South Coast Corporation, La.App., 204 So.2d 615.
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Trump to attend Nasa launch in Florida
NASA is resuming launches from US soil in partnership with SpaceX. The two astronauts in training
Walheim's interview came just before a historic launch in which a private company - SpaceX - will take astronauts into space.
"A new era of human spaceflight is set to begin", NASAsaid in a statement.
In ordinary times, the beaches and roads along Florida's Space Coast would be packed with people eager to witness the first astronaut launch from Florida in nine years. It will also be only the fifth time in US history that NASA astronauts will fly a new spacecraft for the first time. In the interim, NASA has had to pay the Russians tens of millions of dollars per seat to transport USA astronauts in Soyuz capsules. Boeing's hyped-up, unmanned test flight for its Starliner spacecraft in late December, however, infamously did not go as planned as the spacecraft never ended up making it to the ISS. Initially meant to last just one day, the FRR began on Thursday (May 21st), extended to the end of the business day, and continued into Friday (May 22nd).
The mission, NASA's first crewed launch from USA soil since 2011, will launch from Pad 39A of Kennedy Space Center in Florida on Wednesday (May 27) at 4:33 p.m. EDT (2033 GMT).
"At the end we got to a go", NASA administrator Jim Bridenstine told reporters by video of the meticulous Flight Readiness Review, which provided the go-ahead.
For Demo-2, the successful FRR is a crucial pathfinding step to confirming launch, however not the last. This certification and regular operation of Crew Dragon will enable NASA to continue the important research and technology investigations taking place onboard the station, which benefits people on Earth and lays the groundwork for future exploration of the Moon and Mars starting with the agency's Artemis program, which will land the first woman and the next man on the lunar surface in 2024. Reed went on to say that "there'll be constant vigilance and watching of the data and observations as we go through the mission".
As the final flight test for SpaceX, this mission will validate the company's crew transportation system, including the launch pad, rocket, spacecraft, and operational capabilities.
NASA astronauts are scheduled to fly on SpaceX's Crew Dragon spacecraft on May 27, lifting off from the Kennedy Space Center complex at Cape Canaveral.
Jurczyk cited the example of the SpaceX Crew Dragon capsule's parachute system, which had to be reworked and retested to satisfy NASA's safety margin requirements.
The mission passed its Flight Readiness Review, the final major hurdle before lift-off, on Friday. (Gregg Newton/AFP via Getty Images) A Falcon 9 SpaceX heavy rocket lifts off from pad 39A at the Kennedy Space Center in Cape Canaveral, Fla on February 6, 2018. The dry dress rehearsal will encompass every aspect of launch day, from putting on the spacesuits to climbing into the Crew Dragon capsule. It is expected to end just before propellant loading would begin in the countdown. The commercial crew program that supported development of Crew Dragon started in the administration of President Barack Obama, and built on the commercial cargo program started during the administration of President George W. Bush. (SpaceX via AP) The uncrewed SpaceX Crew Dragon spacecraft, with its nose cone open to expose the docking mechanism, approaches the International Space Station's Harmony module on March 4, 2019.
Rochester-area job losses soar in April
Monthly payroll employment estimates are preliminary and subject to revision as more data become available the following month. The number of private sector jobs in NY state is based on a payroll survey of 18,000 NY businesses conducted by the U.S.
BC announces 12 new COVID-19 cases, death toll rises to 152
There were bigger changes in case numbers within health regions because of a statistical change in the way cases are counted. In addition, 60 percent of men have been hospitalized, and 64 percent of people who have needed critical care are male.
Arkansas unemployment rate soars above 10% in April
NY and California, two states greatly affected by the pandemic, had jobless rates of 14.5 percent and 15.5 percent respectively. Seven states recorded unemployment below 10 percent, MarketWatch reports , with CT the lowest with only 7.9 percent.
FBI to review Flynn investigation for misconduct
Judge Sullivan so far has refused, a rare act of defiance by a judge in response to a Justice Department request. The statement said the FBI's review will "complement" that work, and Jensen's examination will take priority.
No new cases of COVID-19 today
Monday, May 25, the Case Counts Dashboard on coronavirus.iowa.gov will not reflect accurate counts during the maintenance period. Numbers from each individual county will be updated below as soon as NewsWatch 12 receives their latest figures.
Former rugby player Allan Makaka dies in grisly road crash
Former Kenyan Sevens star Allan Makaka has died following a road traffic accident, the Kenyan Rugby Union have confirmed. He is said to have rammed into a trailer. "I heard a bang and I went to check out to find the truck had stopped". |
Steigerwald Lake National Wildlife Refuge
Located on the Columbia River, east of Vancouver, Washington, the Steigerwald Lake National Wildlife Refuge consists of historic riverine flood plain habitat, semi-permanent wetlands, cottonwood-dominated riparian corridors, pastures, and remnant stands of Oregon white oak.
The refuge lies partly within the Columbia River Gorge National Scenic Area, and has been designated as the location for a "Gateway to the Gorge" visitor center. This facility is currently in the planning stage with a portion of the construction funds already secured.
The Washington Department of Transportation has estimated that this facility may be used by as many as 100,000 visitors annually, providing the Service with one of the best outreach opportunities in the Pacific Northwest. The refuge also serves as the operational headquarters for the Pierce National Wildlife Refuge.
References
External links
Steigerwald Lake National Wildlife Refuge - U.S. Fish and Wildlife Service
Steigerwald Lake National Wildlife Refuge Map - U.S. Fish and Wildlife Service
Category:National Wildlife Refuges in Washington (state)
Category:Protected areas of Clark County, Washington
Category:Wetlands of Washington (state)
Category:Landforms of Clark County, Washington |
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A couple of days in one of the most fastest developed cities in China: Shanghai（上海）
Moving around China is not difficult especially if there are airports located in our city. Therefore going from Zhuhai to Shanghai is very simple as it takes almost 2 hours and a half to arrive at Shanghai Pudong Airport and the tickets prices are very decent to spend a couple of days in one of the most developed cities in the world. Shanghai is the largest city by population in China and is a city which has massive influence in commerce, culture, finance, media, fashion, technology and last but not least transport. It also has one of the most busiest ports in the world with many imports and exports activity going on.Being in Shanghai for a couple of days, it is inevitable to miss the amazing views of the skyline in The Bund, a real treasure for our eyes.We can see the buildings where the movie Mission Impossible 3 was filmed in the scenes that took part in Shanghai. At night time, this view is just exceptional with the Huangpu River in the middle. We can also visit The Oriental Pearl TV Tower which is one of the main attraction sites in Shanghai that attracts many tourists the entire year.
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Opening a bank account in a foreign country sounds scary, especially if you are in a non English-speaking country like Vietnam. However, having a local bank account is essential, especially if you plan on transferring money from overseas. It is also extremely convenient as you can use local apps such as GrabPay instead of having to carry large amounts of cash around. Fortunately for you, opening a bank account in Vietnam is actually an extremely simple process. No more awkwardly fumbling around in your wallet for cash!
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Last weekend we decided to head to the Yellow Mountains in Anhui province, quite close to Nanjing and Shanghai. The area is well known for its scenery, sunsets, Huangshan Pine trees, hot springs, winter snow, and views of the clouds from above. Huangshan is a frequent subject of traditional Chinese paintings and literature, as well as modern photography. It is a UNESCOWorld Heritage Site, and one of China's major tourist destinations.
To get there, we got the train from the Zhuhai train station (珠海站) towards Guangzhou South Railway Station. This is only a 70 minute ride and really convenient as you can go anywhere across China from there. Our high speed train from 广州南站 to 黄山北站 took 7 hours and was really comfortable and pleasant!
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You must be wondering what we interns do during the weekends after a tiring 5 days of work. Well, we can't possibly be sleeping the weekends away. Like the name suggests, Hello Weekend happens only in the weekends and is a famous flea market here in Ho Chi Minh City that we couldn't miss! It is indeed saying hello to the weekends while we unearth various hidden gems in this growing city of urbanization.
We celebrated the festive occasion by dropping down to the market during Halloween to bask in the spooky season. The area was decorated in pumpkin lanterns which reminded me of the Unique Pumpkin Shaped Lamp on ODM case studies blog. Ho Chi Minh City spontaneously engaged their tourists and locals alike as also seen their efforts at Tous Les Jour with their Halloween custom display design. The excitement could be felt as people buzzed in and out of the markets - carrying bags of their loots. There are even tables and chairs for us to relax after shopping really hard! |
At today’s White House briefing, CNN’s Jim Acosta left the room in a “totally saddened” mood after asked Sarah Sanders if she agrees that the media is the “enemy of the people” and ended up getting dressed down instead. All that caused Brian Stelter to come to the defense of his CNN colleague this way:
Perhaps the press needs to affirm whether or not the *people* are the enemy of the press. You know, the people who are tired of being maligned as racists because they didn’t vote for Hillary or called murderers on networks because they support self-defense. https://t.co/64pqzS5WHj |
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Horseradish peroxidase-catalyzed N-demethylation of aminopyrine and dimethylaniline results in generation of free radical intermediates which can interact with glutathione (GSH) to form a glutathione radical. This can either dimerize to yield glutathione disulfide or react with O2 to form oxygenated products of glutathione. Ethylmorphine is not a substrate in the peroxidase-mediated reaction, and free radical intermediates which react with GSH, are not formed from aminopyrine and dimethylaniline when the horseradish peroxidase/H2O2 system is replaced by liver microsomes and NADPH. Therefore, it appears unlikely that formation of free radical intermediates can be responsible for the depletion of GSH observed during N-demethylation of several drugs in isolated liver cells. |
Q:
Available space not changing when resizing the partition
in my computer I have several partitions. I have one partition with Fedora that had just below 40 Gb of space, and which I wanted to make bigger. Therefore, using gparted, I moved 10 Gb from another partition to the Fedora partition. Now, according to Gparted, I should have just below 50 Gb (see sda 12)
However, from Fedora itself the available space looks different:
This was the same space that was available before I resized the partition. How do I make that space actually available to Fedora?
A:
You've enlarged the partition on disk, but there is a stack of containers which need to be told about the change before you can use it.
First, you need to tell LVM (the logical volume manager) to use the extra space (gparted may have done this already, but this step won't hurt if done multiple times):
pvresize /dev/sda12
You should now see extra free space when you run
vgs
to list the available volume groups (one of which will be using /dev/sda12). You can now add space to the / partition; this involves resizing the logical volume itself, then the contained filesystem. lvextend is used to perform the first step, and it can delegate the second to fsadm itself, so one command should do the trick:
lvextend -r -L+10G fedora/root
Now your root filesystem should show 10GB more free space!
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abstract: 'In this paper, we provide a complete description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds, this description allows us to classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds. Moreover, by following [@PedersenPoonSwann], as an application of the latter result, through these structures we explicitly describe all homogeneous Hermitian metrics which solve the Hermitian-Einstein-Weyl equation on compact homogeneous Hermitian-Weyl manifolds. The main feature of our approach is to employ elements of representation theory of simple Lie algebras to describe Cartan-Ehresmann connections on principal elliptic bundles over complex flag manifolds. From these connections, and from the recent result provided in [@Gauduchon1], which states that any compact homogeneous l.c.K. manifold is Vaisman, we obtain a classification of homogeneous l.c.K. structures on compact homogeneous manifolds. Further, we also present a characterization for Kähler covering of compact homogeneous l.c.K. manifolds as regular locus of certain Stein spaces (with isolated singularity) defined by the complex analytification of HV-varieties, e.g., [@Vinberg], a.k.a. Kostant cones [@WallachI]. This characterization of Kähler covering allows us to describe the Kodaira-type embedding for compact homogeneous l.c.K. manifolds [@Vaismanimmersion] purely in terms of elements of representation theory of semisimple complex algebraic groups.'
author:
- 'Eder M. Correa'
title: 'Principal elliptic bundles and compact homogeneous l.c.K. manifolds'
---
Introduction
============
A locally conformally Kähler manifold (l.c.K.) is a conformal Hermitian manifold $(M,[g],J)$ such that for one (and hence for all) metric $g$ in the conformal class the corresponding fundamental 2-form $\Omega = g(J \otimes {\text{id}})$ satisfies $d\Omega = \theta \wedge \Omega$, where $\theta$ is a closed 1-form. This is equivalent to the existence of an atlas such that the restriction of $g$ to any chart is conformal to a Kähler metric. The 1-form $\theta \in \Omega^{1}(M)$ is called the [*[Lee form]{}*]{} of the Hermitian structure $(g,J)$ and plays an important role in the study of l.c.K. manifolds, especially when $M$ is compact, see for instance [@Gauduchon].
The geometry of l.c.K. manifolds has developed mainly since the 1970s, although, there are early contributions by P. Libermann (going back to 1954), see for instance [@Libermann1], [@Libermann2]. The recent treatment of the subject was initiated by I. Vaisman in 1976. In a long series of papers (see [@Dragomir] and references therein) he established the main properties of l.c.K. manifolds, demonstrated a connection with P. Gauduchon’s standard metrics [@Gauduchon], and recognized the Boothby metric as l.c.K. [@Boothby]. From these, he introduced the notion of [*generalized Hopf manifolds*]{} (g.H.m.), e.g. [@Vaisman]. As pointed out in [@Shells], the name “generalized Hopf manifold” was already used by Brieskorn and van de Ven (see [@Brieskorn]) for some products of homotopy spheres which do not bear Vaisman’s structure. Thus, the generalized Hopf manifolds in the sense of I. Vaisman are known nowadays as [*Vaisman manifolds*]{}.
A l.c.K. manifold $(M,[g],J)$ is said to be Vaisman if its Lee form $\theta $ is parallel with respect to the Levi-Civita connection of $g$. The structure of compact Vaisman manifolds is well-understood, in fact, they are mapping tori of automorphisms of Sasakian manifolds cf. [@Strucvaisman]. Moreover, it was recently proved that every compact homogeneous l.c.K. manifold is Vaisman [@Gauduchon1].
Another important feature of compact Vaisman manifolds is their relation with Einstein-Weyl geometry. Recall that a Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called Weyl connection, see for instance [@Folland]. It is said to be Einstein-Weyl if the symmetric trace-free part of the Ricci tensor of this connection vanishes cf. [@Higa]. As observed by Pedersen, Poon and Swann in [@PedersenPoonSwann], compact l.c.K. manifolds with parallel Lee form describe Hermitian-Einstein-Weyl spaces. Furthermore, under the Vaisman-Sasaki correspondence provided in [@Strucvaisman], the Einstein-Weyl Vaisman manifolds correspond to Sasaki-Einstein manifolds. This latter comment can be illustrated by means of the Hopf fibration and the Hopf surface in the following diagram:
& S\^[3]{} S\^[1]{} &\
S\^[3]{} & & S\^[2]{}
In the above diagram we have a basic example of Einstein-Weyl manifold given by the Hopf surface $S^{3}\times S^{1}$, and Sasaki-Einstein manifold given by $S^{3}$. A Hermitian-Einstein-Weyl structure on the Hopf surface $S^{3}\times S^{1}$ can be obtained from a suitable globally conformally Kähler (Ricci-flat) structure on $S^{3}\times\mathbb{R}$. The key point is that the globally conformally Kähler structure on $S^{3}\times\mathbb{R}$ in question can be completely determined by means of a principal $S^{1}$-connection on the associated Hopf fibration $$S^{1} \hookrightarrow S^{3} \to S^{2},$$ notice that the Hopf fibration connects the flat $S^{1}$-fibration over $S^{3}$ to the principal $T_{\mathbb{C}}^{1}$-fibration over $S^{2}$.
The basic example presented above fits in a broad setting, e.g. [@Kodaira], [@Gauduchon2], [@GauduchonOrnea], [@Belgun], and an interesting question that comes to mind is: What kind of information about the Hermitian-Einstein-Weyl geometry of $S^{3} \times S^{1}$ is encoded in the representation theory which controls the homogeneous Kähler-Einstein geometry of $S^{2}$ ? At first sight, we have the following facts as possible guidelines to approach this last question:
- If we consider the identification $S^{2} \cong \mathbb{C}{\rm{P}}^{1}$, it is straightforward to show that $S^{3} \times \mathbb{R}^{+} \cong \mathscr{O}(-1)^{\times}$, where $\mathscr{O}(-1) \to \mathbb{C}{\rm{P}}^{1}$ is the tautological line bundle;
- From Borel-Weil theorem [@Bott], [@Demazure], it follows that
$H^{0}(\mathbb{C}{\rm{P}}^{1},\mathscr{O}(1))^{\ast} = \mathbb{C}^{2}$,
i.e., the dual space $H^{0}(\mathbb{C}{\rm{P}}^{1},\mathscr{O}(1))^{\ast}$ corresponds to the fundamental irreducible representation of ${\rm{SL}}(2,\mathbb{C})$ defined by the canonical action on $\mathbb{C}^{2}$ cf. [@Baston];
- From the facts above we have
$S^{3} \times S^{1} = \mathscr{O}(-1)^{\times}/\mathbb{Z}$, and $\mathscr{O}(-1)^{\times} = H^{0}(\mathbb{C}{\rm{P}}^{1},\mathscr{O}(1))^{\ast} \backslash \{0\}.$
- Notice that in this context we also have
$\mathbb{C}{\rm{P}}^{1} = {\rm{SL}}(2,\mathbb{C}) \Big / \begin{pmatrix}
\ast & \ast \\
0 & \ast
\end{pmatrix},$
and the Kähler-Einstein structure on $\mathbb{C}{\rm{P}}^{1}$ is determined, up to scale, by $c_{1}(\mathscr{O}(1)) \in H^{2}(\mathbb{C}{\rm{P}}^{1},\mathbb{Z})$.
- We can obtain a Hermitian-Einstein-Weyl structure on $S^{3} \times S^{1} = \mathscr{O}(-1)^{\times}/\mathbb{Z}$ from a suitable principal $T_{\mathbb{C}}^{1}$-connection whose the curvature descends to a Kähler-Einstein metric on $\mathbb{C}{\rm{P}}^{1}$.
Motivated by the previous question and the aforementioned facts, the aim of this work is to study principal elliptic bundles over complex flag manifolds from the perspective of the representation theory which controls their complex and algebraic geometry. Our main goal is providing a complete description of Vaisman structures on principal elliptic bundles over complex flag manifolds and investigate the applications for such a description in the study of compact homogeneous l.c.K. manifolds.
As pointed out in [@Moroianu], by combining [@VaismanII] with [@Strucvaisman] one easily proves that compact homogeneous Vaisman manifolds are mapping tori over the circle with fiber being a compact homogeneous Sasakian manifold, and these ones are total spaces of Boothby-Wang fibrations over compact homogeneous Kähler manifolds [@BW]. These ideas generalize, in some suitable sense, the example described above involving the Hopf surface and the Hopf fibration. Further, by considering the (strong) regularity of the underlying canonical foliation in the homogeneous setting (see for instance [@VaismanII], [@Chen]), it can be shown that compact homogeneous Vaisman manifolds can be realized as principal elliptic bundles over compact homogeneous Kähler manifolds. Thus, by following a recent result provided in [@Gauduchon1], which states that any compact homogeneous l.c.K. manifold is Vaisman, from our description of Vaisman structures on elliptic bundles over complex flag manifolds, we provided a complete classification, up to scale, of homogeneous l.c.K. metrics and homogeneous Hermitian-Einstein-Weyl metrics on compact homogeneous Hermitian manifolds. The key point in our approach, in the last setting mentioned, is to use elements of representation theory of simple Lie algebras in order to achieve an explicit description for the Gauduchon gauge [@Gauduchon] on compact homogeneous l.c.K. manifolds.
Also, we present a complete characterization for Kähler covering of compact homogeneous l.c.K. manifolds as regular locus of Stein spaces (with isolated singularities) defined by the complex analytification of *HV-varieties* cf. [@Vinberg], a.k.a. *Kostant cone* [@WallachI]. This last characterization of Kähler covering allows us to describe the Kodaira-type emebedding of compact homogeneous l.c.K. manifolds provided in [@Vaismanimmersion] purely in terms of elements of representation theory of semisimple complex algebraic groups.
Main results
------------
In this paper we provide the following results:
1. Description of Vaisman structures on elliptic bundles obtained as associated bundles to principal $\mathbb{C}^{\times}$-bundles over flag manifolds;
2. Classification, up to scale, of homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds by means of elements of Lie theory;
3. Concrete description and classification, up to scale, of homogeneous Hermitian-Einstein-Weyl metrics on compact homogeneous Hermitian-Weyl manifolds by means of elements of Lie theory.
4. Description of Kodaira-type embedding of compact homogeneous l.c.K. manifolds into diagonal Hopf manifolds by using representation theory of complex (semi)simple algebraic groups.
In order to give a detailed overview on the results listed above, let us recall some basic facts on the geometry of flag manifolds.
A complex flag manifold $X$ is a compact simply connected homogeneous complex manifold defined by $$X = G^{\mathbb{C}}/P = G/G \cap P,$$ where $G^{\mathbb{C}}$ is a complex simple Lie group with compact real form given by $G$, and $P \subset G^{\mathbb{C}}$ is a parabolic Lie subgroup. Considering ${\text{Lie}}(G^{\mathbb{C}}) = \mathfrak{g}^{\mathbb{C}}$, if we choose a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}^{\mathbb{C}}$, and a simple root system $\Sigma \subset \mathfrak{h}^{\ast}$, up to conjugation, we have that $P = P_{\Theta}$, for some $\Theta \subset \Sigma$, where $P_{\Theta}$ is a parabolic Lie subgroup determined by $\Theta$, see for instance [@Alekseevsky].
In the above setting, it is suitable to denote $X = X_{P}$ in order to emphasize the underlying parabolic Cartan geometry of the pair $(G^{\mathbb{C}},P)$. It is a well-known fact that $X_{P}$ is a compact simply connected homogeneous Hodge manifold (cf. [@BorelK]) whose [*Picard group*]{} can be described by $${\text{Pic}}(X_{P}) = \bigoplus_{\alpha \in \Sigma \backslash \Theta}\mathbb{Z}c_{1}\big ( \mathscr{O}_{\alpha}(1) \big ),$$ such that $\mathscr{O}_{\alpha}(1) = G^{\mathbb{C}}\times_{P}\mathbb{C}$, $\alpha \in \Sigma \backslash \Theta$, is a (homogeneous) holomorphic line bundle obtained as an associated holomorphic vector bundle to the principal $P$-bundle
$P \hookrightarrow G^{\mathbb{C}} \to X_{P}$,
here the underlying $\mathbb{C}$-linear holomorphic representation of $P$ is defined by a holomorphic character $\chi_{\omega_{\alpha}} \colon P \to {\rm{GL}}(1,\mathbb{C}) = \mathbb{C}^{\times}$, see for instance [@BorelH], [@CONTACTCORREA]. For the sake of simplicity, we also denote $$\mathscr{O}_{\alpha}(k) := \underbrace{\mathscr{O}_{\alpha}(1) \otimes \cdots \otimes \mathscr{O}_{\alpha}(1)}_{k{\text{-times}}},$$ for every $k \in \mathbb{Z}$ and every $\alpha \in \Sigma \backslash \Theta$. Thus, given $L \in {\text{Pic}}(X_{P})$, one can write $$\displaystyle L = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(-\ell_{\alpha}),$$ such that $\ell_{\alpha} \in \mathbb{Z}$, $\forall \alpha \in \Sigma \backslash \Theta$. Moreover, we can consider the manifold ${\rm{Tot}}(L^{\times})$ defined by the total space of the principal $\mathbb{C}^{\times}$-bundle
$L^{\times} = \bigg \{ u \in L \ \ \bigg | \ \ ||u|| \neq 0\bigg\}$,
here we consider $||\cdot||$ as being the norm induced by some fixed Hermitian structure on $L$. For the particular case when $L \to X_{P}$ is a negative line bundle, i.e., $c_{1}(L) < 0$, we have an identification $${\rm{Tot}}(L^{\times}) \cong Q(L) \times \mathbb{R}^{+},$$ where $Q(L)$ is the Sasaki manifold defined by the sphere bundle associated to $L$, i.e.,
$Q(L) = \bigg \{ u \in L \ \ \bigg | \ \ ||u|| = 1\bigg\}$.
From this, by following the Vaisman-Sasaki correspondence provided in [@Strucvaisman], and the results provided in [@Tsukada], [@CONTACTCORREA], we obtain the following theorem:
\[Theorem1\] Let $X_{P}$ be a complex flag manifold, associated to some parabolic Lie subgroup $P = P_{\Theta} \subset G^{\mathbb{C}}$, and let $L \in {\text{Pic}}(X_{P})$ be a negative line bundle, such that
$\displaystyle L = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(-\ell_{\alpha}),$
where $\ell_{\alpha} > 0$, $\forall \alpha \in \Sigma \backslash \Theta$. Then, for every $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, we have that the manifold $$\label{quotient}
M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ {\text{where}} \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$$ admits a Vaisman structure completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, defined in coordinates $(z,w) \in L^{\times}|_{U}$ by $$\label{negativepotential1}
{\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w},$$ for some local section $s_{U} \colon U \subset X_{P} \to G^{\mathbb{C}}$, where $v_{\mu(L)}^{+}$ is the highest weight vector of weight $\mu(L)$ associated to the irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu(L)) = H^{0}(X_{P},L^{-1})^{\ast}$.
Under the hypotheses of the above theorem, we have that the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$ provided in Equation \[negativepotential1\] allows us to explicitly describe a Vaisman structure on $M = {\rm{Tot}}(L^{\times})/\Gamma$. In fact, since $M = {\text{Tot}}(L^{\times})/\Gamma$ defines a principal $T_{\mathbb{C}}^{1}$-bundle over $X_{P}$, from the potential \[negativepotential1\] we have a principal connection $\Psi \in \Omega^{1}(M;\mathbb{C})$ locally described by $$\Psi = \frac{1}{2}\big (d^{c} - \sqrt{-1}d\big ) \log({\rm{K}}_{H}\big (z,w \big )) = -\sqrt{-1}\bigg [\partial \log \Big ( \big | \big |s_{U}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{dw}{w}\bigg],$$ for some local section $s_{U} \colon U \subset X_{P} \to G^\mathbb{C}$. From this connection, if one considers the complex structure $J \in {\text{End}}(TM)$, induced from the complex structure on ${\rm{Tot}}(L^{\times})$ via the projection map $\wp \colon {\rm{Tot}}(L^{\times}) \to M$, we obtain a Hermitian metric on $M$ defined by $$\displaystyle g = \frac{1}{2} \Big ( d\Psi({\text{id}} \otimes J) + \Psi \odot \overline{\Psi} \Big ),$$ cf. [@Tsukada]. Thus, we have an associated conformal Hermitian structure $(g,J)$ on $M$ which is in fact l.c.K., such that its Lee form $\theta \in \Omega^{1}(M)$ can be described from ${\rm{K}}_{H}$ (locally) by $$\theta = -\bigg [d\log \Big ( \big | \big |s_{U}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg].$$ It is straightforward to show that the Lee form above is parallel with respect to the Levi-Civita connection induced from $g$, which implies that $(M,g,J)$ is in fact a Vaisman manifold, e.g. [@Gini Proposition 5.4]. Notice that the manifolds described in Theorem \[Theorem1\] are examples of l.c.K. manifolds with potentials cf. [@Vaismanpotential].
An important consequence of Theorem \[Theorem1\] is that it allows us to classify homogeneous l.c.K. structures on compact homogeneous manifolds. Actually, recall that a compact l.c.K. manifold $(M,J,g)$ is said to be homogeneous l.c.K. manifold if it admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$. In this setting, we have $$K = K_{{\text{ss}}}Z(K)_{0},$$ where $K_{{\text{ss}}}$ is a closed, connected, and semisimple Lie subgroup with ${\text{Lie}}(K_{{\text{ss}}}) = [\mathfrak{k},\mathfrak{k}]$, and $Z(K)_{0}$ is the closed connected subgroup defined by the connected component of the identity of the center of $K$ (cf. [@Knapp]). Recently, it was shown in [@Gauduchon1] that any compact homogeneous l.c.K. manifold is Vaisman, since compact homogeneous Vaisman manifolds are defined by principal elliptic bundles over flag manifolds [@VaismanII], the description obtained in Theorem \[Theorem1\] allows us to establish the following result:
\[Theorem2\] Let $(M,g,J)$ be a compact l.c.K. manifold that admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$. Suppose also that $K_{{\text{ss}}}$ is simply connected and has a unique simple component. Then, the following holds:
1. The manifold $M$ is a principal $T_{\mathbb{C}}^{1}$-bundle over a complex flag manifold $X_{P} = K_{{\text{ss}}}^{\mathbb{C}}/P$, for some parabolic Lie subgroup $P = P_{\Theta} \subset K_{{\text{ss}}}^{\mathbb{C}}$. Moreover, $M$ is completely determined by a negative line bundle $L \in {\text{Pic}}(X_{P})$, i.e.,
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ {\text{where}} \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$.
2. The associated l.c.K. structure on $M$ is completely determined by the global Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, defined in coordinates $(z,w) \in L^{\times}|_{U}$ by
${\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w},$
for some local section $s_{U} \colon U \subset X_{P} \to K_{\text{ss}}^{\mathbb{C}}$, where $v_{\mu(L)}^{+}$ is the highest weight vector of weight $\mu(L)$ associated to the irreducible $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$-module $V(\mu(L)) = H^{0}(X_{P},L^{-1})^{\ast}$.
The latter theorem provides a concrete way to describe homogeneous l.c.K. structures by using elements of representation theory of simple Lie algebras. It is worth pointing out that, by following the description of $g$ as in Theorem \[Theorem1\], under the hypotheses of Theorem \[Theorem2\], we obtain a precise description, up to homothety, for the standard Gauduchon metric [@Gauduchon] on any compact homogeneous l.c.K. manifold, notice that $b_{1}(M) = 1$, see for instance [@Vaisman].
Given a compact l.c.K. manifold $(M,[g],J)$, such that $\dim_{\mathbb{R}}(M) = n \geq 6$, without loss of generality one can suppose that the associated Lee form $\theta$ is harmonic [@Gauduchon]. By considering the Levi-Civita connection $\nabla$ obtained from the metric $g$, one can define a Weyl connection by setting $$D = \nabla - \frac{1}{2} \bigg (\theta \odot {\text{id}} - g \otimes A \bigg ),$$ such that $A = \theta^{\sharp}$, see for instance [@Weyl page 125], [@Folland] and [@Higa]. It follows from [@Gauduchon2] that, if the underlying Hermitian-Weyl manifold $(M,[g],D,J)$ is Hermitina Einstein-Weyl, i.e., the symmetrised Ricci tensor of $D$ is a multiple of $g$ at each point, then the Ricci tensor ${\text{Ric}}^{\nabla}$ associate to $\nabla$ satisfies $${\text{Ric}}^{\nabla} = (n-2) \Big (||\theta||^{2}g - \theta \otimes \theta \Big ),$$ cf. [@Gauduchon2 Equation 40]. Moreover, in the latter setting we also have that $\nabla \theta \equiv 0$, which impiles that a compact Hermitian-Einstein-Weyl manifold $(M,[g],D,J)$ is particularly Vaisman.
By following [@PedersenPoonSwann], we have that every compact homogeneous Hermitina Einstein-Weyl manifold can be obtained from a discrete quotient of the maximal root of the canonical bundle of a complex flag manifold $X_{P}$, i.e., a discrete quotient obtained from the complex manifold underlying the total space of the frame bundle associated to the holomorphic line bundle $$K_{X_{P}}^{\otimes \frac{1}{I(X_{P})}} := \mathscr{O}_{X_{P}}(-1),$$ here $I(X_{P}) \in \mathbb{Z}$ denotes the Fano index of $X_{P}$. Combining Theorem \[Theorem2\] with [@PedersenPoonSwann Theorem 4.2], we obtain the following result:
\[Theorem3\] Let $(M,[g],D,J)$ be a compact homogeneous Hermitian-Einstein-Weyl manifold, such that $\dim_{\mathbb{R}}(M) \geq 6$, and let $K$ be the compact connected Lie group which acts on $M$ by preserving the Hermitian-Einstein-Weyl structure. Suppose also that $K_{{\text{ss}}}$ is simply connected and has a unique simple component. Then, we have that $$\label{universalpresentation1}
M = {\rm{Tot}}(\mathscr{O}_{X_{P}}(-\ell)^{\times})/\mathbb{Z},$$ for some $\ell \in \mathbb{Z}_{>0}$, i.e., $M$ is a principal $T_{\mathbb{C}}^{1}$-bundle over a complex flag manifold $X_{P} = K_{{\text{ss}}}^{\mathbb{C}}/P$, for some parabolic Lie subgroup $P = P_{\Theta} \subset K_{{\text{ss}}}^{\mathbb{C}}$. Moreover, the Hermitian-Einstein-Weyl metric $g$ is completely determined by the Lee form $\theta_{g} \in \Omega^{1}(M)$, locally described by $$\label{higgsfield1}
\theta_{g} = -\frac{\ell}{I(X_{P})}\bigg [d\log \Big ( \big | \big |s_{U}v_{\delta_{P}}^{+} \big | \big |^{2}\Big ) + I(X_{P})\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg],$$ for some local section $s_{U} \colon U \subset X_{P} \to K_{\text{ss}}^{\mathbb{C}}$, where $v_{\delta_{P}}^{+}$ is the highest weight vector of weight $\delta_{P}$ associated to the irreducible $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$-module $V(\delta_{P}) = H^{0}(X_{P},K_{X_{P}}^{-1})^{\ast}$, and $I(X_{P})$ is the Fano index of $X_{P}$.
The result above provides a concrete description of $K$-invariant Hermitian-Einstein-Weyl metrics on compact homogeneous manifolds. Moreover, since in the setting above we have that $b_{1}(M) = 1$, and the corresponding [*[Higgs field]{}*]{} $\theta_{g}$ is harmonic $K$-invariant, it follows that, up to homothety, the homogeneous Hermitian Eisntein-Weyl metric described in Theorem \[Theorem3\] is unique on $(M,J)$.
It is worth mentioning that the classification provided by Theorem \[Theorem3\] is consistent with the classification of homogeneous Kähler-Einstein metrics on compact homogeneous manifolds (in the sense of Y. Matsushima [@MATSUSHIMA]), i.e., a compact homogeneous complex manifold admits at most one, up to scale, $K$-invariant Hermitian-Einstein-Weyl structure cf. [@OVconjecture Conjecture 5.1]. The correspondence between $K_{{\text{ss}}}$-invariant Kähler-Einstein metrics and $K$-invariant Hermitian-Einstein-Weyl metrics established by Theorem \[Theorem3\] is codified in the global Kähler potential $$\label{potentialeinstein}
{\rm{K}}_{H} \colon {\rm{Tot}}(\mathscr{O}_{X_{P}}(-\ell)^{\times}) \to \mathbb{R}^{+}, \ \ {\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\delta_{P}}^{+} \big | \big |^{2} \Big)^{\frac{\ell}{I(X_{P})}} w\overline{w},$$ observe that the $1$-form $\eta = \frac{1}{2}d^{c}\log({\rm{K}}_{H})$ defines a contact structure on the sphere bundle of $\mathscr{O}_{X_{P}}(-\ell)$ which descends to a $K_{{\text{ss}}}$-invariant Kähler-Einstein metric $\omega$ on $X_{P}$, i.e., $d\eta = \pi^{\ast}\omega$, such that $$\omega = \frac{\ell\sqrt{-1}}{I(X_{P})}\partial \overline{\partial} \log \Big (\big | \big |s_{U}v_{\delta_{P}}^{+} \big| \big |^{2} \Big ),$$ for more details about the metric above, see for instance [@AZAD] and [@CONTACTCORREA]. Therefore, through the Kähler potential \[potentialeinstein\] one gets a concrete correspondence between the desired $K_{{\text{ss}}}$-invariant Kähler-Einstein metrics and $K$-invariant Hermitian-Einstein-Weyl metrics.
From Theorem \[Theorem2\] one can see that the geometry of compact homogeneous l.c.K. manifolds is closely related to the projective algebraic geometry of complex flag manifolds. Being more precise, since associated to each negative holomorphic line bundle $L \in {\text{Pic}}(X_{P})$ we have a projective embedding[^1] $$\label{negativeembedding}
\iota \colon X_{P} \hookrightarrow {\text{Proj}}\big (H^{0}(X_{P},L^{-1})^{\ast} \big),$$ from Borel-Weil theorem, it follows that $H^{0}(X_{P},L^{-1})^{\ast} = V(\mu(L))$, for some suitable irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu(L))$. Thus, the latter embedding above is given explicitly by $\iota(X_{P}) = \mathbb{P}(G^{\mathbb{C}}\cdot v_{\mu(L)}^{+})$, i.e., $X_{P}$ is embedded as the projectivization of the orbit $G^{\mathbb{C}}\cdot v_{\mu(L)}^{+} \subset V(\mu(L)) \backslash \{0\}$, where $v_{\mu(L)}^{+} \in V(\mu(L))$ denotes the associated highest weight vector of weight $\mu(L)$. The interesting feature of the projective algebraic realization \[negativeembedding\] of $X_{P}$ is that its affine cone $Y \subset V(\mu(L))$ is given by $$Y = \overline{G^{\mathbb{C}} \cdot v_{\mu(L)}^{+}} = G^{\mathbb{C}} \cdot v_{\mu(L)}^{+} \cup \{0\} = {\text{Spec}}\Big ( \bigoplus_{\ell = 0}^{+\infty}V(\ell\mu(L))^{\ast}\Big).$$ The underlying affine algebraic variety above is an example of HV-variety (cone of highest vector) cf. [@Vinberg], a.k.a. Kostant cone [@WallachI]. Considering $Y$ as a complex analytic variety, we have that it defines a Stein space with an isolated singularity at $0 \in Y$. Furthermore, if we denote by $Y_{\text{reg}}$ the regular locus of $Y$, it can be shown through the associated Cartan-Remmert reduction [@GRAUERT] of ${\rm{Tot}}(L)$ that $${\rm{Tot}}(L^{\times}) \cong Y_{\text{reg}},$$ see for instance [@Akhiezer1 Lemma 2]. Thus, the Kähler covering provided in Theorem \[Theorem2\] for compact homogeneous l.c.K. manifolds can be realized as the regular locus of certain HV-varieties.
Through of the above ideas, and the results introduced in [@Vaismanimmersion], [@Vaismanpotential], we provide the following Kodaira-type theorem for compact homogeneous l.c.K. manifold in terms of Lie theoretical elements:
\[Theorem4\] Let $(M,g,J)$ be a compact l.c.K. manifold that admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$. Suppose also that $K_{{\text{ss}}}$ is simply connected and has a unique simple component. Then, there exists a character $\chi \in {\text{Hom}}(T^{\mathbb{C}},\mathbb{C}^{\times})$, for some maximal torus $T^{\mathbb{C}}\subset K_{{\text{ss}}}^{\mathbb{C}}$, and a holomorphic embedding $$\label{Hopfembedding}
\mathscr{R}^{(\Gamma)} \colon M \hookrightarrow {\rm{H}}_{\Gamma} = \Big (V(\chi) \backslash \{0\} \Big )/ \Gamma,$$ such that $V(\chi)$ is an irreducible $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$-module with highest weight vector $v_{\chi}^{+} \in V(\chi)$, and $\Gamma \subset {\rm{GL}}(V(\chi))$ is an infinite cyclic subgroup generated by $\lambda \cdot{\rm{Id}}_{V(\chi)}$, where $\lambda \in \mathbb{C}^{\times}$, with $|\lambda| < 1$. Moreover, we have that $$\mathscr{R}^{(\Gamma)}(M) = Y_{\text{reg}}/\Gamma,$$ such that $Y_{\text{reg}} = K_{{\text{ss}}}^{\mathbb{C}} \cdot v_{\chi}^{+}$, where $$Y = {\text{Spec}}\Big ( \bigoplus_{\ell = 0}^{+\infty}V(\chi^{\ell})^{\ast}\Big).$$ Therefore, every compact homogeneous l.c.K. manifold is biholomorphic to some quotient space of the regular locus of the complex analytification of some HV-variety by a discrete group.
The result above shows that the embedding of compact l.c.K. manifolds provided in [@Vaismanimmersion] takes an explicit form in the homogeneous setting, i.e., under the hypotheses of Theorem \[Theorem4\], we have $$M \cong \Big ( K_{{\text{ss}}}^{\mathbb{C}} \cdot v_{\chi}^{+} \Big) / \Gamma.$$ Notice that Theorem \[Theorem4\] also clarifies the relation between compact homogeneous l.c.K. manifolds and the Borel-Weil theorem. We also point out that Theorem \[Theorem4\] provides a concrete description for Kähler covering of compact homogeneous l.c.K. manifolds purely in terms of elements of Lie theory. In fact, in the setting of Theorem \[Theorem4\], if one denotes by $\mu_{M} = (d\chi)_{e}$ the associated highest weight of $V(\chi)$, we have the Kähler covering $Y$ of $M$ characterized by $$Y = \Big \{ v \in V(\chi) \ \ \Big| \ \ \big( {\rm{C}}_{\mathfrak{k}_{\text{ss}}^{\mathbb{C}}} - c(2\mu_{M}){\rm{Id}}\big)( v \otimes v ) = 0 \Big \},$$ such that ${\rm{C}}_{\mathfrak{k}_{\text{ss}}^{\mathbb{C}}} \in {\mathcal{U}}(\mathfrak{k}_{\text{ss}}^{\mathbb{C}})$ is the [*Casimir element*]{} of $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$ (here ${\mathcal{U}}(\mathfrak{k}_{\text{ss}}^{\mathbb{C}})$ denotes the universal enveloping algebra of $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$), $c(2\mu_{M}) = \kappa ( 2\mu_{M} + 2\varrho, 2\mu_{M})$, where $\kappa$ denotes the Cartan-Killing form of $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$ and $\varrho$ is the weight defined by the half-sum of the positive roots, see for instance [@WallachI], [@Quadric].
Outline of the paper
--------------------
The content and main ideas of this paper are organized as follows:
In Section 2, we shall cover the basic material related to the Vaisman-Sasaki correspondence. In Section 3, we present some basic results on strongly regular Vaisman manifolds realized as principal elliptic bundles over Kähler manifolds. In Section 4, we shall explore some classical results on Hermitian-Weyl geometry and on Hermitian-Einstein-Weyl geometry. In Section 5, we cover some basic results about the realization of compact homogeneous l.c.K. manifolds as principal elliptic bundles over complex flag manifolds. In Section 6, we prove Theorem \[Theorem1\], Theorem \[Theorem2\], Theorem \[Theorem3\] and Theorem \[Theorem4\]. In this last section, we also provide a huge class of examples which illustrate concrete applications of our results.
Vaisman and Sasaki manifolds
============================
In what follows we shall cover the basic generalities about the relation between compact Vaisman manifolds and compact Sasaki manifolds. Further details about the results presented in this section can be found in [@Strucvaisman], [@Gini], [@Blair].
Structure theorem for compact Vaisman manifolds
-----------------------------------------------
Let $(M,g,J)$ be a connected complex Hermitian manifold such that $\dim_{\mathbb{C}}(M) \geq 2$. In what follows we shall denote by $\Omega = g(J \otimes {\text{id}})$ the associated fundamental $2$-form of $(M,g,J)$.
\[DEFLCK\] A Hermitian manifold $(M,g,J)$ is called locally conformally Kähler (l.c.K.) if it satisfies one of the following equivalent conditions:
1. There exists an open cover $\mathscr{U}$ of $M$ and a family of smooth functions $\{f_{U}\}_{U \in \mathscr{U}}$, $f_{U} \colon U \to \mathbb{R}$, such that each local metric $$g_{U} = {\mathrm{e}}^{-f_{U}}g|_{U},$$ is Kählerian, $\forall U \in \mathscr{U}$.
2. There exists a globally defined closed $1$-form $\theta \in \Omega^{1}(M)$ such that $$d\Omega = \theta \wedge \Omega.$$
\[LocalLee\] Notice that the two conditions in the definition above tells us that
$\theta |_{U} = df_{U},$
$\forall U \in \mathscr{U}$, see for instance [@Dragomir]. The closed $1$-form $\theta \in \Omega^{1}(M)$ which satisfies the second condition in the last definition is called the Lee form of a l.c.K. manifold $(M,g,J)$. It is worth observing that, if the Lee form $\theta$ of a l.c.K. manifold $(M,g,J)$ is exact, i.e. $\theta = df$, such that $f \in C^{\infty}(M)$, then it follows that $(M,{\rm{e}}^{-f}g,J)$ is a Kähler manifold. In what follows, unless otherwise stated, we shall assume that $\theta$ is not exact, and $\theta \not \equiv 0$.
An important subclass of l.c.K. manifolds is defined by the parallelism of the Lee form with respect to the Levi-Civita connection of $g$. Being more precise, we have the following definition.
A l.c.K. manifold $(M,g,J)$ is called a Vaisman manifold if $\nabla \theta = 0$, where $\nabla$ is the Levi-Civita connection of $g$.
In order to establish some results which relate Vaisman manifolds to Sasakian manifolds, let us recall some generalities on Sasaki geometry.
\[ContMetric\] Let $(Q,g_{Q})$ be a Riemannian manifold of dimension $2n+1$. A contact metric structure on $(Q,g_{Q})$ is a triple $(\phi,\xi,\eta)$, where $\phi$ is a $(1,1)$-tensor, $\xi$ is a vector field and $\eta$ is a $1$-form, such that:
1. $\eta \wedge (d\eta)^{n} \neq 0$, $\eta(\xi) = 1$,
2. $\phi \circ \phi = - {\rm{id}} + \eta \otimes \xi$,
3. $g_{Q}(\phi \otimes \phi) = g_{Q} - \eta \otimes \eta$,
4. $d\eta = 2g_{Q}(\phi \otimes {\rm{id}})$.
We denote a contact metric structure on $Q$ by $(g_{Q},\phi,\xi,\eta)$. From this, we have the following definition.
A contact metric structure $(g_{Q},\phi,\xi,\eta)$ is called ${\text{K}}$-contact if $\mathscr{L}_{\xi}g_{Q} = 0$, i.e. if $\xi \in \Gamma(TQ)$ is a Killing vector field.
In the setting of ${\text{K}}$-contact structures there is a special subclass which is defined as follows.
A ${\text{K}}$-contact structure $(g_{Q},\phi,\xi,\eta)$ on a smooth manifold $Q$ is called Sasakian if $$\label{sasakicondition}
\big [ \phi , \phi \big ] + d \eta \otimes \xi = 0,$$ where
$\big [ \phi , \phi \big ](X,Y) := \phi^{2}\big [ X,Y\big ] + \big [ \phi X, \phi Y\big ] - \phi \big [\phi X,Y \big] - \phi \big [X,\phi Y \big],$
for every $X,Y \in \Gamma(TQ)$. A Sasaki manifold is a Riemannian manifold $(Q,g_{Q})$ with a $K$-contact structure $(g_{Q},\phi,\xi,\eta)$ which satisfies \[sasakicondition\].
An alternative way to define Sasakian manifolds can be described as follows. Given a ${\text{K}}$-contact structure $(g_{Q},\phi,\xi,\eta)$ on a smooth manifold $Q$, one can consider the manifold defined by its metric cone $$\mathscr{C}(Q) = Q \times \mathbb{R}^{+}.$$ Taking the coordinate $r$ on $\mathbb{R}^{+}$ one can define the warped product Riemannian metric on $\mathscr{C}(Q)$ by setting $$g_{\mathscr{C}} = r^{2}g_{Q} + dr \otimes dr.$$ Furthermore, from $(\phi,\xi,\eta)$ one has an almost-complex structure defined on $\mathscr{C}(Q)$ by setting $$\label{complexcone}
J_{\mathscr{C}}(Y) = \phi(Y) - \eta(Y)r\partial_{r}, \ \ \ \ \ J_{\mathscr{C}}( r\partial_{r}) = \xi.$$
From the above comments, a Sasaki manifold can be defined as follows.
\[sasakikahler\] A contact metric structure $(g_{Q},\phi,\xi,\eta)$ on a smooth manifold $Q$ is called Sasaki if $(\mathscr{C}(Q),g_{\mathscr{C}},J_{\mathscr{C}})$ is a Kähler manifold.
\[realhomolomrphic\] It is worth mentioning that, in the setting of the last definition, by considering the Levi-Civita connection $\nabla$ defined by the metric $g_{\mathscr{C}}$ on the warped product $\mathbb{R}^{+} \times_{r} Q$, it follows from [@ONEILL Page 206] that
1. $\nabla_{r\partial_{r}} ( r\partial_{r}) = r\partial_{r}$, $\nabla_{r\partial_{r}} (X ) = \nabla_{X}( r\partial_{r}) = X$;
2. $\nabla_{X} (Y) = \nabla^{Q}_{X} (Y ) - g_{Q}(X,Y)r\partial_{r}$.
Here $X$ and $Y$ denote vector fields on $Q$, appropriately interpreted also as vector fields on $\mathscr{C}(Q)$, and $\nabla^{Q}$ is the Levi-Civita connection of $g_{Q}$.
Given a Sasakian manifold $(Q,g_{Q})$ with structure tensors $(g_{Q},\phi,\xi,\eta)$, for every $\lambda \in \mathbb{R}^{+}$ we can define a smooth map $\tau_{\lambda} \colon \mathscr{C}(Q) \to \mathscr{C}(Q)$, such that $$\label{dilatation}
\tau_{\lambda}\colon (x,s) \to (x,\lambda s).$$ It is straightforward to show that $\tau_{\lambda}^{\ast}g_{\mathscr{C}} = \lambda^{2}g_{\mathscr{C}}$, i.e. $\tau_{\lambda}$ is a dilatation (homothety). Moreover, the map defined above is also holomorphic with respect to the complex structure \[complexcone\]. Now, we observe that the Kähler form $\omega_{\mathscr{C}} = g_{\mathscr{C}}(J_{\mathscr{C}} \otimes \text{id})$ can be written as $$\displaystyle \omega_{\mathscr{C}} = d \Big (\frac{r^{2}\eta}{2} \Big) = rdr \wedge \eta + \frac{r^{2}}{2}d\eta.$$ Thus, by setting $\psi = \log(r) \colon \mathscr{C}(Q) \to \mathbb{R}$, we obtain
$$\displaystyle \omega_{\mathscr{C}} = {\mathrm{e}}^{2\psi} \bigg ( d\psi \wedge \eta + \frac{d\eta}{2}\bigg).$$
If we consider $\widetilde{\Omega} = {\mathrm{e}}^{-2\psi}\omega_{\mathscr{C}}$, from the identification
$\text{pr}_{1} \times \psi \colon Q \times \mathbb{R}^{+} \to Q \times \mathbb{R}$,
it follows that the map $\tau_{\lambda} \colon \mathscr{C}(Q) \to \mathscr{C}(Q)$ transforms the coordinate $\psi$ by
$\psi \mapsto \psi + \log(\lambda),$
it follows that $\tau_{\lambda}^{\ast} \widetilde{\Omega} = \widetilde{\Omega}$, in fact,
$\displaystyle \tau_{\lambda}^{\ast} \widetilde{\Omega} = d \big(\psi + \log(\lambda) \big) \wedge \eta + \frac{d\eta}{2}.$
Therefore, if we take the manifold $M = \mathscr{C}(Q) / \sim_{\lambda}$, where $`` \sim_{\lambda}"$ stands for the equivalence relation
$(x,s) \sim (x,\lambda s),$
we obtain a l.c.K. structure $(g,J)$ on $M = Q \times S^{1}$, where $J$ is a complex structure induced by $J_{\mathscr{C}}$, and $\wp^{\ast}g = \widetilde{\Omega}({\rm{id}}\otimes J_{\mathscr{C}})$, where $\wp \colon \mathscr{C}(Q) \to \mathscr{C}(Q) / \sim_{\lambda}$, such that $\widetilde{\Omega} = {\mathrm{e}}^{-2\psi}\omega_{\mathscr{C}}$, namely, $$\widetilde{\Omega} = \displaystyle d\psi \wedge \eta + \frac{d\eta}{2}.$$ Notice that if one sets $\widetilde{\theta} = -2d\psi$, it follows that
$ d \widetilde{\Omega} = d({\mathrm{e}}^{-2\psi}\omega_{\mathscr{C}}) = -2{\mathrm{e}}^{-2\psi}d\psi \wedge \omega_{\mathscr{C}} = \widetilde{\theta} \wedge \widetilde{\Omega}$,
which implies that $\wp^{\ast}\theta = -2d\psi$, where $\theta $ is the Lee form associated to the l.c.K. structure $(g,J)$ described above on $M = Q \times S^{1}$. One can also verify that for this last l.c.K. structure the associated Lee form $\theta$ satisfies $\nabla \theta \equiv 0$. In fact, by considering the Levi-Civita connection $\widetilde{\nabla}$ defined by the Riemannian metric $$\label{globalconform}
\widetilde{g} = \widetilde{\Omega}({\rm{id}}\otimes J_{\mathscr{C}}) = g_{Q} + d\psi \otimes d\psi,$$ from [*[Koszul’s]{}*]{} identity one has $$2\widetilde{g}(\widetilde{\nabla}_{X}\widetilde{\theta}^{\sharp},Y) = d\widetilde{\theta}(X,Y) + (\mathscr{L}_{\widetilde{\theta}^{\sharp}}g)(X,Y),$$ $\forall X,Y \in \Gamma(T\mathscr{C}(Q))$, such that $\widetilde{\theta} = \widetilde{g}(\cdot,\widetilde{\theta}^{\sharp})$. Since $d\widetilde{\theta} = 0$, and $$\mathscr{L}_{\frac{\partial}{\partial \psi}}\widetilde{g} = \big (\mathscr{L}_{\frac{\partial}{\partial \psi}}d\psi \big) \otimes d\psi + d\psi \otimes \big (\mathscr{L}_{\frac{\partial}{\partial \psi}}d\psi \big) = 0,$$ it follows that $\widetilde{\nabla}\widetilde{\theta} \equiv 0$. Thus, since the projection $\wp \colon (\mathscr{C}(Q),\widetilde{g}) \to (M,g)$ is a local isometry, it follows that $\nabla \theta \equiv 0$, i.e. $M = Q \times S^{1}$ is a Vaisman manifold.
\[globkahler\] In the above description we implicitly used the fact that there exists a smooth function $f \colon \mathscr{C}(Q) \to \mathbb{R}$, such that $\widetilde{\Omega} = {\mathrm{e}}^{f}\omega_{\mathscr{C}}$, i.e. the metric induced by $\widetilde{\Omega} $ on $\mathscr{C}(Q)$ is globally conformal Kähler, e.g. [@Gini Lemma 5.3].
The last construction above is a quite natural way to obtain Vaisman manifolds by means of Kähler cones over Sasakian manifolds. More generally, given a Sasakian manifold $Q$, if we consider an automorphism of the Sasakian structure $\varphi \colon Q \to Q$, we can define $\varphi_{\lambda} \colon \mathscr{C}(Q) \to \mathscr{C}(Q)$, for some $\lambda \in \mathbb{R}^{+}$, such that
$\varphi_{\lambda} \colon (x,s) \to (\varphi(x),\lambda s).$
The map above also satisfies $\varphi_{\lambda}^{\ast} g_{\mathscr{C}} = \lambda^{2}g_{\mathscr{C}}$, and we can consider the manifold $$M_{\varphi,\lambda} = \mathscr{C}(Q) / \sim_{\varphi,\lambda},$$ where “$\sim_{\varphi,\lambda}$" stands for the equivalence relation $(x,s) \sim (\varphi(x),\lambda s)$. From these, we obtain a l.c.K. manifold defined by $M_{\varphi,\lambda}$. In the last setting above we have the following result.
\[fromsasakitovaisman\] Let $Q$ be a compact Sasakian manifold. Then the l.c.K. manifold $M_{\varphi,\lambda}$ obtained from $\varphi_{\lambda} \colon \mathscr{C}(Q) \to \mathscr{C}(Q)$ as before is Vaisman.
Notice that under the hypotheses of the last Proposition \[fromsasakitovaisman\], if we denote $$\label{grouppresentation}
\Gamma_{\lambda,\varphi} = \bigg \{ \varphi_{\lambda}^{n} \ \ \bigg | \ \ n \in \mathbb{Z} \bigg\},$$ it follows that $M_{\varphi,\lambda} = \mathscr{C}(Q) / \Gamma_{\lambda,\varphi}$ can be endowed with a structure of Vaisman manifold which comes from the globally conformal Kähler structure of the cone $\mathscr{C}(Q) = Q \times \mathbb{R}^{+}$.
\[biholomorphicsubgroups\] It is worth pointing out that the group $\Gamma_{\lambda,\varphi}$ aforementioned is a subgroup of the group $\mathcal{H}(\mathscr{C}(Q))$ of biholomorphisms $f \colon \mathscr{C}(Q) \to \mathscr{C}(Q)$, which satisfies $f^{\ast}g_{\mathscr{C}} = ag_{\mathscr{C}}$, for some $a \in \mathbb{R}^{+}$ (scale factor).
\[Vaismanfromsasaki\] In the general setting, i.e. when $Q$ is not necessarily compact, we can consider the homomorphism $$\label{scalehomomorphism}
\rho_{Q} \colon \mathcal{H}(\mathscr{C}(Q)) \to \mathbb{R}^{+},$$ which assigns to each element of $\mathcal{H}(\mathscr{C}(Q))$ its scale factor. Given a subgroup $\Gamma \subset \mathcal{H}(\mathscr{C}(Q))$ which satisfies $\rho_{Q}(\Gamma) \neq 1$, if we suppose that $f \circ \Phi_{t} = \Phi_{t} \circ f$, $\forall f \in \Gamma$, where $\Phi_{t}$ denotes the flow of $\frac{\partial}{\partial \psi}$, and that $\Gamma$ acts freely and properly discontinuously on $\mathscr{C}(Q)$, it follows that $M_{\Gamma} = \mathscr{C}(Q)/\Gamma$ is a Vaisman manifold. The l.c.K. structure on $M_{\Gamma}$ is obtained from the globally conformal Kähler structure which we have on $\mathscr{C}(Q)$, see for instance [@Gini Proposition 5.4].
Given a Vaisman manifold $M_{\Gamma} = \mathscr{C}(Q)/\Gamma$, obtained as above, we call the pair $(\mathscr{C}(Q),\Gamma)$ a presentation of $M_{\Gamma}$.
It is worthwhile to observe that every compact Vaisman manifold $M$ has a natural presentation $(\widetilde{M} = \mathscr{C}(\widetilde{Q}),\Gamma)$, where $\widetilde{M}$ is the universal covering space of $M$, and $\Gamma = \pi_{1}(M)$, see [@Kamishima]. The universal covering presentation $(\widetilde{M},\pi_{1}(M))$ is also called the [*maximal*]{} presentation of $M$.
The next result provides a complete description of compact Vaisman manifolds in terms of Sasakian manifolds.
\[minimalpresentation\] Let $(M,g,J,\theta)$ be a compact Vaisman manifold. Then $M$ admits a presentation $(\mathscr{C}(Q),\Gamma_{\lambda,\varphi})$, where $\Gamma_{\lambda,\varphi}$ is defined by a Sasakian automorphism $\varphi \colon Q \to Q$ and a dilatation $\tau_{\lambda}$, such that $\lambda \in \mathbb{R}^{+}$, $\lambda >1$. Moreover, we have a Riemannian submersion $f_{\theta} \colon M \to S^{1}$, such that $f_{\theta}^{-1}(c) = Q$, $\forall c \in S^{1}$.
Given a compact Vaisman manifold $M$, the relation between the universal covering space presentation $(\widetilde{M},\pi_{1}(M))$ and the presentation $(\mathscr{C}(Q),\Gamma_{\lambda,\varphi})$ provided by Theorem \[minimalpresentation\] is that $$\mathscr{C}(Q) = \widetilde{M}/\pi_{1}(\mathscr{C}(Q)),$$ thus the Kähler cone $\widetilde{M}$ is just $\mathscr{C}(\widetilde{Q})$, where $\widetilde{Q}$ is the Sasakian manifold given by the universal covering space of $Q$. Moreover, the group $\Gamma_{\lambda,\varphi}$ in Theorem \[minimalpresentation\] is given by the quotient group of $\pi_{1}(M)$ by the subgroup $$\label{Leegroup}
\displaystyle H_{\theta} = \Bigg \{ [\gamma] \in \pi_{1}(M) \ \Bigg | \ \int_{\gamma} \theta = 0 \Bigg \},$$ where $\theta \in \Omega^{1}(M)$ is the associated Lee form, i.e. we have $\Gamma_{\lambda,\varphi} = \pi_{1}(M)/H_{\theta}$.
Given a presentation $(\mathscr{C}(Q),\Gamma)$ of a Vaisman manifold $M$, we have that $\rho_{Q}(\Gamma) \subset \mathbb{R}^{+}$ is a finitely generated subgroup of $\mathbb{R}^{+}$, which implies that it is isomorphic to $\mathbb{Z}^{k}$, where $ 0 \leq k \leq b_{1}(M)$. Notice that if $k = 0$, it implies that $M$ is a Kähler manifold, thus the number $k$ measures the distance of $M$ from being Kähler.
An important fact related to Theorem \[minimalpresentation\] is that $\rho_{Q}(\Gamma_{\lambda,\varphi}) \cong \mathbb{Z}$, thus we have $M = \mathscr{C}(Q)/\mathbb{Z}$, such that $$\mathbb{Z} = \Big \langle (x,s) \mapsto (\varphi(x),\lambda s) \Big \rangle,$$ notice that, since $\lambda >1$, we have $\ker(\rho_{Q}) = \{{\text{id}}\}$. Therefore, the presentation provided in Theorem \[minimalpresentation\] is the minimal one in the sense that $\Gamma_{\lambda,\varphi}$ is the smallest group such that $M = N/\Gamma_{\lambda,\varphi}$, for some Kähler covering $N$. Further discussion about minimal presentations can be found in [@GiniI].
Vaisman manifolds and elliptic fibrations over Kähler manifolds {#Vaismantorusfibration}
===============================================================
This section is devoted to present some basic results related to the canonical foliation of compact Vaisman manifolds. The aim is to provide a precise description in terms of principal bundles and connections for strongly regular compact Vaisman manifolds, as well as for their geometric structures. We also present a detailed exposition of such manifolds as quotient spaces of principal $\mathbb{C}^{\times}$-bundles over compact complex manifolds. Further details on the subject presented in this section can be found in [@Vaisman], [@Chen], [@Tsukada], [@Dragomir].
The canonical foliation of a compact Vaisman manifold
-----------------------------------------------------
Given a compact Vaisman manifold $(M,g,J)$ one can consider the vector field $A = \theta^{\sharp} \in \Gamma(TM)$ (Lee vector field), where $g(X,A) = \theta(X)$, $\forall X \in \Gamma(TM)$. From this, one can take $B = JA$, and considers the anti-Lee form $\vartheta = -\theta \circ J$. The vector fields $A,B \in \Gamma(TM)$ are infinitesimal automorphism of the Hermitian structure, i.e. $$\label{equationfoliation}
\mathscr{L}_{A}J = \mathscr{L}_{B}J = 0, \ {\text{and}} \ \mathscr{L}_{A}g = \mathscr{L}_{B}g = 0.$$ Moreover, one can suppose also that $||\theta|| = ||\vartheta|| = 1$, and $[A,B] = 0$, see for instance [@Vaisman], [@Dragomir Pages 37-39].
The distribution generated by $A,B \in \Gamma(TM)$ defines a foliation $\mathcal{F}$ which is completely integrable, this foliation is called [*canonical ( characteristic) foliation*]{} of $(M,g,J)$ cf. [@Vaisman], [@Tsukada].
As we shall see bellow, the leaves of $\mathcal{F}$ are totally geodesic and locally flat complex submanifolds. Moreover, $\mathcal{F}$ is transversally Kählerian. Thus, $\mathcal{F}$ plays an important role when we investigate topological and complex analytic properties of a compact Vaisman manifold.
Before we state the main theorem which ensures the aforementioned facts about $\mathcal{F}$, let us collect some basic results.
We recall an important result related to regular foliations on compact manifolds.
Let $\mathcal{F}$ be a regular foliation on a compact manifold $M$, then we have the following facts:
1. The leaves of $\mathcal{F}$ are compact submanifolds of $M$;
2. The leaf space $M / \mathcal{F}$ is a (Hausdorff) compact manifold;
3. The natural projection is a smooth map.
In general, given a foliated manifold $(M,\mathcal{F})$ ($\operatorname{rank}{(\mathcal{F})} < \dim(M)$), the topology of the leaf space $M / \mathcal{F}$ might be very complicated, possibly non-Hausdorff. Since we are interested in a more restrictive setting, unless otherwise stated, we shall suppose that the leaf space $M / \mathcal{F}$ of a regular foliation is Hausdorff.
Given a compact Vaisman manifold $(M,g,J)$, with associated Lee form $\theta \in \Omega^{1}(M)$, we denote $$\mathcal{F} = \mathcal{F}_{A} \oplus \mathcal{F}_{B},$$ where $\mathcal{F}_{A}$ is the foliation generated by $A = \theta^{\sharp} \in \Gamma(TM)$, and $\mathcal{F}_{B}$ is the foliation generated by $B = JA \in \Gamma(TM)$.
\[regvaisman\] A compact Vaisman manifold $(M,g,J)$ is called regular, if $\mathcal{F}$ is a regular foliation, and strongly regular, if $\mathcal{F}_{A}$ and $\mathcal{F}$ are both regular.
Now, we consider the following result.
\[flatvaisman\] A compact l.c.K. manifold $(M,g,J)$ is a regular Vaisman if and only if there exists a flat principal $S^{1}$-bundle $\pi \colon M \to M/\mathcal{F}_{A}$, with $Q = M/\mathcal{F}_{A}$ being a connected and compact Sasakian manifold. If this is the case, the restrictions of $\pi$ to the leaves of $\mathcal{F}_{A}$ are covering maps.
\[monodromyvaisman\] It is worth pointing out that from Theorem \[flatvaisman\], it follows that for every flat connection $\beta \in \Omega^{1}(M,\mathfrak{u}(1))$ on a compact Vaisman manifold $M$, we can consider $$\label{holonomyrep}
\text{Hol}_{\beta} \colon \pi_{1}(Q,x_{0}) \to S^{1},$$ where $\text{Hol}_{\beta} \colon [\gamma] \mapsto {\mathrm{e}}^{\int_{\widetilde{\gamma}}\beta}$, denotes the holonomy representation induced by $\beta$, and $\widetilde{\gamma} \colon S^{1} \to M$ denotes the horizontal lift of $\gamma \colon S^{1} \to Q$ with respect to $\beta \in \Omega^{1}(M,\mathfrak{u}(1))$. From this, we obtain that
$M \cong \widetilde{Q} \times_{\text{Hol}_{\beta}} S^{1} = \widetilde{Q} \times S^{1}/\pi_{1}(Q,x_{0})$,
such that $p \colon \widetilde{Q} \to Q$ is the universal covering space of $Q$, it is worth mentioning that the twisted product above is obtained by considering the action of $\pi_{1}(Q,x_{0})$ on $\widetilde{Q}$ by deck transformation, and the action of $\pi_{1}(Q,x_{0})$ on $S^{1}$ through the holonomy representation \[holonomyrep\] induced by $\beta$. Since we have a canonical flat connection $\beta = \sqrt{-1} \theta$, induced by the Lee form $\theta \in \Omega^{1}(M)$, we can realize $M$ as an associated bundle of the (trivial) principal $S^{1}$-bundle defined by the Vaisman manifold $\widetilde{Q} \times S^{1}$, see for instance [@Tondeur Chapter 4].
The relation between Theorem \[minimalpresentation\] and Theorem \[flatvaisman\] is established by means of Tischler’s Theorem [@Tischler] as follows. From Theorem \[flatvaisman\], one can consider a flat connection $\beta \in \Omega^{1}(M,\mathfrak{u}(1))$, let us denote also by $\beta$ the induced closed $1$-form on $M$. Fixing $x_{0} \in M$, and $\widetilde{x}_{0} \in \widetilde{M}$, such that $p(\widetilde{x}_{0}) =x_{0}$, so one obtains the following induced maps from $\beta$:
1. $f_{\beta} \colon \widetilde{M} \to \mathbb{R}$, $f_{\beta}(\widetilde{x}) = \int_{\widetilde{x}_{0}}^{\widetilde{x}}p^{\ast} \beta$, $\forall \widetilde{x} \in \widetilde{M}$,
2. $\mathscr{P}_{\beta} \colon \pi_{1}(M,x_{0}) \to \mathbb{R}$, $\mathscr{P}_{\beta}([\gamma]) = \int_{\gamma} \beta$, $\forall [\gamma] \in \pi_{1}(M,x_{0})$.
The maps above are well defined, and they are related by
$f_{\beta}([\gamma] \cdot \widetilde{x}) = f_{\beta}(\widetilde{x}) + \mathscr{P}_{\beta}([\gamma])$,
$\forall [\gamma] \in \pi_{1}(M,x_{0}) = \text{Deck}(\widetilde{M})$, and $\forall \widetilde{x} \in \widetilde{M}$. Since $\mathscr{P}_{\beta}$ is also a homomorphism, if one considers $H_{\beta} = \ker(\mathscr{P}_{\beta})$, then one obtains an induced map from $f_{\beta}$ on $\widetilde{M}/H_{\beta}$, i.e.
$f_{\beta} \colon \widetilde{M}/H_{\beta} \to \mathbb{R}$.
Now, if one considers the group generated by the periods of $\beta$, i.e. $\text{Per}(\beta) = \mathscr{P}_{\beta}(\pi_{1}(M,x_{0}))$, and $\Gamma_{\beta} = \pi_{1}(M,x_{0})/H_{\beta}$, the last map above descends to
$f_{\beta} \colon M \to \mathbb{R}/\text{Per}(\beta)$.
Now, we have two possibilities:
1. $\text{Per}(\beta) \subset \mathbb{R}$ is infinite cyclic subgroup,
2. $\text{Per}(\beta) \subset \mathbb{R}$ is dense in $\mathbb{R}$,
more details about the facts above can be found in [@Tondeur Chapter 4]. The key point in Theorem \[minimalpresentation\] is that, if we take $\beta = \theta$ in the construction above, we obtain
$f_{\theta} \colon M \to \mathbb{R}/\text{Per}(\theta)$,
and $\text{Per}(\theta) \cong \Gamma_{\theta} \cong \mathbb{Z}$. Thus, for a suitable automorphism $\varphi \colon Q \to Q$, where $Q \subset M$ stands for a typical fiber of the fibration defined by $f_{\theta}$, one can show that
$M \cong Q \times_{\mathbb{Z}}\mathbb{R}$,
where $\mathbb{Z}$ acts on $Q \times \mathbb{R}$, by $n \cdot (x,s) = (\varphi^{n}(x), s + n)$, $\forall (x,s) \in Q \times \mathbb{R}$, and $\forall n \in \mathbb{Z}$. This last action, up to scale factor, turns out to be the action of $\Gamma_{\lambda,\varphi}$ on $Q \times \mathbb{R}$ as in Theorem \[minimalpresentation\].
The next result describes the properties of the leaves of the characteristic foliation $\mathcal{F}$ of a compact Vaisman manifold $(M,g,J)$ previously mentioned.
\[blikekahler\] The characteristic foliation $\mathcal{F}$ is a complex analytic 1-dimensional foliation of $(M,g,J)$ with complex parallelizable leaves which are totally geodesic and locally flat submanifolds of $M$. The metric of $M$ is a bundle-like transversally Kählerian metric with respect to $\mathcal{F}$.
The result above follows essentially from Equation \[equationfoliation\] together with the identities
$\nabla_{A}A = \nabla_{A}B = \nabla_{B}A = \nabla_{B}B = 0,$
the last statement of Theorem \[blikekahler\] follows from Equation \[equationfoliation\] and [@Chen Theorem 6.2].
An important consequence of Theorem \[blikekahler\] is that one can describe locally the Riemannian metric $g$ associated to the underlying Hermitian structure $h = g -\sqrt{-1}\Omega$ by [^2] $$\label{localvaisman}
g = h_{a\overline{b}}dz^{a} \odot d \overline{z}^{b} + \frac{1}{2} \Big (\theta + \sqrt{-1}\vartheta \Big) \odot \Big (\theta - \sqrt{-1} \vartheta \Big),$$ here we consider the $(1,0)$ cobasis $\{dz^{a}, \theta + \sqrt{-1}\vartheta\}$, where $z^{a}$, $a = 1, \ldots, n-1$, are complex coordinates which defines $\mathcal{F}$ (locally) by $dz^{a}=0$, $a = 1, \ldots, n-1$. The bundle-like character of the metric \[localvaisman\] means that the coefficients $h_{a\overline{b}}$ depend only on $z^{a},\overline{z}^{a}$, $a = 1, \ldots, n-1$, see for instance [@Vaisman].
It is worth observing that, by considering a compact Vaisman manifold $(M,g,J)$, since $$\mathscr{L}_{A}\Omega = (\mathscr{L}_{A}g)\big (J\otimes \text{id} \big ) + g\big ((\mathscr{L}_{A}J)\otimes \text{id}\big ),$$ it follows from Equation \[equationfoliation\] and Cartan’s magic formula that $$0 = \mathscr{L}_{A}\Omega = ||\theta||^{2}\Omega - \theta \wedge \vartheta + d\vartheta.$$ Thus, since $||\theta|| = 1$, it follows that $\Omega = -d\vartheta + \theta \wedge \vartheta$. From Equation \[localvaisman\], we conclude that $$-d\vartheta = \sqrt{-1} h_{a\overline{b}}dz^{a} \wedge d \overline{z}^{b},$$ so the transversal Kähler structure with respect to the characteristic foliation $\mathcal{F}$ is determined by the anti-Lee form $\vartheta \in \Omega^{1}(M)$.
\[canonicalvaisman\] As we have seen previously, we have a canonical globally conformal Kähler structure $(\widetilde{\Omega}, J_{\mathscr{C}})$ on a Kähler cone $\mathscr{C}(Q)$ over a Sasakian manifold $Q$, such that $$\label{globalconformal1}
\widetilde{\Omega} = \frac{d\eta}{2} + \displaystyle d\psi \wedge \eta, \ \ {\text{and}} \ \ \widetilde{g} = g_{Q} + d\psi \otimes d\psi.$$ see Equation \[globalconform\]. Now, by considering $$\label{structure}
\widetilde{\theta} = -2d\psi, \ \ {\text{and}} \ \ \displaystyle g_{Q} = \frac{1}{2}d\eta({\rm{id}}\otimes \phi) + \eta \otimes \eta,$$ it follows that $$\label{globalconformal}
\widetilde{g} = \frac{1}{2}d\eta({\rm{id}}\otimes \phi) + \frac{1}{2} \Big (\eta - \sqrt{-1}d\psi \Big) \odot \Big (\eta + \sqrt{-1} d\psi \Big).$$ Therefore, if we suppose $Q$ as being compact, from Proposition \[fromsasakitovaisman\] we have that the l.c.K. manifold $M_{\varphi,\lambda} = (\mathscr{C}(Q), \Gamma_{\varphi,\lambda})$, presented by some $\Gamma_{\varphi,\lambda} \subset \mathcal{H}(\mathscr{C}(Q))$, is a compact Vaisman manifold with Vaisman structure $(\Omega, J, \theta)$ induced from $(\widetilde{\Omega}, J_{\mathscr{C}}, \widetilde{\theta})$, notice that $\widetilde{\nabla}\widetilde{\theta} \equiv 0$, where $\widetilde{\nabla}$ is the Levi-Civita connection associated to $\widetilde{g}$.
The next result provides a precise description of compact strongly regular Vaisman manifolds in terms of principal toroidal bundles [@Vaisman], [@Chen].
\[vaismanclass\] Let $(M,g,J)$ be a compact connected strongly regular Vaisman manifold. Then the projection
$\pi \colon M \to M/\mathcal{F},$
defines a principal $T_{\mathbb{C}}^{1}$-bundle over a compact Hodge manifold $N = M/\mathcal{F}$. Moreover, $M$ also defines a flat principal $S^{1}$-bundle over a compact Sasaki manifold $Q = M/\mathcal{F}_{A}$, which in turn can be realized as a principal $S^{1}$-bundle over $N$ whose Chern class differs only by a torsion element from the Chern class of the induced principal $T_{\mathbb{C}}^{1}$-bundle defined by $M$.
The converse of Theorem \[vaismanclass\] is given by the following result.
\[vaismanbundle\] Let $\pi \colon M \to N$ be a holomorphic principal $T_{\mathbb{C}}^{1}$-bundle over a compact Kähler manifold. If $M$ admits a connection $\Psi$ with the following properties:
1. $\Psi \circ J = \sqrt{-1} \Psi$,
2. there exists a Kähler form $\omega$ on $N$, such that $d\Psi = c \pi^{\ast} \omega$ for a nonzero real constant $c$,
then $M$ admits a Vaisman structure $(\Omega,J,\theta)$, such that the leaves of the associate characteristic foliation $\mathcal{F}$ coincides with the fibers of the bundle.
Notice that, under the hypotheses of Theorem \[vaismanbundle\], one can write $\Psi \in \Omega^{1}(M;\mathbb{C})$ as $$\label{connectiontorus}
\Psi = \Re(\Psi) + \sqrt{-1} \Im(\Psi).$$ It follows from condition (1) of Theorem \[vaismanbundle\] that $$\label{connectioncond1}
\Re(\Psi) \circ J = - \Im(\Psi).$$ Moreover, from condition (2) of Theorem \[vaismanbundle\] it can be easily derived that $$\label{connectioncond2}
d\big (\Re(\Psi) \big) = c\pi^{\ast} \omega, \ \ {\text{and}} \ \ d\big (\Im(\Psi) \big) = 0,$$ for a basic Kähler form $\omega \in \Omega^{1,1}(N)$.
Therefore, from $\Psi \in \Omega^{1}(M;\mathbb{C})$ one can define the Riemannian metric on $M$ by setting $$\label{bundlemetric}
g = \frac{1}{2} \Big (d\Psi({\text{id}} \otimes J) + \Psi \odot \overline{\Psi} \Big).$$ It is straightforward to see that the fundamental form $\Omega = g(J\otimes {\rm{id}})$ is given by $$\Omega = \frac{c}{2}\pi^{\ast}\omega - \Im(\Psi) \wedge \Re(\Psi) \Longrightarrow d\Omega = 2\Im(\Psi) \wedge \Omega.$$ Hence, from the aforementioned fundamental $2$-form, it follows that the Lee form $\theta \in \Omega^{1}(M)$ associated to the l.c.K. structure defined by the connection $\ref{connectiontorus}$ is given by $$\label{Leefrombundle}
\theta = 2\Im(\Psi).$$ Also, it can be easily shown that $\nabla \theta = 0$, where $\nabla$ is the Levi-Civita connection associated to the Riemannian metric defined in \[bundlemetric\], see [@Tsukada Theorem 4.1] for further details.
The results which we have briefly described so far provide an equivalence between the class of compact strongly regular Vaisman manifolds and the class of holomorphic principal $T_{\mathbb{C}}^{1}$-bundles over compact Hodge manifolds. Therefore, in the setting of compact strongly regular Vaisman manifolds we have the following diagram of fibrations:
((Q),\_,J\_ ) & (M,,J,) & (Q,g\_[Q]{},,,)\
& (N,,J’ )
It is worth mentioning that the Riemannian geometry which underlies the diagram above is a fruitful ground to study Kaluza-Klein theory and Riemannian submersions, e.g. [@KK], [@Falcitelli].
\[toyexample\] A well-known example which illustrates the situation described above is provided by the following diagram
S\^[3]{} & S\^[3]{} S\^[1]{} & S\^[3]{}\
& S\^[2]{}
If one considers the identification $S^{2} \cong \mathbb{C}P^{1}$, and $S^{3}\times \mathbb{R} \cong {\text{Tot}}(\mathscr{O}_{\mathbb{C}P^{1}}(-1)^{\times})$, where $``{\text{Tot}}"$ stands for the manifold underlying the total space of the principal $\mathbb{C}^{\times}$-bundle, the last diagram can be rewritten as
(\_[P\^[1]{}]{}(-1)\^) & (\_[P\^[1]{}]{}(-1)\^)/ & S\^[3]{}\
& P\^[1]{}
As we have mentioned in the introduction, this example is a prototype for our approach from the perspective o homogeneous geometry and Lie theory. We shall analyze it again after developing some ideas.
Negative line bundles and principal elliptic bundles {#negativeelliptic}
----------------------------------------------------
As we have seen, every compact strongly regular Vaisman manifold can be realized as the total space of a holomorphic principal $T_{\mathbb{C}}^{1}$-fibration over a compact Hodge manifold. In this subsection we shall discuss how to construct such fibrations by means of negative line bundles over compact Hodge manifolds. The idea is to describe in details a natural generalization for the prototype provided in the last diagram of the previous subsection.
Let us recall some basic facts and generalities related to holomorphic line bundles over compact complex manifolds.
\[Griffithsnegative\] Let $L \in {\text{Pic}}(X)$ be a holomorphic line bundle over a compact complex manifold $X$. We say that $L$ is (Griffiths) negative if there exists a system of trivializations $f_{i} \colon L|_{U_{i}} \to U_{i} \times \mathbb{C}$ (with transition functions $g_{ik}$) and a Hermitian metric $H$ on $L$, such that
1. $H(f_{i}^{-1}(z,w),f_{i}^{-1}(z,w)) = h_{i}(z)|w|^{2}$, where $h_{i} \colon U_{i} \to \mathbb{R}^{+}$ is a smooth function,
2. ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, ${\rm{K}}_{H}(u) = H(u,u)$, is strictly plurisubharmonic.
Under the hypotheses of the definition above the following characterization will be important for our approach.
\[Gnegative\] A line bundle $L \in {\text{Pic}}(X)$ is (Griffiths) negative if only if there exists a system of positive smooth functions $\varrho_{i} \colon U_{i} \to \mathbb{R}^{+}$, such that:
1. $-\log(\varrho_{i})$ is strictly plurisubharmonic on $U_{i}$,
2. $\varrho_{i} = |g_{ik}| \cdot \varrho_{k}$ on $U_{i} \cap U_{k}$.
In the setting of the proposition above, we have the Chern class of $L \in {\text{Pic}}(X)$ represented by the real $(1,1)$-form $\omega \in \Omega^{1,1}(X)$ defined locally by $$\label{connection}
\displaystyle \omega|_{U_{i}} = \frac{ \sqrt{-1}}{\pi} \partial \overline{\partial}\log(\varrho_{i}).$$ Thus, since for $L \in {\text{Pic}}(X)$ Griffiths negative we have $-\log(\varrho_{i})$ strictly plurisubharmonic on $U_{i}$, it follows that $\omega$ is a negative $(1,1)$-form, which implies that $L^{-1}$ is positive in the sense of Kodaira. It is straightforward to show that a positive line bundle $L$ (in the sense of Kodaira) has its inverse $L^{-1}$ Griffiths negative.
\[griffithsequiv\] Notice that, given a Griffiths negative line bundle $L \in {\text{Pic}}(X)$ with a Hermitian structure $H$, since on each open set $L|_{U_{i}}$ we have
$H(f_{i}^{-1}(z,w),f_{i}^{-1}(z,w)) = h_{i}(z)|w|^{2},$
by setting $h_{i} = \frac{1}{\varrho_{i}^{2}}$, from the Chern connection $\nabla |_{U_{i}} = d + H^{-1}\partial H $ induced form the Hermitian structure $H$, it it follows that $c_{1}(L,\nabla) = [\omega]$, where $\omega$ is given in Equation \[connection\]. It is worth pointing out that this description of $c_{1}(L)$ does not depend on the choice of the Hermitian structure. For further details on this subject we suggest [@NEGATIVELINEBUNDLE], [@DANIEL].
Given a compact complex manifold $X$, and a negative line bundle $L \in {\text{Pic}}(X)$, by fixing a Hermitian metric as in Remark \[griffithsequiv\] on $L$, one can consider the sphere bundle $$\label{spherebundle}
Q(L) = \bigg \{ u \in L^{\times} \ \bigg | \ \sqrt{H(u,u)} = 1 \bigg \}.$$ By taking a Chern connection $\nabla$ on $L$ induced from $H$, we obtain a system of $(1,0)$-forms $A_{i} \in \Omega^{1,0}(U_{i})$, such that $\nabla |_{i} = d + A_{i} $. If one sets $\mathcal{A}_{i} = \frac{1}{2}(A_{i} - \overline{A}_{i})$, on each $U_{i} \subset X$, then one obtains a connection $1$-form $\eta' \in \Omega^{1}(Q(L);\mathfrak{u}(1))$, locally given by $$\label{principalconnection}
\displaystyle \eta' = \pi^{\ast}\mathcal{A}_{i} + \sqrt{-1}d\sigma_{i},$$ such that $\pi \colon Q(L) \to X$ is the natural principal $S^{1}$-bundle projection, see [@KN Proposition 1.4]. Moreover, it follows that $$\frac{\sqrt{-1}}{2 \pi} d\eta' = \frac{\sqrt{-1}}{2\pi} \pi^{\ast} F_{\nabla},$$ where $F_{\nabla}$ is the curvature of $\nabla$. Since $L$ is negative, one can define a contact structure on $Q(L)$ by setting $\eta = -\sqrt{-1} \eta'$.
Now, we consider the following result.
\[almostcircle\] Let $Q$ be a principal ${\rm{U}}(1)$-bundle over a complex manifold $(N,J')$. Suppose we have a connection $1$-form $\sqrt{-1}\eta$ on $Q$ such that $d\eta = \pi^{\ast}\omega$. Here $\pi$ denotes the projection of $Q$ onto $N$, and $\omega$ is a $2$-form on $N$ satisfying
$\omega(J'X,J'Y) = \omega(X,Y),$
for $X,Y \in \Gamma(TN)$. Then, we can define a $(1,1)$-tensor field $\phi$ on $M$ and a vector field $\xi$ on $M$ such that $(\phi,\xi,\eta)$ is a normal almost contact structure on $M$.
Let us recall that an almost contact manifold is a (2n+1)-dimensional smooth manifold $M$ endowed with structure tensors $(\phi, \xi,\eta)$, such that $\phi \in {\text{End}}(TM)$, $\xi \in \Gamma(TM)$, and $\eta \in \Omega^{1}(M)$, satisfying $$\label{almostcontact}
\phi \circ \phi = - {\rm{id}} + \eta \otimes \xi, \ \ \eta(\xi) = 1.$$ An almost contact structure is said to be normal if additionally satisfies $$\label{normality}
\big [ \phi,\phi \big ] + d\eta \otimes \xi = 0,$$ where $[\phi,\phi]$ is the Nijenhuis torsion of $\phi$ (see Equation \[sasakicondition\]).
If we apply Theorem \[almostcircle\] above on $Q(L)$ defined in \[spherebundle\], we obtain a Sasakian structure $(g_{Q(L)},\phi,\xi,\eta)$ on $Q(L)$, see [@Blair Chapter 6], such that $$\label{sasakistructure}
\eta = -\sqrt{-1} \eta', \ \ \xi = \displaystyle \frac{\partial}{\partial \sigma}, \ \ \phi(X) := \begin{cases}
(J'\pi_{\ast}X)^{H}, \ \ \ {\text{if}} \ \ X \bot \xi.\\
\ \ \ \ \ 0 \ \ \ \ \ \ \ , \ \ \ {\text{if}} \ \ X \parallel \xi . \\
\end{cases}$$ Here we denote by $(J'\pi_{\ast}X)^{H}$ the horizontal lift of $J'\pi_{\ast}X$ relative to the connection $\sqrt{-1}\eta \in \Omega^{1}(Q(L);\mathfrak{u}(1))$, and $$\label{Sasakimetric}
g_{Q(L)} = \frac{1}{2}d\eta(\text{id}\otimes \phi) + \eta \otimes \eta.$$ Notice that in the definition of the above metric we have $d\eta = \pi^{\ast}\omega_{X}$, where $\omega_{X} \in \Omega^{1,1}(X)$ is a Kähler form, such that $-\omega_{X} \in 2\pi c_{1}(L)$.
As we have seen so far, on can associate to every negative line bundle $L$ over a compact complex manifold $X$ a Sasaki manifold $Q(L)$, with structure tensors $(g_{Q(L)},\phi,\xi,\eta)$ completely determined by a principal connection $\sqrt{-1}\eta \in \Omega^{1}(Q(L); \mathfrak{u}(1))$. Moreover, we have $$\label{coneandtot}
{\text{Tot}}(L^{\times}) \cong Q(L) \times \mathbb{R}^{+},$$ such that the above isomorphism is defined by $$u \in {\text{Tot}}(L^{\times}) \mapsto \Bigg (\frac{u}{\sqrt{{\rm{K}}_{H}(u)}}, \sqrt{{\rm{K}}_{H}(u)} \Bigg) \in Q(L) \times \mathbb{R}^{+},$$ where ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, ${\rm{K}}_{H}(u) = H(u,u)$, see Definition \[Griffithsnegative\].
It is straightforward to show that the identification \[coneandtot\] also provides an identification as complex manifolds between ${\text{Tot}}(L^{\times})$ and $(\mathscr{C}(Q(L)),J_{\mathscr{C}})$, here $J_{\mathscr{C}}$ is defined as in \[complexcone\]. In fact, we have $(\mathscr{C}(Q(L)),J_{\mathscr{C}})$ as a holomorphic principal $\mathbb{C}^{\times}$-bundle over $X$ for which the underlying $\mathbb{C}^{\times}$-action is defined by the flow of the holomorphic vector field $\xi -\sqrt{-1}J_{\mathscr{C}}(\xi)$[^3], so the isomorphism \[coneandtot\] is an isomorphism of holomorphic principal $\mathbb{C}^{\times}$-bundles. Furthermore, by taking $r^{2} = {\rm{K}}_{H}$, from this last identification, we obtain $$\label{gpotential}
\frac{\sqrt{-1}}{2} \partial \overline{\partial} {\rm{K}}_{H} = d\Big(\frac{r^{2}\eta}{2}\Big),$$ which implies that ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$ defines a global Kähler potential. Actually, locally we have
${\rm{K}}_{H}(f_{i}^{-1}(z,w)) = {\rm{K}}_{H}(z,w) = h_{i}(z)w\overline{w} = {\mathrm{e}}^{\log(h_{i}(z))}w\overline{w}$,
where $f_{i} \colon L|_{U_{i}} \to U_{i} \times \mathbb{C}$ is a local trivialization. Thus, it follows that $$\displaystyle \partial \overline{\partial} {\rm{K}}_{H} = {\mathrm{e}}^{\log(h_{i})}|w|^{2}\bigg [ \partial \log(h_{i}) \wedge \overline{\partial} \log(h_{i}) + \Big (\frac{dw}{w} \Big) \wedge \overline{\partial}\log(h_{i}) + \partial \log(h_{i}) \wedge \Big (\frac{d \overline{w}}{\overline{w}} \Big ) + \partial \overline{\partial} \log(h_{i}) + \Big (\frac{dw}{w} \Big) \wedge \Big(\frac{d \overline{w}}{\overline{w}} \Big ) \bigg].$$
Now, since we have $r^{2} = {\rm{K}}_{H}$, we obtain $$\label{polar}
w = r {\mathrm{e}}^{-\frac{\log(h_{i})}{2}}{\mathrm{e}}^{\sqrt{-1}\sigma_{i}},$$ which implies that
$\displaystyle \frac{dw}{w} = \frac{dr}{r} - \frac{d\log(h_{i})}{2} + \sqrt{-1}d\sigma_{i}$, $\displaystyle \frac{d \overline{w}}{\overline{w}} = \frac{dr}{r} - \frac{d\log(h_{i})}{2} - \sqrt{-1}d\sigma_{i}$,
from this we have
$\displaystyle \partial \overline{\partial} {\rm{K}}_{H} = r^{2} \bigg [ \frac{dr}{r} \wedge \big ( (\overline{\partial} - \partial)\log(h_{i}) -2 \sqrt{-1}d\sigma_{i} \big) + \partial \overline{\partial} \log(h_{i}) \bigg]$.
Since $h_{i} = \frac{1}{\varrho_{i}^{2}}$, see Remark \[griffithsequiv\], it follows from Equation \[connection\], and Equation \[principalconnection\], that
$\displaystyle \frac{\sqrt{-1}}{2} \partial \overline{\partial} {\rm{K}}_{H} = rdr \wedge \eta + \frac{r^{2}}{2}d\eta = d\Big(\frac{r^{2}\eta}{2} \Big)$.
Hence, we have the equality given in Equation \[gpotential\].
From the last computations, given a negative line bundle $L \to X$, over a compact complex manifold $X$, we can use the characterization \[coneandtot\] to construct elliptic fibrations over $X$ as follows. Let $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| < 1$, from this we can consider the infinite cyclic group $$\label{cyclicgroup}
\Gamma_{\lambda} = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}.$$ Since $L^{\times}$ defines a principal $\mathbb{C}^{\times}$-bundle over $X$, we can consider the action of $\Gamma_{\lambda} \subset \mathbb{C}^{\times}$ on ${\text{Tot}}(L^{\times})$ by restriction. From the identification ${\text{Tot}}(L^{\times}) = \mathscr{C}(Q(L))$, it follows that $$\label{principalholomorphic}
u \cdot a \mapsto \bigg ( \frac{u}{\sqrt{{\rm{K}}_{H}(u)}} \frac{a}{|a|}, |a|\sqrt{{\rm{K}}_{H}(u)} \bigg),$$ $\forall a \in \Gamma_{\lambda}$, and $\forall u \in {\text{Tot}}(L^{\times})$. Therefore, since $\lambda = |\lambda|{\mathrm{e}}^{ \sqrt{-1}\sigma(\lambda)}$, we have that the action of $\Gamma_{\lambda}$ on ${\text{Tot}}(L^{\times})$ corresponds to the action of $\Gamma_{|\lambda|,\sigma(\lambda)}$ on $\mathscr{C}(Q(L))$ (see \[grouppresentation\]), where $\sigma(\lambda) \colon Q(L) \to Q(L)$ is defined by
$\sigma(\lambda) \colon x \mapsto x \cdot {\mathrm{e}}^{ \sqrt{-1}\sigma(\lambda)}$,
$\forall x \in Q(L)$, i.e. $\sigma(\lambda)$ is an automorphism, which preserves the Sasakian structure of $Q(L)$, induced by the flow of $\xi$. From this, we can consider $\varphi_{\lambda} \colon \mathscr{C}(Q(L)) \to \mathscr{C}(Q(L))$, such that $\varphi_{\lambda}(x,s) = (\sigma(\lambda)(x), |\lambda|s)$, $\forall (x,s) \in \mathscr{C}(Q(L))$, and describe the group $\Gamma_{\lambda}$ as
$\Gamma_{\lambda} = \bigg \{ \varphi_{\lambda}^{n} \in {\rm{Diff}}\big(\mathscr{C}(Q(L)) \big) \ \ \bigg | \ \ n \in \mathbb{Z}\bigg \}$.
It is straightforward to check that [^4] $ \Gamma_{\lambda} \subset \mathcal{H}(\mathscr{C}(Q(L)))$, see Remark \[biholomorphicsubgroups\], and also that $\rho_{Q(L)}(\Gamma_{\lambda}) \neq 1$, see Remark \[Vaismanfromsasaki\]. Now, if we consider $\psi = \log(r) \colon \mathscr{C}(Q(L)) \to \mathbb{R}$, we can check that every $\varphi_{\lambda}^{n} \in \Gamma_{\lambda}$ transforms the coordinate $\psi$ by
$\psi \mapsto \psi + n \log(|\lambda|)$.
Thus, by taking the flow $\Phi_{t}$ of $\frac{\partial}{\partial \psi}$, it follows that
$\Phi_{t} \big (\varphi_{\lambda}^{n}(x,\psi) \big) = \big(x\cdot {\mathrm{e}}^{ \sqrt{-1}\sigma(\lambda)}, \psi + n \log(|\lambda|) + t \big) = \varphi_{\lambda}^{n}\big (\Phi_{t}(x,\psi) \big)$,
which implies that $\Phi_{t} \circ \varphi_{\lambda}^{n} = \varphi_{\lambda}^{n} \circ \Phi_{t}$. Therefore, we have that $$M_{\Gamma_{\lambda}} = {\text{Tot}}(L^{\times})/\Gamma_{\lambda} = \big (\mathscr{C}(Q(L)),\Gamma_{\lambda} \big ),$$ defines a Vaisman manifold, see Remark \[Vaismanfromsasaki\] or [@Gini Proposition 5.4].
Now, let us briefly discuss the aspects of $M_{\Gamma_{\lambda}}$ as a principal bundle over $X$. At first, consider $$\displaystyle \varpi = \frac{\log(\lambda)}{2\pi \sqrt{-1}},$$ here as before we have $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| < 1$. From this, it follows that the imaginary part of $\varpi$ is a positive real number. Now, we can take the lattice $$\Lambda = \mathbb{Z} + \varpi \mathbb{Z},$$ and consider the complex elliptic curve $$\mathbb{E}(\Lambda) = \mathbb{C} / \Lambda.$$ From the natural projection map $\pi_{\Lambda} \colon \mathbb{C} \to \mathbb{E}(\Lambda)$, we can define a homomorphism $h \colon \mathbb{C}^{\times} \to \mathbb{E}(\Lambda)$, such that $$h(w) = \pi_{\Lambda} \bigg ( \frac{\log(w)}{2\pi \sqrt{-1}}\bigg),$$ $\forall w \in \mathbb{C}^{\times}$. It follows that $\ker(h) = \Gamma_{\lambda}$, which implies that $$\mathbb{E}(\Lambda) \cong \mathbb{C}^{\times}/\Gamma_{\lambda}.$$ Therefore, since that $L^{\times}$ is a principal $\mathbb{C}^{\times}$-bundle over $X$, it follows that $$\label{ellipticprincipal}
M_{\Gamma_{\lambda}} = L^{\times}/\Gamma_{\lambda} = \big (L^{\times} \times \mathbb{E}(\Lambda) \big) / \mathbb{C}^{\times}.$$ Thus, we have $M_{\Gamma_{\lambda}}$ realized as an $\mathbb{E}(\Lambda)$-bundle over $X$, e.g. [@KN]. Notice that, since $\Gamma_{\lambda} \subset \mathbb{C}^{\times}$ is a normal subgroup, and $\mathbb{E}(\Lambda) = T_{\mathbb{C}}^{1}$, we have that $M_{\Gamma_{\lambda}}$ is in fact a principal $T_{\mathbb{C}}^{1}$-bundle over $X$, see for instance [@Gund Proposition 2.2.20].
\[mainremark\] In the above setting, in order to provide an explicit description of the Vaisman structure on $M_{\Gamma_{\lambda}}$ in terms of its principal bundle features, we proceed as follows: Considering $M_{\Gamma_{\lambda}} $ as a principal elliptic bundle over $X$, we can obtain a connection on $M_{\Gamma_{\lambda}}$, satisfying the the conditions of Theorem \[vaismanbundle\], from a principal $\mathbb{C}^{\times}$-connection on $L^{\times}$. In order to see that, we define $\widetilde{\Psi} \in \Omega^{1}(L^{\times};\mathbb{C})$, such that $$\label{potentiallee}
\Re(\widetilde{\Psi}) = \frac{1}{2}d^{c}\log({\rm{K}}_{H}), \ \ {\text{and}} \ \ \Im(\widetilde{\Psi}) = -\frac{1}{2}d \log({\rm{K}}_{H}),$$ where $d^{c} = \sqrt{-1}( \overline{\partial} - \partial)$. Notice that from the differential $1$-forms in \[potentiallee\] we can recover the canonical globally conformally Kähler structure on ${\text{Tot}}(L^{\times})$ given in Remark \[canonicalvaisman\]. Actually, if we consider ${\text{Tot}}(L^{\times}) = \mathscr{C}(Q(L))$, by setting $r^{2} = {\rm{K}}_{H}$, it follows that $\widetilde{\theta} = -2d\psi$, where $\psi = \log(r)$ (see Equation \[structure\]). Moreover, we have $$\label{contactlee}
\displaystyle d^{c}\log({\rm{K}}_{H}(z,w)) = \sqrt{-1}( \overline{\partial} - \partial) \log(h_{i}(z)) - \frac{\sqrt{-1}}{|w|^{2}}\big (\overline{w}dw - wd\overline{w} \big),$$ where $(z,w)$ are local coordinates in $L^{\times}|_{U_{i}}$, see Definition \[Griffithsnegative\]. Since $w \in \mathbb{C}^{\times}$, it can be represented by polar coordinates as in Equation \[polar\], which provides
$\displaystyle \frac{\sqrt{-1}}{|w|^{2}}\big (\overline{w}dw - wd\overline{w} \big) = -2d\sigma_{i}$.
Therefore, by rearranging expression \[contactlee\], from Equation \[principalconnection\] we can write $$\label{contactlocal}
\displaystyle d^{c}\log({\rm{K}}_{H}(z,w)) = 2 \bigg [ \frac{\sqrt{-1}}{2}(\partial -\overline{\partial})\log(\varrho_{i}^{2}) + d\sigma_{i} \bigg ] = 2\eta.$$ Hence, we obtain from $\widetilde{\Psi} \in \Omega^{1}(L^{\times};\mathbb{C})$, defined as \[potentiallee\], that $$\displaystyle \widetilde{\Omega} = \frac{1}{2} d \big (\Re(\widetilde{\Psi})\big ) - \Im(\widetilde{\Psi}) \wedge \Re(\widetilde{\Psi}),$$ see Equation \[globalconformal1\]. From this, we see that the differential $1$-forms in \[potentiallee\] define completely the globally conformally Kähler structure on ${\text{Tot}}(L^{\times}) = \mathscr{C}(Q(L))$ presented in Remark \[canonicalvaisman\].
Now, in order to verify that $\widetilde{\Psi}$ defined above is a principal $\mathbb{C}^{\times}$-connection, we notice that:
1. $a^{\ast} \widetilde{\Psi} = \widetilde{\Psi}$, $\forall a \in \mathbb{C}^{\times}$,
2. $v \mapsto \Re(v) \xi + \Im(v)J_{\mathscr{C}}(\xi)$, $\forall v \in \mathbb{C} = {\text{Lie}}(\mathbb{C}^{\times})$.
The item (1) above follows from \[principalholomorphic\], i.e. $a = ({\mathrm{e}}^{ \sqrt{-1}\sigma(a)}, \tau_{|a|})$, where ${\mathrm{e}}^{ \sqrt{-1}\sigma(a)}$ preserves the underlying Sasakian structure of $Q(L)$, and $\tau_{|a|} \colon \mathscr{C}(Q(L)) \to \mathscr{C}(Q(L))$ is a homothety of dilation factor $|a|$, see \[dilatation\]. Hence, we have
$a^{\ast}\eta = \eta$, and $a^{\ast}(d\psi) = d(\psi + \log(|a|)).$
The item (2) follows from the identification $ \mathbb{C}^{\times} \cong S^{1} \times \mathbb{R}^{+}$, and also from \[principalholomorphic\], it is straightforward to check that the map in (2) is the infinitesimal action of ${\text{Lie}}(\mathbb{C}^{\times})$. Thus, we have $\widetilde{\Psi}(\widetilde{v}) = v$, where $\widetilde{v}$ is the fundamental vector field corresponding to $v \in {\text{Lie}}(\mathbb{C}^{\times})$. Notice that, since $\xi$ and $r\partial = J_{\mathscr{C}}$ are real holomorphic, the flow $\Phi_{t}^{v} \colon (\mathscr{C}(Q(L)),J_{\mathscr{C}}) \to (\mathscr{C}(Q(L)),J_{\mathscr{C}})$, associated to the fundamental vector field $\widetilde{v}$, defines a biholomorphism for any $v \in {\text{Lie}}(\mathbb{C}^{\times})$.
It is worth mentioning that, under the identification $(T \mathscr{C}(Q(L)),J_{\mathscr{C}}) \cong T^{1,0} \mathscr{C}(Q(L))$, provided by the map
$X \in \big (T \mathscr{C}(Q(L)),J_{\mathscr{C}} \big ) \mapsto X -\sqrt{-1}J_{\mathscr{C}}(X) \in T^{1,0} \mathscr{C}(Q(L)),$
the infinitesimal action of ${\text{Lie}}(\mathbb{C}^{\times})$ given in item (2) also can be described as $$v \in {\text{Lie}}(\mathbb{C}^{\times}) \mapsto v(\xi - \sqrt{-1}J_{\mathscr{C}}(\xi)) \in T^{1,0} \mathscr{C}(Q(L)).$$ Thus, the connection $\widetilde{\Psi} \in \Omega(L^{\times};\mathbb{C})$ corresponds precisely to the dual of the holomorphic vector field $\xi - \sqrt{-1}J_{\mathscr{C}}(\xi)$ as an element in $\Omega^{1,0}(\mathscr{C}(Q(L)))$.
Now, since $\eta = - \frac{dr}{r} \circ J_{\mathscr{C}}$, and $\widetilde{\Psi} = \eta - \sqrt{-1}\frac{dr}{r}$, it follows that $$\label{vaismanconnection}
\displaystyle \widetilde{\Psi} \circ J_{\mathscr{C}} = \eta \circ J_{\mathscr{C}} - \sqrt{-1} \frac{dr}{r} \circ J_{\mathscr{C}} = \sqrt{-1} \widetilde{\Psi}.$$ From the theory of connections on principal bundles [@Nomizu Section 10], we have that the principal $\mathbb{C}^{\times}$-connection $\widetilde{\Psi} \in \Omega^{1}(L^{\times};\mathbb{C})$ described above induces a unique principal $T_{\mathbb{C}}^{1}$-connection $\Psi \in \Omega^{1}(L^{\times}/\Gamma_{\lambda};\mathbb{C})$, such that $$\label{pullback}
\wp^{\ast}\Psi = \widetilde{\Psi},$$ where $\wp \colon L^{\times} \to L^{\times}/\Gamma_{\lambda}$ is the natural projection. Moreover, since the induced complex structure $J$ on ${\text{Tot}}(L^{\times})/\Gamma_{\lambda}$ satisfies $\wp_{\ast} \circ J_{\mathscr{C}} = J \circ \wp_{\ast}$, it follows that $\Psi \circ J = \sqrt{-1}\Psi$. Now, if we denote by $p_{\Lambda} \colon L^{\times}/\Gamma_{\lambda} \to X$ the projection map, it follows that $$d\Psi = p_{\Lambda}^{\ast} \omega_{X},$$ such that $\omega_{X} = -\sqrt{-1}\partial \overline{\partial}\log(\varrho_{i}^{2})$ is a Kähler form on $X$, see Equation \[contactlocal\], and Equation \[connection\]. Notice also that, in this case, we have $d\eta = \pi^{\ast}\omega_{X}$. Therefore, by applying Theorem \[vaismanbundle\], we have a Vaisman structure $(g,J)$ on $M_{\Gamma_{\lambda}}$, such that $J$ is the complex structure induced from $J_{\mathscr{C}}$, and $$\displaystyle g = \frac{1}{2} \Big (d\Psi({\text{id}} \otimes J) + \Psi \odot \overline{\Psi} \Big ),$$ where $\Psi \in \Omega^{1}(M_{\Gamma_{\lambda}};\mathbb{C})$ is a principal $T_{\mathbb{C}}^{1}$-connection. Moreover, the Lee form $\theta \in \Omega^{1}(M_{\Gamma_{\lambda}})$ in this case is given by $\theta = 2\Im(\Psi)$, see Equation \[Leefrombundle\], and we have $$\label{globalpotlee}
\wp^{\ast}\theta = 2\Im(\widetilde{\Psi}) = - d \log({\rm{K}}_{H}),$$ thus we obtain $\widetilde{\theta} = -2d\psi = \wp^{\ast}\theta$, where $\psi = \log(r)$. It is worth pointing out that, from Equation \[pullback\], we have $$\wp^{\ast}\Omega = \displaystyle d\psi \wedge \eta + \frac{d\eta}{2}, \ \ {\text{and}} \ \ \wp^{\ast}g = g_{Q(L)} + d\psi \otimes d\psi,$$ where $g_{Q(L)}$ is the Sasaki metric given in \[Sasakimetric\]. Therefore, we have that, in this case, the Vaisman structure obtained from the connection $\Psi \in \Omega^{1}(M_{\Gamma_{\lambda}};\mathbb{C})$ is exactly the canonical Vaisman structure obtained from the Kähler cone $\mathscr{C}(Q(L))$, see Remark \[canonicalvaisman\].
Before we move on to our next topic, let us summarize the results which we have covered so far. As we have seen, for a pair $(X,L)$ composed by a compact complex manifold $X$, and a negative holomorphic line bundle $L \in {\text{Pic}}(X)$, one can associate to a diagram
(L\^) & (L\^)/& Q(L)\
& X
for some choice of $\Gamma = \Gamma_{\lambda}$, such that $\lambda \in \mathbb{C}^{\times}$, $|\lambda|<1$. Moreover, we have a principal $\mathbb{C}^{\times}$-connection $1$-form on $L^{\times}$ $$\label{connectionlee}
\widetilde{\Psi} = \frac{1}{2}\big (d^{c} - \sqrt{-1}d\big )\log({\rm{K}}_{H}) = -\sqrt{-1} \partial \log({\rm{K}}_{H}),$$ which encodes all the information about the associated Riemannian geometry which underlies the above diagram. Being more precise, we have that the Kähler potential ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, allows us to equip the previous diagram with geometric structures as follows:
((L\^),\_,J\_ ) & ((L\^)/,g,J ) & (Q(L),g\_[Q(L)]{},,,)\
& (X,\_[X]{})
The geometries obtained from ${\rm{K}}_{H}$ are:
- $\big({\text{Tot}}(L^{\times}),\omega_{\mathscr{C}},J_{\mathscr{C}} \big)$ and $\big (X,\omega_{X} \big)$ (Kähler manifolds);
- $\big ({\text{Tot}}(L^{\times})/\Gamma,g,J \big )$ (l.c.K. manifold);
- $\big (Q(L),g_{Q(L)},\phi,\xi,\eta \big )$ (Sasaki manifold).
As one can see, the Kähler potential ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$ plays an important role in the study of the relations between the geometries described above. Thus, an important task to be considered in the homogeneous setting will be to provide a concrete description for such potentials, we shall do this later. Also, observe that the latter diagram provides the appropriate generalization for Example \[toyexample\].
Generalities on Hermitian-Weyl geometry
=======================================
In this section, we provide a brief account on Weyl manifolds and Einstein-Weyl metrics. In what follows, by following [@Gauduchon2] and [@PedersenPoonSwann], we shall provide a complete characterization of Hermitian-Einstein-Weyl manifolds, with strongly regular underlying Vaisman structure, in terms of the geometry of certain Calabi-Yau cones associated to Sasaki-Einstein manifolds. As we shall see, this last characterization fits in the context of principal elliptic bundles described in the previous section, which allows us to use connections and curvatures of principal bundles to describe solutions of the Hermitian-Einstein-Weyl equation on compact Hermitian-Weyl manifolds.
Hermitian-Weyl structures and l.c.K. manifolds
----------------------------------------------
Recall that a conformal structure on $M$ is an equivalence class [^5] $[g ]$ of Riemann metrics on $M$, such that $$[g ] = \Big \{ {\rm{e}}^{f}g \ \ \Big | \ \ f \in C^{\infty}(M) \Big \}.$$ A manifold with a conformal structure is called a conformal manifold.
Given a conformal manifold $(M,[g])$, we have the following notion of compatible connection with the conformal class $ [g]$.
\[weylconnectiondef\] A Weyl connection $D$ on a conformal manifold $(M,[g])$ is a torsion-free connection which preserves the conformal class $[g]$. In this last setting, we say that $D$ defines a Weyl structure on $(M,[g])$ and that $(M,[g],D)$ is a Weyl manifold.
In the above definition by preserving the conformal class it means that for each $g' \in [g]$, we have a $1$-form $\theta_{g'}$ (Higgs field) such that $$\label{higgs}
Dg' = \theta_{g'} \otimes g'.$$ The Equation \[higgs\] is conformally invariant in the sense that, if $h = {\mathrm{e}}^{f}g'$, $f \in C^{\infty}(M)$, then $$\label{conformalchange}
\theta_{h} = \theta_{g'} + df.$$ From this, under a conformal change $g \mapsto {\mathrm{e}}^{f}g$, we have $\theta_{g} \mapsto \theta_{g} + df$, $\forall f \in C^{\infty}(M)$. Conversely, if we start with a Riemannian manifold $(M,g)$ and a fixed $1$-form $\theta \in \Omega^{1}(M)$ (with $A = \theta^{\sharp}$), we can define a connection by setting $$\label{WeylLevi}
D = \nabla^{g} - \frac{1}{2} \bigg (\theta \odot {\text{id}} - g \otimes A \bigg ),$$ where $\nabla^{g}$ is the Levi-Civita connection associated to $g$ [^6]. It is straightforward to show that the aforementioned connection defines a Weyl structure on $(M,[g])$ which satisfies $Dg = \theta \otimes g$.
It is worth mentioning that, for any Riemannian manifold $(M^{n},g)$, by fixing a metric connection $\nabla$, i.e. $\nabla g \equiv 0$, under a conformal change $\widetilde{g} = {\rm{e}}^{-2\varphi}g$, one obtains, respectively, the following relations for the Levi-Civita connection $\widetilde{\nabla}$ of $\widetilde{g}$ and its Ricci tensor ${\text{Ric}}^{\widetilde{\nabla}} $ :
1. $\widetilde{\nabla} = \nabla - \big ( d\varphi \odot {\text{id}} - g \otimes \nabla \varphi\big)$;
2. ${\text{Ric}}^{\widetilde{\nabla}} = {\text{Ric}}^{\nabla} + \big ( \triangle \varphi - (n-2)||\nabla \varphi ||^{2}\big)g + (n-2){\rm{e}}^{-\varphi} \nabla^{2}({\rm{e}}^{\varphi})$,
see for instance [@Khunel], [@Besse]. Therefore, given a l.c.K. manifold $(M^{n},g,J)$, with $n = 2m$, by considering its Lee form $\theta$, since locally $g_{U} = {\rm{e}}^{-f_{U}}g|_{U}$ and $\theta|_{U} = df_{U}$, see Definition \[DEFLCK\], we obtain from (1) the following (local) expression for the Levi-Civita connection $\nabla^{U}$ of the Kähler manifold $(U,g_{U} = {\rm{e}}^{-f_{U}}g|_{U})$ $$\label{localweyl}
\nabla^{U} = \nabla - \frac{1}{2} \bigg (\theta \odot {\text{id}} - g \otimes A \bigg ),$$ see Equation \[WeylLevi\]. Moreover, since $\triangle \varphi = - \delta^{g}(d\varphi)$ and ${\rm{e}}^{-\varphi} \nabla^{2}({\rm{e}}^{\varphi}) = d\varphi \otimes d\varphi + \nabla(d\varphi)$, by considering $\varphi = \frac{f_{U}}{2}$ in the previous expression (2), the Ricci tensor ${\text{Ric}}^{U}$ of the (local) Kähler metric $g_{U}$ is given by $${\text{Ric}}^{\nabla^{U}} = {\text{Ric}}^{\nabla} - \frac{1}{2}\bigg [\delta^{g}(\theta) + \frac{(n-2)}{2}||\theta||^{2}\bigg]g + \frac{(n-2)}{2}\bigg (\nabla \theta + \frac{\theta \otimes \theta}{2}\bigg),$$ here we consider $||\omega|| = \sqrt{g(\omega^{\sharp},\omega^{\sharp})}$, $\forall \omega \in \Omega^{1}(M)$.
On a Weyl manifold $(M,[g],D)$, Weyl introduced the [*distance curvature function*]{}, a 2-form defined by $d\theta_{g}$. From Equation \[conformalchange\], we see that this last definition does not depend on of the representative of $[g]$. Also, if $d\theta_{g} = 0$, it follows that the cohomology class $[\theta_{g}] \in H_{DR}^{1}(M)$ does not depend on of the representative of $[g]$. In this latter case, the Weyl structure is called a closed Weyl structure.
An equivalent way to define a Weyl manifold is the following. A Weyl structure on a conformal manifold $(M,[g])$ also can be seen as a mapping $\mathscr{W} \colon [g] \to \Omega^{1}(M)$, satisfying $$\mathscr{W}({\rm{e}}^{f}g) = \mathscr{W}(g) - df, \ \ \forall f \in C^{\infty}(M).$$ From the above map we say that a linear connection $D$ on $M$ is compatible with the Weyl structurte $\mathscr{W}$ if $$\label{weylconnection}
D g + \mathscr{W}(g)\otimes g = 0.$$ Therefore, we can define a Weyl manifold as a triple $(M,[g],\mathscr{W})$. From this last definition we can show that there exists a unique torsion free linear connection $D$ which satisfies Equation \[weylconnection\], i.e. a Weyl connection as in Definition \[weylconnectiondef\]. Further details about the above comments can be found in [@Folland].
Let $(M,[g],D)$ be a Weyl manifold, in what follows we shall fix a representative $g$ for the conformal class, and consider the $1$-form $\theta_{g}$ which defines its Higgs field, i.e. which satisfies $Dg = \theta_{g} \otimes g$.
We say that a Weyl manifold $(M,[g],D)$ is a Hermitian-Weyl manifold if it admits an almost complex structure $J \in {\text{End}}(TM)$, which satisfies:
1. $g(JX,JY) = g(X,Y)$, $\forall X,Y \in \Gamma(TM);$
2. $DJ = 0.$
Notice that, since $D$ is a torsion-free connection, the second condition in the last definition implies that $J$ is integrable. Therefore, particularly, a Hermitian-Weyl manifold is a complex manifold. Also, from the first condition in the last definition, it follows that a Hermitian-Weyl manifold is particularly a Hermitian manifold.
Given a Hermitian-Weyl manifold $(M,[g],D,J)$, we can consider the fundamental $2$-form associated to the underlying Hermitian structure, i.e.
$\Omega = g(J \otimes {\text{id}})$,
it is straightforward to check that $d\Omega = \theta_{g} \wedge \Omega$. Moreover, since $\Omega$ is non-degenerate, we have an isomorphism[^7] between $\bigwedge^{2}T^{\ast}M$ and $\bigwedge^{2n-2}T^{\ast}M$ given by the multiplication by $\Omega^{n-2}$, i.e.
$\Omega^{n-2} \colon \omega \mapsto \omega \wedge \Omega^{n-2}$.
Hence, provided that $\dim_{\mathbb{R}}(M) \geq 6$, in particular, it follows that $\Omega \colon \bigwedge^{2}T^{\ast}M \to \bigwedge^{4}T^{\ast}M$ is injective, so we obtain $$\label{closedhiggs}
0 = d(d\Omega) = d\theta_{g} \wedge \Omega,$$ which implies that $d\theta_{g} = 0$. Thus, locally, we can write $\theta_{g} = df$. Therefore, we can take an open cover $\mathscr{U}$ of $M$, and a family of smooth functions $\{f_{U}\}_{U \in \mathscr{U}}$, $f_{U} \colon U \to \mathbb{R}$, such that
$\theta_{g}|_{U} = df_{U}$,
$\forall U \in \mathscr{U}$. From this, we can set $g_{U} = {\mathrm{e}}^{-f_{U}}g|_{U}$, and then obtain
$\theta_{g_{U}} = \theta_{g}|_{U} - df_{U} = 0,$
which implies that $d( {\mathrm{e}}^{-f_{U}}\Omega|_{U}) = 0$, $\forall U \in \mathscr{U}$. The conclusion from this last computation is that a Hermitian-Weyl manifold $(M,[g],D,J)$, with $\dim_{\mathbb{R}}(M) \geq 6$, is in fact a l.c.K. manifold $(M,g,J)$ with Lee form defined by the Higgs field $\theta_{g}$. Conversely, given a l.c.K. manifold $(M,\Omega,J,\theta)$ of (real) dimension at least $4$, we can consider the Levi-Civita connection $\nabla^{g}$ associated to $g = \Omega({\rm{id}}\otimes J)$, and the Lee form $\theta \in \Omega^{1}(M)$, from thes we can define a Weyl connection $D$, see expression \[WeylLevi\], such that $Dg = \theta_{g}\otimes g$, with $\theta_{g} = \theta$. Notice that, the Weyl connection obtained from this latter procedure is exactly the same as gluing the locally defined connections \[localweyl\].
An important result which summarizes the above discussion is given by the following proposition.
\[HweylVaisman\] Any Hermitian-Weyl manifold of (real) dimension at least $6$ is l.c.K.. Conversely, a l.c.K. manifold of (real) dimension at least $4$ admits a compatible Hermitian-Weyl structure.
\[gauduchongauge\] For a compact Hermitian-Weyl manifold $(M,[g],D,J)$ Gauduchon [@Gauduchon] showed that, up to homothety, there is a unique choice of metric $g_{0}$ in the conformal class such that the corresponding $1$-form $\theta_{g_{0}}$ is co-closed. The metric $g_{0} \in [g]$ is called Gauduchon’s metric ( a.k.a. Gauduchon’s gauge). It follows from Proposition \[HweylVaisman\] that any compact Vaisman manifold admits a Hermitian-Weyl structure uniquely determined by the Gauduchon metric.
Hermitian-Einstein-Weyl manifolds
---------------------------------
The Einstein-Weyl equations are a conformally invariant generalization of the Einstein equations. Given a Weyl manifold $(M,[g],D)$, if one considers the curvature tensor $R^{D}$ of the affine connection $D$, one can define $${\text{Ric}}^{D}(X,Y) = {\rm{Tr}} \big \{ Z \mapsto R^{D}(X,Z)Y \big \}.$$ Since $D$ is not a metric connection, it follows that its Ricci curvature is not necessarily symmetric, thus we say that $(M,[g],D)$ is Einstein-Weyl if $$\label{EinsteinWeyl}
{\rm{Sym}}({\text{Ric}}^{D}) = \lambda g,$$ for some representative $g$, where $\lambda$ is a smooth function, and $${\rm{Sym}}({\text{Ric}}^{D})(X,Y) = \frac{1}{2} \big ( {\text{Ric}}^{D}(X,Y) + {\text{Ric}}^{D}(Y,X) \big),$$ is the symmetric part of ${\text{Ric}}^{D}$. It is worthwhile to observe that, unlike the Einstein equations in manifolds with dimension at least three [@Besse], the function factor $\lambda$ in Equation \[EinsteinWeyl\] is not constant.
\[einsteinweyl\] The relation between the curvatures associated to $D$ and $\nabla^{g}$ (Eq. \[WeylLevi\]) can be described as follows. At first, we consider the scalar curvatures of $D$ with respect to $g$, i.e. $$\label{Scalweyl}
{\text{Scal}}_{g}^{D} = {\rm{Tr}}_{g}({\text{Ric}}^{D}).$$ Now, we can verify that the Ricci curvature ${\text{Ric}}^{\nabla}$, and the scalar curvature ${\text{Scal}}_{g} = {\rm{Tr}}_{g}({\text{Ric}}^{\nabla})$, associated to the Levi-Civita connection $\nabla^{g}$ satisfy $$\label{Scals}
{\text{Scal}}_{g}^{D} = {\text{Scal}}_{g} + 2(n-1)\delta^{g}(\theta_{g}) - (n-1)(n-2)||\theta_{g}||^{2},$$ $$\label{ricciweyl}
{\text{Ric}}^{D} = {\text{Ric}}^{\nabla} + \delta^{g}(\theta_{g}) \cdot g - (n-2) \Big ( \nabla^{g}(\theta_{g}) + ||\theta_{g}||^{2}g - \theta_{g} \otimes \theta_{g} \Big ).$$ Moreover, for an Einstein-Weyl structure one has $$\label{einsteinweyleq}
{\text{Ric}}^{D} = \frac{{\text{Scal}}_{g}^{D}}{n}g - \frac{n-2}{2}d\theta_{g},$$ thus Equation \[EinsteinWeyl\] can be written as ${\rm{Sym}}({\text{Ric}}^{D}) = \frac{{\text{Scal}}_{g}^{D}}{n}g$, see for instance [@Gauduchon2]. It is worth observing that, from Equation \[einsteinweyleq\], it follows that ${\text{Ric}}^{D}$ is symmetric if and only if $D$ is a closed Weyl structure, i.e. $d\theta_{g} = 0$.
Given a compact Hermitian-Weyl manifold $(M,[g],D,J)$ of (real) dimension at least $6$, by fixing the Gauduchon gauge $g_{0} \in [g]$, it follows that $\theta_{g_{0}}$ is co-closed, so we have $$0 = \delta^{g_{0}}\theta_{g_{0}} = -\frac{1}{2}\big (\mathscr{L}_{A}g_{0}\big),$$ see for instance [@Besse]. Thus, it follows that that $A = \theta_{g_{0}}^{\sharp}$ is a Killing vector field cf. [@Tod]. As we have seen, $(M,g_{0},J)$ is also l.c.K. manifold, which implies that $d\theta_{g_{0}} = 0$, see Equation \[closedhiggs\]. Therefore, since $$2g_{0}(\nabla_{X}A,Y) = d\theta_{g_{0}}(X,Y) + \big(\mathscr{L}_{A}g_{0}\big)(X,Y),$$ $\forall X,Y \in \Gamma(TM)$, it follows that $(M,g_{0},J)$ is Vaisman, i.e. $\nabla \theta_{g_{0}} \equiv 0$. Moreover, under the assumption that the associated Weyl structure is Einstein-Weyl, it follows that ${\text{Scal}}_{g_{0}}^{D} = 0$ and ${\text{Ric}}^{D} \equiv 0$. In fact, since $\theta_{g_{0}}$ is parallel, it follows that $||\theta_{g_{0}}||$ is constant, so from [*[Bochner’s]{}*]{} formula we obtain $$0 = -\frac{1}{2}\triangle ||\theta_{g_{0}}||^{2} = -g_{0}(\triangle \theta_{g_{0}},\theta_{g_{0}}) + ||\nabla \theta_{g_{0}}||^{2} + {\text{Ric}}^{\nabla}(A,A),$$ which implies from Equation \[ricciweyl\] and Equation \[einsteinweyleq\] that $$0 = {\text{Ric}}^{\nabla}(A,A) = {\text{Ric}}^{D}(A,A) = \frac{{\text{Scal}}_{g}^{D}}{n}||A||^{2},$$ so we have ${\text{Scal}}_{g_{0}}^{D} = 0$ and ${\text{Ric}}^{D} \equiv 0$. Hence, if $(M,[g],D,J)$ is a compact Hermitian-Einstein-Weyl manifold such that $\dim_{\mathbb{R}}(M) \geq 6$, under the assumption that $\theta_{g_{0}}$ is not exact, it follows that the underlying l.c.K. manifold $(M,\theta_{g_{0}},J)$ is in fact Vaisman, and from Equation \[ricciweyl\], and Equation \[einsteinweyleq\], the Einstein-Weyl condition turns out to be $$\label{Ric}
{\text{Ric}}^{\nabla} = (n-2) \Big (||\theta_{g_{0}}||^{2}g_{0} - \theta_{g_{0}} \otimes \theta_{g_{0}}\Big ).$$ In a more general setting we have the following result.
\[EinsteinWeylGauduchon\] Let $D$ be an Einstein-Weyl structure on a compact, oriented conformal manifold $(M,[g])$ of dimension $>2$. Let $g_{0} \in [g]$ be the Gauduchon metric associated to $D$ and $\theta_{g_{0}}$ the corresponding Higgs field. If $\theta_{g_{0}}$ is closed, but not exact, then
1. $\theta_{g_{0}}$ is $\nabla^{g_{0}}$-parallel, i.e. $\nabla^{g_{0}}\theta_{g_{0}} \equiv 0$ (in particular, $\theta_{g_{0}}$ is also $g_{0}$-harmonic).
2. ${\text{Ric}}^{D} \equiv 0$.
\[universalsplitting\] An important consequence of Theorem \[EinsteinWeylGauduchon\] is the following. Under the hypotheses of theorem above we have that $\theta_{g_{0}}$ has constant norm. Thus, after a normalization $||\theta_{g_{0}}|| = 1$, since ${\text{Ric}}^{D} \equiv 0$, it follows from Equation \[ricciweyl\] that $$\label{tangentEW}
{\text{Ric}}^{\nabla}(A) = 0, \ \ {\text{and}} \ \ {\text{Ric}}^{\nabla}(X) = (n-2)X, \ \forall X \in \langle A \rangle^{\perp},$$ notice that, similarly to the compact Hermitina-Einstein-Weyl case, Equation \[Ric\] also holds. Now, given any $\beta \in \Omega^{1}(M)$, if we consider the Bochner’s formula $$\label{Widentity}
-\frac{1}{2}\triangle ||\beta||^{2} = ||\nabla \beta||^{2} + {\text{Ric}}^{\nabla}(\beta^{\sharp},\beta^{\sharp}) - g_{0}(\triangle\beta,\beta),$$ it follows from Equation \[tangentEW\] and Equation \[Widentity\], that $$\label{equivharmonic}
\triangle\beta \equiv 0 \Longleftrightarrow \beta^{\sharp} = c_{0} A ,$$ for some constant $c_{0} \in \mathbb{R}$. In fact, we have $$\label{laplacianint}
0 = -\int_{M}\frac{1}{2}\triangle ||\beta||^{2}dV_{g_{0}} = \int_{M}\Big(||\nabla \beta||^{2} + {\text{Ric}}^{\nabla}(\beta^{\sharp},\beta^{\sharp}) - g_{0}(\triangle\beta,\beta) \Big)dV_{g_{0}},$$ so if $\triangle \beta = 0$, since from Equation \[Ric\] we have ${\text{Ric}}^{\nabla}(\beta^{\sharp},\beta^{\sharp}) = (n-2)||\beta^{\sharp} - \theta_{g_{0}}(\beta^{\sharp})A||^{2}$, it follows from Equation \[laplacianint\] that
$\displaystyle 0 = \int_{M}\Big(||\nabla \beta||^{2} + (n-2)||\beta^{\sharp} - \theta_{g_{0}}(\beta^{\sharp})A||^{2} \Big)dV_{g_{0}},$
which implies $\beta^{\sharp} = \theta_{g_{0}}(\beta^{\sharp})A$ and $\nabla \beta = 0$. From the parallelism of $\beta$ and $\theta_{g_{0}}$ we have that $\theta_{g_{0}}(\beta^{\sharp})$ is a real constant. Conversely, if $\beta^{\sharp} = c_{0}A$, for some real constant $c_{0} \in \mathbb{R}$, it is easy to see that $\triangle \beta = 0$. Therefore, the equivalence \[equivharmonic\] implies that $$1 = \dim\mathscr{H}^{1}(M)= \dim({\text{Iso}}(M,g_{0})),$$ where $\mathscr{H}^{1}(M)$ is the space of harmonic 1-forms, and ${\text{Iso}}(M,g_{0})$ is the associated isometry group.
Now, still in the above setting, by passing to the universal cover $\wp \colon (\widetilde{M},g) \to (M,g_{0})$, where $g = \wp^{\ast}g_{0}$, we have $f \in C^{\infty}(\widetilde{M})$, which satisfies $df = \wp^{\ast}\theta_{g_{0}}$, so we obtain [^8]
$||\nabla f||_{g} = ||A||_{g_{0}} = 1$.
Hence, from the Cheeger-Gromoll Splitting Theorem [@Cheeger], see also [@Eschenburg], we have that $(\widetilde{M},g)$ is isometric to a product $$\big (\widetilde{Q} \times \mathbb{R}, g_{\widetilde{Q}} + dt\otimes dt\big).$$ Another way to describe the above splitting is by observing that $f \colon (\widetilde{M},g) \to \mathbb{R}$ is a distance function, i.e. $||\nabla f||_{g} =1$. Thus, since $\nabla f$ is also parallel, we have ${\text{Hess}}(f) \equiv 0$, which implies that $f$ is a linear affine function. We can consider an isometry induced by the flow of $\nabla f$ from $\widetilde{Q} \times \mathbb{R}$ to $\widetilde{M}$, where $\widetilde{Q} = f^{-1}(0)$ is a totally geodesic hypersurface, notice also that $\widetilde{Q}$ is a simply connected compact Einstein manifold with positive scalar curvature (see Equation \[tangentEW\]). It follows from Equation \[Ric\] and the last comments that the metric $g$ on $\widetilde{M}$ can be written as $$\label{liftmetric}
g = \frac{1}{n-2} \wp^{\ast}{\text{Ric}}^{\nabla} + df \otimes df.$$
The next result provides a unified setting which takes into account the aspects of the structure of principal elliptic fibration which underlies compact Hermitian-Einstein-Weyl manifolds of real dimension $\geq 6$.
\[HEW\] Each compact Hermitian-Einstein-Weyl manifold $(M,[g],D,J)$ ($\dim_{\mathbb{R}}(M) \geq 6$) which is strongly regular Vaisman manifold, arises as a fibration over a compact Kähler-Einstein manifold $X$ ($\dim_{\mathbb{R}}(X) = \dim_{\mathbb{R}}(M) - 2$) of positive scalar curvature. Moreover, $M$ is obtained as a discrete quotient of the Ricci-flat Kähler structure on a principal $\mathbb{C}^{\times}$-bundle associated to a maximal root of the canonical bundle of $X$.
The result above can be obtained by observing the following facts. Under the hypotheses of the last theorem, by considering the underlying strongly regular Vaisman manifold $(M,\theta_{g_{0}})$, where $g_{0} \in [g]$ is the associated Gauduchon gauge, we have that its universal covering space $\widetilde{M}$ is isometric to Riemannian cone over a compact Sasaki manifold[^9] $\widetilde{Q}$, i.e. $\widetilde{M} = \widetilde{Q} \times \mathbb{R}$. Moreover, by considering the covering projection $\wp \colon \widetilde{M} \to M$, and $f \in C^{\infty}(\widetilde{M})$ such that $df = \wp^{\ast}\theta_{g_{0}}$, from Equation \[liftmetric\], we have $$g = \frac{1}{2(m-2)} \wp^{\ast}{\text{Ric}}^{\nabla} + df \otimes df = g_{\widetilde{Q}} + df \otimes df,$$ here we suppose $\dim_{\mathbb{R}}(M) = 2m$. The above expression provides that $${\text{Ric}}^{\widetilde{\nabla}} = 2(m-1)g_{\widetilde{Q}},$$ where $g_{\widetilde{Q}}$ is given by the restriction of $g$ to the totally geodesic hypersurface $\widetilde{Q} = f^{-1}(0)$, and $\widetilde{\nabla}$ is its associated metric connection. Hence, we have that $\widetilde{Q}$ is in fact Sasaki-Einstein. Thus, by taking the change of coordinates $r = {\rm{e}}^{-f}$, and identifying $\widetilde{M} = \mathscr{C}(\widetilde{Q})$, it follows that $(\widetilde{M} = \mathscr{C}(\widetilde{Q}),g_{CY})$ is Kähler Ricci-flat [@BoyerGalicki], where $g_{CY}$ is defined by $$g_{CY} = {\rm{e}}^{-2f}\bigg (\frac{1}{2(m-2)} \wp^{\ast}{\text{Ric}}^{\nabla} + df \otimes df \bigg) = r^{2}g_{\widetilde{Q}} + dr\otimes dr.$$ Notice that the Lee form $\widetilde{\theta}$ associated to the globally conformal Kähler metric $g = {\rm{e}}^{2f}g_{CY}$ satisfies $$\widetilde{\theta} = 2df = 2\wp^{\ast}\theta_{g_{0}},$$ see Equation \[globalpotlee\], it follows that, up to scale, the Hermitian-Einstein-Weyl structure on $M$ is completely determined by the Kähler Ricci-flat structure on $\widetilde{M} = \mathscr{C}(\widetilde{Q})$.
Now, by considering the associated Boothby-Wang fibration [@BW] defined by
$\pi \colon \widetilde{Q} \to \widetilde{Q} /{\rm{U}}(1) = X,$
it follows that $X$ is a Kähler-Einstein Fano manifold with Kähler-Einstein metric $\omega_{X}$ determined by a connection $\eta' \in \Omega^{1}(\widetilde{Q};\sqrt{-1}\mathbb{R})$, i.e. $d\eta' = \sqrt{-1}\pi^{\ast}\omega_{X}$. Thus, since $X$ is simply connected [@Kobayashi], it follows from [@Tsukada Theorem 4.1] that $\widetilde{M} = L^{\times}$, where
$L = K_{X}^{\otimes \frac{\ell}{I(X)}}$,
for some positive integer $\ell \in \mathbb{Z}$, here $I(X)$ is the maximal integer such that $\frac{1}{I(X)}c_{1}(X) \in H^{2}(X,\mathbb{Z})$, i.e. the [*Fano index*]{} of $X$. Therefore, since $\widetilde{Q}$ is simply connected, it follows that $\ell = 1$, see [@Kobayashi1], [@Blair Corollary 2.1].
\[canonicalroot\] In analogy to the tautogical line bundle over projective spaces, given a compact simply connected Kähler manifold $X$, we also shall denote the maximal root of the canonical bundle of $X$ by $$\mathscr{O}_{X}(-1) := K_{X}^{\otimes \frac{1}{I(X)}},$$ note that, with the above convention, we have $\mathscr{O}_{X}(-1)^{\otimes \ell} = \mathscr{O}_{X}(-\ell)$, $\forall \ell \in \mathbb{Z}$.
Compact homogeneous Vaisman manifolds
=====================================
In this section, we provide a description of compact homogeneous l.c.K. manifolds and explain their relation with complex flag manifolds. In order to do so, we use the fact that compact homogeneous l.c.K. manifolds are Vaisman [@Gauduchon], and then we apply the results on compact homogeneous Vaisman manifolds introduced in [@VaismanII]. For further details on compact homogeneous l.c.K. manifolds, we suggest [@Gauduchon] and references therein.
Principal elliptic bundles over flag manifolds {#compacthomogeneoussetting}
----------------------------------------------
Before we move on to the study of homogenous manifolds, let us recall some basic facts which summarize some ideas developed in the previous sections. As we have seen in the previous sections, given a compact Vaisman manifold $(M,J,\theta)$ with strongly regular canonical foliation $\mathcal{F} = \mathcal{F}_{A} \oplus \mathcal{F}_{B}$, we obtain the following principal bundles:
$ S^{1} \hookrightarrow (M,J,\theta) \to M/\mathcal{F}_{A} = Q$, and $T_{\mathbb{C}}^{1} \hookrightarrow (M,J,\theta) \to M/\mathcal{F} = X$,
where the first one fibration is a flat $S^{1}$-principal bundle over a Sasaki manifold $Q$, and the second one is a $T_{\mathbb{C}}^{1}$-principal bundle over a compact Hodge manifold $X$. Moreover, we also have a principal $S^{1}$-bundle defined by
$S^{1} \hookrightarrow Q \to X$,
where $Q$ is a compact Sasaki manifold.
\[withouttorsion\] Under the assumption that $X$ is simply connected, we have that the Chern class of $M$, as a $T_{\mathbb{C}}^{1}$-principal bundle over $X$, coincides with the Chern class of $Q$ as a principal $S^{1}$-bundle, see [@Chen]. Thus, in this last case, we can realize $(M,J,\theta)$ as a quotient space
$M = {\text{Tot}}(L^{\times})/\Gamma$,
where $L \in \text{Pic}(X)$ is a negative line bundle, such that
$L = Q \times_{{\rm{U}}(1)} \mathbb{C}$,
and $\Gamma = \Gamma_{\lambda} \subset \mathbb{C}^{\times}$ is a cyclic group obtained from $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| < 1$, see \[cyclicgroup\] and [@Tsukada Theorem 4.2].
In what follows we shall further explore the basic ideas described above in the setting of homogeneous spaces. Our exposition is essentially based on [@Gauduchon1], for more details about the background in Lie theory, we suggest [@Knapp], [@Humphreys], see also [@PARABOLICTHEORY], [@Akhiezer].
Let $(M,J,g)$ be a compact l.c.K. manifold, we say that $(M,J,g)$ is a homogeneous l.c.K. manifold if it admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$.
In the above setting, if we denote by $\tau \colon K \to {\rm{Diff}}(M)$ the underlying Lie group action, for each $X \in \mathfrak{k}$ we can associate to a vector field $\delta \tau (X) \in \Gamma(TM)$ defined by $$\delta \tau(X)_{p} = \frac{d}{dt}\Big|_{t=o}\tau \big (\exp(tX) \big )\cdot p, \ \ \ \forall p \in M.$$ The map $\delta \tau \colon \mathfrak{k} \to \Gamma(TM)$ defines an infinitesimal action which satisfies $$\big [\delta \tau(X),\delta \tau (Y) \big ] = - \delta \tau \big(\big [X,Y\big] \big ),$$ $\forall X,Y \in \mathfrak{k}$, so it follows that $\delta \tau \colon \mathfrak{k} \to \Gamma(TM)$ is an anti-homomorphism of Lie algebras. Moreover, since by definition the action is effective, we have that $\delta \tau $ is injective.
Given a homogeneous l.c.K. manifold $(M,g,J)$, since $\mathscr{L}_{\delta \tau (X)}J = \mathscr{L}_{\delta \tau (X)}g = 0$, $\forall X \in \mathfrak{k}$, it follows that $$\mathscr{L}_{\delta \tau (X)} \Omega = \big (\mathscr{L}_{\delta \tau (X)}g\big )\big (J\otimes \text{id} \big ) + g\big ((\mathscr{L}_{\delta \tau (X)}J)\otimes \text{id}\big ) = 0, \ \ \forall X \in \mathfrak{k},$$ where $\Omega$ is the fundamental $2$-form of $(M,J,g)$. Since $d\Omega = \theta \wedge \Omega$ and $\mathscr{L}_{\delta \tau (X)} d\Omega = 0$, it implies[^10] that $\mathscr{L}_{\delta \tau (X)}\theta = 0$, $\forall X \in \mathfrak{k}$. Therefore, we have
$0 = d\theta\big (\delta \tau (X),\delta \tau(Y) \big ) = \mathscr{L}_{\delta \tau (X)}\big(\theta(\delta \tau (Y))\big) - \mathscr{L}_{\delta \tau (Y)}\big(\theta(\delta \tau (X))\big) - \theta\big(\big [\delta \tau(X),\delta \tau(Y) \big] \big),$
$\forall X,Y \in \mathfrak{k}$. Thus, since $\theta(\delta \tau (X))$ is constant, $ \forall X \in \mathfrak{k}$, we obtain $$\label{leefoliation}
\theta\big(\big [\delta \tau(X),\delta \tau(Y) \big] \big) = 0, \ \ \forall X,Y \in \mathfrak{k},$$ which implies that the vector fields $\delta \tau \big ([\mathfrak{k},\mathfrak{k}] \big ) \subset \Gamma(TM)$ are tangent to the foliation defined by the kernel of $\theta$. Notice that, since $K$ is compact, it follows that its Lie algebra $\mathfrak{k}$ is reductive, so we have $$\label{reductivealgebra}
\mathfrak{k} = \mathcal{Z}(\mathfrak{k}) \oplus \big [ \mathfrak{k},\mathfrak{k}\big ],$$ where $\mathcal{Z}(\mathfrak{k})$ denotes the center of $\mathfrak{k}$, and $[\mathfrak{k},\mathfrak{k}]$ its semisimple part, e.g. [@Knapp]. Therefore, it follows from Equation \[leefoliation\] that a compact semisimple Lie group cannot act transitively on a l.c.K. manifold by keeping its underlying Lee form invariant.
Let $(M,g,J)$ be a homogeneous l.c.K. manifold, and consider $${\text{Inv}} = \Big \{ X \in \Gamma(TM) \ \ \Big| \ \ X \ \ {\text{is}} \ \ K{\text{-invariant}}\Big \},$$ the subspace above is in fact a Lie subalgebra of $\Gamma(TM)$. Further, since each element of ${\text{Inv}}$ commutes with all elements in $\delta \tau(\mathfrak{k})$, it follows that $${\text{Inv}} \cap \delta \tau(\mathfrak{k}) = \delta \tau (\mathcal{Z}(\mathfrak{k})).$$ As we have seen above, it must exists $T \in \mathcal{Z}(\mathfrak{k})$, such that $\theta(\delta \tau(T)) \not\equiv 0$, and we can suppose that $\theta(\delta \tau(T)) = 1$. In [@Gauduchon1] it was shown that, for any chosen $p \in M$, we have $${\text{Inv}} = N_{\mathfrak{k}}(\mathfrak{k}_{p})/\mathfrak{k}_{p}, \ \ {\text{and}} \ \ \operatorname{rank}({\text{Inv}}) = 2,$$ such that $N_{\mathfrak{k}}(\mathfrak{k}_{p})$ is the normalizer of the isotropy subalgebra $\mathfrak{k}_{p} = \big \{X \in \mathfrak{k} \ \big | \ \delta \tau(X)_{p} = 0 \big \}$. Therefore, it follows that is isomorphic either to the abelian Lie algebra $\mathbb{R} \oplus \mathbb{R}$ or to the unitary Lie algebra $\mathfrak{u}(2) = \mathbb{R} \oplus \mathfrak{su}(2)$. From this, it can be shown that in both cases we can write $$\label{Leekilling}
J\delta \tau(T) = a\delta \tau(T) + bJA,$$ with $a,b \in \mathbb{R}$, such that $b \neq 0$, where $\theta = g(\cdot,A)$. Therefore, since $J$ is integrable, we have
$0 = N_{J}(X,Y) = J\big( \mathscr{L}_{X}J\big) Y - \big( \mathscr{L}_{JX}J\big)Y$,
$\forall X,Y \in \Gamma(TM)$, where $N_{J}$ is the Nijenhuis torsion of $J$, and from Equation \[Leekilling\] it follows that $A$ preserves the complex structure $J$. Also, notice that
$\mathscr{L}_{JA}\Omega = \iota_{JA}d\Omega + d\iota_{JA}\Omega = -d\theta + \theta(JA)\Omega + \theta \wedge \theta = 0$,
which implies from Equation \[Leekilling\] that $A$ preserves $\Omega$, so $A$ is a Killing vector field. Now, from Koszul’s formula we obtain $$2g(\nabla_{X}A,Y) = d\theta(X,Y) + \big (\mathscr{L}_{A}g\big)(X,Y) = 0,$$ $\forall X,Y \in \Gamma(TM)$, which implies that $\nabla \theta \equiv 0$. The ideas briefly described above lead to the following result.
\[LCKisVaisman\] Any compact homogeneous l.c.K. manifold $(M,g,J)$ is Vaisman.
\[parallellee\] Notice that from Theorem \[LCKisVaisman\], and the above comments, we have that the Harmonic representative and the $K$-invariant representative[^11] in $H^{1}(M,\mathbb{R})$ for the Lee form are the same, i.e. a $K$-invariant l.c.K. metric is the Gauduchon metric in its conformal class.
Just like in the case of compact homogeneous contact manifolds [@BW], we have the following result related to the regularity of the canonical foliation of compact homogeneous Vaisman manifolds.
\[Vaismanregular\] Any compact homogeneous Vaisman manifold $(M,g,J)$ is strongly regular. Moreover, the following facts hold:
1. Associated to $M$ we have two principal fiber bundles given by
$ S^{1} \hookrightarrow (M,g,J) \to M/\mathcal{F}_{A} = Q$, and $T_{\mathbb{C}}^{1} \hookrightarrow (M,g,J) \to M/\mathcal{F} = X$,
where $Q$ is a compact homogeneous Sasaki manifold, and $X$ is a compact simply connected homogeneous Hodge manifold.
2. The manifold $Q$ above defines a Boothby-Wang fibration over $X$, i.e. $Q$ is a $S^{1}$-principal bundle over $X$ whose Euler class defines an invariant Hodge metric on $X$.
In the setting of the last theorem, since the base manifold $X$ is simply connected, it follows from Theorem \[vaismanclass\] that the principal fiber bundles
$T_{\mathbb{C}}^{1} \hookrightarrow M \to X$, and $ S^{1} \hookrightarrow Q \to X$,
are defined by the same characteristic class in $H^{2}(X,\mathbb{Z})$. Therefore, we can realize $M$ as a quotient space $$M = {\text{Tot}}(L^{\times})/\Gamma,$$ where $L$ is a negative line bundle over $X$ which satisfies $Q(L) = Q$ (see \[spherebundle\]) and $\Gamma \subset \mathbb{C}^{\times}$ is a cyclic group, see Remark \[withouttorsion\].
\[bettinumber\] Another important fact concerned with the setting above is that, since the base Hodge manifold $X$ is simply connected, its odd Betti numbers vanish [@BorelH], [@Borel]. Thus it can be shown from the Gysin sequence for the fibrations described in Theorem \[Vaismanregular\] that $b_{1}(M) = 1$, see for instance [@Vaisman].
Given a compact homogeneous l.c.K. manifold $(M,g,J)$, if we consider the decomposition \[reductivealgebra\], it follows that the compact reductive Lie group $K$ can be written as $$K = K_{{\text{ss}}}Z(K)_{0},$$ where $K_{{\text{ss}}}$ is a closed, connected and semisimple Lie subgroup with ${\text{Lie}}(K_{{\text{ss}}}) = [\mathfrak{k},\mathfrak{k}]$, and $Z(K)_{0}$ is the closed connected subgroup defined by the connected component of the identity of the center, see for instance [@Knapp]. Observing that $J\delta \tau(T) \in {\text{inv}}$, it follows from equation \[Leekilling\] that $A,B\in {\text{inv}}$, recall that $B = JA$. Thus, since $\operatorname{rank}({\text{Inv}}) = 2$ and $[A,B] = 0$, we obtain an effective and transitive Lie group action of $K_{{\text{ss}}}$ on the leaf space $M/\mathcal{F}$, induced from $\tau \colon K \to {\rm{Diff}}(M)$, such that $$K/Z(K)_{0} \times M/\mathcal{F} \to M/ \mathcal{F}, \ \ \big (kZ(K)_{0},[p] \big ) \mapsto \big [\tau(k)\cdot p \big ]$$ $\forall kZ(K)_{0} \in K/Z(K)_{0}$, and $\forall [p] \in M/ \mathcal{F}$. Therefore, the compact simply connected homogeneous Hodge manifold $X$ can be realized as a quotient space $$\label{HomogeneousHd}
X = K_{{\text{ss}}}/H,$$ where $H$ is compact, connected and equal to the centralizer of a torus of $K_{{\text{ss}}}$, see for instance [@BorelK].
Considering the description provided in \[HomogeneousHd\], if we take the decomposition of $K_{{\text{ss}}}$ into simple Lie groups, i.e. $$K_{{\text{ss}}} = G_{1} \times \cdots \times G_{r},$$ then we have $H = H_{1} \times \cdots \times H_{r}$, such that $H_{j} = H \cap G_{j}$ is a centralizer of a torus in $G_{j}$, $j = 1,\ldots, r$, and $$\label{decompositionflag}
X = G_{1}/H_{1} \times \cdots \times G_{r}/H_{r}.$$ see for instance [@BorelK]. If we consider the complexification $\mathfrak{k}^{\mathbb{C}} = \mathfrak{k} \otimes \mathbb{C}$, by choosing a [*Cartan subalgebra*]{} $\mathfrak{h} \subset \mathfrak{k}^{\mathbb{C}}$, we can consider the associated root system $\Pi = \Pi^{+} \cup \Pi^{-}$, which induces a triangular decompostion $$\mathfrak{k}^{\mathbb{C}} = \bigg ( \sum_{\alpha \in \Pi^{+}}\mathfrak{k}_{\alpha}\bigg) \oplus \mathfrak{h} \oplus \bigg ( \sum_{\alpha \in \Pi^{-}}\mathfrak{k}_{\alpha}\bigg).$$ Then, if we take the [*Borel subalgebra*]{} $$\mathfrak{b} = \bigg ( \sum_{\alpha \in \Pi^{+}}\mathfrak{k}_{\alpha}\bigg) \oplus \mathfrak{h},$$ it can be shown that there exists a parabolic Lie subalgebra $\mathfrak{p} \subset \mathfrak{k}^{\mathbb{C}}$, which contains $\mathfrak{b}$, such that its normalizer $P = N_{K_{ss}^{\mathbb{C}}}(\mathfrak{p})$ (parabolic Lie subgroup) satisfies $$H = P \cap K_{ss}.$$ Moreover, it follows from the Iwasawa decomposition of $K_{ss}^{\mathbb{C}}$ that $$\label{cartesian}
X = K_{ss}^{\mathbb{C}}/P.$$ From these, we have $K_{ss}^{\mathbb{C}} = G_{1}^{\mathbb{C}}\times \cdots \times G_{r}^{\mathbb{C}}$, and then $$\label{flagdecompgeral}
X = G_{1}^{\mathbb{C}}/P_{1} \times \cdots \times G_{r}^{\mathbb{C}}/P_{r},$$ such that $P_{j} = P\cap G_{j}^{\mathbb{C}}$ is a parabolic Lie subgroup of $G_{j}^{\mathbb{C}}$, $j = 1,\ldots,r$. Further, we also can write $$\label{parabolicdecomp}
P = P_{1} \times \cdots \times P_{r},$$ notice that we have $H_{j} = P_{j} \cap G_{j}$, for every $j = 1,\ldots,r$. In the setting above, if we denote by $\Sigma \subset \mathfrak{h}^{\ast}$ the associated simple root system defined by the pair $(\mathfrak{k}^{\mathbb{C}},\mathfrak{h})$, it follows that $$\mathfrak{h} = \mathfrak{h}_{1} \times \cdots \times \mathfrak{h}_{r}, \ \ {\text{and}} \ \ \Sigma = \Sigma_{1}\cup \cdots \cup \Sigma_{r},$$ such that $\mathfrak{h}_{j} \subset \mathfrak{g}_{j}^{\mathbb{C}}$ is a Carta subalgebra, and $\Sigma_{j} \subset \mathfrak{h}_{j}^{\ast}$, $j = 1,\ldots,r$, is a simple root system. Now, we notice that the parabolic Lie subgroup $P \subset K_{ss}^{\mathbb{C}}$ is uniquely determined by $\Theta_{P} \subset \Sigma$. Actually, we have $\mathfrak{p} = \mathfrak{p}_{\Theta_{P}}$, such that $$\mathfrak{p}_{\Theta_{P}} = \bigg ( \sum_{\alpha \in \Pi^{+}}\mathfrak{k}_{\alpha}\bigg) \oplus \mathfrak{h} \oplus \bigg ( \sum_{\alpha \in \langle \Theta_{P} \rangle^{-}}\mathfrak{k}_{\alpha}\bigg),$$ where $\langle \Theta_{P} \rangle^{-} = \langle \Theta_{P} \rangle \cap \Pi^{-}$. Furthermore, from \[parabolicdecomp\] we obtain $$\Theta_{P} = \Theta_{P_{1}} \cup \cdots \cup \Theta_{P_{r}},$$ such that $\mathfrak{p}_{\Theta_{P_{j}}} = \mathfrak{p}_{j} \subset \mathfrak{g}_{j}^{\mathbb{C}}$ is a parabolic Lie subalgebra defined by $$\mathfrak{p}_{j} = \bigg ( \sum_{\alpha \in \Pi_{j}^{+}}\mathfrak{g}_{\alpha}^{(j)}\bigg) \oplus \mathfrak{h}_{j} \oplus \bigg ( \sum_{\alpha \in \langle \Theta_{P_{j}} \rangle^{-}}\mathfrak{g}_{\alpha}^{(j)}\bigg),$$ for every $j = 1,\ldots,r$.
The ideas above show that each factor in the decomposition \[flagdecompgeral\] bears the same Lie-theoretical features as the product manifold $X$ described in \[cartesian\]. Moreover, given a negative line bundle $L \to X$, it follows that $$L = \text{pr}_{1}^{\ast} L_{1} \otimes \cdots \otimes \text{pr}_{r}^{\ast} L_{r},$$ where $\text{pr}_{j} \colon X \to G_{j}^{\mathbb{C}}/P_{j}$, $j = 1,\ldots,r$, and $L_{j} \in {\text{Pic}}(G_{j}^{\mathbb{C}}/P_{j})$, $j = 1,\ldots,r$. Therefore, from Remark \[withouttorsion\], and the fact that ${\text{Pic}}(G_{j}^{\mathbb{C}}/P_{j})$ can be described purely in terms of Lie-theoretical elements (e.g. [@Bott], [@BorelH]), the study of compact homogeneous l.c.K. manifolds reduces to the study of principal elliptic bundles defined by negative line bundles over a compact simply connected homogeneous Hodge manifold $X$, such that $$X = G^{\mathbb{C}}/P = G/G \cap P,$$ where $G^{\mathbb{C}}$ is a connected simply connected complex simple Lie group with compact real form given by $G$, and $P \subset G^{\mathbb{C}}$ is a parabolic Lie subgroup. In order to emphasize the parabolic Lie subgroup which defines a complex flag manifold, as well as the parabolic geometry defined by the pair $(G^{\mathbb{C}},P)$, we shall denote $X_{P} = G^{\mathbb{C}}/P$.
Proof of main results and examples
==================================
In this section, we provide a complete proof for our main results. In order to do so, we start by collecting some results which allow us to describe geometric structures associated to negative line bundles over flag manifolds by using elements of representation theory of Lie algebras and Lie groups. Then, we combine this with the results covered in the previous sections, particularly, Theorem \[HEW\], Theorem \[vaismanbundle\], Theorem \[LCKisVaisman\] and Theorem \[Vaismanregular\], to prove Theorem \[Theorem1\], Theorem \[Theorem2\] and Theorem \[Theorem3\]. In order to prove Theorem \[Theorem4\], we start introducing some results about the background in algebraic geometry which underlies the study of the projective algebraic realization of flag manifolds, and then we prove the result. Throughout this section we also provide several examples which illustrate each one of our main results.
Vaisman structures on principal elliptic bundles over flag manifolds
--------------------------------------------------------------------
In this subsection we develop a study of principal elliptic fibrations over complex flag manifolds. Our main purpose is providing a concrete description of compact homogeneous Vaisman manifold by following Theorem \[vaismanbundle\].
Notice that, by following the results covered in Subsection \[negativeelliptic\], given a complex flag manifold $X_{P} = G^{\mathbb{C}}/P$ and a negative line bundle $L \in {\text{Pic}}(X_{P})$, we can associate to the pair $(X_{P},L)$ the following diagram:
((L\^ ),\_,J\_ ) & ((L\^)/,g,J ) & (Q(L),g\_[Q(L)]{},,,)\
& (X\_[P]{},\_[X\_[P]{}]{})
for a suitable $\Gamma \subset \mathbb{C}^{\times}$. Moreover, the geometric structures defined on each manifold in the diagram above are completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, ${\rm{K}}_{H}(u) = H(u,u), \forall u \in {\text{Tot}}(L^{\times})$, such that $H$ is some Hermitian structure on $L$. In fact, as we have seen, the fundamental object which allows to compute the geometric structures on each manifold of the last diagram is the principal $\mathbb{C}^{\times}$-connection $\widetilde{\Psi} \in \Omega^{1}({\text{Tot}}(L^{\times} );\mathbb{C})$ given by $$\widetilde{\Psi} = \frac{1}{2}\big (d^{c} - \sqrt{-1}d\big )\log({\rm{K}}_{H}) = -\sqrt{-1} \partial \log({\rm{K}}_{H}).$$ Hence, since the principal bundles in the last diagram are classified by the same characteristic class in $H^{2}(X_{P},\mathbb{Z})$, in order to describe the Vaisman structures in the homogeneous setting we need to describe the unique $G$-invariant element in the characteristic class of elliptic fibrations over $X_{P}$.
### Line bundles over complex flag manifolds {#subsec3.1}
In order to study principal elliptic fibrations defined by negative line bundles, let us collect some general facts about line bundles over flag manifolds.
Let $\mathfrak{g}^{\mathbb{C}}$ be a complex simple Lie algebra, by fixing a Cartan subalgebra $\mathfrak{h}$ and a simple root system $\Sigma \subset \mathfrak{h}^{\ast}$, we have a decomposition of $\mathfrak{g}^{\mathbb{C}}$ given by
$\mathfrak{g}^{\mathbb{C}} = \mathfrak{n}^{-} \oplus \mathfrak{h} \oplus \mathfrak{n}^{+}$,
where $\mathfrak{n}^{-} = \sum_{\alpha \in \Pi^{-}}\mathfrak{g}_{\alpha}$ and $\mathfrak{n}^{+} = \sum_{\alpha \in \Pi^{+}}\mathfrak{g}_{\alpha}$, here we denote by $\Pi = \Pi^{+} \cup \Pi^{-}$ the root system associated to the simple root system $\Sigma = \{\alpha_{1},\ldots,\alpha_{l}\} \subset \mathfrak{h}^{\ast}$.
Let us denote by $\kappa$ the Cartan-Killing form of $\mathfrak{g}^{\mathbb{C}}$. Now, given $\alpha \in \Pi^{+}$, we have $h_{\alpha} \in \mathfrak{h}$, such that $\alpha = \kappa(\cdot,h_{\alpha})$. We can choose $x_{\alpha} \in \mathfrak{g}_{\alpha}$ and $y_{\alpha} \in \mathfrak{g}_{-\alpha}$, such that $[x_{\alpha},y_{\alpha}] = h_{\alpha}$. Moreover, for every $\alpha \in \Sigma$, we can set $$h_{\alpha}^{\vee} = \frac{2}{\kappa(h_{\alpha},h_{\alpha})}h_{\alpha},$$ from this we have the fundamental weights $\{\omega_{\alpha} \ | \ \alpha \in \Sigma\} \subset \mathfrak{h}^{\ast}$, where $\omega_{\alpha}(h_{\beta}^{\vee}) = \delta_{\alpha \beta}$, $\forall \alpha, \beta \in \Sigma$. We denote by $$\Lambda_{\mathbb{Z}_{\geq 0}}^{\ast} = \bigoplus_{\alpha \in \Sigma}\mathbb{Z}_{\geq 0}\omega_{\alpha},$$ the set of integral dominant weights of $\mathfrak{g}^{\mathbb{C}}$.
From the Lie algebra representation theory, given $\mu \in \Lambda_{\mathbb{Z}_{\geq 0}}^{\ast}$ we have an irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu)$ with highest weight $\mu$, we denote by $v_{\mu}^{+} \in V(\mu)$ the highest weight vector associated to $\mu \in \Lambda_{\mathbb{Z}_{\geq 0}}^{\ast}$.
Now, let $G^{\mathbb{C}}$ be a connected, simply connected, and complex Lie group with simple Lie algebra $\mathfrak{g}^{\mathbb{C}}$, and consider $G \subset G^{\mathbb{C}}$ as being a compact real form of $G^{\mathbb{C}}$. Given a parabolic Lie subgroup $P \subset G^{\mathbb{C}}$, without loss of generality we can suppose that
$P = P_{\Theta}$, for some $\Theta \subseteq \Sigma$.
Recall that, by definition, we have $P_{\Theta} = N_{G^{\mathbb{C}}}(\mathfrak{p}_{\Theta})$, where ${\text{Lie}}(P_{\Theta}) = \mathfrak{p}_{\Theta} \subset \mathfrak{g}^{\mathbb{C}}$ is a parabolic Lie subalgebra given by
$\mathfrak{p}_{\Theta} = \mathfrak{n}^{+} \oplus \mathfrak{h} \oplus \mathfrak{n}(\Theta)^{-}$, with $\mathfrak{n}(\Theta)^{-} = \displaystyle \sum_{\alpha \in \langle \Theta \rangle^{-}} \mathfrak{g}_{\alpha}$,
and $N_{G^{\mathbb{C}}}(\mathfrak{p}_{\Theta})$ is its normalizer in $G^{\mathbb{C}}$.
In what follows it will be useful to consider the following basic chain of Lie subgroups
$T^{\mathbb{C}} \subset B \subset P \subset G^{\mathbb{C}}$.
For each element in the aforementioned chain of Lie subgroups we have the following characterization:
- $T^{\mathbb{C}} = \exp(\mathfrak{h})$; (complex torus)
- $B = N^{+}T^{\mathbb{C}}$, where $N^{+} = \exp(\mathfrak{n}^{+})$; (Borel subgroup)
- $P = P_{\Theta} = N_{G^{\mathbb{C}}}(\mathfrak{p}_{\Theta})$, for some $\Theta \subset \Sigma \subset \mathfrak{h}^{\ast}$. (parabolic subgroup)
The following theorem allows us to describe all $G$-invariant Kähler structures on $X_{P}$.
\[AZADBISWAS\] Let $\omega \in \Omega^{1,1}(X_{P})^{G}$ be a closed invariant real $(1,1)$-form, then we have
$\pi^{\ast}\omega = \sqrt{-1}\partial \overline{\partial}\varphi$,
where $\pi \colon G^{\mathbb{C}} \to X_{P}$, and $\varphi \colon G^{\mathbb{C}} \to \mathbb{R}$ is given by
$\varphi(g) = \displaystyle \sum_{\alpha \in \Sigma \backslash \Theta}c_{\alpha}\log||gv_{\omega_{\alpha}}^{+}||$,
with $c_{\alpha} \in \mathbb{R}_{\geq 0}$, $\forall \alpha \in \Sigma \backslash \Theta$. Conversely, every function $\varphi$ as above defines a closed invariant real $(1,1)$-form $\omega_{\varphi} \in \Omega^{1,1}(X_{P})^{G}$. Moreover, if $c_{\alpha} > 0$, $\forall \alpha \in \Sigma \backslash \Theta$, then $\omega_{\varphi}$ defines a Kähler form on $X_{P}$.
\[innerproduct\] It is worth pointing out that the norm $|| \cdot ||$ in the last theorem is a norm induced from some fixed $G$-invariant inner product $\langle \cdot, \cdot \rangle_{\alpha}$ on $V(\omega_{\alpha})$, $\forall \alpha \in \Sigma \backslash \Theta$.
By means of theorem above we can describe the unique $G$-invariant element in each integral class in $H^{2}(X_{P},\mathbb{Z})$. In fact, consider the associated principal $P$-bundle $P \hookrightarrow G^{\mathbb{C}} \to X_{P}$. In terms of $\check{C}$ech cocycles, if we take an open covering $X_{P} = \bigcup_{i \in I}U_{i}$, we can write
$G^{\mathbb{C}} = \Big \{(U_{i})_{i \in I}, \psi_{ij} \colon U_{i} \cap U_{j} \to P \Big \}$.
Now, given a fundamental weight $\omega_{\alpha} \in \Lambda_{\mathbb{Z}_{\geq 0}}^{\ast}$, by considering $\chi_{\omega_{\alpha}} \in {\text{Hom}}(T^{\mathbb{C}},\mathbb{C}^{\times})$, such that $d(\chi_{\omega_{\alpha}})_{e} = \omega_{\alpha}$, we can take the homomorphism $\chi_{\omega_{\alpha}} \colon P \to \mathbb{C}^{\times}$, defined via holomorphic extension of $\chi_{\omega_{\alpha}}$. From this last homomorphism, we consider $\mathbb{C}_{-\omega_{\alpha}}$ as a $P$-space with the action $pz = \chi_{\omega_{\alpha}}(p)^{-1}z$, $\forall p \in P$, and $\forall z \in \mathbb{C}$. Therefore, we obtain an associated holomorphic line bundle $\mathscr{O}_{\alpha}(1) = G^{\mathbb{C}} \times_{P}\mathbb{C}_{-\omega_{\alpha}}$, which can be described in terms of $\check{C}$ech cocycles by $$\label{linecocycle}
\mathscr{O}_{\alpha}(1) = \Big \{(U_{i})_{i \in I},\chi_{\omega_{\alpha}}^{-1} \circ \psi_{i j} \colon U_{i} \cap U_{j} \to \mathbb{C}^{\times} \Big \},$$ thus $\mathscr{O}_{\alpha}(1) = \{g_{ij}\} \in \check{H}^{1}(X_{P},\mathscr{O}_{X_{P}}^{\ast})$, with $g_{ij} = \chi_{\omega_{\alpha}}^{-1} \circ \psi_{i j}$, where $i,j \in I$. Notice that we can also realize $\mathscr{O}_{\alpha}(1)$ as a quotient space of $G^{\mathbb{C}} \times \mathbb{C}_{-\omega_{\alpha}}$ by the equivalence relation $``\sim"$ defined by $$\label{eqrelationbundle}
(g,z) \sim (h,w) \Longleftrightarrow \exists p \in P, \ {\text{such that}} \ h = gp \ {\text{and}} \ w = p^{-1}z = \chi_{\omega_{\alpha}}(p)z.$$ Therefore, for a typical element $u \in \mathscr{O}_{\alpha}(1)$ we have $u = [g,z]$, for some $(g,z) \in G^{\mathbb{C}} \times \mathbb{C}_{-\omega_{\alpha}}$.
\[parabolicdec\] Notice that, if we have a parabolic Lie subgroup $P \subset G^{\mathbb{C}}$, such that $P = P_{\Theta}$, the decomposition $$P_{\Theta} = \big[P_{\Theta},P_{\Theta} \big]T(\Sigma \backslash \Theta)^{\mathbb{C}},$$ see for instance [@Akhiezer Proposition 8], shows us that ${\text{Hom}}(P,\mathbb{C}^{\times}) = {\text{Hom}}(T(\Sigma \backslash \Theta)^{\mathbb{C}},\mathbb{C}^{\times})$. Therefore, if we take $\omega_{\alpha} \in \Lambda_{\mathbb{Z}_{\geq 0}}^{\ast}$, such that $\alpha \in \Theta$, we obtain $\mathscr{O}_{\alpha}(1) = X_{P} \times \mathbb{C}$, i.e. the associated holomorphic line bundle $\mathscr{O}_{\alpha}(1)$ is trivial.
Given $\mathscr{O}_{\alpha}(1) \in {\text{Pic}}(X_{P})$, such that $\alpha \in \Sigma \backslash \Theta$, as described previously, if we consider an open covering $X_{P} = \bigcup_{i \in I} U_{i}$ which trivializes both $P \hookrightarrow G^{\mathbb{C}} \to X_{P}$ and $ \mathscr{O}_{\alpha}(1) \to X_{P}$, such that $\alpha \in \Sigma \backslash \Theta$, by taking a collection of local sections $(s_{i})_{i \in I}$, such that $s_{i} \colon U_{i} \to G^{\mathbb{C}}$, we can define $h_{i} \colon U_{i} \to \mathbb{R}^{+}$ by setting $$\label{functionshermitian}
h_{i} = {\mathrm{e}}^{-2\pi \varphi_{\omega_{\alpha}} \circ s_{i}} = \frac{1}{||s_{i}v_{\omega_{\alpha}}^{+}||^{2}},$$ for every $i \in I$. These functions $(h_{i})_{i \in I}$ satisfy $h_{j} = |\chi_{\omega_{\alpha}}^{-1} \circ \psi_{ij}|^{2}h_{i}$ on $U_{i} \cap U_{j} \neq \emptyset$, here we have used that $s_{j} = s_{i}\psi_{ij}$ on $U_{i} \cap U_{j} \neq \emptyset$, and $pv_{\omega_{\alpha}}^{+} = \chi_{\omega_{\alpha}}(p)v_{\omega_{\alpha}}^{+}$, for every $p \in P$, such that $\alpha \in \Sigma \backslash \Theta$. Hence, we have a collection of functions $(h_{i})_{i \in I}$ which satisfies on $U_{i} \cap U_{j} \neq \emptyset$ the following relation $$\label{collectionofequ}
h_{j} = |g_{ij}|^{2}h_{i},$$ such that $g_{ij} = \chi_{\omega_{\alpha}}^{-1} \circ \psi_{i j}$, where $i,j \in I$.
From the collection of smooth functions described above, we can define a Hermitian structure $H$ on $\mathscr{O}_{\alpha}(1)$ by taking on each trivialization $f_{i} \colon L_{\chi_{\omega_{\alpha}}} \to U_{i} \times \mathbb{C}$ a metric defined by $$\label{hermitian}
H(f_{i}^{-1}(x,v),f_{i}^{-1}(x,w)) = q_{i}(x) v\overline{w},$$ for $(x,v),(x,w) \in U_{i} \times \mathbb{C}$. The Hermitian metric above induces a Chern connection $\nabla = d + \partial \log H$ with curvature $F_{\nabla}$ satisfying $$\displaystyle \frac{\sqrt{-1}}{2\pi}F_{\nabla} \Big |_{U_{i}} = \frac{\sqrt{-1}}{2\pi} \partial \overline{\partial}\log \Big ( \big | \big | s_{i}v_{\omega_{\alpha}}^{+}\big | \big |^{2} \Big).$$ Hence, by considering the $G$-invariant $(1,1)$-form $\Omega_{\alpha} \in \Omega^{1,1}(X_{P})^{G}$, which satisfies $$\label{basicforms}
\pi^{\ast}\Omega_{\alpha} = \sqrt{-1}\partial \overline{\partial} \varphi_{\omega_{\alpha}},$$ where $\pi \colon G^{\mathbb{C}} \to G^{\mathbb{C}} / P = X_{P}$, and $\varphi_{\omega_{\alpha}}(g) = \displaystyle \frac{1}{2\pi}\log||gv_{\omega_{\alpha}}^{+}||^{2}$, $\forall g \in G^{\mathbb{C}}$, we have $$c_{1}(\mathscr{O}_{\alpha}(1)) = \big [ \Omega_{\alpha}\big].$$ From the ideas described above we have the following result.
\[C8S8.2Sub8.2.3P8.2.6\] Let $X_{P}$ be a flag manifold associated to some parabolic Lie subgroup $P = P_{\Theta}\subset G^{\mathbb{C}}$. Then, we have $$\label{picardeq}
{\text{Pic}}(X_{P}) = H^{1,1}(X_{P},\mathbb{Z}) = H^{2}(X_{P},\mathbb{Z}) = \displaystyle \bigoplus_{\alpha \in \Sigma \backslash \Theta}\mathbb{Z}\big [\Omega_{\alpha} \big ].$$
Let us sketch the proof. The last equality on the right side of $\ref{picardeq}$ follows from the following facts:
- $\pi_{2}(X_{P}) \cong \pi_{1}(T(\Sigma \backslash \Theta)^{\mathbb{C}}) = \mathbb{Z}^{\#(\Sigma \backslash \Theta)}$, where
$$T(\Sigma \backslash \Theta)^{\mathbb{C}} = \exp \Big \{ \displaystyle \sum_{\alpha \in \Sigma \backslash \Theta}a_{\alpha}h_{\alpha} \ \Big | \ a_{\alpha} \in \mathbb{C} \Big \};$$
- Since $X_{P}$ is simply connected, it follows that $H_{2}(X_{P},\mathbb{Z}) \cong \pi_{2}(X_{P})$ (Hurewicz’s theorem);
- By taking $\mathbb{P}_{\alpha}^{1} \hookrightarrow X_{P}$, such that
$$\mathbb{P}_{\alpha}^{1} = \overline{\exp(\mathfrak{g}_{-\alpha})x_{0}} \subset X_{P},$$ $\forall \alpha \in \Sigma \backslash \Theta$, where $x_{0} = eP \in X_{P}$, it follows that
$\Big \langle c_{1}(\mathscr{O}_{\alpha}(1)), \big [ \mathbb{P}_{\beta}^{1}\big] \Big \rangle = \displaystyle \int_{\mathbb{P}_{\beta}^{1}} c_{1}(\mathscr{O}_{\alpha}(1)) = \delta_{\alpha \beta},$
$\forall \alpha,\beta \in \Sigma \backslash \Theta$. Hence, we obtain
$\pi_{2}(X_{P}) = \displaystyle \bigoplus_{\alpha \in \Sigma \backslash \Theta} \mathbb{Z}\big [ \mathbb{P}_{\alpha}^{1}\big],$ and $H^{2}(X_{P},\mathbb{Z}) = \displaystyle \bigoplus_{\alpha \in \Sigma \backslash \Theta} \mathbb{Z} c_{1}(\mathscr{O}_{\alpha}(1))$.
Therefore, $H^{1,1}(X_{P},\mathbb{Z}) = H^{2}(X_{P},\mathbb{Z})$. Now, from the Lefschetz theorem on (1,1)-classes [@DANIEL p. 133], and the fact that $0 = b_{1}(X_{P}) = {\text{rk}}({\text{Pic}}^{0}(X_{P}))$, we obtain the first equality in \[picardeq\].
From Theorem \[AZADBISWAS\] and Proposition \[C8S8.2Sub8.2.3P8.2.6\], given a negative line bundle $L \in {\text{Pic}}(X_{P})$, we have $$\label{negativedecomp}
L = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(1)^{\otimes \langle c_{1}(L),[ \mathbb{P}_{\alpha}^{1}] \rangle},$$ where $\big \langle c_{1}(L),\big [ \mathbb{P}_{\alpha}^{1} \big] \big \rangle < 0$, $\forall \alpha \in \Sigma \backslash \Theta$. For the sake of simplicity, we denote $\ell_{\alpha} = - \big \langle c_{1}(L),\big [ \mathbb{P}_{\alpha}^{1} \big] \big \rangle$, and $\mathscr{O}_{\alpha}(1)^{\otimes k} = \mathscr{O}_{\alpha}(k)$, for every $k \in \mathbb{Z}$, $\forall \alpha \in \Sigma \backslash \Theta$. Thus, we can rewrite \[negativedecomp\] as $$L = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(-\ell_{\alpha}).$$ From the characterization above, if we consider the weight $\mu(L) \in \Lambda_{\mathbb{Z}_{\geq 0}}^{\ast}$ defined by $$\label{negativeweight}
\mu(L) = - \sum_{\alpha \in \Sigma \backslash \Theta} \big \langle c_{1}(L),\big [ \mathbb{P}_{\alpha}^{1} \big] \big \rangle \omega_{\alpha} = \sum_{\alpha \in \Sigma \backslash \Theta} \ell_{\alpha} \omega_{\alpha},$$ we can associate to $L$ an irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu(L))$ with highest weight vector $v_{\mu(L)}^{+} \in V(\mu(L))$, such that $$\displaystyle V(\mu(L)) \subset \bigotimes_{\alpha \in \Sigma \backslash \Theta}V(\omega_{\alpha})^{\otimes \ell_{\alpha}}, \ \ {\text{and}} \ \ \displaystyle v_{\mu(L)}^{+} = \bigotimes_{\alpha \in \Sigma \backslash \Theta}(v_{\omega_{\alpha}}^{+})^{\otimes \ell_{\alpha}},$$ see for instance [@PARABOLICTHEORY p. 186].
Notice that, associated to the weight \[negativeweight\] we have a character $\chi^{(L)} \colon P \to \mathbb{C}^{\times}$, such that $$\label{negativecharacter}
\chi^{(L)} = \prod_{\alpha \in \Sigma \backslash \Theta}\chi_{\omega_{\alpha}}^{\ell_{\alpha}}.$$ Therefore, in terms of $\check{C}$ech cocycles we have $$\label{negaticecocycle}
L = \Big \{(U_{i})_{i \in I},\chi^{(L)} \circ \psi_{i j} \colon U_{i} \cap U_{j} \to \mathbb{C}^{\times} \Big \},$$ thus $L = \{g_{ij}\} \in \check{H}^{1}(X_{P},\mathscr{O}_{X_{P}}^{\ast})$, with $g_{ij} = \chi^{(L)} \circ \psi_{i j}$, where $i,j \in I$.
\[proj\] It is worthwhile to observe that, in the setting above, the ample line bundle $L^{-1} \in {\text{Pic}}(X_{P})$ is in fact very ample, i.e. we have a projective embedding
$\iota \colon X_{P} \hookrightarrow \mathbb{P}(V(\mu(L))) = {\text{Proj}}\big (H^{0}(X_{P},L^{-1})^{\ast} \big),$
see for instance [@Flaginterplay Page 193], [@PARABOLICTHEORY Theorem 3.2.8], [@TAYLOR]. Thus, we have the identification $$\label{tautological}
L \cong \iota^{\ast}\mathscr{O}_{\mathbb{P}(V(\mu(L))}(-1) = \Big \{ \big ([x],v \big ) \in X_{P} \times V(\mu(L)) \ \ \Big | \ \ v \in \langle x \rangle_{\mathbb{C}}\Big \},$$ where $\mathscr{O}_{\mathbb{P}(V(\mu(L))}(-1)$ is the tautological line bundle over $\mathbb{P}(V(\mu(L))$, here we consider the geometric realization $V(\mu(L)) = H^{0}(X_{P},L^{-1})^{\ast}$ via Borel-Weil theorem.
### Vaisman structrures on compact homogeneous Hermitian manifolds
By following the ideas which we have established in the previous sections, now we are able to prove our main result about Vaisman structures on principal elliptic bundles over flag manifolds.
\[Maintheo1\] Let $X_{P}$ be a complex flag manifold, associated to some parabolic Lie subgroup $P = P_{\Theta} \subset G^{\mathbb{C}}$, and let $L \in {\text{Pic}}(X_{P})$ be a negative line bundle, such that
$\displaystyle L = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(-\ell_{\alpha}),$
where $\ell_{\alpha} > 0$, $\forall \alpha \in \Sigma \backslash \Theta$. Then, for every $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, we have that the manifold $$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ {\text{where}} \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$$ admits a Vaisman structure completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, defined in coordinates $(z,w) \in L^{\times}|_{U}$ by $$\label{negativepotential}
{\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w},$$ for some local section $s_{U} \colon U \subset X_{P} \to G^{\mathbb{C}}$, where $v_{\mu(L)}^{+}$ is the highest weight vector of weight $\mu(L)$ associated to the irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu(L)) = H^{0}(X_{P},L^{-1})^{\ast}$.
The proof follows from the results which we have described so far in the previous sections. Actually, under the hypotheses therorem above, if we consider the characterization
$L = \Big \{(U_{i})_{i \in I},\chi^{(L)} \circ \psi_{i j} \colon U_{i} \cap U_{j} \to \mathbb{C}^{\times} \Big \},$
see Remark \[negaticecocycle\], we can define the Hermitian structure on $L \to X_{P}$ by setting $$\label{hermiotianinvariant}
\displaystyle H \big (f_{i}^{-1}(z,v),f_{i}^{-1}(z,w) \big ) = \Big ( \prod_{\alpha \in \Sigma \backslash \Theta}\big | \big |s_{i}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{2\ell_{\alpha}}\Big)v\overline{w},$$ here as before we consider local trivializations $(L|_{U_{i}},f_{i})$ and local sections $s_{i} \colon U_{i} \to G^{\mathbb{C}}$, $\forall i \in I$. Since we have $s_{j} = s_{i}\psi_{ij}$, $f_{i} \circ f_{j}^{-1}(z,w) = (z,g_{ij}(z)w)$ on $U_{i} \cap U_{j} \neq \emptyset$, and $pv_{\omega_{\alpha}}^{+} = \chi_{\omega_{\alpha}}(p)v_{\omega_{\alpha}}^{+}$, for every $p \in P$, such that $\alpha \in \Sigma \backslash \Theta$, it follows from Equation \[negativecharacter\] that
$\displaystyle H \big (f_{j}^{-1}(z,v),f_{j}^{-1}(z,w) \big ) = \big |\chi^{(L)}(\psi_{ij}(z)) \big |^{2}\Big ( \prod_{\alpha \in \Sigma \backslash \Theta}\big | \big |s_{i}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{2\ell_{\alpha}}\Big)v\overline{w} = H \big (f_{i}^{-1}(z,g_{ij}(z)v),f_{i}^{-1}(z,g_{ij}(z)w) \big ),$
$\forall (z,w),(z,v) \in U_{i}\cap U_{j} \times \mathbb{C}$. Hence, from the Hermitian structure above we can define a Kähler potential ${\rm{K}}_{H} \colon {\text{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, by setting ${\rm{K}}_{H}(u) = H(u,u)$. From this, locally on $L^{\times}|_{U_{i}}$ we have
$\displaystyle {\rm{K}}_{H}(z,w) = {\rm{K}}_{H}(f_{i}^{-1}(z,w)) = \Big ( \prod_{\alpha \in \Sigma \backslash \Theta}\big | \big |s_{i}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{2\ell_{\alpha}}\Big)w\overline{w}$.
Now, in order to obtain Equation \[negativepotential\], we proceed as follows: Since we have $$\displaystyle V(\mu(L)) \subset \bigotimes_{\alpha \in \Sigma \backslash \Theta}V(\omega_{\alpha})^{\otimes \ell_{\alpha}}, \ \ {\text{and}} \ \ \displaystyle v_{\mu(L)}^{+} = \bigotimes_{\alpha \in \Sigma \backslash \Theta}(v_{\omega_{\alpha}}^{+})^{\otimes \ell_{\alpha}},$$ we can take a $G$-invariant inner product on $V(\mu(L))$ induced from a $G$-invariant inner product $\langle \cdot, \cdot \rangle_{\alpha}$ on each factor $V(\omega_{\alpha})$, $\forall \alpha \in \Sigma \backslash \Theta$, such that[^12] $$\label{innerinv}
\Big \langle \bigotimes_{\alpha \in \Sigma \backslash \Theta} \big(v_{1}^{(\alpha)} \otimes \cdots \otimes v_{\ell_{\alpha}}^{(\alpha)}), \bigotimes_{\alpha \in \Sigma \backslash \Theta} \big(w_{1}^{(\alpha)} \otimes \cdots \otimes w_{\ell_{\alpha}}^{(\alpha)}) \Big \rangle = \prod_{ \alpha \in \Sigma \backslash \Theta}\big \langle v_{1}^{(\alpha)},w_{1}^{(\alpha)} \big \rangle_{\alpha}\cdots \big \langle v_{\ell_{\alpha}}^{(\alpha)},w_{\ell_{\alpha}}^{(\alpha)} \big \rangle_{\alpha},$$ see Remark \[innerproduct\]. Considering the norm on $V(\mu(L))$ induced by the inner product above, we can rewrite
$\displaystyle {\rm{K}}_{H}\big (z,w \big ) = \Big ( \prod_{\alpha \in \Sigma \backslash \Theta}\big | \big |s_{i}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{2\ell_{\alpha}}\Big)w\overline{w} = \Big (\big | \big |s_{i}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w}$.
It is straightforward to see that the Kähler potential above defines a Vaisman structure on the manifold $M = {\text{Tot}}(L^{\times})/\Gamma$. In fact, if we consider a principal $\mathbb{C}^{\times}$-connection $\widetilde{\Psi} \in \Omega^{1}(L^{\times};\mathbb{C})$, given by
$\widetilde{\Psi} = \frac{1}{2}\big (d^{c} - \sqrt{-1}d\big )\log({\rm{K}}_{H}) = -\sqrt{-1} \partial \log({\rm{K}}_{H})$,
by regarding $M$ as principal elliptic bundle over $X_{P}$, namely,
$M = L^{\times}/ \Gamma = \big (L^{\times} \times \mathbb{E}(\Lambda) \big) / \mathbb{C}^{\times},$
such that $\mathbb{E}(\Lambda) = \mathbb{C}^{\times}/\Gamma$, we have an induced principal $\mathbb{E}(\Lambda)$-connection $\Psi \in \Omega^{1}(M;\mathbb{C})$, which satisfies
$\wp^{\ast}\Psi = \widetilde{\Psi}$,
where $\wp \colon {\text{Tot}}(L^{\times}) \to {\text{Tot}}(L^{\times})/\Gamma$ is the projection map. Now, by considering the complex structure $J \in {\text{End}}(TM)$ induced from ${\text{Tot}}(L^{\times})$, it follows that $\Psi \circ J = \sqrt{-1}\Psi$. Furthermore, from the definition of ${\rm{K}}_{H}$, it follows that (locally) on $M|_{U_{i}}$ we have
$\displaystyle d\Psi = \sum_{\alpha \in \Sigma \backslash \Theta}\ell_{\alpha}\sqrt{-1} \partial\overline{\partial}\log \Big (\big | \big |s_{i}v_{\omega_{\alpha}}^{+} \big | \big |^{2}\ \Big) = 2\pi \Big (\sum_{\alpha \in \Sigma \backslash \Theta}\ell_{\alpha}\Omega_{\alpha}|_{U_{i}} \Big) = 2\pi p_{\Lambda}^{\ast} \big (c_{1}(L^{-1})|_{U_{i}} \big),$
where $p_{\Lambda} \colon M \to X_{P}$ is the bundle projection. Since $c_{1}(L^{-1})$ defines a $G$-invariant Kähler metric on $X_{P}$, see Theorem \[AZADBISWAS\], we conclude the proof by applying Theorem \[vaismanbundle\].
\[Ginvinner\] Notice that, given any inner product $\langle \cdot ,\cdot \rangle_{0}$ on $V(\mu(L))$ we can construct a $G$-invariant inner product $\langle \cdot , \cdot \rangle$ on $V(\mu(L))$ by setting $$\big \langle v_{1},v_{2} \big \rangle = \int_{G}\big\langle gv_{1}, gv_{2} \big \rangle_{0}d\nu,$$ $\forall v_{1},v_{2} \in V(\mu(L))$, here $d\nu$ denotes the Haar measure on $G$ normalized, so $\int_{G} 1 \cdot d\nu = 1$. Under the hypotheses of the last theorem, by taking any $G$-invariant inner product $\langle \cdot , \cdot \rangle$ on $V(\mu(L))$, and considering its induced norm $||\cdot||$, we can also define a Hermitian structure on $L$ by gluing
$\displaystyle H \big (f_{i}^{-1}(z,v),f_{i}^{-1}(z,w) \big ) = \Big (\big | \big |s_{i}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) v\overline{w}$.
The computation to construct a Vaisman structure on the manifold ${\text{Tot}}(L^{\times})/\Gamma$ is the same as in the case of the Hermitian structure defined in \[hermiotianinvariant\]. Also, notice that two different $G$-invariant inner products on $V(\mu(L))$ induce the same conformal structure on ${\text{Tot}}(L^{\times})/\Gamma$ by the procedure given in the proof of the last theorem.
\[metricdescription\] An important feature of theorem above is that it provides a concrete description for Vaisman structures on principal elliptic bundles by means of elements of representation theory of simple Lie algebras.
As we have seen, under the hypotheses of Theorem \[Maintheo1\], we obtain an explicit description for a principal $\mathbb{E}(\Lambda)$-connection on $M = {\text{Tot}}(L^{\times})/\Gamma$ in terms of representation theory, i.e. we have $\Psi \in \Omega^{1}(M;\mathbb{C})$ locally described by $$\Psi = -\sqrt{-1}\bigg [\partial \log \Big ( \big | \big |s_{i}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{dw}{w}\bigg],$$ for some local section $s_{i} \colon U_{i} \subset X_{P} \to G^\mathbb{C}$. From this connection we have a Riemannian metric on $M$ defined by
$\displaystyle g = \frac{1}{2} \Big ( d\Psi({\text{id}} \otimes J) + \Psi \odot \overline{\Psi} \Big )$.
Moreover, from the identification ${\text{Tot}}(L^{\times}) =\mathscr{C}(Q(L))$, as we have seen in Subsection \[negativeelliptic\], associated to the Vaismans structure $(M,J,g)$ we have the Lee form $\theta \in \Omega^{1}(M)$ given in terms of ${\rm{K}}_{H}$ (locally) by $$\label{leelocal}
\theta = -\bigg [d\log \Big ( \big | \big |s_{i}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg],$$ see for instance Equation \[globalpotlee\]. It is worthwhile to observe that, we can also describe the anti-Lee form $\vartheta = -\theta \circ J \in \Omega^{1}(M)$ by $$\vartheta = -\bigg [d^{c}\log \Big ( \big | \big |s_{i}v_{\mu(L)}^{+} \big | \big |^{2}\Big )- \frac{\sqrt{-1}}{|w|^{2}}\big (\overline{w}dw - wd\overline{w} \big) \bigg ],$$ for the above description see Equation \[connectiontorus\], and Equation \[connectionlee\].
By combining the previous theorem with the result provided in [@Gauduchon1], we obtain the following theorem.
\[Theo1\] Let $(M,g,J)$ be a compact l.c.K. manifold that admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$. Suppose also that $K_{{\text{ss}}}$ is simply connected and has a unique simple component. Then, the following holds:
1. The manifold $M$ is a principal $T_{\mathbb{C}}^{1}$-bundle over a complex flag manifold $X_{P} = K_{{\text{ss}}}^{\mathbb{C}}/P$, for some parabolic Lie subgroup $P = P_{\Theta} \subset K_{{\text{ss}}}^{\mathbb{C}}$. Moreover, $M$ is completely determined by a negative line bundle $L \in {\text{Pic}}(X_{P})$, i.e.,
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ {\text{where}} \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$.
2. The associated l.c.K. structure on $M$ is completely determined by the global Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, defined in coordinates $(z,w) \in L^{\times}|_{U}$ by
${\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w},$
for some local section $s_{U} \colon U \subset X_{P} \to K_{\text{ss}}^{\mathbb{C}}$, where $v_{\mu(L)}^{+}$ is the highest weight vector of weight $\mu(L)$ associated to the irreducible $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$-module $V(\mu(L)) = H^{0}(X_{P},L^{-1})^{\ast}$.
The result above can be obtained as follows. For the first item, we notice that from Theorem \[LCKisVaisman\], Theorem \[Vaismanregular\], and Theorem \[vaismanclass\], it follows that $M$ is a principal $T_{\mathbb{C}}^{1}$-bundle over a complex flag manifold $X_{P} = K_{{\text{ss}}}^{\mathbb{C}}/P$. Since $X_{P}$ has no torsion elements in $H^{2}(X_{P},\mathbb{Z})$, it follows that there exists a negative line bundle $L \in {\text{Pic}}(X_{P})$ such that
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ {\text{where}} \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, see for instance Theorem \[vaismanclass\], and [@Tsukada Theorem 4.2].
For the second item, we notice that, since $$\label{bettiharmonic}
\dim\mathscr{H}^{1}(M) = b_{1}(M) = 1,$$ where $\mathscr{H}^{1}(M)$ is the space of harmonic 1-forms, see Remark \[bettinumber\], it follows that there is $f \in C^{\infty}(M)$ such that the Lee form $\theta_{h} \in \Omega^{1}(M)$ of $h = {\rm{e}}^{-f}g$ is a multiple of the $1$-form \[leelocal\], i.e. $$\label{scalelee}
\theta_{h} = - c_{0}\bigg [d\log \Big ( \big | \big |s_{i}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg],$$ for some real constant $c_{0} \in \mathbb{R}$. In fact, by considering $H_{DR}^{1}(M) = \mathbb{R}[\theta]$, such that $\theta \in \Omega^{1}(M)$ is given by the expression \[leelocal\], it follows that $\theta_{g} = c_{0}\theta + df$, for some $f \in C^{\infty}(M)$ and some real constant $c_{0} \in \mathbb{R}$. Thus, after a conformal change $g \mapsto h = {\rm{e}}^{-f}g$, we obtain $\theta_{g} \mapsto \theta_{h} = \theta_{g} -df$. By considering \[scalelee\], we can define a connection $\Psi_{h} \in \Omega^{1}(M;\mathbb{C})$ by setting $$\displaystyle \Psi_{h} = \frac{c_{0}}{\sqrt{-1}}\bigg [\partial \log \Big ( \big | \big |s_{i}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{dw}{w}\bigg],$$ notice that $\Psi_{h} = -\vartheta_{h} + \sqrt{-1}\theta_{h}$, such that $\vartheta_{h} = -\theta_{h} \circ J$. On the other hand, we have a connection on $M$ defined by $\Psi_{g} = -\vartheta_{g} + \sqrt{-1}\theta_{g}$, such that $\vartheta_{g} = -\theta_{g} \circ J$. Hence, we obtain
$\Psi_{g} = \Psi_{h} + d^{c}f + \sqrt{-1}df.$
Since both $d\Psi_{g}$ and $d\Psi_{h}$ descend to a real $K_{{\text{ss}}}$-invariant $(1,1)$-form on $X_{P}$, from the uniqueness of $K_{{\text{ss}}}$-invariant representative for the Chern class of the principal $T_{\mathbb{C}}^{1}$-bundle $M$ over $X_{P}$, we obtain that $d\Psi_{g} = d\Psi_{h}$. Therefore, we have $$\sqrt{-1} \partial \overline{\partial} f = \frac{1}{2}dd^{c}f = 0.$$ Since $M$ is a connected compact complex manifold, it follows that $f$ is constant, so we obtain $\theta_{g} = \theta_{h}$. Now, from the bundle-like feature of $g$, see Theorem \[blikekahler\], it follows that
$g = d\Psi_{g}({\text{id}} \otimes J) + \frac{1}{2}\Psi_{g} \odot \overline{\Psi_{g}}$,
see Equation \[localvaisman\]. Notice that, since $\Psi_{g} = -\vartheta_{g} + \sqrt{-1}\theta_{g}$, such that $\vartheta_{g} = -\theta_{g} \circ J$, and since $d\Psi_{g}$ descends to a Kähler metric on $X_{P}$, we must have $c_{0}>0$. Moreover, from the above description of $g$, we have
$\Omega = g(J \otimes {\rm{id}}) = d(\theta_{g} \circ J) - \theta_{g} \wedge (\theta_{g} \circ J),$ and $d\Omega = \theta_{g} \wedge \Omega.$
Hence, we obtain that the Vaisman structure $(g,J,\theta_{g})$ on $M$ is completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$ defined in the item (2).
\[affinecone\] Cones over smooth projective varieties provide a simple class of isolated singularities. The relation between Theorem \[Theo1\] and analytic affine cones over flag varieties can be described as follows. In the setting of the last theorem, if we consider the pair $(X_{P},L)$, such that $L \in {\text{Pic}}(X_{P})$ is a negative line bundle, we have a projective embedding given by
$\iota \colon X_{P} \hookrightarrow \mathbb{P}(V(\mu(L)))$, $\iota \colon gP \mapsto \big [gv_{\mu(L)}^{+} \big], \ \forall gP \in X_{P}$,
see Remark \[proj\]. Considering the projection $\pi \colon V(\mu(L)) \backslash \{0\} \to \mathbb{P}(V(\mu(L)))$, we have the analytic affine cone over $X_{P}$ given by $$C(X_{P},L) = \pi^{-1}(X_{P}) \cup \{0\} = G^{\mathbb{C}}\cdot v_{\mu(L)}^{+} \cup \{0\}.$$ The analytic affine cone $C(X_{P},L) \subset V(\mu(L))$ is a (closed) analytic affine variety which has an isolated singularity at $0 \in C(X_{P},L)$. Since a closed analytic subvariety of a Stein space is a Stein space [@GRAUERT1 Chapter V, $\mathsection$ 1], we have that $C(X_{P},L)$ is a singular Stein space. Moreover, the regular locus of $C(X_{P},L)$, which we denote by $C(X_{P},L)_{{\text{reg}}} = \pi^{-1}(X_{P})$, defines a principal $\mathbb{C}^{\times}$-bundle over $X_{P} \subset \mathbb{P}(V(\mu(L)))$, which is identified with the frame bundle of the dual to the hyperplane line bundle $\mathscr{O}_{\mathbb{P}(V(\mu(L))}(1)$ restrict to $X_{P}$. Thus, by considering the identification \[tautological\], it follows that $$C(X_{P},L)_{{\text{reg}}} \cong {\rm{Tot}}(L^{\times}) \Longrightarrow G^{\mathbb{C}}\cdot v_{\mu(L)}^{+} \cong {\rm{Tot}}(L^{\times}).$$ From the characterization above, we have a resolution ${\mathscr{R}} \colon {\rm{Tot}}(L) \to C(X_{P},L)$ (a.k.a. Cartan-Remmert reduction, see for instance instance [@GRAUERT]), such that ${\mathscr{R}}$ contracts to $0$ the zero section $X_{P} \subset L$. In this situation ${\mathscr{R}}$ is the blow up of $0$ in $C(X_{P},L)$, and the exceptional divisor $X_{P}$ is the projective tangent cone at $0$ in $C(X_{P},L)$. Hence, we can realize $ {\rm{Tot}}(L) = {\text{Bl}}_{0}(C(X_{P},L))$.
As we can see, from Theorem \[Theo1\] we obtain a constructive method to describe Vaisman manifolds by means of regular locus of analytic affine cones over flag manifolds. Furthermore, we have a concrete description, in terms of the potential ${\rm{K}}_{H} \colon C(X_{P},L)_{{\text{reg}}} \to \mathbb{R}$, for Kähler metrics defined on the regular locus of analytic affine cones over flag manifolds. We shall further explore later the ideas described above taking into account basic elements of the algebraic geometry.
In order to prove our next result, let us recall some basic facts related to the canonical bundle o flag manifolds. In the context of complex flag manifolds, the anticanonical line bundle can be described as follows. Consider the identification
$\mathfrak{m} = \displaystyle \sum_{\alpha \in \Pi^{+} \backslash \langle \Theta \rangle^{+}} \mathfrak{g}_{-\alpha} = T_{x_{0}}^{1,0}X_{P}$,
where $x_{0} = eP \in X_{P}$. We have the following characterization for $T^{1,0}X_{P}$ as an associated holomoprphic vector bundle to the principal $P$-bundle $P \hookrightarrow G^{\mathbb{C}} \to X_{P}$:
$T^{1,0}X_{P} = G^{\mathbb{C}} \times_{P} \mathfrak{m}$,
such that the twisted product on the right side above is obtained from the isotropy representation ${\rm{Ad}} \colon P \to {\rm{GL}}(\mathfrak{m})$. Let us introduce $\delta_{P} \in \mathfrak{h}^{\ast}$ by setting
$\delta_{P} = \displaystyle \sum_{\alpha \in \Pi^{+} \backslash \langle \Theta \rangle^{+}} \alpha$.
Since $P = [P,P]T(\Sigma \backslash \Theta)^{\mathbb{C}}$, a straightforward computation shows that $$\label{charactercanonical}
\det \circ {\rm{Ad}} = \chi_{\delta_{P}}^{-1},$$ which implies that $$\label{canonicalbundleflag}
K_{X_{P}}^{-1} = \det \big(T^{1,0}X_{P} \big) =\det \big ( G^{\mathbb{C}} \times_{P} \mathfrak{m} \big )= L_{\chi_{\delta_{P}}}.$$ Moreover, since the holomorphic character associated to $\delta_{P}$ can be written as $$\chi_{\delta_{P}} = \displaystyle \prod_{\alpha \in \Sigma \backslash \Theta} \chi_{\omega_{\alpha}}^{\langle \delta_{P},h_{\alpha}^{\vee} \rangle},$$ it implies that $$K_{X_{P}} = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(-\ell_{\alpha}),$$ such that $\ell_{\alpha} = \langle \delta_{P}, h_{\alpha}^{\vee} \rangle, \forall \alpha \in \Sigma \backslash \Theta$.
Notice that, from the above description, the fano index for a complex flag manifold $X_{P}$ is given by $$I(X_{P}) = {\text{gcd}} \Big ( \langle \delta_{P}, h_{\alpha}^{\vee} \rangle \ \Big | \ \alpha \in \Sigma \backslash \Theta \Big ),$$ here we suppose $P = P_{\Theta} \subset G^{\mathbb{C}}$, for some $\Theta \subset \Sigma$. Thus, $I(X_{P})$ can be completely determined from the Cartan matrix of $\mathfrak{g}^{\mathbb{C}}$.
Given a complex flag manifold $X_{P}$, associated to some parabolic Lie subgroup $P \subset G^{\mathbb{C}}$, we can consider the invariant Kähler metric $\rho_{0} \in \Omega^{1,1}(X_{P})^{G}$, such that $$\label{riccinorm}
\rho_{0}|_{U} = \sqrt{-1}\partial \overline{\partial} \log \Big (\big | \big |s_{U}v_{\delta_{P}}^{+} \big| \big |^{2} \Big ),$$ for some local section $s_{U} \colon U \subset X_{P} \to G^{\mathbb{C}}$, notice that $\mu(K_{X_{P}}) = \delta_{P}$. It is not difficult to see that
$\displaystyle c_{1}(X_{P}) = \Big [ \frac{\rho_{0}}{2\pi}\Big]$,
and by the uniqueness of $G$-invariant representative of $c_{1}(X_{P})$, see for instance [@Algmodels page 13], it follows that ${\text{Ric}}^{\nabla}(\rho_{0}) = \rho_{0}$, i.e. $\rho_{0} \in \Omega^{1,1}(X_{P})^{G}$ defines a Kähler-Einstein metric, see also [@MATSUSHIMA].
Now, let us recall a well-known result which will play an important role in the proof of next theorem. Given a Sasaki manifold $(M,g)$, with structure tensors $(\phi,\xi,\eta)$, since $M$ is a ${\text{K}}$-contact manifold, we have that $\frac{d\eta}{2}$ defines a symplectic form on the distribution $\mathscr{D} = \ker(\eta)$, which satisfies $\mathscr{L}_{\xi}(\frac{d\eta}{2}) = 0$. Let us denote the transversal Riemannian metric on $\mathscr{D}$ by $$g^{T} = \frac{1}{2}d\eta({\rm{id}}\otimes \phi),$$ notice that $J = \phi|_{\mathscr{D}}$ defines an almost complex structure on $\mathscr{D} = \ker(\eta)$, which is in fact integrable due to the Sasaki condition \[sasakicondition\], thus $\frac{d\eta}{2}$ is a Kähler form. The relation between the Ricci curvature of $g^{T}$ and $g$ is given by the following theorem.
\[RicT\] Let $(M,g)$ be a ${\text{K}}$-contact manifold of dimension $2n+1$ with structure tensors $(\phi,\xi,\eta)$. Then, the following identities hold:
1. ${\text{Ric}}_{g}(X,Y) = {\text{Ric}}_{g^{T}}(X,Y) - 2g(X,Y)$, $\forall X,Y \in \mathscr{D} = \ker{\eta}$;
2. ${\text{Ric}}_{g}(X,\xi) = 2n\eta(X)$, $\forall X \in TM.$
\[homothetic\] In the setting of theorem above, if we have ${\text{Ric}}(g^{T}) = \lambda_{0}g^{T}$, for some real constant $\lambda_{0} > 0$, we can take $a > 0$, such that $$a = \frac{\lambda_{0}}{2(n+1)},$$ and define $$g_{a} = ag + (a^{2} - a)\eta \otimes \eta, \ \ \eta' = a \eta, \ \ \xi' = \frac{1}{a}\xi, \ \ \phi' = \phi.$$ The change above is called $\mathscr{D}$-[*homothetic transformation*]{} [@Tanno], and the resulting structure tensors $(g_{a},\phi',\xi',\eta')$ also defines a ${\text{K}}$-contact structure on $M$. It is straightforward to see that $g_{a}^{T} = ag^{T}$, and since ${\text{Ric}}(ag^{T}) = {\text{Ric}}(g^{T})$, from the first item of Theorem \[RicT\] we obtain
${\text{Ric}}_{g_{a}}(X,Y) = {\text{Ric}}_{g_{a}^{T}}(X,Y) - 2g_{a}(X,Y) = 2(n+1)g_{a}^{T} - 2g_{a}(X,Y) = 2ng_{a}(X,Y),$
$\forall X,Y \in \mathscr{D} = \ker{\eta}$, here in the last equality we used that $g_{a}|_{\mathscr{D}} = g_{a}^{T}$. Since in the second item of Theorem \[RicT\] remains essentially the same, namely, ${\text{Ric}}_{g_{a}}(X,\xi') = 2n\eta'(X)$, $\forall X \in TM.$, we have ${\text{Ric}}(g_{a}) = 2ng_{a}$.
A compact Hermitian-Weyl manifold $(M,[g],D,J)$ is said to be homogeneous if it admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$.
Now, we are able to prove the following result.
\[Theo2\] Let $(M,[g],D,J)$ be a compact homogeneous Hermitian-Einstein-Weyl manifold, such that $\dim_{\mathbb{R}}(M) \geq 6$, and let $K$ be the compact connected Lie group which acts on $M$ by preserving the Hermitian-Einstein-Weyl structure. Suppose also that $K_{{\text{ss}}}$ is simply connected and has a unique simple component. Then, we have that $$\label{universalpresentation}
M = {\rm{Tot}}(\mathscr{O}_{X_{P}}(-\ell)^{\times})/\mathbb{Z},$$ for some $\ell \in \mathbb{Z}_{>0}$, i.e. $M$ is a principal $T_{\mathbb{C}}^{1}$-bundle over a complex flag manifold $X_{P} = K_{{\text{ss}}}^{\mathbb{C}}/P$, for some parabolic Lie subgroup $P = P_{\Theta} \subset K_{{\text{ss}}}^{\mathbb{C}}$. Moreover, the Hermitian-Einstein-Weyl metric $g$ is completely determined by the Lee form $\theta_{g} \in \Omega^{1}(M)$, locally described by $$\label{higgsfield}
\theta_{g} = -\frac{\ell}{I(X_{P})}\bigg [d\log \Big ( \big | \big |s_{U}v_{\delta_{P}}^{+} \big | \big |^{2}\Big ) + I(X_{P})\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg],$$ for some local section $s_{U} \colon U \subset X_{P} \to K_{\text{ss}}^{\mathbb{C}}$, where $v_{\delta_{P}}^{+}$ is the highest weight vector of weight $\delta_{P}$ associated to the irreducible $\mathfrak{k}_{\text{ss}}^{\mathbb{C}}$-module $V(\delta_{P}) = H^{0}(X_{P},K_{X_{P}}^{-1})^{\ast}$, and $I(X_{P})$ is the Fano index of $X_{P}$.
From Proposition \[HweylVaisman\], Theorem \[LCKisVaisman\], and Theorem \[Vaismanregular\], it follows that $M$ is a principal $T_{\mathbb{C}}^{1}$-bundle over a complex flag manifold $X_{P} = K_{{\text{ss}}}^{\mathbb{C}}/P$, for some parabolic Lie subgroup $P = P_{\Theta} \subset K_{{\text{ss}}}^{\mathbb{C}}$. Moreover, we have that $M$ is a flat principal $S^{1}$-bundle over a compact homogeneous Sasaki manifold $Q$, see Theorem \[minimalpresentation\].
Since the universal covering space $\widetilde{Q}$ of $Q$ is a compact homogeneous Sasaki-Einstein manifold, see Remark \[universalsplitting\] and Theorem \[HEW\], it follows that $Q$ is also compact homogeneous Sasaki-Einstein manifold, so $\pi_{1}(Q)$ is finite. Now, from the long exact sequence of homotopy groups
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since $X_{P}$ is simply connected, we have that $\pi_{1}(Q)$ is trivial or a cyclic group, in the latter case we have $\pi_{1}(Q) = \mathbb{Z}_{\ell}$, for some $\mathbb{\ell} \in \mathbb{Z}_{>0}$, so we obtain $Q = \widetilde{Q}/\mathbb{Z}_{\ell}$. From Theorem \[HEW\], we have $\widetilde{Q} = Q(\mathscr{O}_{X_{P}}(-1))$, where $\mathscr{O}_{X_{P}}(-1)$ denotes the maximal root of the canonical bundle of $X_{P}$. Thus, if $Q = \widetilde{Q}/\mathbb{Z}_{\ell}$, it implies that $Q = Q(\mathscr{O}_{X_{P}}(-\ell))$.
At first, let us suppose that $\pi_{1}(Q)$ is trivial. In this case we have $M = \widetilde{Q} \times S^{1}$, see Remark \[monodromyvaisman\], and the associated presentation for $M$ is given by $$M = \big ({\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times}),\mathbb{Z} \big).$$ Under the considerations above, it follows from Theorem \[HEW\] that the Hermitian-Einstein-Weyl structure on $M$ is obtained from the Kähler Ricci-flat structure on the cone ${\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})$. In order to explicitly describe this Hermitian-Einstein-Weyl structure, we observe the following. Since in this case we have
$\displaystyle \mathscr{O}_{X_{P}}(-1) = \bigotimes_{\alpha \in \Sigma \backslash \Theta}\mathscr{O}_{\alpha}(-1)^{\otimes \frac{\langle \delta_{P}, h_{\alpha}^{\vee}\rangle}{I(X_{P})}},$
we can consider the Kähler potential ${\rm{K}}_{H} \colon \displaystyle \mathscr{O}_{X_{P}}(-1)^{\times} \to \mathbb{R}^{+}$, described locally by $$\label{Kpotencialcanonical}
\displaystyle {\rm{K}}_{H}\big (z,w \big ) = \Big ( \prod_{\alpha \in \Sigma \backslash \Theta}\big | \big |s_{i}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{\frac{2\langle \delta_{P}, h_{\alpha}^{\vee}\rangle}{I(X_{P})}}\Big)w\overline{w} = \Big (\big | \big |s_{i}(z)v_{\delta_{P}}^{+} \big | \big |^{\frac{2}{I(X_{P})}} \Big) w\overline{w}.$$ From the Kähler potential $r^{2} = {\rm{K}}_{H}$ above we obtain a Kähler form
$\displaystyle \omega_{\mathscr{C}} = \frac{\sqrt{-1}}{2} \partial \overline{\partial} {\rm{K}}_{H} = d\Big(\frac{r^{2}\eta}{2}\Big) = {\mathrm{e}}^{2\psi} \bigg ( d\psi \wedge \eta + \frac{d\eta}{2}\bigg)$,
such that $\psi = \log(r)$, and
$\displaystyle \eta = \frac{1}{2} d^{c}\log({\rm{K}}_{H}(z,w)) = \frac{1}{2I(X_{P})}d^{c}\log \Big( \big | \big |s_{i}(z)v_{\delta_{P}}^{+} \big | \big |^{2}\Big) + d\sigma_{i}$.
Notice that $d\eta = \pi^{\ast}\big(\frac{\rho_{0}}{I(X_{P})}\big)$, where $\pi \colon Q(\mathscr{O}_{X_{P}}(-1) ) \to X_{P}$ denotes the associated bundle projection.
Now, we observe that, if we consider the Sasaki structure $(g_{S},\phi,\xi,\eta)$ on $Q(\mathscr{O}_{X_{P}}(-1) )$, induced as in \[sasakistructure\], where the Sasaki metric $g_{S}$ on $Q(\mathscr{O}_{X_{P}}(-1) )$ is given by
$g_{S} = \frac{1}{2}d\eta(\text{id}\otimes \phi) + \eta \otimes \eta$,
denoting by $g_{S}^{T}$ the Kähler (transversal) metric induced by $\frac{\rho_{0}}{2I(X_{P})}$ on $X_{P}$, it follows from Theorem \[RicT\] that
1. ${\text{Ric}}_{g_{S}}(X,Y) = {\text{Ric}}_{g_{S}^{T}}(X,Y) - 2g_{S}(X,Y)$, $\forall X,Y \in \mathscr{D} \cong TX_{P}$,
2. ${\text{Ric}}_{g_{S}}(X,\xi) = 2\dim_{\mathbb{C}}(X_{P})\eta(X)$, $\forall X \in TQ(\mathscr{O}_{X_{P}}(-1) ).$
Thus, since ${\text{Ric}}(g_{S}^{T}) = 2I(X_{P})g_{S}^{T}$, see Equation \[riccinorm\], by considering the $\mathscr{D}$-homothetic transformation defined by $$\label{rescaleconst}
a = \frac{I(X_{P})}{\dim_{\mathbb{C}}(X_{P}) + 1},$$ we obtain a Sasaki structure $(g_{a},\phi,\frac{1}{a}\xi,a\eta)$ on $Q(\mathscr{O}_{X_{P}}(-1) )$, with rescaled metric $g_{a}$ given by $$\label{metricsasakieinstein}
g_{a} = \displaystyle \frac{I(X_{P})}{\dim_{\mathbb{C}}(X_{P})+1} \Bigg ( \frac{1}{2}d \eta ({\rm{id}} \otimes \phi) + \frac{I(X_{P})}{\dim_{\mathbb{C}}(X_{P})+1}\eta \otimes \eta \Bigg ),$$ cf. [@CONTACTCORREA Theorem 2]. The metric above satisfies ${\text{Ric}}(g_{a}) = 2\dim_{\mathbb{C}}(X_{P})g_{a}$, see Remark \[homothetic\], so it defines the appropriate Sasaki-Einstein structure to be considered. From the Sasaki-Einstein structure $(g_{a},\phi,\frac{1}{a}\xi,a\eta)$, we can consider the Ricci-flat metric on ${\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})$ defined by $$g_{CY} = r^{2}g_{a} + dr\otimes dr,$$ see [@BoyerGalicki Corollary 11.1.8], notice that $r^{2} = {\rm{K}}_{H}$, such that $ {\rm{K}}_{H}$ is defined in \[Kpotencialcanonical\]. From the $\mathscr{D}$-homothetic transformation induced by \[rescaleconst\], the complex structure $\mathscr{J} \in {\text{End}}(T({\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})))$ becomes $$\label{complexchange}
\mathscr{J}(Y)= \phi(Y) - a\eta(Y)r\partial_{r}, \ \ \ \ \ \mathscr{J}( r\partial_{r}) = \frac{1}{a}\xi.$$ By taking $\psi = \log(r)$, we obtain $g_{CY} = {\rm{e}}^{2\psi}\big(g_{a} + d\psi\otimes d\psi\big)$, and $$\omega_{CY} = g_{CY}(\mathscr{J}\otimes {\rm{id}}) = a \bigg ( rdr \wedge \eta + \frac{r^{2}}{2}d\eta \bigg) = a{\rm{e}}^{2\psi}\bigg ( d\psi \wedge \eta + \frac{d\eta}{2}\bigg).$$ From this, we can consider the globally conformally Kähler structure $(\widetilde{\Omega}, \mathscr{J},\widetilde{\theta})$ on ${\rm{Tot}}(\mathscr{O}_{X_{P}}(-1))$, such that $$\displaystyle \widetilde{\Omega} = {\rm{e}}^{-2\psi}\omega_{CY} = a \bigg (d\psi \wedge \eta + \frac{d\eta}{2} \bigg),$$ notice that $d\widetilde{\Omega} = (-2d\psi) \wedge \widetilde{\Omega}$, so we have $\theta = -2d\psi$. Therefore, we obtain a precise description of the Hermitian-Einstein-Weyl structure $(\Omega,J,\theta)$ on $M = {\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})/\mathbb{Z}$ in terms of the Ricci-flat structure of ${\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})$, i.e. we have $$\label{VaismanRicciflat}
\displaystyle \wp^{\ast}\Omega = a \bigg (d\psi \wedge \eta + \frac{d\eta}{2} \bigg), \ \ J \circ \wp_{\ast} = \wp_{\ast} \circ \mathscr{J}, \ \ {\text{and}} \ \ \wp^{\ast}\theta = -2d\psi,$$ such that $\wp \colon {\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times}) \to M$ is the associated projection map. By denoting $\theta = \theta_{g}$, where $g = \Omega({\rm{id}}\otimes J)$, we have locally $$\label{Leericciflat}
\theta_{g} = -\frac{1}{I(X_{P})}\bigg [d\log \Big ( \big | \big |s_{U}v_{\delta_{P}}^{+} \big | \big |^{2}\Big ) + I(X_{P})\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg],$$ for some local section $s_{U} \colon U \subset X_{P} \to K_{\text{ss}}^{\mathbb{C}}$.
In order to see that the expression \[Leericciflat\] defines completely the Hermitian-Einstein-Weyl structure on $M$, we proceed as follows: By considering the Kähler Ricci-flat structure $(\omega_{CY},\mathscr{J})$ on ${\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})$, we notice that, since $\psi = \log(r)$, it follows from \[complexchange\] that $$-\widetilde{\theta} \circ \mathscr{J} = 2d\psi \circ \mathscr{J} = 2\frac{dr}{r} \circ \mathscr{J} = -2a\eta.$$ Thus, since $\wp^{\ast}\theta_{g} = \widetilde{\theta}$, the anti-Lee form $\vartheta_{g} = -\theta_{g} \circ J$ associated to the Hermitian-Einstein-Weyl structure $(\Omega,J,\theta)$ on $M$ described in \[VaismanRicciflat\] is locally given by $$\label{antiLee}
\vartheta_{g} = -\frac{1}{\dim_{\mathbb{C}}(X_{P}) + 1}\bigg [d^{c}\log \Big ( \big | \big |s_{U}v_{\delta_{P}}^{+} \big | \big |^{2}\Big ) - I(X_{P})\frac{\sqrt{-1}}{|w|^{2}}\big (\overline{w}dw - wd\overline{w} \big) \bigg],$$ cf. Remark \[mainremark\]. Therefore, considering $M$ as a principal $T_{\mathbb{C}}^{1}$-bundle over $X_{P}$, one can define a principal $T_{\mathbb{C}}^{1}$-connection $\Psi_{g} = - \vartheta_{g} +\sqrt{-1}\theta_{g} \in \Omega^{1}(M;\mathbb{C})$. It is straightforward to see that $\Psi_{g} \circ J = \sqrt{-1}\Psi_{g}$, and $d\Psi_{g} = p_{\Lambda}^{\ast}\omega$ for some Kähler form $\omega \in \Omega^{1,1}(X_{P})^{G}$. Thus, since $\wp^{\ast}\theta_{g} = -2\psi$ and $\wp^{\ast}\vartheta_{g} = -2a\eta$, from Theorem \[vaismanbundle\], one obtains $$\label{HEWmetric}
g = \Omega({\rm{id}}\otimes J) = \frac{1}{4}\Big(d\Psi_{g} + \frac{1}{2}\Psi_{g}\odot\overline{\Psi_{g}}\Big),$$ where $(\Omega,J,\theta_{g})$ is described in \[VaismanRicciflat\], i.e. $$\wp^{\ast}g = g_{a} + d\psi\otimes d\psi = {\rm{e}}^{-2\psi}g_{CY}.$$ Hence, since $\Psi_{g} = \theta_{g} \circ J + \sqrt{-1}\theta_{g}$, we have $(\Omega,J,\theta_{g})$ completely determined by $\theta_{g}$. The expression above shows how the Hermitian-Einstein-Weyl metric $g$ on $M$ is conformally related to the Ricci-flat Kähler metric $g_{CY}$ on ${\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})$.
Now, as we have seen, in the general case we have $Q = \widetilde{Q}/\mathbb{Z}_{\ell}$, and $M = \widetilde{Q} \times_{\mathbb{Z}_{\ell}}S^{1}$. In this case, the associated (minimal) presentation is given by
$M = \big ({\rm{Tot}}(\mathscr{O}_{X_{P}}(-\ell)^{\times}), \mathbb{Z} \big).$
The proof for item (2) in this case is essentially the same as the previous case. In fact, we just need to consider the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(\mathscr{O}_{X_{P}}(-\ell)^{\times}) \to \mathbb{R}$, locally described by
$\displaystyle {\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{i}(z)v_{\delta_{P}}^{+} \big | \big |^{\frac{2\ell}{I(X_{P})}} \Big)w\overline{w},$
and take a $\mathscr{D}$-homothetic transformation on the Sasaki structure of $Q$ defined by $$\label{rescaleconsttop}
a = \frac{ \text{ord}(\pi_{1}(Q)) I(X_{P})}{\dim_{\mathbb{C}}(X_{P}) + 1},$$ here we denote $\ell = \text{ord}(\pi_{1}(Q))$. From this, proceeding exactly as in the previous case, we obtain the desired expression for the Lee form provided in item (2) and a complete description of the Hermitian-Einstein-Weyl metric in terms of this 1-form.
It is worth pointing out that, the expression \[antiLee\] above shows how the $\mathscr{D}$-homothetic transformation in the underlying Sasaki structure defined by \[rescaleconst\] afects the Vaisman structure on $M = {\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})/\mathbb{Z}$ provided by Theorem \[Maintheo1\]. We also point out that, in the general case $M = {\rm{Tot}}(\mathscr{O}_{X_{P}}(-\ell)^{\times})/\mathbb{Z}$, the anti-Lee form is given by $$\vartheta_{g} = -\frac{\text{ord}(\pi_{1}(Q))}{\dim_{\mathbb{C}}(X_{P}) + 1}\bigg [d^{c}\log \Big ( \big | \big |s_{U}v_{\delta_{P}}^{+} \big | \big |^{2}\Big ) - I(X_{P})\frac{\sqrt{-1}}{|w|^{2}}\big (\overline{w}dw - wd\overline{w} \big) \bigg],$$ here, as before, we denote $\ell = \text{ord}(\pi_{1}(Q))$.
In the setting of Theorem \[Theo2\] we have a quite interesting underlying relation with Hamilton-Jacobi theory [@Calin Chapter 7] which can be briefly formulated as follows: Under the hypotheses of Theorem \[Theo2\], by denoting $ \widetilde{M} = {\rm{Tot}}(\mathscr{O}_{X_{P}}(-1)^{\times})$ we can consider the smooth function $f \in C^{\infty}(\widetilde{M})$ given by $$f(z,w) = \frac{1}{2}\log \big ({\rm{K}}_{H}(z,w) \big ), \ \ {\text{where}} \ \ {\rm{K}}_{H}(z,w) = \Big (\big | \big |s_{i}(z)v_{\delta_{P}}^{+} \big | \big |^{\frac{2}{I(X_{P})}} \Big) w\overline{w},$$ notice that $f = \psi = \log(r)$. Considering the Riemannian metric on $\widetilde{M}$ defined by $\widetilde{g} = {\rm{e}}^{-2f}g_{CY}$, namely,
$\widetilde{g} = g_{a} + df\otimes df$,
it follows that $||\nabla f||_{\widetilde{g}} = 1$[^13]. Thus, by taking the gradient flow $\gamma(u,\cdot) \colon \mathbb{R} \to \widetilde{M}$ of $f$, through a point $u \in \widetilde{M}$, we have that $$\label{afinnelinear}
f(\gamma(u,t)) = f(u) + t, \ \ \forall t \in \mathbb{R}.$$ We notice that $\nabla f$ is a Killing vector field, and from [*[Koszul’s]{}*]{} identity we have $$2\widetilde{g}(\widetilde{\nabla}_{X}(\nabla f),Y) = d (df)(X,Y) + (\mathscr{L}_{\nabla f} g)(X,Y) = 0,$$ $\forall X,Y \in \Gamma(T\widetilde{M})$, i.e. $\nabla f$ is parallel. Now, we consider the natural Lagrangian ${\rm{L}} \colon T\widetilde{M} \to [0,+\infty)$ defined by $${\rm{L}}(u;v) = \frac{1}{2}\widetilde{g}_{u}(v,v),$$ such that $(u;v) \in T\widetilde{M}$. Since $\nabla f$ is parallel, it follows that the gradient flow of $f$ is a solution of the Euler-Lagrange equation $$\frac{d}{dt}\bigg ( \frac{d{\rm{L}}}{d\varphi}\bigg) - \frac{d{\rm{L}}}{d\dot{\varphi}} = 0,$$ here we consider $\varphi(t) = \gamma(u,t)$, $\forall t \in \mathbb{R}$, and $u \in \widetilde{M}$. Therefore, if we define ${\rm{S}} \colon \mathbb{R} \times \widetilde{M} \to \mathbb{R}$ by setting $${\rm{S}}(t,u) = \frac{1}{4}f\big(\gamma(u,t)\big),$$ since $||\nabla f||_{\widetilde{g}} = 1$, from Equation \[afinnelinear\] it is straightforward to see that $${\rm{S}}\big (t,\varphi(t) \big) = S_{0} + \int_{t_{0}}^{t}{\rm{L}}\big(\varphi(s);\dot{\varphi}(s)\big)ds, \ \ {\text{such that}} \ \ S_{0} = S\big(t_{0},\varphi(t_{0})\big),$$ where $\varphi(t) = \gamma(u,t)$, $\forall t \in \mathbb{R}$. Hence, we have that ${\rm{S}}$ is a solution of the Hamilton-Jacobi equation $$\label{HamiltonJacobi}
\frac{\partial S}{\partial t} + {\rm{H}}\bigg (\varphi;\frac{\partial S}{\partial \varphi}\bigg) = 0,$$ with the initial condition $S_{0} = S\big(t_{0},\varphi(t_{0})\big)$, where ${\rm{H}} \colon T\widetilde{M} \to [0,+\infty)$ is the Hamiltonian associated to the Lagrangian ${\rm{L}}$, i.e. $${\rm{H}}(u;v) = \frac{1}{2}\widetilde{g}_{u}(v,v), \ \ \forall (u;v) \in T\widetilde{M}.$$ Actually, in order to check that Equation \[HamiltonJacobi\] holds, we observe that $$\frac{\partial {\rm{S}}}{\partial t} = {\rm{L}}(\varphi;\dot{\varphi}) - \widetilde{g}_{\varphi}\bigg(\frac{\partial S}{\partial \varphi},\dot{\varphi}\bigg), \ \ {\text{and}} \ \
{\rm{H}}\bigg (\varphi;\frac{\partial S}{\partial \varphi}\bigg) = \widetilde{g}_{\varphi}\bigg(\frac{\partial S}{\partial \varphi},\dot{\varphi}\bigg) - {\rm{L}}(\varphi;\dot{\varphi}),$$ further details about the ideas above can be found in [@Calin Theorem 7.9].
### Examples of Vaisman structures and Hermitian-Einstein-Weyl metrics
In this subsection we shall provide some examples which illustrate the results presented in the previous subsection. In what follows we fix the notations and conventions used in Subsection \[subsec3.1\].
In order to do some local computations, it will be useful for us to consider the analytic cellular decomposition of complex flag manifolds by means Schubert cells [@MONOMIAL $\S$ 3.1]. Being more precise, in what follows we shall consider the open set defined by the “opposite" big cell in $X_{P}$. This open set is a distinguished coordinate neighbourhood $U \subset X_{P}$ of $x_{0} = eP \in X_{P}$ defined by the maximal Schubert cell. A brief description for the opposite big cell can be done as follows: Let $\Pi = \Pi^{+} \cup \Pi^{-}$ be the root system associated to a simple root system $\Sigma \subset \mathfrak{h}^{\ast}$, from this we can define the opposite big cell $U \subset X_{P}$ by
$U = B^{-}x_{0} = R_{u}(P_{\Theta})^{-}x_{0} \subset X_{P}$,
where $B^{-} = \exp(\mathfrak{h} \oplus \mathfrak{n}^{-})$, and
$R_{u}(P_{\Theta})^{-} = \displaystyle \prod_{\alpha \in \Pi^{-} \backslash \langle \Theta \rangle^{-}}N_{\alpha}^{-}$, (opposite unipotent radical)
with $N_{\alpha}^{-} = \exp(\mathfrak{g}_{\alpha})$, $\forall \alpha \in \Pi^{-} \backslash \langle \Theta \rangle^{-}$. It is worth mentioning that the opposite big cell defines a contractible open dense subset in $X_{P}$, thus the restriction of any vector bundle over this open set is trivial. For further results about Schubert cells and Schubert varieties we suggest [@MONOMIAL].
\[basicmodel\] By keeping the notation of Subsection \[subsec3.1\], consider $G^{\mathbb{C}}$ as being a simple Lie group, and take $\Theta = \Sigma \backslash \{\alpha\}$, for some fixed $\alpha \in \Sigma$. Let us denote by $P_{\Theta} = P_{\omega_{\alpha}}$, such that $\omega_{\alpha} \in \Lambda_{\mathbb{Z}_{\geq 0}}^{\ast}$, the parabolic Lie subgroup associated to $\Theta \subset \Sigma$. From this, we can consider the flag manifold $$X_{P_{\omega_{\alpha}}} = G^{\mathbb{C}}/P_{\omega_{\alpha}}.$$ From Proposition \[C8S8.2Sub8.2.3P8.2.6\], we have that ${\text{Pic}}(X_{P_{\omega_{\alpha}}}) = \mathbb{Z}c_{1}(\mathscr{O}_{\alpha}(1))$, a straightforward computation shows that $$\label{maximalparabolic}
I(X_{P_{\omega_{\alpha}}}) = \langle \delta_{P_{\omega_{\alpha}}},h_{\alpha}^{\vee} \rangle, \ \ \ {\text{and}} \ \ \ K_{X_{P_{\omega_{\alpha}}}}^{ \otimes \frac{1}{ \langle \delta_{P_{\omega_{\alpha}}},h_{\alpha}^{\vee} \rangle}} = \mathscr{O}_{\alpha}(-1).$$ Hence, given a negative line bundle $L \in {\text{Pic}}(X_{P_{\omega_{\alpha}}})$, it follows that
$L = \mathscr{O}_{\alpha}(-\ell)$,
for some integer $\ell \in \mathbb{Z}$, such that $\ell>0$. Now, we can associate to $L$ an irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu(L))$ with highest weight vector $v_{\mu(L)}^{+} \in V(\mu(L))$, such that
$\displaystyle V(\mu(L)) \subset V(\omega_{\alpha})^{\otimes \ell}$, $\displaystyle v_{\mu(L)}^{+} = \underbrace{v_{\omega_{\alpha}}^{+} \otimes \cdots \otimes v_{\omega_{\alpha}}^{+}}_{\ell{\text{-times}}}$,
notice that $\mu(L) = \ell\omega_{\alpha}$, see Equation \[negativeweight\]. From the above data we can consider the manifold $$M = {\rm{Tot}}(\mathscr{O}_{\alpha}(-\ell)^{\times})/\Gamma, \ \ \ {\text{where}} \ \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$$ for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$. Applying Theorem \[Theo1\] we obtain a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, which can be describe in coordinates $(z,w) \in \mathscr{O}_{\alpha}(-\ell)^{\times}|_{U}$ by $$\label{potentialmaxparabolic}
{\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{2\ell} \Big) w\overline{w},$$ for some local section $s_{U} \colon U \subset X_{P_{\omega_{\alpha}}} \to G^{\mathbb{C}}$. By means of the Kähler potential above we can describe the l.c.K. structure $(\Omega, J, \theta)$ on $M$ from the Lee form $$\displaystyle \theta = -\ell\bigg [d\log \Big ( \big | \big |s_{U}v_{\omega_{\alpha}}^{+} \big | \big |^{2}\Big ) + \frac{1}{\ell}\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg],$$ recall that $\vartheta = -\theta \circ J$, and $\Omega = -\frac{1}{4}(d\vartheta - \theta \wedge \vartheta)$, see Remark \[metricdescription\] and Remark \[mainremark\].
If we consider $\ell = 1$, it follows that $L = \mathscr{O}_{\alpha}(-1)$, i.e. $L$ is the maximal root of the canonical bundle of $X_{P_{\omega_{\alpha}}}$, see Equation \[maximalparabolic\]. In this particular case, by applying Theorem \[Theo2\] on a compact Homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}(\mathscr{O}_{\alpha}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}}^{\mathbb{C}} = G^{\mathbb{C}}$, we have the Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ completely determined by $$\theta_{g} = -\frac{1}{\langle \delta_{P_{\omega_{\alpha}}},h_{\alpha}^{\vee} \rangle}\Bigg [d\log \Big ( \big | \big |s_{U}v_{ \delta_{P_{\omega_{\alpha}}}}^{+} \big | \big |^{2}\Big ) + \langle \delta_{P_{\omega_{\alpha}}},h_{\alpha}^{\vee} \rangle\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg],$$ recall that $g = \Omega({\rm{id}} \otimes J)$, see Equation \[HEWmetric\]. Notice that, in this latter setting, the complex structure $J \in {\text{End}}(TM)$ is obtained from a complex structure on ${\rm{Tot}}(\mathscr{O}_{\alpha}(-1)^{\times})$ after a suitable $\mathscr{D}$-homothetic transformation induced from $$a = \frac{\langle \delta_{P_{\omega_{\alpha}}},h_{\alpha}^{\vee} \rangle}{\dim_{\mathbb{C}}(X_{P_{\omega_{\alpha}}}) + 1},$$ on the Sasaki structure defined on the unitary frame bundle of $\mathscr{O}_{\alpha}(-1)$, see for instance \[complexchange\].
The ideas above provide a constructive method to describe a huge class of concrete examples of Vaisman manifolds and homogeneous Hermitian-Einstein-Weyl manifolds associated to complex flag manifolds with Picard number equal to one. In what follows we shall further explore the application of the previous ideas in a class of concrete examples provided by complex Grassmannians.
\[Grassmanianexample\] Consider $G^{\mathbb{C}} = {\rm{SL}}(n+1,\mathbb{C})$, by fixing the Cartan subalgebra $\mathfrak{h} \subset \mathfrak{sl}(n+1,\mathbb{C})$ given by diagonal matrices whose the trace is equal to zero, we have the set of simple roots given by $$\Sigma = \Big \{ \alpha_{l} = \epsilon_{l} - \epsilon_{l+1} \ \Big | \ l = 1, \ldots,n\Big\},$$ here we consider $\epsilon_{l} \colon {\text{diag}}\{a_{1},\ldots,a_{n+1} \} \mapsto a_{l}$, $ \forall l = 1, \ldots,n+1$. Therefore, the set of positive roots is given by $$\Pi^+ = \Big \{ \alpha_{ij} = \epsilon_{i} - \epsilon_{j} \ \Big | \ i<j \Big\},$$ notice that $\alpha_{l} = \alpha_{ll+1}$, $\forall l = 1, \ldots,n$. Considering $\Theta = \Sigma \backslash \{\alpha_{k}\}$ and $P = P_{\omega_{\alpha_{k}}}$, we have the complex Grassmannian manifold
${\rm{Gr}}(k,\mathbb{C}^{n+1}) = {\rm{SL}}(n+1,\mathbb{C})/P_{\omega_{\alpha_{k}}}.$
A straightforward computation shows that ${\text{Pic}}({\rm{Gr}}(k,\mathbb{C}^{n+1})) = \mathbb{Z}c_{1}(\mathscr{O}_{\alpha_{k}}(1))$, and we can also show that $$\label{fanograssmannian}
\big \langle \delta_{P_{\omega_{\alpha_{k}}}},h_{\alpha_{k}}^{\vee} \big \rangle = n+1 \Longrightarrow K_{{\rm{Gr}}(k,\mathbb{C}^{n+1})}^{\otimes \frac{1}{n+1}} = \mathscr{O}_{\alpha_{k}}(-1).$$ Now, given a negative line bundle $L \in {\text{Pic}}({\rm{Gr}}(k,\mathbb{C}^{n+1}))$, we have $L = \mathscr{O}_{\alpha_{k}}(-\ell)$, for some integer $\ell >0$. From the above data and the previous example, we can consider the manifold defined by
$M = {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-\ell)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$. Applying Theorem \[Theo1\] we obtain a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, which can be described in coordinates $(z,w) \in \mathscr{O}_{\alpha_{k}}(-\ell)^{\times}|_{U}$ by
${\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\omega_{\alpha_{k}}}^{+} \big | \big |^{2\ell} \Big) w\overline{w}.$
In order to provide a concrete description for the potential above, we proceed as follows: Since we have
$V(\omega_{\alpha_{k}}) = \bigwedge^{k}(\mathbb{C}^{n+1}), \ \ \ {\text{and}} \ \ \ v_{\omega_{\alpha_{k}}}^{+} = e_{1} \wedge \ldots \wedge e_{k}$,
we fix the canonical basis $e_{i_{1}}\wedge \cdots \wedge e_{i_{k}}$, $\{i_{1} < \ldots < i_{k}\} \subset \{1, \ldots, n+1\}$, for $V(\omega_{\alpha_{k}}) = \bigwedge^{k}(\mathbb{C}^{n+1})$. By taking the coordinate neighborhood defined by the opposite big cell $U = R_{u}(P_{\omega_{\alpha_{k}}})^{-}x_{0} \subset {\rm{Gr}}(k,\mathbb{C}^{n+1})$, from the parameterization
$Z \in \mathbb{C}^{(n+1-k)k} \mapsto n(Z)x_{0} = \begin{pmatrix}
\ 1_{k} & 0_{k,n+1-k} \\
Z & 1_{n+1-k}
\end{pmatrix}x_{0},$
here we identified $\mathbb{C}^{(n+1-k)k} \cong {\rm{M}}_{n+1-k,k}(\mathbb{C})$ and consider $Z = (z_{ij})$, we can take the local section $s_{U} \colon U \subset {\rm{Gr}}(k,\mathbb{C}^{n+1})\to {\rm{SL}}(n+1,\mathbb{C})$ defined by
$s_{U}(n(Z)x_{0}) = n(Z) = \begin{pmatrix}
\ 1_{k} & 0_{k,n+1-k} \\
Z & 1_{n+1-k}
\end{pmatrix}.$
The local section above allows us to write locally $$\label{potentialgrassman}
{\rm{K}}_{H}\big (Z,w \big ) = \Bigg (\sum_{I} \bigg | \det_{I} \begin{pmatrix}
\ 1_{k} \\
Z
\end{pmatrix} \bigg |^{2} \Bigg )^{\ell}w\overline{w},$$ where the sum above is taken over all $k \times k$ submatrices whose the lines are labeled by the set of index $I = \{i_{1} < \ldots < i_{k}\} \subset \{1, \ldots, n+1\}$. By means of the Kähler potential above we can describe the l.c.K. structure $(\Omega, J, \theta)$ on $M$ from the Lee form $$\displaystyle \theta = -\ell\Bigg [d\log \Bigg (\sum_{I} \bigg | \det_{I} \begin{pmatrix}
\ 1_{k} \\
Z
\end{pmatrix} \bigg |^{2} \Bigg ) + \frac{1}{\ell}\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg].$$ From Equation \[fanograssmannian\], and Theorem \[Theo2\], for the particular case $\ell = 1$, if we consider a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}} = {\rm{SU}}(n+1)$, the homogeneous Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ is completely determined by the Lee form $$\theta_{g} = -\Bigg [d\log \Bigg (\sum_{I} \bigg | \det_{I} \begin{pmatrix}
\ 1_{k} \\
Z
\end{pmatrix} \bigg |^{2} \Bigg ) + \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg],$$ notice that $g = \Omega({\rm{id}} \otimes J)$, and $\Omega = -d\vartheta_{g} + \theta_{g} \wedge \vartheta_{g}$, with $\vartheta_{g} = -\theta_{g} \circ J$.
A particular low dimensional example of the construction above is provided by ${\rm{Gr}}(2,\mathbb{C}^{4}) = {\rm{SL}}(4,\mathbb{C})/P_{\omega_{\alpha_{2}}}$ (Klein quadric). In this case we have
$V(\omega_{\alpha_{2}}) = \bigwedge^{2}(\mathbb{C}^{4})$ and $v_{\omega_{\alpha_{2}}}^{+} = e_{1} \wedge e_{2}$,
here we fix the basis $\{e_{i} \wedge e_{j}\}_{i<j}$ for $V(\omega_{\alpha_{2}}) = \bigwedge^{2}(\mathbb{C}^{4})$. Similarly to the previous computations, we consider the open set defined by the opposite big cell $U = B^{-}x_{0} \subset {\rm{Gr}}(2,\mathbb{C}^{4})$. This open set is parameterized by local coordinates $z \mapsto n(z)x_{0} \in U$ given by
$z = (z_{1},z_{2},z_{3},z_{4}) \in \mathbb{C}^{4} \mapsto\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0 \\
z_{1} & z_{3} & 1 & 0 \\
z_{2} & z_{4} & 0 & 1
\end{pmatrix} x_{0} \in U = B^{-}x_{0}.$
Given a negative line bundle $L \in {\text{Pic}}({\rm{Gr}}(2,\mathbb{C}^{4}))$, we have $L = \mathscr{O}_{\alpha_{2}}(-\ell)$, for some integer $\ell >0$. From the above data and the previous ideas we can consider the manifold
$M = {\rm{Tot}}(\mathscr{O}_{\alpha_{2}}(-\ell)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$. Applying Theorem \[Theo1\] we obtain a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{2}}(-\ell)^{\times}) \to \mathbb{R}^{+}$, which in turn can be described in coordinates $(z,w) \in \mathscr{O}_{\alpha_{2}}(-\ell)^{\times}|_{U}$ by
${\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{\omega_{\alpha_{2}}}^{+} \big | \big |^{2\ell} \Big) w\overline{w}.$
Thus, by considering the local section $s_{U} \colon U \subset {\rm{Gr}}(2,\mathbb{C}^{4})\to {\rm{SL}}(4,\mathbb{C})$, such that $s_{U}(n(z)x_{0}) = n(z)$, we obtain the concrete expression $$\label{C8S8.3Sub8.3.2Eq8.3.21}
{\rm{K}}_{H}\big (z,w \big ) = \displaystyle \Bigg (1+ \sum_{k = 1}^{4}|z_{k}|^{2} + \bigg |\det \begin{pmatrix}
z_{1} & z_{3} \\
z_{2} & z_{4}
\end{pmatrix} \bigg |^{2} \Bigg)^{\ell}w\overline{w}.$$ By means of the Kähler potential above we can describe the l.c.K. structure $(\Omega, J, \theta)$ on $M$ from the Lee form $$\displaystyle \theta = -\ell\Bigg [d\log \Bigg (1+ \sum_{k = 1}^{4}|z_{k}|^{2} + \bigg |\det \begin{pmatrix}
z_{1} & z_{3} \\
z_{2} & z_{4}
\end{pmatrix} \bigg |^{2} \Bigg) + \frac{1}{\ell}\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg].$$ It is worthwhile to observe that in this case we have the Fano index of ${\rm{Gr}}(2,\mathbb{C}^{4})$ given by $I({\rm{Gr}}(2,\mathbb{C}^{4})) = 4$, thus we obtain
$K_{{\rm{Gr}}(2,\mathbb{C}^{4})}^{\otimes \frac{1}{4}} = \mathscr{O}_{\alpha_{2}}(-1).$
As before, from Theorem \[Theo2\], for the particular case $\ell = 1$, for a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}} = {\rm{SU}}(4)$, the homogeneous Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ is completely determined by the Lee form $$\theta_{g} = -\Bigg [d\log \Bigg (1+ \sum_{k = 1}^{4}|z_{k}|^{2} + \bigg |\det \begin{pmatrix}
z_{1} & z_{3} \\
z_{2} & z_{4}
\end{pmatrix} \bigg |^{2} \Bigg) + \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg].$$ It is worth pointing out that, in this particular case, the compact simply connected Sasaki manifold defined by the sphere bundle of $\mathscr{O}_{\alpha_{2}}(-1)$ is given by the Stiefel manifold, namely, the underlying homogeneous Boothby-Wang fibration is given by the Tits fibration
$S^{1} \hookrightarrow \mathscr{V}_{2}(\mathbb{R}^{6}) \to {\rm{Gr}}(2,\mathbb{C}^{4}),$
such that $\mathscr{V}_{2}(\mathbb{R}^{6})$ is the Stiefel manifold. Therefore, we obtain $M = \mathscr{V}_{2}(\mathbb{R}^{6}) \times S^{1}$.
Now, let us briefly describe a low dimensional particular example of the construction provided in Example \[basicmodel\]. The example below is a classic example in the literature of Hermitian non-Kähler manifolds.
\[exampleP1\] Another particular case of the previous construction is given by principal $T_{\mathbb{C}}^{1}$-bundles over the projective space $\mathbb{C}{\rm{P}}^{1}$.
In what follows we shall consider the representation of $\mathfrak{sl}(2,\mathbb{C})$ as a differential operator algebra, further details about this representation can be found in [@TAYLOR].
Consider $G^{\mathbb{C}} = {\rm{SL}}(2,\mathbb{C})$, and fix the triangular decomposition for $\mathfrak{sl}(2,\mathbb{C})$ given by
$\mathfrak{sl}(2,\mathbb{C}) = \Big \langle x = \begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix} \Big \rangle_{\mathbb{C}} \oplus \Big \langle h = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} \Big \rangle_{\mathbb{C}} \oplus \Big \langle y = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix} \Big \rangle_{\mathbb{C}}$.
Notice that the decomposition above is codified in the simple root system and the root system given, respectively, by $\Sigma = \{\alpha\}$, and $\Pi = \{\alpha,-\alpha\}$. Also, notice that we have the set of linear representations of $\mathfrak{sl}(2,\mathbb{C})$ parameterized by
$\Lambda_{\mathbb{Z}_{\geq 0}}^{\ast} = \mathbb{Z}_{\geq 0}\omega_{\alpha} $.
By choosing $\Theta = \Sigma$, we have $P_{\Theta} = B$ (Borel subgroup), and then we obtain
$X_{B} = {\rm{SL}}(2,\mathbb{C})/B = \mathbb{C}{\rm{P}}^{1}$.
The analytic cellular decomposition in this case is given by
$X_{B} = \mathbb{C}{\rm{P}}^{1} = N^{-}x_{0} \cup \pi \Big ( \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix} \Big ),$
such that $x_{0} = eB$, and
$N^{-} = \Bigg \{ \begin{pmatrix}
1 & 0 \\
z & 1
\end{pmatrix} \ \ \Bigg | \ \ z \in \mathbb{C} \Bigg \}$.
From the cellular decomposition above we take the open set defined by the opposite big cell $U = N^{-}x_{0} \subset X_{B}$, and the local section $s_{U} \colon U \subset \mathbb{C}{\rm{P}}^{1} \to {\rm{SL}}(2,\mathbb{C})$ defined by $$s_{U}(nx_{0}) = n, \ \ \forall n \in N^{-}.$$ It is worthwhile to observe that in this case we have the open set $U = N^{-}x_{0} \subset \mathbb{C}{\rm{P}}^{1}$ parameterized by
$z \in \mathbb{C} \mapsto \begin{pmatrix}
1 & 0 \\
z & 1
\end{pmatrix} x_{0} \subset \mathbb{C}{\rm{P}}^{1}.$
Further, we have ${\text{Pic}}(\mathbb{C}{\rm{P}}^{1}) = \mathbb{Z}c_{1}(\mathscr{O}_{\alpha}(1))$, so we obtain $$I(\mathbb{C}{\rm{P}}^{1}) = \langle \delta_{B},h_{\alpha}^{\vee} \rangle = 2 \Longrightarrow K_{\mathbb{C}{\rm{P}}^{1}}^{\otimes \frac{1}{2}} = \mathscr{O}_{\alpha}(-1).$$ Now, in order to describe the representation theory data required in our approach, we may think $\mathfrak{sl}(2,\mathbb{C})$ as the operator algebra generated by $$x = X\frac{\partial}{\partial Y}, \ \ y = Y\frac{\partial}{\partial X}, \ \ h = X\frac{\partial}{\partial X} - Y\frac{\partial}{\partial Y}.$$ From this, we can consider for each integer $\ell > 0$ the irredicible $\mathfrak{sl}(2,\mathbb{C})$-module $$\label{polyrepresentation}
V(\ell \omega_{\alpha}) = \Big \{ P(X,Y) \in \mathbb{C}\big[X,Y \big] \ \ \Big | \ \ P(X,Y) \ {\text{is homogeneous and }} {\text{deg}}(P) = \ell \Big \}.$$ The weight spaces of $h \in \mathfrak{sl}(2,\mathbb{C})$ are the 1-dimensional spaces spanned by the elements. $$P_{k}(X,Y) = X^{\ell - k}Y^{k}, \ \ k = 0,\ldots,\ell.$$ It is straightforward to see that the highest weight vector is given by $v_{\ell \omega_{\alpha}}^{+} = P_{0}(X,Y) \in V(\ell\omega_{\alpha})$. Now, by considering the Iwasawa decomposition
$n(z) = \begin{pmatrix}
1 & 0 \\
z & 1
\end{pmatrix} = g \begin{pmatrix}
\sqrt{1+|z|^{2}} & 0 \\
0 & (\sqrt{1+|z|^{2}})^{-1}
\end{pmatrix} \begin{pmatrix}
1 & \frac{\overline{z}}{1+|z|^{2}} \\
0 & 1
\end{pmatrix},$
for some $g \in {\rm{SU}}(2)$, and $z \in \mathbb{C}$, it follows that
$n(z)v_{\ell \omega_{\alpha}}^{+} = (1+|z|^{2})^{\frac{\ell}{2}} gv_{\ell \omega_{\alpha}}^{+}.$
Hence, given a negative line bundle $L \in {\text{Pic}}(\mathbb{C}{\rm{P}}^{1})$, such that $L = \mathscr{O}_{\alpha}(-\ell)$, for some integer $\ell >0$, we can consider the manifold
$M = {\rm{Tot}}(\mathscr{O}_{\alpha}(-\ell)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$.
Considering a norm $||\cdot||$ in $V(\ell \omega_{\alpha})$ induced from some ${\rm{SU}}(2)$-invariant inner product, we have from Theorem \[Maintheo1\] a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, which is defined in coordinates $(n(z),w) \in L^{\times}|_{U}$ by $$\label{Kpotential}
{\rm{K}}_{H}\big (n(z),w \big ) = (1+|z|^{2})^{\ell}w\overline{w},$$ here we consider $||v_{\ell \omega_{\alpha}}^{+}|| = 1$, see Equation \[potentialmaxparabolic\]. By means of the Kähler potential above we can describe the l.c.K. structure $(\Omega, J, \theta)$ on $M$ from the Lee form $$\label{leeformhopf}
\displaystyle \theta = -\ell\Bigg [d\log \big ( 1+|z|^{2}\big ) + \frac{1}{\ell}\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg] = - \ell\frac{zd\overline{z} +\overline{z}dz}{1+|z|^{2}} - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ From Theorem \[Theo2\], for the particular case $\ell = 1$, if we consider a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}(\mathscr{O}_{\alpha}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}} = {\rm{SU}}(2)$, it follows that $M = {\rm{Tot}}(\mathscr{O}_{\alpha}(-1)^{\times})/\mathbb{Z} = S^{3} \times S^{1}$, notice that the associated underlying homogeneous Boothby-Wang fibration defined by the spehere bundle of $\mathscr{O}_{\alpha}(-1)$ is given by the Hopf fibration
$S^{1} \hookrightarrow S^{3} \to \mathbb{C}{\rm{P}}^{1}.$
Moreover, the homogeneous Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ is completely determined by the Lee form $$\displaystyle \theta_{g} = - \frac{zd\overline{z} +\overline{z}dz}{1+|z|^{2}} - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}},$$ see Equation \[higgsfield\]. A remarkable result of Kodaira [@Kodaira] asserts that any compact complex manifold which is homeomorphic to $S^{3} \times S^{1}$ is complex analytically homeomorphic to $(\mathbb{C}^{2}\backslash \{0\})/\Gamma$, where $\Gamma$ is an infinite cyclic group generated by a complex analytic automorphism $a \colon \mathbb{C}^{2}\backslash \{0\} \to \mathbb{C}^{2}\backslash \{0\}$, such that $$a \colon \big (z_{1},z_{2} \big ) \mapsto \big (sz_{1} + \lambda z_{2}^{m},tz_{2} \big),$$ with $m$ being a positive integer and $s,\lambda,t$ being complex numbers satisfying $$0 < |s| \leq |t| < 1, \ \ \big ( t^{m} - s \big)\lambda = 0.$$ The manifolds $(\mathbb{C}^{2}\backslash \{0\})/\Gamma$ defined as above are known as *primary Hopf surface*. The example which we have described above belongs to the particular class of primary Hopf surfaces of class $1$, i.e., the class of primary Hopf surfaces which satisfy: $\lambda = 0$ and $s = t$. For a description of Locally conformally Kähler metrics on primary Hopf surfaces we suggest [@GauduchonOrnea], [@Belgun].
In 1948, H. Hopf [@HOPF] gave the first examples of compact complex manifolds which are non-Kähler by showing that $S^{2m+1} \times S^{1}$ admits a complex structure for any positive integer $m$. These structures are obtained from the quotient of $\mathbb{C}^{m+1} \backslash \{0\}$ by a holomorphic and totally discontinuous action of $\mathbb{Z}$. The construction which we have described above by meas of representation theory of Lie algebras can be naturally generalized to $\mathbb{C}{\rm{P}}^{m}$. Actually, by considering $\mathbb{C}{\rm{P}}^{m} = {\rm{Gr}}(1,\mathbb{C}^{m+1})$, from Example \[Grassmanianexample\] we obtain
$\mathbb{C}{\rm{P}}^{m} = {\rm{SL}}(m+1,\mathbb{C})/P_{\omega_{\alpha_{1}}}.$
Thus, it follows that $${\text{Pic}}(\mathbb{C}{\rm{P}}^{m}) = \mathbb{Z}c_{1}\big(\mathscr{O}_{\alpha_{1}}(1)\big) \Longrightarrow K_{\mathbb{C}{\rm{P}}^{m}}^{\otimes \frac{1}{m+1}} = \mathscr{O}_{\alpha_{1}}(-1).$$ Since in this case we have $V(\omega_{\alpha_{1}}) = \mathbb{C}^{m+1}$ and $v_{\omega_{\alpha_{1}}}^{+} = e_{1}$, if we take the open set defined by the opposite big cell $U = R_{u}(P_{\omega_{\alpha_{1}}})^{-}x_{0} \subset \mathbb{C}{\rm{P}}^{m}$, where $x_0=eP_{\omega_{\alpha_{1}}}$ (trivial coset), we have that $U = R_{u}(P_{\Theta})^{-}x_{0}$ can be parameterized by
$(z_{1},\ldots,z_{m}) \in \mathbb{C}^{m} \mapsto \begin{pmatrix}
1 & 0 &\cdots & 0 \\
z_{1} & 1 &\cdots & 0 \\
\ \vdots & \vdots &\ddots & \vdots \\
z_{m} & 0 & \cdots &1
\end{pmatrix}x_{0} \in U = R_{u}(P_{\omega_{\alpha_{1}}})^{-}x_{0}$.
Fro the sake of simplicity, we denote the matrix above by $n(z) \in {\rm{SL}}(m+1,\mathbb{C})$. From this, we can take a local section $s_{U} \colon U \subset \mathbb{C}{\rm{P}}^{m} \to {\rm{SL}}(m+1,\mathbb{C})$, defined by $$s_{U}(n(z)x_{0}) = n(z) \in {\rm{SL}}(m+1,\mathbb{C}),$$ such that $z = (z_{1},\ldots,z_{m}) \in \mathbb{C}^{m}$. Thus, by considering a negative line bundle $L \in {\text{Pic}}(\mathbb{C}{\rm{P}}^{m})$, since $L = \mathscr{O}_{\alpha_{1}}(-\ell)$, for some integer $\ell > 0$, we have ${\rm{K}}_{H} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-\ell)^{\times}) \to \mathbb{R}^{+}$, which is described in local coordinates $(z,w) \in \mathscr{O}_{\alpha_{1}}(-\ell)^{\times}|_{U}$ by
$\displaystyle {\rm{K}}_{H}(z,w) = \bigg (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \bigg )^{\ell}w\overline{w},$
see Equation \[potentialgrassman\]. Therefore, by considering the manifold
$M = {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-\ell)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, we can describe the (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ from the Lee form $$\displaystyle \theta = -\ell\Bigg [d\log \bigg (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \bigg )+ \frac{1}{\ell}\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\Bigg] = - \ell \sum_{i = 1}^{m}\frac{z_{i}d\overline{z}_{i} +\overline{z}_{i}dz_{i}}{\big (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \big)} - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ Similarly to the previous example, from Theorem \[Theo2\], for the particular case $\ell = 1$, if we consider a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}} = {\rm{SU}}(m+1)$, it follows that $M = {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-1)^{\times})/\mathbb{Z} = S^{2m+1} \times S^{1}$, notice that the associated underlying homogeneous Boothby-Wang fibration defined by the sphere bundle of $\mathscr{O}_{\alpha_{1}}(-1)$ is given by the complex Hopf fibration
$S^{1} \hookrightarrow S^{2m+1} \to \mathbb{C}{\rm{P}}^{m}.$
Further, the homogeneous Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ is completely determined by the Lee form $$\displaystyle \theta_{g} = - \sum_{i = 1}^{m}\frac{z_{i}d\overline{z}_{i} +\overline{z}_{i}dz_{i}}{\big (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \big)} - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ Therefore, we obtain a complete description of the homogeneous Hermitian-Einstein-Weyl structure on the Hopf manifold $S^{2m+1} \times S^{1}$ by means of elements of Lie theory.
I is worth mentioning that the previous ideas used in the description of the Vaisman structure on $M = S^{2m+1} \times S^{1}$ also can holds for any manifold of the form
$M = \mathbb{L}(m;\ell) \times S^{1},$
where $\mathbb{L}(m;\ell) = S^{2m+1}/\mathbb{Z}_{\ell}$ (Lens space), $\forall \ell \in \mathbb{Z}_{>0}$. Actually, by considering $\mathscr{O}_{\alpha_{1}}(-\ell) \to \to \mathbb{C}{\rm{P}}^{m}$, it is not difficult to see that the underlying homogeneous Boothby-Wang fibration defined by the sphere bundle of $\mathscr{O}_{\alpha_{1}}(-\ell)$ is given by the complex Hopf fibration
$S^{1} \hookrightarrow \mathbb{L}(m;\ell) \to \mathbb{C}{\rm{P}}^{m}.$
In order to equip $M = \mathbb{L}(m;\ell) \times S^{1}$ with a (Vaisman) l.c.K. structure, we just need to observe that ${\text{Tot}}(\mathscr{O}_{\alpha_{1}}(-\ell)) \to M$ defines a Kähler covering, so we can apply Theorem \[Theo2\].
Our next example is also a constructive model obtained as a particular case from the results established in the previous section.
\[examplefullflag\] Recall that a full flag manifold is defined as the homogeneous space given by $G/T$, where $G$ is a compact simple Lie group, and $T \subset G$ is a maximal torus. For the sake of simplicity, let us suppose also that $G$ is simply connected.
Considering the root system $\Pi = \Pi^{+} \cup \Pi^{-}$ associated to the pair $(G,T)$ [@Knapp], from the complexification $G^{\mathbb{C}}$ of $G$ we have an identification $$G/T \cong G^{\mathbb{C}}/B,$$ where $B \subset G^{\mathbb{C}}$ is a Borel subgroup such that $B \cap G = T$. From Proposition \[C8S8.2Sub8.2.3P8.2.6\], we have $${\text{Pic}}(G/T) = \bigoplus_{\alpha \in \Sigma}\mathbb{Z}c_{1}\big (\mathscr{O}_{\alpha}(1) \big ).$$ Therefore, given a negative line bundle $L \in {\text{Pic}}(G/T)$, it follows that
$\displaystyle L = \bigotimes_{\alpha \in \Sigma}\mathscr{O}_{\alpha}(-\ell_{\alpha})$,
such that $\ell_{\alpha} > 0$, $\forall \alpha \in \Sigma$. From Theorem \[Theo2\], we have a Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, defined in coordinates $(z,w) \in L^{\times}|_{U}$ by $$\label{kahlerpotentialfull}
\displaystyle {\rm{K}}_{H}\big (z,w \big ) = \Big ( \prod_{\alpha \in \Sigma}\big | \big |s_{U}(z)v_{\omega_{\alpha}}^{+} \big | \big |^{2\ell_{\alpha}}\Big)w\overline{w} = \Big ( \big | \big |s_{U}(z)v_{\mu(L)}^{+} \big | \big |^{2}\Big)w\overline{w},$$ for some local section $s_{U} \colon U \subset G^{\mathbb{C}}/B \to G^{\mathbb{C}}$, where $v_{\mu(L)}^{+}$ is the highest weight vector of weight $\mu(L)$ for the irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\mu(L)) = H^{0}(G/T,L^{-1})^{\ast}$, see Equation \[negativeweight\]. Hence, by considering the manifold
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, from the Kähler potential described above we have a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$, such that the Lee form $\theta $ is locally given by $$\displaystyle \theta = -\bigg [d\log \Big ( \big | \big |s_{U}v_{\mu(L)}^{+} \big | \big |^{2}\Big ) + \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg].$$ Now, if we consider $$\label{fullweight}
\displaystyle \varrho = \frac{1}{2} \sum_{\alpha \in \Pi^{+}}\alpha,$$ it follows that $\delta_{B} = 2\varrho$. Therefore, from Equation \[canonicalbundleflag\], since $I(G/T) = 2$ [@Humphreys $\S$ 13.3], it follows that $$\label{canonicalfull}
K_{G/T} = L_{\chi_{2\varrho}}^{-1} \Longrightarrow \mathscr{O}_{G/T}(-1) = L_{\chi_{\varrho}}^{-1},$$ recall that $K_{G/T}^{\otimes \frac{1}{I(G/T)}} = \mathscr{O}_{G/T}(-1)$. Thus, if we consider a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}( \mathscr{O}_{G/T}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}} = G$, the homogeneous Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ is completely determined by the Lee form locally describe by[^14] $$\label{leefull}
\displaystyle \theta_{g} = -\frac{1}{2}\bigg [d\log \Big ( \big | \big |s_{U}v_{2\varrho}^{+} \big | \big |^{2}\Big ) + 2\frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}\bigg] = -d\log \Big ( \big | \big |s_{U}v_{\varrho}^{+} \big | \big |^{2}\Big ) - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}},$$ for some local section $s_{U} \colon U \subset G/T \to G^{\mathbb{C}}$, where $v_{\varrho}^{+}$ denotes the highest weight vector of weight $\varrho$ for the irreducible $\mathfrak{g}^{\mathbb{C}}$-module $V(\varrho)$. Notice that $g = \Omega({\rm{id}} \otimes J)$, see Equation \[HEWmetric\].
In order to illustrate the ideas described above by means of a concrete example, consider $G = {\rm{SU}}(n+1)$ and the full flag manifold
${\rm{SU}}(n+1)/T^{n} = {\rm{SL}}(n+1,\mathbb{C})/B,$
here we fix the same Lie-theoretical data for $\mathfrak{sl}(n+1,\mathbb{C})$ as in Example \[Grassmanianexample\]. By taking a negative line bundle $L \in {\text{Pic}}({\rm{SU}}(n+1)/T^{n})$, we can consider the manifold
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$. From Theorem \[Theo1\], we have a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ completely determined by the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, which can be described as follows. As we have seen in Example \[Grassmanianexample\], in this particular case we have the irreducible fundamental $\mathfrak{sl}(n+1,\mathbb{C})$-modules given by
$V(\omega_{\alpha_{k}}) = \bigwedge^{k}(\mathbb{C}^{n+1}), \ \ \ {\text{and}} \ \ \ v_{\omega_{\alpha_{k}}}^{+} = e_{1} \wedge \ldots \wedge e_{k},$
such that $k = 1,\ldots,n$. Let $U = R_{u}(B)^{-}x_{0} \subset{\rm{SL}}(n+1,\mathbb{C})/B$ be the opposite big cell. This open set is parameterized by the holomorphic coordinates $$z \in \mathbb{C}^{\frac{n(n+1)}{2}} \mapsto n(z)x_{0} = \begin{pmatrix}
1 & 0 & 0 & \cdots & 0 \\
z_{21} & 1 & 0 & \cdots & 0 \\
z_{31} & z_{32} & 1 &\cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
z_{n+1,1} & z_{n+1,2} & z_{n+1,3} & \cdots & 1
\end{pmatrix}x_{0},$$ where $ n = n(z) \in N^{-}$ and $z = (z_{ij}) \in \mathbb{C}^{\frac{n(n+1)}{2}}$. From this, we define for each subset $I = \{i_{1} < \cdots < i_{k}\} \subset \{1,\cdots,n+1\}$, with $1 \leq k \leq n$, the following polynomial functions $\det_{I} \colon {\rm{SL}}(n+1,\mathbb{C}) \to \mathbb{C}$, such that
$$\textstyle{\det_{I}}(g) = \det \begin{pmatrix}
g_{i_{1}1} & g_{i_{1}2} & \cdots & g_{i_{1} k} \\
g_{i_{2}1} & g_{i_{2}2} & \cdots & g_{i_{2} k} \\
\vdots & \vdots& \ddots & \vdots \\
g_{i_{k}1} & g_{i_{k}2} & \cdots & g_{i_{k}k}
\end{pmatrix},$$ for every $g \in {\rm{SL}}(n+1,\mathbb{C})$. Now, we observe that, for every $g \in {\rm{SL}}(n+1,\mathbb{C})$ we have
$g(e_{1} \wedge \ldots \wedge e_{l}) = \displaystyle \sum_{i_{1} < \cdots < i_{l}} \textstyle{\textstyle{\det_{I}}}(g)e_{i_{1}} \wedge \ldots \wedge e_{i_{l}},$
notice that the sum above is taken over $I = \{i_{1} < \cdots < i_{l}\} \subset \{1,\cdots,n+1\}$, with $1 \leq l \leq n$. By taking the local section $s_{U} \colon U \subset {\rm{SL}}(n+1,\mathbb{C})/B \to {\rm{SL}}(n+1,\mathbb{C})$, such that $s_{U}(n^{-}(z)x_{0}) = n(z)$, and by supposing that
$\displaystyle L = \bigotimes_{k = 1}^{n}\mathscr{O}_{\alpha_{k}}(-\ell_{k}),$
with $\ell_{k} > 0$, for $k = 1,\ldots,k$, it follows from Equation \[kahlerpotentialfull\] that the Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$, defined in coordinates $(z,w) \in L^{\times}|_{U}$, is given by $$\label{sl-potential}
\displaystyle {\rm{K}}_{H}\big (z,w \big ) = \Bigg [\prod_{k = 1}^{n}\Bigg ( \sum_{i_{1} < \cdots < i_{k}} \bigg | \textstyle{\det_{I}} \begin{pmatrix}
1 & 0 & \cdots & 0 \\
z_{21} & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
z_{n+1,1} & z_{n+1,2} & \cdots & 1
\end{pmatrix} \bigg |^{2}\Bigg)^{\ell_{k}} \Bigg ] w\overline{w}.$$ Hence, from the Kähler potential above, the Lee form associated to the (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ provides by Theorem \[Maintheo1\] is given locally by $$\displaystyle \theta = -\sum_{k = 1}^{n}\ell_{k}d\log\Bigg ( \sum_{i_{1} < \cdots < i_{k}} \bigg | \textstyle{\det_{I}} \begin{pmatrix}
1 & 0 & \cdots & 0 \\
z_{21} & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
z_{n+1,1} & z_{n+1,2} & \cdots & 1
\end{pmatrix} \bigg |^{2}\Bigg) - \displaystyle \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ Moreover, if we consider a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}(\mathscr{O}_{{\rm{SU}}(n+1)/T^{n}}(-1)^{\times})/\mathbb{Z}$, such that $K_{\text{ss}} = {\rm{SU}}(n+1)$, we have from Theorem \[Theo2\] a Hermitian-Einstein-Weyl structure $(\Omega,J,\theta_{g})$ on $M$ completely determined by the Lee form $$\displaystyle \theta_{g} = -\sum_{k = 1}^{n}d\log\Bigg ( \sum_{i_{1} < \cdots < i_{k}} \bigg | \textstyle{\det_{I}} \begin{pmatrix}
1 & 0 & \cdots & 0 \\
z_{21} & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
z_{n+1,1} & z_{n+1,2} & \cdots & 1
\end{pmatrix} \bigg |^{2}\Bigg) - \displaystyle \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}},$$ in the expression above we used that the description given in \[canonicalfull\], and the characterization $$\label{sumweights}
\displaystyle \varrho = \omega_{\alpha_{1}} + \cdots + \omega_{\alpha_{n}},$$ see for instance [@Humphreys $\S$ 13.3].
Let us consider the low-dimensional particular example of the construction above provided by the the Wallach space [@Wallach] $$W_{6} = {\rm{SU}}(3)/T^{2}.$$ Since in this case we have
${\text{Pic}}(W_{6}) = \mathbb{Z}c_{1}\big (\mathscr{O}_{\alpha_{1}}(1)\big) \oplus \mathbb{Z}c_{1}\big (\mathscr{O}_{\alpha_{2}}(1)\big),$
given a negative line bundle $L \in {\text{Pic}}(W_{6})$, we have a Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(L^{\times}) \to \mathbb{R}^{+}$ given by
$\displaystyle {\rm{K}}_{H}\big (z,w \big ) = \Bigg [ \bigg ( 1 + \sum_{i = 2}^{3}|z_{i1}|^{2} \bigg )^{\ell_{1}} \bigg (1 + |z_{32}|^{2} + \bigg | \det \begin{pmatrix}
z_{21} & 1 \\
z_{31} & z_{32}
\end{pmatrix} \bigg |^{2} \bigg )^{\ell_{2}} \Bigg ]w\overline{w},$
see Equation \[sl-potential\]. From this, by considering the manifold
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, we have from Theorem \[Theo1\] a (Vaisman) l.c.K. structure $(\Omega, J, \theta)$ on $M$ with associated Lee form (locally) described by $$\theta = -\ell_{1}d\log\bigg ( 1 + \sum_{i = 2}^{3}|z_{i1}|^{2} \bigg ) - \ell_{2}d\log \bigg (1 + |z_{32}|^{2} + \bigg | \det \begin{pmatrix}
z_{21} & 1 \\
z_{31} & z_{32}
\end{pmatrix} \bigg |^{2} \bigg ) - \displaystyle \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ Now, if we consider a compact homogeneous Hermitian-Einstein-Weyl $K$-manifold of the form $M = {\rm{Tot}}\big (\mathscr{O}_{{\rm{SU}}(3)/T^{2}}(-1)^{\times}\big )/\mathbb{Z}$, such that $K_{\text{ss}} = {\rm{SU}}(3)$, it follows that $$M = X_{1,1} \times S^{1},$$ such that $X_{1,1}$ is the Aloff-Wallach space [@Aloff] defined by $$X_{1,1} = {\rm{SU}}(3)/{\rm{U}}(1).$$ Actually, recall that $M$ is a flat principal $S^{1}$-bundle over the simply connected Sasaki manifold defined by the sphere bundle of $\mathscr{O}_{{\rm{SU}}(3)/T^{2}}(-1)$, which is defined by
$S^{1} \hookrightarrow X_{1,1} \to W_{6}.$
Therefore, from Theorem \[Theo2\], we have a Hermitian-Einstein-Weyl structure $(\Omega, J, \theta_{g})$ on $M$ completely determined by Lee form $$\theta_{g} = -d\log\bigg ( 1 + \sum_{i = 2}^{3}|z_{i1}|^{2} \bigg ) - d\log \bigg (1 + |z_{32}|^{2} + \bigg | \det \begin{pmatrix}
z_{21} & 1 \\
z_{31} & z_{32}
\end{pmatrix} \bigg |^{2} \bigg ) - \displaystyle \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}},$$ notice that $g = \Omega({\rm{id}} \otimes J)$. Hence, we obtain a concrete description for the homogeneous Hermitian-Einstein-Weyl metric on $M = X_{1,1} \times S^{1}$.
Kodaira-type embedding via Borel-Weil theorem
---------------------------------------------
In this subsection, we study the following result in the setting of compact homogeneous Vaisman manifolds:
Let $M$ be a compact Vaisman manifold. Then $M$ can be embedded into a Vaisman-type Hopf manifold $H = \big (\mathbb{C}^{n} \backslash \{0\} \big)/ \langle O \rangle$ (cf. [@Kamishima]), where $O \in {\rm{End}}(\mathbb{C}^{n})$ is a diagonal linear operator on $\mathbb{C}^{n}$ with all eigenvalues $\lambda_{1}, \ldots,\lambda_{n}$, satisfying $|\lambda_{j}|<1$, $j=1,\ldots,n$.
The main purpose is to show that the result aforementioned takes a concrete form in the homogeneous context, which is essentially the content of our Theorem \[Theorem4\]. This concrete formulation provides a connection between compact homogeneous Hermitian-Weyl manifolds and a certain class of affine algebraic varieties (cone of highest vector).
### Kähler covering and HV-varieties
In this subsection, and throughout the next subsections, we shall assume the basic background on semisimple complex algebraic groups. The details about most of the results which we shall use can be found in [@HumphreysLAG], [@Flaginterplay].
Let $G^{\mathbb{C}}$ be a connected, simply connected, simple complex algebraic group, and let $V(\mu)$ be an irreducible representation of $G^{\mathbb{C}}$ with highest weight $\mu$ and highest weight vector $v_{\mu}^{+} \in V(\mu)$. From this basic data we can construct a projective algebraic variety by taking
$X = G^{\mathbb{C}} \cdot [v_{\mu}^{+}] \subset \mathbb{P}(V(\mu)),$
i.e. the $G^{\mathbb{C}}$-orbit in the associated projective representation. In this latter setting, a HV-variety $Y$ is defined by the affine cone of $X \subset \mathbb{P}(V(\mu))$, so we have[^15] $$\label{hv-varieties}
Y = {\text{Affcone}}(X) = \overline{G^{\mathbb{C}} \cdot v_{\mu}^{+}} = G^{\mathbb{C}} \cdot v_{\mu}^{+} \cup \{0\}.$$ The affine algebraic variety underlying $Y$ is characterized by a system of quadrics defined by $${\rm{C}}_{\mathfrak{g}^{\mathbb{C}}}\big ( v \otimes v \big) = \kappa \big ( 2\mu + 2\varrho, 2\mu\big )\big ( v \otimes v \big),$$ such that ${\rm{C}}_{\mathfrak{g}^{\mathbb{C}}} \in {\mathcal{U}}(\mathfrak{g}^{\mathbb{C}})$ is the Casimir element of $\mathfrak{g}^{\mathbb{C}}$, where ${\mathcal{U}}(\mathfrak{g}^{\mathbb{C}})$ denotes the universal enveloping algebra of $\mathfrak{g}^{\mathbb{C}}$, and $\varrho$ is the weight defined by the half-sum of the positive roots, see for instance [@Quadric]. For more details about HV-varieties from the point of view of algebraic geometry we suggest [@Vinberg]. It is worth mentioning that the algebraic variety described above is a particular example of Kostant cone presented in [@WallachI §4.3.3]
It will be suitable for us to consider the underlying analytic spaces associated to HV-varieties as described above, as well as the complex analytic Lie groups underlying semisimple complex algebraic groups, so we point out the following facts:
1. Any complex algebraic group is a smooth algebraic variety, so any complex algebraic group is a complex Lie group. Further, any complex algebraic subgroup of a complex algebraic group is a complex Lie subgroup (cf. [@TAYLOR $\mathsection$ 15.1]).
2. Any complex semisimple Lie group admits a unique structure of affine algebraic group which is compatible with its given structure of complex Lie group, e.g. [@TAYLOR Corollary 15.8.6].
3. Given a complex algebraic variety $Y$, if we consider its complex analytification $Y^{\text{an}}$, then we have the following equality of set points
${\text{Sing}}(Y) = {\text{Sing}}(Y^{\text{an}}),$
see for instance [@TAYLOR Proposition 13.3.6].
From the facts above, given a HV-variety $Y$ as in \[hv-varieties\], we have that $Y^{\text{an}} = C(X,L)$, such that $\mu = \mu(L)$ for a suitable negative line bundle $L \in {\text{Pic}}(X)$ (see \[negativeweight\]), i.e. the complex analytic variety underlying $Y$ is the analytic variety presented in Remark \[affinecone\] as the analytic affine cone over a complex flag manifold. It is worth mentioning that, in this latter setting, we have from Borel-Weil theorem that $$\label{speccone}
Y = {\text{Spec}}\Big( \bigoplus_{k = 0}^{+\infty}H^{0}(X,-kL)\Big) = {\text{Spec}}\Big ( \bigoplus_{k = 0}^{+\infty}V(k\mu(L))^{\ast}\Big),$$ here we denote $-kL = (L^{-1})^{\otimes k}$, $\forall k \in \mathbb{Z}_{\geq 0}$.
\[kahlerpotentialalgebraiclie\] For the sake of simplicity, unless otherwise stated, in what follows we shall not distinguish a complex affine variety from its complex analytification. Also, in the GAGA setting of analogies [@Serre], [@TAYLOR Chapter 13], we shall not distinguish a complex projective algebraic variety from its analytification.
As we have seen in Remark \[affinecone\], given a complex flag manifold $X_{P} = G^{\mathbb{C}}/P$ and a negative line bundle $L \in {\text{Pic}}(X_{P})$, we have a biholomorphism
${\mathscr{R}} \colon {\rm{Tot}}(L^{\times}) \to C(X_{P},L)_{{\text{reg}}},$
where $C(X_{P},L)$ is the HV-variety associated to $v_{\mu(L)}^{+} \in V(\mu(L))$. Since $C(X_{P},L)_{{\text{reg}}} = G^{\mathbb{C}}\cdot v_{\mu(L)}^{+} \subset V(\mu(L))$, one can consider the function
$F \colon C(X_{P},L)_{{\text{reg}}} \to \mathbb{R}_{>0}$, such that $F(v) = ||v||^{2}$, $\forall v \in C(X_{P}.L)_{{\text{reg}}}$,
here we have the norm $|| \cdot || \colon V(\mu(L)) \to \mathbb{R}_{\geq 0}$ induced from the $G$-invariant inner product defined in \[innerinv\]. In order to understand the function $F$, let us introduce some basic facts. If we consider the decomposition $$V(\mu(L)) = \mathbb{C}\cdot v_{\mu(L)}^{+} \oplus V', \ \ {\text{such that}} \ \ V' = \bigoplus_{\mu' \prec \mu(L)}V(\mu(L))_{\mu'},$$ we can define $l_{0} \in V(\mu(L))^{\ast}$ by setting
$l_{0}(v_{\mu(L)}^{+} ) = 1$, and $l_{0} (V') = \{0\}$.
From this, by denoting $f \colon C(X_{P},L)_{{\text{reg}}} \to \mathbb{C}$ the restriction $f = l_{0}|_{C(X_{P},L)_{{\text{reg}}}}$, we can take the open (dense) set defined by $D(f) = C(X_{P},L)_{{\text{reg}}} \backslash {\rm{div}}(f)$[^16]. Considering the natural projection $\pi \colon V(\mu(L)) \backslash \{0\} \to \mathbb{P}(V(\lambda))$, it is straightforward to show that
$D(f) = \pi^{-1}(U)$, where $U = B^{-}\cdot [v_{\mu(L)}^{+} ] = R_{u}(P)^{-}\cdot[v_{\mu(L)}^{+} ] \subset X_{P}$ (opposite big cell).
In fact, in order to see that, consider the Bruhat decomposition $$G^{\mathbb{C}} = \bigsqcup_{d \in W}N^{-}n_{d}B,$$ where $W \cong N_{G^{\mathbb{C}}}(T^{\mathbb{C}})/T$ is the Weyl group defined by the pair $(G^{\mathbb{C}},T^{\mathbb{C}})$, and $n_{d} \in N_{G^{\mathbb{C}}}(T^{\mathbb{C}})$ stands for some fixed representative of $d \in W$. From this, given $g \in G^{\mathbb{C}}$, it follows that $g = xn_{d}b$, such that $x \in N^{-}$, $n_{d} \in N_{G^{\mathbb{C}}}(T^{\mathbb{C}})$ and $b \in B$. Thus, we have that
$f(gv_{\mu(L)}^{+}) = \chi^{(L)}(b)f(xv_{d(\mu(L))})$,
such that $\chi^{(L)} \colon B \to \mathbb{C}^{\times}$ is the extension of the character associated to $\mu(L)$ (see \[negativecharacter\]). Since $x \cdot V' = V'$, $\forall x \in N^{-}$, see for instance [@HumphreysLAG $\mathsection$ 27.2], from the latter equation above we obtain $$f(gv_{\mu(L)}^{+})\neq 0 \Longleftrightarrow d(\mu(L)) = \mu(L) \Longleftrightarrow [gv_{\mu(L)}^{+} ]= [xv_{\mu(L)}^{+}],$$ which implies that $D(f) = \pi^{-1}(U)$. Hence, denoting $\dim_{\mathbb{C}}(X_{P}) = n$, if we consider coordinates
$z \in \mathbb{C}^{n} \mapsto [n(z)v_{\mu(L)}] \in U \subset X_{P}$,
such that $n(z)\in R_{u}(P)^{-}$, given $v \in D(f)$, we have
$v = v(z,w) = w n(z)v_{\mu(L)}^{+}$,
for some $(z,w) \in \mathbb{C}^{d}\times\mathbb{C}^{\times}$. Therefore, by taking the restriction $F|_{D(f)}$, we get $$F(v(z,w)) = \Big (\big | \big |n(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w} = \Big (\big | \big |s_{U}(z)v_{\mu(L)}^{+} \big | \big |^{2} \Big) w\overline{w} = {\rm{K}}_{H}(z,w),$$ where $s_{U} \colon U \subset X_{P} \to G^{\mathbb{C}}$ is a local section, such that $s(n(z)x_{0}) = n(z)$, $\forall n(z)x_{0} \in U$. From this, we conclude that the Kähler potential presented in Equation \[negativepotential\] is obtained by the composition: $$\label{potentialreduction}
{\rm{K}}_{H} = F \circ {\mathscr{R}} = \mathscr{R}^{\ast}(F).$$ Therefore, the Vaisman structure obtained in Theorem \[Maintheo1\] can be explicitly described by means of a pair $(Y,||\cdot||^{2}\big |_{Y})$, such that $Y = \overline{G^{\mathbb{C}} \cdot v_{\mu(L)}^{+}}$ is a HV-variety and $||\cdot|| \colon V(\mu(L)) \to \mathbb{R}_{\geq0}$ is a norm induced by some $G$-invariant inner product on $V(\mu(L))$ cf. Remark \[Ginvinner\].
Let us illustrate the previous ideas by means of the classical example provided by the Hopf surface. In order to do so, we consider the same notation and conventions as in Example \[exampleP1\], i.e. $G^{\mathbb{C}} = {\rm{SL}}(2,\mathbb{C})$ and
$X_{B} = {\rm{SL}}(2,\mathbb{C})/B = \mathbb{C}{\rm{P}}^{1}$.
As we have seen, we can consider the analytic cellular decomposition
$X_{B} = \mathbb{C}{\rm{P}}^{1} = N^{-}x_{0} \cup \pi \Big ( \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix} \Big ),$
such that $x_{0} = eB$, and
$N^{-} = \Bigg \{ \begin{pmatrix}
1 & 0 \\
z & 1
\end{pmatrix} \ \ \Bigg | \ \ z \in \mathbb{C} \Bigg \}$,
notice that the opposite big cell is given by $U = N^{-}x_{0} \subset X_{B}$. Now, since ${\text{Pic}}(X_{B}) = \mathbb{Z}c_{1}(\mathscr{O}_{\alpha}(1))$, if we consider the very ample line bundle $\mathscr{O}_{\alpha}(1) \to X_{B}$, it follows from Borel-Weil theorem that
$H^{0}(\mathbb{C}{\rm{P}}^{1},\mathscr{O}_{\alpha}(1))^{\ast} = V(\omega_{\alpha})$,
observe that $V(\omega_{\alpha}) = \mathbb{C}^{2}$ and $v_{\omega_{\alpha}}^{+} = e_{1}$ (see Equation \[polyrepresentation\]). From this, when one considers the projective embedding $X_{B} \hookrightarrow \mathbb{P}(V(\omega_{\alpha}))$, it turns out that
$Y = C(X_{B},\mathscr{O}_{\alpha}(-1)) = \overline{{\rm{SL}}(2,\mathbb{C}) \cdot v_{\omega_{\alpha}}^{+}} = \mathbb{C}^{2},$
so it follows that $C(X_{B},\mathscr{O}_{\alpha}(-1))_{\text{reg}} = \mathbb{C}^{2} \backslash \{0\}$, notice that the projective embedding in this particular case is in fact a bihomomorphism, i.e. $X_{B} \cong \mathbb{P}(\mathbb{C}^{2})$. From the last biholomorphism, it is straightforward to see that the opposite big cell is defined by
$U = \Big \{ [1:z] \in \mathbb{P}(\mathbb{C}^{2}) \ \ \Big | \ \ z \in \mathbb{C}\Big \}$.
Thus, if one considers the natural projection map $\pi \colon \mathbb{C}^{2}\backslash \{0\} \to \mathbb{P}(\mathbb{C}^{2})$, it follows that
$\pi^{-1}(U) = \Big \{ w(1,z) \in \mathbb{C}^{2}\backslash \{0\} \ \Big | \ w \in \mathbb{C}^{\times}, \ z \in \mathbb{C} \Big \} \subset C(X_{B},\mathscr{O}_{\alpha}(-1))_{\text{reg}}.$
Now, by taking the norm $||\cdot||$ in $\mathbb{C}^{2}$ induced from the canonical inner product, we can consider $F \colon C(X_{B},\mathscr{O}_{\alpha}(-1))_{\text{reg}} \to \mathbb{R}_{>0}$, such that $F(z_{1},z_{2}) = |z_{1}|^{2} + |z_{2}|^{2}$. Then, the restriction $F|_{\pi^{-1}(U)}$ is given by
$F(w,wz) = (1+|z|^{2})w\overline{w},$
see Equation \[Kpotential\]. The function $F\colon C(X_{B},\mathscr{O}_{\alpha}(-1))_{\text{reg}} \to \mathbb{R}_{>0}$ defines a globally conformal Kähler structure on $ C(X_{B},\mathscr{O}_{\alpha}(-1))_{\text{reg}} = \mathbb{C}^{2} \backslash \{0\}$, for which the associated Hermitian metric is defined by $$\displaystyle g_{0} = \frac{dz_{1}\odot d\overline{z_{1}} + dz_{2}\odot d\overline{z_{2}}}{|z_{1}|^{2} + |z_{2}|^{2}},$$ see for instance [@GauduchonOrnea]. Notice that in this case the Lee form is given by $\theta_{0} = -d\log F(z_{1},z_{2})$, so we obtain $$\theta_{0}|_{\pi^{-1}(U)} = - \frac{zd\overline{z} +\overline{z}dz}{1+|z|^{2}} - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}},$$ see Equation \[leeformhopf\]. Hence, the standard approach to describe the canonical l.c.K. structure on $S^{3}\times S^{1}$, via globally conformal Kähler metric $g_{0}$ on $\mathbb{C}^{2}\backslash \{0\}$, can be naturally recovered by using elements of representation theory of ${\rm{SL}}(2,\mathbb{C})$. It is worth mentioning that the ideas above can be naturally generalized to $S^{2n+1}\times S^{1}$.
### Kodaira-type embedding for compact homogeneous l.c.K. manifolds
From the description of the previous subsection for Kähler covering of Vaisman manifolds defined by negative line bundles $L \in {\text{Pic}}(X_{P})$, i.e. manifolds given by $$\label{remebergamma}
M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ \ {\text{where}} \ \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$$ for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, we can provide a concrete realization (in terms of Lie theory) for the immersion of such manifolds into certain Hopf manifolds (cf. [@Kamishima]). In order to do so, let us collect some basic facts.
As we have seen, by considering the manifold \[remebergamma\], we have an associated HV-variety $Y = \overline{G^{\mathbb{C}} \cdot v_{\mu(L)}^{+}}$. From this, we can describe the map $\mathscr{R} \colon {\rm{Tot}}(L^{\times}) \to Y_{{\text{reg}}}$ explicitly by[^17] $$\label{explicityRemmert}
\mathscr{R} \colon [g,z] \mapsto zgv_{\mu(L)}^{+},$$ here we used the characterization of $L$ as $G^{\mathbb{C}} \times \mathbb{C}/\sim$, such that $[g,z]$ is defined by
$(g,z) \sim (h,w) \Longleftrightarrow \exists p \in P \ {\text{such that}} \ h = gp \ {\text{and}} \ w = p^{-1}z = \chi^{(L)}(p)z,$
see Equation \[eqrelationbundle\].
\[matrixcoefitients\] We observe that the explicit description given in \[explicityRemmert\] is obtained by the following procedure. At first, from the representation theory of semisimple complex algebraic groups, it follows that $$H^{0}(X_{P},L^{-1}) = \Big \{ f \in \mathscr{O}(G^{\mathbb{C}}) \ \ \Big | \ \ f(gp) = \chi^{(L)}(p)f(g), \forall g \in G^{\mathbb{C}},\forall p \in P\Big \},$$ where $\mathscr{O}(G^{\mathbb{C}})$ is the coordinate ring of $G^{\mathbb{C}}$, see for instance [@MONOMIAL Page 26]. From the characterization above, if we fix a basis $v_{0} = v_{\mu(L)}^{+}, v_{1}, \ldots,v_{N}$ for $V(\mu(L))$, here we suppose $\dim_{\mathbb{C}}(V(\mu(L))) = N+1$, and consider its dual basis $l_{0}, \ldots,l_{N}$ for $V(\mu(L))^{\ast} = H^{0}(X_{P},L^{-1})$, it follows that $$gv_{\mu(L)}^{+} = l_{0}\big(gv_{\mu(L)}^{+}\big)v_{0} + \cdots + l_{N}\big(gv_{\mu(L)}^{+}\big)v_{N}.$$ Therefore, we obtain a set of regular functions $\mathscr{R}_{0},\mathscr{R}_{1},\ldots,\mathscr{R}_{N} \in H^{0}(X_{P},L^{-1})$, such that $$\mathscr{R}_{j} \colon G^{\mathbb{C}} \to \mathbb{C}, \ \ \mathscr{R}_{j}(g) = l_{j}\big(gv_{\mu(L)}^{+}\big), \forall g \in G^{\mathbb{C}}, \ \ \forall j = 0,\ldots,N.$$ Now, also from the theory of complex semisimple algebraic groups, we have that there exists a homomorphism $\zeta^{(L)} \colon \mathbb{C}^{\times} \to T^{\mathbb{C}} \subset G^{\mathbb{C}}$ (1-parameter subgroup, see [@Flaginterplay]), such that $$\chi^{(L)}\big(\zeta^{(L)}(w)\big) = w, \ \ \forall w \in \mathbb{C}^{\times}.$$ From the data obtained above, we can take $\mathscr{R} \colon {\rm{Tot}}(L^{\times}) \to V(\mu(L))$, such that $$\label{remmertreductionmap}
\mathscr{R}\big ([g,w] \big) = \sum_{j = 0}^{N}\mathscr{R}_{j}\big (g\zeta^{(L)}(w)\big)v_{j},$$ $\forall [g,w] \in {\rm{Tot}}(L^{\times})$. Therefore, we obtain
$\mathscr{R}\big ([g,w] \big) = \displaystyle \sum_{j = 0}^{N}\mathscr{R}_{j}\big (g\zeta^{(L)}(w) \big)v_{j} = \chi^{(L)}\big(\zeta^{(L)}(w)\big) \Big ( \sum_{j = 0}^{N}\mathscr{R}_{j}\big (g \big)v_{j}\Big) = wgv_{\mu(L)}^{+},$
notice that the last equation on the right side above shows that the characterization \[remmertreductionmap\] does not depend on the choice of the basis for $V(\mu(L))$.
It is worth pointing out that, under the hypotheses above, the Kodaira embedding of $X_{P}$ relative to the positive line bundle $L^{-1} \in {\text{Pic}}(X_{P})$ also can be explicitly described by $$\label{embeddingflags}
gP \in X_{P} \mapsto \big [ \mathscr{R}_{0}(g) : \ldots : \mathscr{R}_{N}(g) \big ] \in \mathbb{C}{\rm{P}}^{N} \cong \mathbb{P}(V(\mu(L))).$$
If we identify $X_{P}$ with its image under the projective embedding
$\iota \colon X_{P} \hookrightarrow \mathbb{P}(V(\mu(L))) = {\text{Proj}}\big (H^{0}(X_{P},L^{-1})^{\ast} \big),$
we have a principal $\mathbb{C}^{\times}$-bundle $\mathbb{C}^{\times}\hookrightarrow Y_{{\text{reg}}} \to X_{P}$. The associated action of $\mathbb{C}^{\times}$ is naturally defined by $$\label{actionreglocus}
Y_{{\text{reg}}} \times \mathbb{C}^{\times} \to Y_{{\text{reg}}}, \ \ {\text{such that }} \ \ (gv_{\mu(L)}^{+},w) \mapsto wgv_{\mu(L)}^{+},$$ it is straightforward to see that the the associated (left) action of $G^{\mathbb{C}}$ on $Y_{{\text{reg}}}$ commutes with the action above. Moreover, by considering ${\rm{Tot}}(L^{\times})$ and $Y_{{\text{reg}}}$ as $\mathbb{C}^{\times}$-spaces, it follows that the map \[explicityRemmert\] is equivariant, i.e. $\mathscr{R}(u\cdot w) = \mathscr{R}(u)\cdot w$, $\forall u \in {\rm{Tot}}(L^{\times})$, and $\forall w \in \mathbb{C}^{\times}$.
We also observe that $\Gamma \subset \mathbb{C}^{\times}$ (\[remebergamma\]) can be represented as a subgroup of diagonal matrices of ${\rm{GL}}(V(\mu(L)))$, i.e. we have the identification $$\label{lineargamma}
\Gamma \cong \Bigg \{ \lambda^{n} \cdot {\rm{Id}}_{V(\mu(L))} \in {\rm{GL}}(V(\mu(L))) \ \ \Bigg | \ \ n \in \mathbb{Z} \Bigg\}.$$ Notice that, from the characterization above, the action of $\Gamma$ on $Y_{{\text{reg}}}$ coincides with the action of $\Gamma$ as subgroup of $\mathbb{C}^{\times}$ via \[actionreglocus\], so we shall not distinguish these two actions. The key point of the characterization \[lineargamma\] is that it allows to define an action of $\Gamma$ on the whole vector space $V(\mu(L))$.
Let $O \in {\rm{GL}}(n,\mathbb{C})$ be a linear operator acting on $\mathbb{C}^{n}$ with all eigenvalues satisfying $|\lambda_{i}| < 1$. Denote by $\langle O \rangle$ the cyclic group generated by $O$. The quotient $$\Big (\mathbb{C}^{n} \backslash \{0\} \Big)/ \langle O \rangle,$$ is called [*linear Hopf manifold*]{}. If $O$ is diagonalizable, then $(\mathbb{C}^{n} \backslash \{0\}) / \langle O \rangle$ is called a [*diagonal Hopf manifold*]{}.
From the definition above we can define by means of the linear realization \[lineargamma\] of $\Gamma$ the diagonal Hopf manifold $${\rm{H}}_{\Gamma} = \Big ( V(\mu(L)) \backslash \{0\} \Big )/ \Gamma.$$ Now we can prove the following result.
\[Techlemma\] Let $L \in {\text{Pic}}(X_{P})$ be a negative line bundle and $M = {\rm{Tot}}(L^{\times})/\Gamma$, defined as in \[remebergamma\]. Then, we have an injective holomorphic immersion $$\mathscr{R}^{(\Gamma)} \colon M \to {\rm{H}}_{\Gamma} = \Big ( V(\mu(L)) \backslash \{0\} \Big )/ \Gamma,$$ such that $\mathscr{R}^{(\Gamma)}(M) = Y_{\text{reg}}/\Gamma$, where $Y_{\text{reg}} = G^{\mathbb{C}} \cdot v_{\mu(L)}^{+}$.
Consider the biholomorphic map $\mathscr{R} \colon {\rm{Tot}}(L^{\times}) \to Y_{{\text{reg}}}$, defined in \[explicityRemmert\]. Since
$\mathscr{R}(u\cdot w) = \mathscr{R}(u)\cdot w$, $\forall u \in {\rm{Tot}}(L^{\times})$, and $\forall w \in \mathbb{C}^{\times}$,
we have that $\mathscr{R}$ induces a map $\mathscr{R}^{(\Gamma)} \colon M \to Y_{\text{reg}}/\Gamma \subset {\rm{H}}_{\Gamma}$. Now, consider the diagram
[[Tot]{}]{}(L\^) & & Y\_\
M & & Y\_/
It is straightforward to see that $\mathscr{R}^{(\Gamma)} \circ \wp_{1} = \wp_{2} \circ \mathscr{R}$, since $\mathscr{R}$ is a biholomorphism and $\wp_{2}$ is a local biholomorphism (covering map), we have that $\mathscr{R}^{(\Gamma)}$ is a holomorphic immersion. Also, we have that $\mathscr{R}^{(\Gamma)}(M) = Y_{\text{reg}}/\Gamma$ from the fact that $\mathscr{R}$ is bijective and $\wp_{2}$ is surjective (onto). In order to see that $\mathscr{R}^{(\Gamma)}$ is injective, observe that
$\mathscr{R}^{(\Gamma)}(\wp_{1}(u_{1})) = \mathscr{R}^{(\Gamma)}(\wp_{1}(u_{2})) \Longleftrightarrow \wp_{2}(\mathscr{R}(u_{1})) = \wp_{2}(\mathscr{R}(u_{2})).$
The equality on the right side above is equivalent to $\mathscr{R}(u_{1}) = \mathscr{R}(u_{2})\cdot w$, for some $w \in \Gamma$. Thus, we obtain from the equivariance of $\mathscr{R}$ that
$\mathscr{R}(u_{1}) = \mathscr{R}(u_{2}) \cdot w = \mathscr{R}(u_{2} \cdot w) \Longleftrightarrow u_{1} = u_{2} \cdot w \Longleftrightarrow \wp_{1}(u_{1}) = \wp_{1}(u_{2})$,
so we have the desired result.
Now, from the previous ideas we have the following theorem.
\[Theo4\] Let $(M,g,J)$ be a compact l.c.K. manifold that admits an effective and transitive smooth (left) action of a compact connected Lie group $K$, which preserves the metric $g$ and the complex structure $J$. Suppose also that $K_{{\text{ss}}}$ is simply connected and has a unique simple component. Then, there exists a character $\chi \in {\text{Hom}}(T^{\mathbb{C}},\mathbb{C}^{\times})$, for some maximal torus $T^{\mathbb{C}}\subset K_{{\text{ss}}}^{\mathbb{C}}$, and a holomorphic embedding $$\mathscr{R}^{(\Gamma)} \colon M \hookrightarrow {\rm{H}}_{\Gamma} = \Big ( V(\chi) \backslash \{0\} \Big)/ \Gamma,$$ such that $V(\chi)$ is an irreducible $\mathfrak{k}_{{\text{ss}}}^{\mathbb{C}}$-module with highest weight vector $v_{\chi}^{+} \in V(\chi)$, and $\Gamma \subset {\rm{GL}}(V(\chi))$ is an infinite cyclic subgroup generated by $\lambda \cdot{\rm{Id}}_{V(\chi)}$, where $\lambda \in \mathbb{C}^{\times}$, with $|\lambda| < 1$. Moreover, we have that $$\mathscr{R}^{(\Gamma)}(M) = Y_{\text{reg}}/\Gamma,$$ such that $Y_{\text{reg}} = K_{{\text{ss}}}^{\mathbb{C}} \cdot v_{\chi}^{+}$, where $$\label{algebraiccovering}
Y = {\text{Spec}}\Big ( \bigoplus_{\ell = 0}^{+\infty}V(\chi^{\ell})^{\ast}\Big).$$ Therefore, every compact homogeneous l.c.K. manifold is biholomorphic to some quotient space of the regular locus of the complex analytification of some HV-variety by a discrete group.
The proof follows from the following facts:
1. From Theorem \[Theo1\] it follows that there exists a parabolic Lie subgroup $P \subset K_{{\text{ss}}}^{\mathbb{C}}$, $T^{\mathbb{C}}\subset P$, and a negative line bundle $L \in {\text{Pic}}(K_{{\text{ss}}}^{\mathbb{C}}/P)$ such that
$M = {\rm{Tot}}(L^{\times})/\Gamma, \ \ {\text{where}} \ \ \Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \},$
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$. Moreover, $L \to K_{{\text{ss}}}^{\mathbb{C}}/P$ is determined by $\chi^{(L)} \in {\text{Hom}}(T^{\mathbb{C}},\mathbb{C}^{\times})$, such that $(d\chi^{(L)})_{e} = \mu(L)$ is a weight of $\mathfrak{k}_{ss}^{\mathbb{C}}$.
2. By taking $\chi = \chi^{(L)}$, and denoting $V(\chi) = V(\mu(L))$, from Lemma \[Techlemma\] we have a holomorphic injective immersion
$\mathscr{R}^{(\Gamma)} \colon M \hookrightarrow {\rm{H}}_{\Gamma} = \Big (V(\chi) \backslash \{0\} \Big )/ \Gamma$, such that $\mathscr{R}^{(\Gamma)}(M) = Y_{\text{reg}}/\Gamma$.
Since $M$ is compact and ${\rm{H}}_{\Gamma}$ is Hausdorff, it follows that $\mathscr{R}^{(\Gamma)}$ is proper, so it defines a holomorphic embedding. Notice that the algebraic description \[algebraiccovering\] follows from the fact that $(d\chi^{(L)})_{e} = \mu(L)$, see Equation \[speccone\]. For the last statement of theorem, i.e. when $K_{\text{ss}}$ has more than one simple component, the result follows from the items (1)-(2) above and the comments at the end of Subsection \[compacthomogeneoussetting\].
Notice that, under the hypotheses of Theorem \[Theo4\], if we suppose $\dim_{\mathbb{C}}(V(\chi)) = N+1$, by fixing a $K_{\text{ss}}$-invariant inner product $\langle \cdot,\cdot \rangle$ on $V(\chi)$, and taking an orthonormal basis $v_{0} = v_{\chi}^{+},v_{1},\ldots,v_{N}$ for $V(\chi)$, from Remark \[explicityRemmert\] we have $\mathscr{R} \colon {\rm{Tot}}(L^{\times}) \to V(\chi)$, such that
$\mathscr{R}([g,w]) = \big (\mathscr{R}_{0}\big (g\zeta^{(L)}(w) \big),\ldots, \mathscr{R}_{N}\big (g\zeta^{(L)}(w) \big)\big) \in \mathbb{C}^{N+1} \cong V(\chi)$
for every $[g,w] \in {\rm{Tot}}(L^{\times})$. From this, if we denote by $[v]_{\Gamma} \in {\rm{H}}_{\Gamma}$, the class of $v \in V(\chi)$ in the quotient space ${\rm{H}}_{\Gamma} = \big (V(\chi) \backslash \{0\} \big )/ \Gamma$, we obtain an explicit description for the embedding $\mathscr{R}^{(\Gamma)} \colon M \hookrightarrow {\rm{H}}_{\Gamma}$ given by
$\mathscr{R}^{(\Gamma)}\big (\wp_{1}([g,w]) \big ) = \Big [\mathscr{R}_{0}\big (g\zeta^{(L)}(w) \big),\ldots, \mathscr{R}_{N}\big (g\zeta^{(L)}(w) \big) \Big]_{\Gamma} \in \Big (\mathbb{C}^{N+1} \backslash \{0\} \Big )/ \Gamma$
for every $[g,w] \in {\rm{Tot}}(L^{\times})$, here we consider the projection $\wp_{1} \colon {\rm{Tot}}(L^{\times}) \to M$. Therefore, once we know concretely the component functions
$\mathscr{R}_{0},\mathscr{R}_{1},\ldots,\mathscr{R}_{N} \colon K_{\text{ss}}^{\mathbb{C}} \to \mathbb{C}$,
we can describe explicitly the embedding provided by Theorem \[Theo4\].
Examples of embedding
---------------------
This subsection is devoted to describe in details some interesting examples which illustrate the result provided in Theorem \[Theo4\], as well as discuss its relation with other interesting topics (e.g. ALE spaces, exotic spheres, conifold deformation, etc.). As we have seen, likewise complex flag manifolds can be embedded into a projective space by using elements of representation theory, we also have a Kodaira-type embedding for compact homogeneous l.c.K. manifolds which incorporates elements of representation theory. Therefore, the purpose of the examples in the next subsections is to investigate how some aspects of algebraic geometry which arise from the study of flag varieties can be used to understand homogeneous l.c.K. geometry.
### Plücker relations and HV-varieties
As in Example \[Grassmanianexample\], consider $G^{\mathbb{C}} = {\rm{SL}}(n+1,\mathbb{C})$, and fix the Cartan subalgebra $\mathfrak{h} \subset \mathfrak{sl}(n+1,\mathbb{C})$ given by diagonal matrices whose the trace is equal to zero. Associated to the pair $(\mathfrak{sl}(n+1,\mathbb{C}),\mathfrak{h} )$ we have a simple root system $\Sigma \subset \mathfrak{h}$ defined by $$\Sigma = \Big \{ \alpha_{l} = \epsilon_{l} - \epsilon_{l+1} \ \Big | \ l = 1, \ldots,n\Big\},$$ here we consider the linear functionals $\epsilon_{l} \colon {\text{diag}}\{a_{1},\ldots,a_{n+1} \} \mapsto a_{l}$, $ \forall l = 1, \ldots,n+1$. If we take $\Theta = \Sigma \backslash \{\alpha_{k}\}$ and $P = P_{\omega_{\alpha_{k}}}$, we obtain the complex Grassmannian manifold
${\rm{Gr}}(k,\mathbb{C}^{n+1}) = {\rm{SL}}(n+1,\mathbb{C})/P_{\omega_{\alpha_{k}}}.$
Since in this case we have
${\text{Pic}}({\rm{Gr}}(k,\mathbb{C}^{n+1})) = \mathbb{Z}c_{1}(\mathscr{O}_{\alpha_{k}}(1)),$
when one considers the projective embedding induced from the very ample line bundle $\mathscr{O}_{\alpha_{k}}(1) \to {\rm{Gr}}(k,\mathbb{C}^{n+1})$, we have
${\rm{Gr}}(k,\mathbb{C}^{n+1}) \hookrightarrow \mathbb{P}(\bigwedge^{k}(\mathbb{C}^{n+1})),$
it is worth recalling that
$H^{0}\big ({\rm{Gr}}(k,\mathbb{C}^{n+1}),\mathscr{O}_{\alpha_{k}}(1) \big )^{\ast} = \bigwedge^{k}(\mathbb{C}^{n+1}), \ \ \ {\text{and}} \ \ \ v_{\omega_{\alpha_{k}}}^{+} = e_{1} \wedge \ldots \wedge e_{k}$.
Therefore, if we fix in $\bigwedge^{k}(\mathbb{C}^{n+1})$ the canonical basis $$e_{j_{1}} \wedge \ldots \wedge e_{j_{k}}, \ \ 1 \leq j_{1} < j_{2} < \cdots < j_{k} \leq n+1,$$ and consider the inner product on $\bigwedge^{k}(\mathbb{C}^{n+1})$ defined by
$\langle u_{1} \wedge \ldots \wedge u_{k}, v_{1} \wedge \ldots \wedge v_{k} \rangle = \det \begin{pmatrix}
\langle u_{1},v_{1} \rangle & \cdots &\langle u_{1},v_{k} \rangle \\
\vdots & \ddots & \vdots \\
\langle u_{k},v_{1} \rangle & \cdots & \langle u_{1},v_{1} \rangle
\end{pmatrix},$
such that $\langle \cdot , \cdot \rangle$ is the canonical inner product on $\mathbb{C}^{n+1}$, we obtain a (dual) basis for $(\bigwedge^{k}(\mathbb{C}^{n+1}))^{\ast}$ defined by
$l_{j_{1}\cdots j_{k}} \colon u_{1} \wedge \ldots \wedge u_{k} \mapsto \langle u_{1} \wedge \ldots \wedge u_{k},e_{j_{1}} \wedge \ldots \wedge e_{j_{k}} \rangle$, $1 \leq j_{1} < j_{2} < \cdots < j_{k} \leq n+1$.
From this, by proceeding as in Remark \[matrixcoefitients\], we obtain polynomial functions $\mathscr{R}_{j_{1}\cdots j_{k}} \colon {\rm{SL}}(n+1,\mathbb{C}) \to \mathbb{C}$, $1 \leq j_{1} < j_{2} < \cdots < j_{k} \leq n+1$, such that $$\mathscr{R}_{j_{1}\cdots j_{k}}(g) = l_{j_{1}\cdots j_{k}}(gv_{\omega_{\alpha_{k}}}^{+}) = \det \begin{pmatrix}
g_{j_{1}1} & \cdots & g_{j_{k}1} \\
\vdots & \ddots & \vdots \\
g_{j_{1}k} & \cdots & g_{j_{k}k}
\end{pmatrix}, \ \ \ \forall g = (g_{ij}) \in {\rm{SL}}(n+1,\mathbb{C}).$$ For the sake of simplicity, let us denote $J = \{j_{1} < j_{2} < \cdots < j_{k}\}$ and $\mathscr{R}_{J} \in \mathbb{C}[{\rm{SL}}(n+1,\mathbb{C})]$. Also, we denote $J_{0} = \{1 < 2 < \cdots < k\}$ and consider the partial order $$J' \geq J \Longleftrightarrow j'_{s} \geq j_{s}, \ \ \forall s = 1,\ldots,k,$$ such that $J = \{j_{1} < \cdots < j_{k}\}$ and $J' = \{j'_{1} < \cdots < j'_{k}\}$, notice that $J \geq J_{0}$, $\forall J= \{j_{1} < \cdots < j_{k}\}$. Further, let us denote $$N := \# \Big \{ J \ \ \Big | \ \ J = \{j_{1} < \cdots < j_{k}\}, 1 \leq j_{1} < \cdots < j_{k} \leq n+1 \Big \} = \binom{n+1}{k},$$ notice that $\dim_{\mathbb{C}}(V(\omega_{\alpha_{k}})) = \dim_{\mathbb{C}}(\bigwedge^{k}(\mathbb{C}^{n+1})) = N$.
From the data above, we can describe explicitly the holomorphic map $\mathscr{R} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times}) \to \bigwedge^{k}(\mathbb{C}^{n+1})$ (see Remark \[matrixcoefitients\]) by $$\label{explicitRemmert}
\mathscr{R}\big ([g,w]\big) = \sum_{J}\mathscr{R}_{J}(g \zeta^{\omega_{\alpha_{k}}}(w))e_{j_{1}} \wedge \ldots \wedge e_{j_{k}} = w\Bigg (\sum_{J} \det \begin{pmatrix}
g_{j_{1}1} & \cdots & g_{j_{k}1} \\
\vdots & \ddots & \vdots \\
g_{j_{1}k} & \cdots & g_{j_{k}k}
\end{pmatrix}e_{j_{1}} \wedge \ldots \wedge e_{j_{k}} \Bigg )$$ for every $[g,w] \in {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times})$, where $\zeta^{\omega_{\alpha_{k}}} \colon \mathbb{C}^{\times} \to T^{\mathbb{C}} \subset {\rm{SL}}(n+1,\mathbb{C})$ satisfies $\chi_{\omega_{\alpha_{k}}}(\zeta^{\omega_{\alpha_{k}}}(w)) = w$, $\forall w \in \mathbb{C}^{\times}$. Also, as we have seen, the image of the map above is the regular locus of the HV-veriety
$Y = \overline{{\rm{SL}}(n+1,\mathbb{C}) \cdot v_{\omega_{\alpha_{k}}}^{+}} \subset \bigwedge^{k}(\mathbb{C}^{n+1})$,
i.e. $Y_{{\text{reg}}} = {\rm{SL}}(n+1,\mathbb{C}) \cdot v_{\omega_{\alpha_{k}}}^{+}$. Moreover, since we have $Y$ as the affine cone over the image of ${\rm{Gr}}(k,\mathbb{C}^{n+1})$ under the projective embedding defined by $\mathscr{O}_{\alpha_{k}}(1)$, the affine variety $Y$ is given by the zero locus of the same set of polynomials which defines ${\rm{Gr}}(k,\mathbb{C}^{n+1})$ inside of $\mathbb{P}(\bigwedge^{k}(\mathbb{C}^{n+1}))$. Actually, the projective embedding of ${\rm{Gr}}(k,\mathbb{C}^{n+1})$ can be explicitly described by $$\label{projgrassmannian}
gP_{\omega_{\alpha_{k}}} \in {\rm{Gr}}(k,\mathbb{C}^{n+1}) \mapsto \big [\mathscr{R}_{J_{0}}(g): \ldots : \mathscr{R}_{J_{N-1}}(g)\big] \in \mathbb{C}{\rm{P}}^{N-1} \cong \mathbb{P}(\textstyle{\bigwedge^{k}(\mathbb{C}^{n+1}))},$$ here we consider $J_{0} \leq J_{1} \leq \cdots \leq J_{N}$.
It is worth pointing out that, under the hypotheses above, we can consider coordinates $Z_{j_{1},\cdots,j_{k}}$, $1 \leq j_{1} < \cdots < j_{k} \leq n+1$, in $V(\omega_{\alpha_{k}}) = \bigwedge^{k}(\mathbb{C}^{n+1})$. Moreover, for an arbitrary subset of distinct indexes $\{j_{1},\ldots,j_{k}\} \subset \{1,\ldots,n+1\}$, we set
$Z_{j_{1},\cdots,j_{k}} = (-1)^{|\sigma|} Z_{j_{\sigma(1)},\cdots,j_{\sigma(k)}}$, with $\sigma \in S_{k}$, such that $j_{\sigma(1)} < \cdots < j_{\sigma(k)}$,
and for an arbitrary subset $\{j_{1},\ldots,j_{k}\} \subset \{1,\ldots,n+1\}$, we set $Z_{j_{1},\cdots,j_{k}} = 0$, if $j_{1},\ldots,j_{k}$ are not distinct.
Under the previous hypotheses and considering the remark above, we have the following well-known result, see for instance [@MONOMIAL].
\[Pluckerelation\] The image of the complex Garssmannian ${\rm{Gr}}(k,\mathbb{C}^{n+1})$ under the projective embedding ${\rm{Gr}}(k,\mathbb{C}^{n+1}) \hookrightarrow \mathbb{C}{\rm{P}}^{N-1}$ defined in \[projgrassmannian\] consists the zeros in $\mathbb{C}{\rm{P}}^{N-1}$ of the following quadratic polynomials $$P_{a_{1},\cdots,a_{k-1};b_{1},\cdots,b_{k+1}} = \sum_{\ell = 1}^{k+1}(-1)^{\ell}Z_{a_{1},\dots,a_{k-1},a_{\ell}}Z_{b_{1},\ldots,b_{\ell},\ldots,{b}_{k+1}},$$ such that $1 \leq a_{1} < \cdots < a_{k-1} \leq n+1$, $1 \leq b_{1} < \cdots < b_{k+1} \leq n+1$.
In the setting of Theorem \[Pluckerelation\] the relation obtained by the equation $P_{a_{1},\cdots,a_{k-1};b_{1},\cdots,b_{k+1}} = 0$, for some $1 \leq a_{1} < \cdots < a_{k-1} \leq n+1$, $1 \leq b_{1} < \cdots < b_{k+1} \leq n+1$, is usually called a [*Plücker relation*]{}.
From Theorem \[Pluckerelation\], if we consider the subset $T \subset \mathbb{C} \big [Z_{J_{0}},\ldots,Z_{J_{N-1}}]$, such that $$T = \Big \{ P_{a_{1},\cdots,a_{k-1};b_{1},\cdots,b_{k+1}} \ \ \Big | \ \ 1 \leq a_{1} < \cdots < a_{k-1} \leq n+1, 1 \leq b_{1} < \cdots < b_{k+1} \leq n+1\ \ \Big\},$$ so we have $$Y = \overline{{\rm{SL}}(n+1,\mathbb{C}) \cdot v_{\omega_{\alpha_{k}}}^{+}} = \mathcal{Z}(T) \subset \textstyle{\bigwedge^{k}(\mathbb{C}^{n+1})}.$$ Therefore, if one considers the manifold
$M = {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, from Lemma \[Techlemma\] have a holomorphic immersion
$\mathscr{R}^{(\Gamma)} \colon M \to {\rm{H}}_{\Gamma} = \Big (\bigwedge^{k}(\mathbb{C}^{n+1}) \backslash \{0\} \Big )/ \Gamma$,
such that $\mathscr{R}^{(\Gamma)}(M) = \mathcal{Z}(T)_{\text{reg}}/\Gamma$. Moreover, from the description \[explicitRemmert\], it follows that $$\mathscr{R}^{(\Gamma)}\big (\wp_{1}([g,w]) \big ) = \Big [\mathscr{R}_{J_{0}}\big (g\zeta^{\omega_{\alpha_{k}}}(w) \big),\ldots, \mathscr{R}_{J_{N-1}}\big (g\zeta^{\omega_{\alpha_{k}}}(w) \big) \Big]_{\Gamma},$$ $\forall [g,w] \in {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times})$, here we consider the projection $\wp_{1} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times}) \to {\rm{Tot}}(\mathscr{O}_{\alpha_{k}}(-1)^{\times})/\Gamma$.
In order to make more clear the ideas above, let us provide some concrete computations on the classical low dimensional example provided by the Klein quadric ${\rm{Gr}}(2,\mathbb{C}^{4}) = {\rm{SL}}(4,\mathbb{C})/P_{\omega_{\alpha_{2}}}.$ In this particular case, from the very ample line bundle $\mathscr{O}_{\alpha_{2}}(1) \to {\rm{Gr}}(2,\mathbb{C}^{4})$, we have the Plücker embedding given by
${\rm{Gr}}(2,\mathbb{C}^{4}) \hookrightarrow \mathbb{P}(\bigwedge^{2}(\mathbb{C}^{4})) \cong \mathbb{C}{\rm{P}}^{5}$.
From the previous ideas, we have the holomorphic map $\mathscr{R} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{2}}(-1)^{\times}) \to \bigwedge^{2}(\mathbb{C}^{4})$ explicitly described by $$\mathscr{R}(\big[g,w\big]) = w\Bigg ( \sum_{1 \leq i_{1}<i_{2}\leq 4}\det \begin{pmatrix}
g_{i_{1}1} & g_{i_{1}2} \\
g_{i_{2}1} & g_{i_{2}2}
\end{pmatrix}e_{i_{1}} \wedge e_{i_{2}} \Bigg ).$$ From this, by fixing coordinates $Z_{i_{1}i_{2}}$, $1 \leq i_{1}<i_{2}\leq 4$, on $\bigwedge^{2}(\mathbb{C}^{4}) \cong \mathbb{C}^{6}$, the closure of the image of the map $\mathscr{R} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{2}}(-1)^{\times}) \to \bigwedge^{2}(\mathbb{C}^{4})$ is the HV-variety $Y \subset \bigwedge^{2}(\mathbb{C}^{4})$ defined by $$Y = \overline{{\rm{SL}}(4,\mathbb{C}) \cdot v_{\omega_{\alpha_{2}}}^{+}} = \Big \{ Z_{12}Z_{34} - Z_{13}Z_{24} + Z_{14}Z_{23} = 0\Big \}.$$ Hence, since ${\rm{Tot}}(\mathscr{O}_{\alpha_{2}}(-1)^{\times}) \cong \mathscr{V}_{2}(\mathbb{R}^{6}) \times \mathbb{R}$, we obtain from Theorem \[Theo4\] a Kodaira-type embedding for the Hermitian-Einstein-Weyl manifold $M = \mathscr{V}_{2}(\mathbb{R}^{6}) \times S^{1} \hookrightarrow {\rm{H}}_{\Gamma} = \Big (\bigwedge^{2}(\mathbb{C}^{4}) \backslash \{0\} \Big )/ \mathbb{Z}$ explicitly described by $$\mathscr{R}^{(\Gamma)}\big (\wp_{1}([g,w]) \big ) = \Bigg [\det \begin{pmatrix}
g_{11} & g_{12} \\
g_{21} & g_{22}
\end{pmatrix}w, \ldots, \det \begin{pmatrix}
g_{31} & g_{32} \\
g_{41} & g_{42}
\end{pmatrix}w, \Bigg ]_{\mathbb{Z}},$$ $\forall [g,w] \in {\rm{Tot}}(\mathscr{O}_{\alpha_{2}}(-1)^{\times})$. Moreover, the image of the holomorphic map above is given by $$\mathscr{R}^{(\Gamma)}(\mathscr{V}_{2}(\mathbb{R}^{6}) \times S^{1}) = \mathcal{Z}(P)_{\text{reg}}/\mathbb{Z},$$ where $\mathcal{Z}(P) =\{ P = 0\} \subset \mathbb{C}^{6}$, such that $P = Z_{12}Z_{34} - Z_{13}Z_{24} + Z_{14}Z_{23} \in \mathbb{C}[\mathbb{C}^{6}]$. Thus, we obtain a concrete description for the embedding of $\mathscr{V}_{2}(\mathbb{R}^{6}) \times S^{1}$ into the linear Hopf manifold ${\rm{H}}_{\Gamma} = \Big (\bigwedge^{2}(\mathbb{C}^{4}) \backslash \{0\} \Big )/ \mathbb{Z}$.
### Cones over projective complex hyperquadrics
Let $V = \mathbb{K}^{N}$ be a $\mathbb{K}$-vector space ($\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$). By fixing the canonical symmetric bilinear form $q$ on $V$, such that $q(x,y) = \sum_{k = 1}^{N}x_{k}y_{k}$, $\forall x,y \in V$, we have the associated [*Clifford algebra*]{}[^18] $${\rm{Cl}}(V,q) = \mathcal{T}(V)/\mathcal{I}_{q}(V),$$ where $ \mathcal{T}(V)$ denotes the tensor algebra of $V$ and $\mathcal{I}_{q}(V) \subset {\rm{T}}(V)$ is the two-sided ideal generated by elements of the form $$u\otimes v + v \otimes u + 2q(u,v) \cdot 1, \ \ \forall u,v \in V.$$ The unital associative algebra ${\rm{Cl}}(V,q) $ admits a $\mathbb{Z}_{2}$-grading induced by the the extension (via universal property) of the morphism[^19] of quadratic vector spaces $\Xi \colon (V,q) \to (V,q)$, $\Xi(v) = -v$, $\forall v \in V$, such that
${\rm{Cl}}(V,q) = {\rm{Cl}}^{\overline{0}}(V,q)\oplus {\rm{Cl}}^{\overline{1}}(V,q),$
where ${\rm{Cl}}^{\overline{0}}(V,q) = \{x \in {\rm{Cl}}(V,q) \ | \ \Xi(x) = x\}$ and ${\rm{Cl}}^{\overline{1}}(V,q) = \{x \in {\rm{Cl}}(V,q) \ | \ \Xi(x) = -x\}$. Now, we consider the [*Clifford group*]{} $\Gamma(V)$, which is a subgroup of the group of invertible elements ${\rm{Cl}}(V,q)^{\times}$ given by $$\Gamma(V) = \Big \{ x \in {\rm{Cl}}(V,q)^{\times} \ \ \Big | \ \ \Xi(x)vx^{-1} \in V, \forall v \in V\Big \}.$$ The group above defines a representation ${\rm{A}} \colon \Gamma(V) \to {\rm{GL}}(V)$, such that ${\rm{A}}_{x}(v) = \Xi(x)vx^{-1}$, $\forall (x,v) \in \Gamma(V) \times V$. Let us consider the canonical basis $e_{1},\ldots,e_{N}$ for $V$ to identify ${\rm{GL}}(V) \cong {\rm{GL}}(N,\mathbb{K})$.
It is straightforward to see that $\ker(A) = \mathbb{C}^{\times}$ and
$q\big ({\rm{A}}_{x}(u),{\rm{A}}_{x}(v) \big ) = q \big (u,v \big ),$
so we obtain the exact sequence
1 & \^ & (V) & [[O]{}]{}(N,) & 1.
Also, for every $x \in V$, such that $q(x,x) \neq 0$ (non-isotropic), we have $${\rm{A}}_{x}(v) = v - \frac{q(x,v)}{q(x,x)}x = {\rm{R}}_{x}(v),$$ i.e. ${\rm{A}}_{x} \in {\rm{O}}(N,\mathbb{K})$ is the reflection ${\rm{R}}_{x} \colon V \to V$ induced by $x \in V$. Therefore, from Cartan-Dieudonné theorem [@Meinrenken], it follows that every $x \in \Gamma(V)$ can be written as product of non-isotropic elements of $V$, i.e.
$x = x_{1}\cdots x_{k}$,
with $x_{1}, \ldots,x_{k} \in V$ satisfying $q(x_{j},x_{j}) \neq 0$, $\forall j = 1,\ldots,k$, such that $k \leq \dim_{\mathbb{K}}(V)$. From this, we can define a norm homomorphism $\mathscr{N} \colon \Gamma(V) \to \mathbb{K}^{\times}$, by setting $\mathscr{N}(v) = q(v,v)$, for every non-isotropic $v \in V$. By considering the [*special Clifford group*]{} $$S\Gamma(V) = \Gamma(V) \cap {\rm{Cl}}^{\overline{0}}(V,q),$$ we obtain the [*spin group*]{} as being $${\rm{Spin}}(V) := S\Gamma(V) \cap \ker(\mathscr{N}).$$ Observe that, since the normalization $\mathscr{N}(x) = 1$ specifies $x \in \Gamma(V)$ up to sign, one can write the exact sequence
1 & \_[2]{} & [[Spin]{}]{}(V) & [[SO]{}]{}(N,) & 1,
which gives ${\rm{Spin}}(V)$ as a two-sheeted covering of ${\rm{SO}}(N,\mathbb{K})$. The restriction map ${\rm{A}} \colon {\rm{Spin}}(V) \to {\rm{SO}}(N,\mathbb{K})$ allows to define an isomorphism
${\rm{A}}_{\ast} \colon \mathfrak{spin}(V) \to \bigwedge^{2} (V) \cong \mathfrak{so}(N,\mathbb{K}),$
notice that the isomorphism on the right side above is obtained by considering
$(x\wedge y) \colon v \mapsto q(x,v)y - q(y,v)x$,
$\forall x,y,v \in V$, for the inverse map see [@Meinrenken]. Further, notice that, in terms of basis, we have
${\rm{A}}_{\ast} \colon e_{i}\cdot e_{j} \mapsto 2(e_{i}\wedge e_{j})$,
$\forall 1 \leq i < j \leq n$. For the details omitted in the brief exposition provided above, see [@Meinrenken], [@Lawson] and [@Onishchik].
The ideas above show that, for the particular case $\mathbb{K} = \mathbb{C}$, we have ${\rm{Spin}}(\mathbb{C}^{N}) = {\rm{Spin}}_{N}(\mathbb{C})$, so we can consider the elements of Lie theory associated to the classical (matrix) Lie algebra $$\mathfrak{so}(N,\mathbb{C}) = \Big \{ X \in \mathfrak{sl}(N,\mathbb{C}) \ \ \Big | \ \ X + X^{T} = 0\Big \},$$ to describe the embedding of flag manifolds associated to the simple Lie group ${\rm{Spin}}_{N}(\mathbb{C})$ via representation theory. Hence, let us introduce some basic Lie theoretical data associated to $\mathfrak{so}(N,\mathbb{C})$.
At first, we observe that we have to consider two cases, i.e. we have the following description, see for instance [@Brocker], [@Sepanski]:
1. For $N = 2n$, such that $n > 1$, we consider the Cartan subalgebra $\mathfrak{h}_{2n} \subset \mathfrak{so}(2n,\mathbb{C})$ defined by
$\mathfrak{h}_{2n} = \Bigg \{ {\text{diag}} \Bigg (\begin{pmatrix}
\ \ 0 & z_{1} \\
-z_{1} & 0
\end{pmatrix}, \ldots, \begin{pmatrix}
\ \ 0 & z_{n} \\
-z_{n} & 0
\end{pmatrix} \Bigg) \ \ \Bigg | \ \ \ z_{i} \in \mathbb{C} \Bigg \};$
2. For $N = 2n + 1$, such that $n>1$, we consider the Cartan subalgebra $\mathfrak{h}_{2n+1} \subset \mathfrak{so}(2n+1,\mathbb{C})$ defined by
$\mathfrak{h}_{2n+1} = \Bigg \{ {\text{diag}} \Bigg (\begin{pmatrix}
\ \ 0 & z_{1} \\
-z_{1} & 0
\end{pmatrix}, \ldots, \begin{pmatrix}
\ \ 0 & z_{n} \\
-z_{n} & 0
\end{pmatrix}, 0\Bigg) \ \ \Bigg | \ \ \ z_{i} \in \mathbb{C} \Bigg \}.$
In the first case, by considering the linear functionals
$\epsilon_{l} \colon {\text{diag}} \Bigg (\begin{pmatrix}
\ \ 0 & z_{1} \\
-z_{1} & 0
\end{pmatrix}, \ldots, \begin{pmatrix}
\ \ 0 & z_{n} \\
-z_{n} & 0
\end{pmatrix} \Bigg) \mapsto -\sqrt{-1}z_{l}$, $ \forall l = 1, \ldots,n$,
we can describe the associated set of simple roots $\Sigma_{2n} \subset \mathfrak{h}_{2n}^{\ast}$ by
$\Sigma_{2n} = \Big \{ \alpha_{1} = \epsilon_{1} - \epsilon_{2}, \ldots, \alpha_{n-1} = \epsilon_{n-1} - \epsilon_{n}, \alpha_{n} = \epsilon_{n-1} + \epsilon_{n}\Big\}$.
Similarly, in the second case we can consider linear functionals
$\epsilon_{l} \colon {\text{diag}} \Bigg (\begin{pmatrix}
\ \ 0 & z_{1} \\
-z_{1} & 0
\end{pmatrix}, \ldots, \begin{pmatrix}
\ \ 0 & z_{n} \\
-z_{n} & 0
\end{pmatrix},0 \Bigg) \mapsto -\sqrt{-1}z_{l}$, $ \forall l = 1, \ldots,n$,
so we can describe the associated set of simple roots $\Sigma_{2n+1} \subset \mathfrak{h}_{2n+1}^{\ast}$ by
$\Sigma_{2n+1} = \Big \{ \alpha_{1} = \epsilon_{1} - \epsilon_{2}, \ldots, \alpha_{n-1} = \epsilon_{n-1} - \epsilon_{n}, \alpha_{n} = \epsilon_{n}\Big\}$.
\[lowdimspin\] It is worth mentioning that, for $N = 4$, we have that ${\rm{Spin}}_{4}(\mathbb{C}) \cong {\rm{SL}}(2,\mathbb{C}) \times {\rm{SL}}(2,\mathbb{C})$. In fact, in this particular case we have
$\Sigma_{4} = \Big \{ \alpha_{1} = \epsilon_{1} - \epsilon_{2}, \alpha_{2} = \epsilon_{1} + \epsilon_{2}\Big\}$,
and the isomorphism $\mathfrak{spin}_{4}(\mathbb{C}) \cong \mathfrak{sl}(2,\mathbb{C}) \times \mathfrak{sl}(2,\mathbb{C})$ is obtained from $$\label{isomatrixspin}
\displaystyle x_{\alpha_{i}} \longleftrightarrow \begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix}, \ \ h_{\alpha_{i}}^{\vee} = \frac{2}{\kappa(h_{\alpha_{i}},h_{\alpha_{i}})}h_{\alpha_{i}} \longleftrightarrow \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}, \ \ y_{\alpha_{i}} \longleftrightarrow \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix},$$ such that $i = 1,2$. From this, in what follows we shall consider $N = 2n$, such that $n >2$, and the case $N = 4$ will be considered separately.
Another low dimensional example which is worth observing is the case $N = 3$, i.e., the case ${\rm{Spin}}_{3}(\mathbb{C})$. Actually, in this case we have
$\Sigma_{3} = \Big \{ \alpha_{1} = \epsilon_{1} - \epsilon_{2}\Big\}$,
and by considering a correspondence similar to \[isomatrixspin\], we obtain $\mathfrak{spin}_{3}(\mathbb{C}) \cong \mathfrak{sl}(2,\mathbb{C})$, which implies that ${\rm{Spin}}_{3}(\mathbb{C}) \cong {\rm{SL}}(2,\mathbb{C})$. Also, by looking at the Dynkin diagram of $\mathfrak{spin}_{3}(\mathbb{C})$, it is straightforward to see that ${\rm{Spin}}_{6}(\mathbb{C}) \cong {\rm{SL}}(4,\mathbb{C})$.
From the comments and the basic data above, if we take $\Theta = \Sigma_{N} \backslash \{\alpha_{1}\}$, we obtain a maximal Lie parabolic subgroup $P_{\omega_{\alpha_{1}}} \subset {\rm{Spin}}_{N}(\mathbb{C})$, which defined the complex flag manifold $$X_{P_{\omega_{\alpha_{1}}}} = {\rm{Spin}}_{N}(\mathbb{C})/P_{\omega_{\alpha_{1}}} = {\rm{Spin}}(N)/{\rm{Spin}}(2)\times {\rm{Spin}}(N-2),$$ such that ${\rm{Spin}}(N) \subset {\rm{Spin}}_{N}(\mathbb{C})$ is a maximal compact subgroup (compact real form) associated to the exact sequence
1 & \_[2]{} & [[Spin]{}]{}(N) & [[SO]{}]{}(N,) & 1,
i.e. ${\rm{Spin}}(N)$ is a two-sheeted covering of ${\rm{SO}}(N,\mathbb{R})$. Hence, since $\mathbb{Z}_{2} = \{\pm 1\} \subset {\rm{Spin}}(2)\times {\rm{Spin}}(N-2)$ is normal in ${\rm{Spin}}(N)$, we also have the characterization
$X_{P_{\omega_{\alpha_{1}}}} = {\rm{SO}}(N,\mathbb{R})/{\rm{SO}}(2,\mathbb{R}) \times {\rm{SO}}(N-2,\mathbb{R}).$
Notice that for the particular case ${\rm{Spin}}_{6}(\mathbb{C}) \cong {\rm{SL}}(4,\mathbb{C})$ the construction above provides
$X_{P_{\omega_{\alpha_{1}}}} = {\rm{Spin}}(6)/{\rm{Spin}}(2)\times {\rm{Spin}}(4) = {\rm{SU}}(4)/{\rm{U}}(1)\times{\rm{SU}}(2)\times {\rm{SU}}(2),$
see Remark \[lowdimspin\], so we have that $X_{P_{\omega_{\alpha_{1}}}} = {\rm{Gr}}(2,\mathbb{C}^{4})$ (Klein quadric). Hence, since this example is covered by the content on Plücker embedding developed in the previous subsection, unless otherwise stated, in what follows for $N = 2n$ we shall suppose $n\neq3$.
It is worth pointing out that the compact simply connected Lie group ${\rm{Spin}}(N) = {\rm{Spin}}(\mathbb{R}^N)$ can be described by
${\rm{Spin}}(N) = \Big \{ x_{1} \cdots x_{k} \in S\Gamma(\mathbb{R}^{N}) \ \ \Big | \ \ x_{1}, \ldots,x_{k} \in S^{N-1}, \ \ 1 \leq k \leq N \Big\},$
it follows from the fact that, for the particular case of $\mathbb{K} = \mathbb{R}$ in the previous construction, the bilinear form $q$ coincides with the canonical inner product in $\mathbb{R}^{N}$.
As we have seen in the previous sections, the isomorphism classes of holomorphic line bundles over $X_{P_{\omega_{\alpha_{1}}}}$ are described by $${\text{Pic}}(X_{P_{\omega_{\alpha_{1}}}}) = \mathbb{Z}c_{1}\big (\mathscr{O}_{\alpha_{1}}(1)\big),$$ and relative to the positive line bundle $\mathscr{O}_{\alpha_{1}}(1) \to X_{P_{\omega_{\alpha_{1}}}}$, we have a projective embedding
$\iota \colon X_{P_{\omega_{\alpha_{1}}}} \hookrightarrow \mathbb{P}(V(\omega_{\alpha_{1}})) = {\text{Proj}}\big (H^{0}(X_{P_{\omega_{\alpha_{1}}}},\mathscr{O}_{\alpha_{1}}(1))^{\ast} \big).$
From the representation theory of classical Lie algebras, we have that $V(\omega_{\alpha_{1}}) = \mathbb{C}^{N}$, with highest weight vector $v_{\omega_{\alpha_{1}}}^{+} = e_{1} - \sqrt{-1}e_{2}$, e.g. [@Brocker]. Thus, the image of $X_{P_{\omega_{\alpha_{1}}}}$ under the projective embedding above is the complex Quadric $$\mathcal{Q}_{N-1} = \Big \{ [z_{1} : \ldots :z_{N}] \in \mathbb{C}{\rm{P}}^{N-1} \ \ \Big | \ \ z_{1}^{2} + \cdots + z_{N}^{2} = 0 \Big \}.$$ In fact, given $[z] \in {\rm{Spin}}_{N}(\mathbb{C}) \cdot [v_{\omega_{\alpha_{1}}}^{+}] \subset \mathbb{C}{\rm{P}}^{N-1}$, it follows that $z = {\rm{A}}_{x}(v_{\omega_{\alpha_{1}}}^{+})$, for some $x \in {\rm{Spin}}_{N}(\mathbb{C}) $, so we obtain
$q \big (z,z \big ) = q \big ({\rm{A}}_{x}(v_{\omega_{\alpha_{1}}}^{+}),{\rm{A}}_{x}(v_{\omega_{\alpha_{1}}}^{+}) \big) = q \big (v_{\omega_{\alpha_{1}}}^{+},v_{\omega_{\alpha_{1}}}^{+} \big)= 0$,
which implies that ${\rm{Spin}}_{N}(\mathbb{C}) \cdot [v_{\omega_{\alpha_{1}}}^{+}] \subset \mathcal{Q}_{N-1}$. On the other hand, if we denote $z \in \mathbb{C}^{N} \backslash \{0\}$ by $z = u + \sqrt{-1}v$, such that $u,v \in \mathbb{R}^{N}$, it is straightforward to see that $$[z] = [u+\sqrt{-1}v] \in \mathcal{Q}_{N-1} \Longleftrightarrow |u| = |v| \ \ {\text{and}} \ \ \langle u,v\rangle = 0,$$ here we consider the canonical inner product $\langle \cdot,\cdot \rangle$ on $\mathbb{R}^{N}$ and its induced norm $| \cdot |$. Thus, given $[z] = [u+\sqrt{-1}v] \in \mathcal{Q}_{N-1}$, we can suppose that $u,v \in S^{N-1}$ and obtain (by completion) an orthonormal basis $f_{1} = u, f_{2} = -v,f_{3},\ldots f_{N}$ for $\mathbb{R}^{N}$, so we get $A \in {\rm{SO}}(N,\mathbb{R})$ with these columns vectors, i.e. $A(e_{j}) = f_{j}$, for every $j = 1,\ldots, N$. Therefore, since $A = {\rm{A}}_{x}$, for some $x \in {\rm{Spin}}(N) \subset {\rm{Spin}}_{N}(\mathbb{C})$, by considering the extension $A \colon \mathbb{C}^{N} \to \mathbb{C}^{N}$, we have that
$A(v_{\omega_{\alpha_{1}}}^{+}) = A(e_{1}) - \sqrt{-1}A(e_{2}) = f_{1} - \sqrt{-1}f_{2} = u+\sqrt{-1}v$,
and we obtain $[z] = [ u+\sqrt{-1}v] = [{\rm{A}}_{x}(v_{\omega_{\alpha_{1}}}^{+})] \in {\rm{Spin}}_{N}(\mathbb{C}) \cdot [v_{\omega_{\alpha_{1}}}^{+}] \Longrightarrow \mathcal{Q}_{N-1} = {\rm{Spin}}_{N}(\mathbb{C}) \cdot [v_{\omega_{\alpha_{1}}}^{+}]$.
Now, consider the real Stiefel manifold [@Hatcher] $$\label{Stiefeldefinition}
\mathscr{V}_{2}(\mathbb{R}^{N}) = \Big \{ (u,v) \in \mathbb{R}^{N}\times \mathbb{R}^{N} \ \ \Big | \ \ u,v \in \mathbb{R}^{N} {\text{are orthonormal}}\Big\} \cong {\rm{SO}}(N,\mathbb{R})/{\rm{SO}}(N-2,\mathbb{R}),$$ notice that $\mathscr{V}_{2}(\mathbb{R}^{N}) \subset S^{N-1} \times S^{N-1}$. As we can see, the manifold $\mathscr{V}_{2}(\mathbb{R}^{N})$ defines a principal $S^{1}$-bundle $S^{1} \hookrightarrow \mathscr{V}_{2}(\mathbb{R}^{N}) \to \mathcal{Q}_{N}$ whose the projection map can be described by $$(u,v) \in \mathscr{V}_{2}(\mathbb{R}^{N}) \mapsto [u+\sqrt{-1}v] \in \mathcal{Q}_{N-1}.$$ On the other hand, we have the affine cone over $\mathcal{Q}_{N-1} \subset \mathbb{C}{\rm{P}}^{N-1}$ given by $$K\mathcal{Q}_{N-1} = \Big \{ (z_{1},\ldots, z_{N}) \in \mathbb{C}^{N} \ \ \Big | \ \ z_{1}^{2} + \cdots + z_{N}^{2} = 0 \Big \}.$$ Notice that, from \[explicityRemmert\], we obtain a description of ${\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-1)^{\times})$ as the regular locus of the analytification of the affine algebraic variety defined by $K\mathcal{Q}_{N-1} \subset \mathbb{C}^{N}$, i.e. we have a biholomorphism induced by the Cartan-Remmert reduction, namely,
$\mathscr{R} \colon {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-1)^{\times}) \to (K\mathcal{Q}_{N-1})_{{\text{reg}}}.$
Thus, if we take the homogeneous quadratic polynomial $P \colon \mathbb{C}^{N} \to \mathbb{C}$, such that
$P(z_{1},\ldots,z_{N}) = z_{1}^{2} + \cdots + z_{N}^{2},$
it follows that $K\mathcal{Q}_{N-1} = \mathcal{Z}(P) = \{P = 0\}$, so we have $$\label{linkdescription}
\mathscr{V}_{2}(\mathbb{R}^{N}) \cong \mathcal{Z}(P) \cap S^{2N-1},$$ which means that $\mathscr{V}_{2}(\mathbb{R}^{N})$ can be identified with the link associate do the hypersurface $\mathcal{Z}(P) \subset \mathbb{C}^{N}$ (e.g. [@Seade]).
From the above description, if one considers the manifold
$M = {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-1)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, since $\mathscr{V}_{2}(\mathbb{R}^{N})$ is $(N-3)$-connected [@Hatcher], it follows that
$M = {\rm{Tot}}(\mathscr{O}_{\alpha_{1}}(-1)^{\times})/\Gamma \cong \mathscr{V}_{2}(\mathbb{R}^{N}) \times S^{1}\big.$
for every $N >3$. Applying Theorem \[Theo4\] one obtains an embedding $$\label{embeddingstiefel}
\mathscr{R}^{(\Gamma)} \colon \mathscr{V}_{2}(\mathbb{R}^{N}) \times S^{1} \hookrightarrow {\rm{H}}_{\Gamma} = \Big ( \mathbb{C}^{N} \backslash \{0\} \Big)/ \Gamma,$$ such that $\mathscr{R}^{(\Gamma)}(\mathscr{V}_{2}(\mathbb{R}^{N}) \times S^{1}) = \mathcal{Z}(P)_{{\text{reg}}}/\Gamma$. It shows that the Kähler covering of the compact Vaisman manifold $\mathscr{V}_{2}(\mathbb{R}^{N}) \times S^{1}$ is determined by the regular locus of the Stein variety $\mathcal{Z}(P) = \{P = 0\} \subset \mathbb{C}^{N}$.
For $N = 3$, we have ${\rm{Spin}}_{3}(\mathbb{C}) \cong {\rm{SL}}(2,\mathbb{C})$, so the affine cone
$K\mathcal{Q}_{2} = \Big \{ (z_{1},z_{2}, z_{3}) \in \mathbb{C}^{3} \ \ \Big | \ \ z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0 \Big \},$
can be seen as a cone associated to some suitable embedding $\mathbb{C}{\rm{P}}^{1} \hookrightarrow \mathbb{C}{\rm{P}}^{2}$. Being more precise, if one considers the linear complex change of variables
$X = z_{1} + \sqrt{-1}z_{2}$, $Y = z_{1} - \sqrt{-1}z_{2}$, $Z = z_{3},$
it follows that $K\mathcal{Q}_{2} = \mathcal{Z}(Z^{2} + XY) $. Then, by considering the isomorphism $\mathbb{C}^{3} \cong \mathfrak{sl}(2,\mathbb{C})$ defined by
$ \big (X,Y,Z\big) \longleftrightarrow \begin{pmatrix}
Z & \ X\\
Y & -Z
\end{pmatrix}$,
we obtain the following characterization: $$\label{nipontentcone}
K\mathcal{Q}_{2} = \Bigg \{ \begin{pmatrix}
Z & \ X\\
Y & -Z
\end{pmatrix} \in \mathfrak{sl}(2,\mathbb{C}) \ \ \Bigg | \ \ Z^{2} + XY = 0 \Bigg \}.$$ On the other hand, since $X_{B} = {\rm{SL}}(2,\mathbb{C})/B = \mathbb{C}{\rm{P}}^{1}$, it follows that
$\mu(K_{\mathbb{C}{\rm{P}}^{1}}) = 2\varrho,$
see Equation \[canonicalfull\] and Equation \[sumweights\]. Hence, the embedding obtained from the very ample line bundle $K_{\mathbb{C}{\rm{P}}^{1}}^{-1} \to \mathbb{C}{\rm{P}}^{1}$ provides that $\mathbb{C}{\rm{P}}^{1} \cong {\rm{SL}}(2,\mathbb{C}) \cdot [v_{2\varrho}^{+}]$, where $v_{2\varrho}^{+}$ is the highest weight vector associated to $V(2\varrho)$. Now, since we can consider $$V(2\varrho) = (\mathfrak{sl}(2,\mathbb{C}), {\text{ad}}), \ \ {\text{and}} \ \ v_{2\varrho}^{+} = \begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix},$$ where ${\text{ad}} \colon \mathfrak{sl}(2,\mathbb{C}) \to \mathfrak{gl}(\mathfrak{sl}(2,\mathbb{C}))$ is the adjoint representation, it is straightforward to see that the image of $\mathbb{C}{\rm{P}}^{1} \hookrightarrow \mathbb{P}(\mathfrak{sl}(2,\mathbb{C}))$ $$\mathbb{C}{\rm{P}}^{1} \cong \Bigg \{ \Bigg [\begin{pmatrix}
Z & \ X\\
Y & -Z
\end{pmatrix} \Bigg ] \in \mathbb{P}(\mathfrak{sl}(2,\mathbb{C})) \ \ \Bigg | \ \ Z^{2} + XY = 0 \Bigg \}.$$ From the above description we see that, in this particular case, we have $${\rm{SO}}(3) = \mathscr{V}_{2}(\mathbb{R}^{3}) \cong \mathcal{Z}(Z^{2} + XY) \cap S^{5},$$ so the link described by the previous procedure is not simply connected. the above description also can be seen as a consequence of the fact that $K_{\mathbb{C}{\rm{P}}^{1}}^{-1} = L_{\chi_{\varrho}}^{\otimes 2}$, such that $L_{\chi_{\varrho}} = \mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(1)$, which implies the following relation of sphere bundles: $$\label{linkA1sing}
Q(K_{\mathbb{C}{\rm{P}}^{1}}) = Q(L_{\chi_{\varrho}})/\mathbb{Z}_{2} \cong S^{3}/\mathbb{Z}_{2}.$$ From this, if we consider the manifold the manifold
$M = {\rm{Tot}}(K_{\mathbb{C}{\rm{P}}^{1}}^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, it follows that
$M = {\rm{Tot}}(K_{\mathbb{C}{\rm{P}}^{1}}^{\times})/\Gamma \cong S^{3} \times_{\mathbb{Z}_{2}} S^{1},$
so applying Theorem \[Theo4\] one obtains an embedding $$\mathscr{R}^{(\Gamma)} \colon S^{3} \times_{\mathbb{Z}_{2}} S^{1} \hookrightarrow {\rm{H}}_{\Gamma} = \Big (\mathfrak{sl}(2,\mathbb{C}) \backslash \{0\} \Big)/ \Gamma.$$ It is worth pointing out that, for the particular case that $\lambda \in \mathbb{R}^{+}$, such that $\lambda <1$, it follows that
$M = {\rm{Tot}}(K_{\mathbb{C}{\rm{P}}^{1}}^{\times})/\Gamma \cong {\rm{SO}}(3) \times S^{1},$
so from Theorem \[Theo4\] we obtain an identification ${\rm{SO}}(3) \times S^{1} \cong \mathcal{Z}(Z^{2} + XY)_{{\text{reg}}}/\Gamma$.
Let $H \subset {\rm{SU}}(2) \subset {\rm{SL}}(2,\mathbb{C})$ be a finite subgroup, the [*Kleinian singularity*]{} attached to $H$ is the quotient singularity defined by $\mathscr{V}(H) = \mathbb{C}^{2}/H$. The orbit space $\mathscr{V}(H)$ is an analytic variety (cf. [@Cartan], [@Lamotke]) with an isolated singularity at the image of $(0,0) \in \mathbb{C}^{2}$, which we denote by $o \in \mathscr{V}(H)$. An important result of Felix Klein [@Klein] shows that the $\mathscr{V}(H)$ can be viewed as a hypersurface $$\mathscr{V}(H) \cong \Big \{ (X,Y,Z) \in \mathbb{C}^{3} \ \Big | \ P(X,Y,Z) = 0\Big\},$$ such that $P$ is the relation between three fundamental generators $\mathfrak{I}_{1},\mathfrak{I}_{2},\mathfrak{I}_{3}$ of the invariant ring $\mathbb{C}[z_{1},z_{2}]^{H}$ of $\mathbb{C}^{2}$. In other words, we have a map $\mathfrak{I} \colon \mathbb{C}^{2} \to \mathbb{C}^{3}$, such that
$\mathfrak{I}(z_{1},z_{2}) = \big (\mathfrak{I}_{1}(z_{1},z_{2}),\mathfrak{I}_{2}(z_{1},z_{2}),\mathfrak{I}_{3}(z_{1},z_{2}) \big)$,
which factorizes over the quotient $\mathscr{V}(H) = \mathbb{C}^{2}/H$. The image of $\mathfrak{I}$ is the zero set of $P$, i.e. $\operatorname{im}(\mathfrak{I}) = \mathcal{Z}(P)$, so we have an isomorphism of varieties $\mathscr{V}(H) \cong \mathcal{Z}(P)$. From the classification of finite subgroups $H \subset {\rm{SU}}(2)$, see for instance [@Miller], if one considers the particular case that $H = \mathbb{Z}_{n}$ (finite cyclic), it turns out that $\mathfrak{I}_{1} = z_{1}^{n}$, $\mathfrak{I}_{2} = -z_{2}^{n}$, $\mathfrak{I}_{3} = z_{1}z_{2}$, so we obtain $$\mathscr{V}(\mathbb{Z}_{n}) \cong \mathcal{Z}(Z^{n} + XY).$$ Therefore, we have the previous cone $K\mathcal{Q}_{2}$ is given by the Kleinian singularity $\mathscr{V}(\mathbb{Z}_{2}) \cong \mathbb{C}^{2}/\mathbb{Z}_{2}$.
Given a general Kleinian singularity $\mathscr{V}(H)$, since $H \subset {\rm{SU}}(2)$ preserves the usual Hermitian metric on $\mathbb{C}^{2}$, it follows that it preserves all spheres centred at $(0,0) \in \mathbb{C}^{2}$, so the radial deformation retract of the punctured unit ball $\{z \in \mathbb{C}^{2} \ | \ 0<|z| \leq 1\}$ onto the boundary sphere $S^{3}$ induces a deformation retract of $V_{1} = \{[z] \in \mathscr{V}(H) \ | \ 0<|z| \leq 1\}$ onto the boundary $\partial V_{1} = S^{3}/H$. Therefore, we have $S^{3}/H$ as the link of the isolated singularity $o \in \mathscr{V}(H)$ (see Equation \[linkA1sing\]).
Given an isolated singularity $\mathscr{V}(\mathbb{Z}_{n})$, associated to some $\mathbb{Z}_{n} \subset {\rm{SU}}(2)$, at first sight, we can always resolve[^20] the singularity $o \in \mathscr{V}(\mathbb{Z}_{n})$ by taking the Cartan-Remmert reduction $$\mathscr{R} \colon \mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(-n) \to \mathscr{V}(\mathbb{Z}_{n}),$$ here one can consider the underlying smooth complex surface
${\rm{Tot}}\big ( \mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(-n) \big) = \Big \{ ([z],w) \in \mathbb{C}{\rm{P}}^{1} \times \mathbb{C}^{2} \ \Big | \ w_{1}z_{2}^{n} = w_{2}z_{1}^{n} \Big \},$
and $\mathscr{R} \colon ([z_{1}:z_{1}],(w_{1},w_{2})) \mapsto (t_{1},t_{2})$, such that $t_{i}^{n} = w_{i}$ and $t_{1}z_{2} = t_{2}z_{1}$, see for instance [@Lamotke]. From the adjunction formula, it can be shown that $\mathscr{R}^{\ast}K_{\mathscr{V}(\mathbb{Z}_{n})} = K_{\mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(-n)}$ if and only if $n = 2$ cf. [@CONTACTCORREA Proposition 5.7]. Moreover, in the latter case, the manifold $K_{\mathbb{C}{\rm{P}}^{1}} = \mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(-2)$ admits a complete Calabi-Yau metric provided by the Calabi Ansatz [@CALABIANSATZ], a.k.a. Eguchi-Hanson metric [@EGUCHIHANSON].
In order to understand the precise relation between the Hermitian-Einstein-Weyl metric on ${\rm{SO}}(3) \times S^{1}$ provided by Theorem \[Theo2\] and the aforementioned Calabi-Yau metric on $K_{\mathbb{C}{\rm{P}}^{1}}$, we observe the following. If we consider the identification $\mathscr{V}(\mathbb{Z}_{2}) = K\mathcal{Q}_{3}$, it follows that $$\mathscr{V}(\mathbb{Z}_{2}) \backslash \{o\} = {\text{Ad}}\big({\rm{SL}}(2,\mathbb{C})\big) \cdot\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix},$$ see Equation \[nipontentcone\]. From this, given a matrix $\mathcal{W} \in \mathscr{V}(\mathbb{Z}_{2}) \backslash \{o\}$, it is straightforward to check that $$\label{matrizregular}
\mathcal{W} = \begin{pmatrix}
z_{1}z_{2} & \ \ z_{1}^{2}\\
z_{2}^{2} & - z_{1}z_{2}
\end{pmatrix},$$ i.e. $\mathcal{W} = \mathcal{W}(z_{1},z_{2})$, where $(z_{1},z_{2}) \in \mathbb{C}^{2}\backslash \{(0,0)\}$, so if one considers the inner product on $\mathfrak{sl}(2,\mathbb{C})$ defined by the paring [^21]:
$(\mathcal{W}_{1},\mathcal{W}_{2}) \mapsto \Re\big({\rm{Tr}}(\mathcal{W}_{1}\mathcal{W}_{2}^{\dagger})\big )$,
$\forall \mathcal{W}_{1},\mathcal{W}_{2} \in \mathfrak{sl}(2,\mathbb{C})$, by following Remark \[kahlerpotentialalgebraiclie\], one can take $F \colon \mathscr{V}(\mathbb{Z}_{2}) \backslash \{o\} \to \mathbb{R}^{+}$, such that
$F(\mathcal{W}) = {\rm{Tr}}(\mathcal{W}\mathcal{W}^{\dagger}).$
From Equation \[matrizregular\], it turns out that $F(\mathcal{W}(z_{1},z_{2})) = (|z_{1}|^{2} + |z_{2}|^{2})^{2}$; moreover, if we take the smooth function ${\rm{K}}_{{\text{EH}}} \colon \mathscr{V}(\mathbb{Z}_{2}) \backslash \{o\} \to \mathbb{R} $ definded by $${\rm{K}}_{{\text{EH}}}(\mathcal{W}) = \sqrt{1+4{\rm{Tr}}(\mathcal{W}\mathcal{W}^{\dagger})} + \frac{1}{2} \log \Bigg [ \frac{4{\rm{Tr}}(\mathcal{W}\mathcal{W}^{\dagger})}{\big(1+\sqrt{1+4{\rm{Tr}}(\mathcal{W}\mathcal{W}^{\dagger})}\big)^{2}}\Bigg],$$ from the biholomorphism $\mathscr{R} \colon K_{\mathbb{C}{\rm{P}}^{1}}^{\times} \to \mathscr{V}(\mathbb{Z}_{n}) \backslash \{o\} $, it follows from Equation \[potentialreduction\] that $$\label{potentialeguchi}
\mathscr{R}^{\ast}{\rm{K}}_{{\text{EH}}} = \frac{1}{2}\log \big ({\rm{K}}_{H} \big) + \Upsilon\big ({\rm{K}}_{H}\big ) = \frac{1}{2}\log \big( {\rm{e}}^{2\Upsilon({\rm{K}}_{H})}{\rm{K}}_{H}\big),$$ where $\Upsilon \colon \mathbb{R}^{+} \to \mathbb{R}$, such that $\Upsilon(x) = \sqrt{1+4x} - \log(\frac{1+\sqrt{1+4x}}{2})$ cf. [@CALABIANSATZ]. The potential above defines a $2$-form
$ \displaystyle \omega_{{\text{EH}}} = \sqrt{-1}\partial \overline{\partial} (\mathscr{R}^{\ast}{\rm{K}}_{{\text{EH}}}) = \frac{\sqrt{-1}}{2}\partial \overline{\partial} \log \big ({\rm{K}}_{H} \big) + \sqrt{-1}\partial \overline{\partial}\Upsilon\big ({\rm{K}}_{H}\big ),$
on $K_{\mathbb{C}{\rm{P}}^{1}}^{\times}$ which extends smoothly and unique to $K_{\mathbb{C}{\rm{P}}^{1}}$ defining the Eguchi-Hanson Calabi-Yau metric. The Calabi-Yau manifold $(K_{\mathbb{C}{\rm{P}}^{1}},\omega_{{\text{EH}}})$ is an example of asymptotically locally Euclidean (ALE) space, for more details on the constriction of ALE spaces see [@Gibbons], [@Hitchin] and [@Kronheimer].
By considering the Higgs field $\theta_{g}$ on ${\rm{SO}}(3) \times S^{1}$ associated to the Hermian-Einstein-Weyl structure $([g],D,J)$ provided by Theorem \[Theo2\], from Equation \[potentialeguchi\] we have (locally) the following relation $$\frac{1}{2}\theta_{g} = -\frac{1}{2}d\log \big( {\rm{K}}_{H}(z,w) \big) = - d\Big ( \mathscr{R}^{\ast}{\rm{K}}_{{\text{EH}}}(z,w)\Big) +d \Upsilon(z,w),$$ notice that ${\rm{K}}_{H}(z,w) = (1 + |z|^{2})^{2}w\overline{w}$ (see Example \[exampleP1\]), which implies that (locally) $$\mathscr{R}^{\ast}{\rm{K}}_{{\text{EH}}}(z,w) = \sqrt{1+4(1 + |z|^{2})^{2}w\overline{w}} + \frac{1}{2} \log \Bigg [ \frac{4(1 + |z|^{2})^{2}w\overline{w}}{\big(1+\sqrt{1+4(1 + |z|^{2})^{2}w\overline{w}}\big)^{2}}\Bigg],$$ cf. [@Lindstrom]. The ideas above provide a unified background in terms of representation theory for the Hermitian-Einstein-Weyl geometry on ${\rm{SO}}(3) \times S^{1}$ and the Calabi-Yau geometry on $K_{\mathbb{C}{\rm{P}}^{1}} = \mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(-2)$.
It is worth mentioning that the description provided in \[linkdescription\] for the Stiefel manifold $\mathscr{V}_{2}(\mathbb{R}^{N})$ fits in a more general context. Actually, given a $(d+1)$-tuple $a = (a_{0},\ldots,a_{d})$, with $a_{j} \in \mathbb{Z}_{>0}$, such that $a_{j} \geq 2$, $j = 0,\ldots,d$, we can consider the [*Brieskorn-Pham polynomial*]{} $P \colon \mathbb{C}^{d+1} \to \mathbb{C}$ given by
$P(z) = z_{0}^{a_{0}} + \cdots + z_{d}^{a_{d}}$, $\forall z \in \mathbb{C}^{d+1}$.
see for instance [@Pham], [@Brieskorn1]. From this, we obtain a complex hypersurface with an isolated singularity at $0 \in \mathbb{C}^{d}$ defined by $$V(a) = \mathcal{Z}(P) = \Big \{ z_{0}^{a_{0}} + \cdots + z_{d}^{a_{d}} = 0\Big\},$$ the singular affine variety defined above is an example of singular [*Brieskorn variety*]{} [@Brieskorn1], [@Brieskorn2]. The example above of singular affine variety plays an important role in the study homotopy spheres and exotic spheres. Being more precise, in [@Hirzebruch] Hirzebruch showed that links (a.k.a. [*Brieskorn-Pham manifolds*]{}) of the form $$\label{Brieskornlink}
\Sigma(a) = V(a) \cap S^{2d+1},$$ can sometimes be homeomorphic, but not diffeomorphic to standard spheres. Moreover, as shown by Egbert V. Brieskorn in [@Brieskorn2], links of the form $$\Sigma(2,2,2,3,6k-1) = \Big \{ z_{0}^{2} + z_{1}^{2} + z_{2}^{2} + z_{3}^{3} + z_{4}^{6k-1} = 0 \Big\} \cap S^{9},$$ with $k = 1, \ldots,28$, realize explicitly all the distinct smooth structures on the $7$-sphere $S^{7}$ classified by Michel Kervaire and Milnor in [@Kervaire], see also [@Milnor]. From the brief comments above, supposing $d = N$, we have that $$\mathscr{V}_{2}(\mathbb{R}^{N}) = \Sigma(2, \ldots, 2),$$ further, it follows that $\Sigma(2, \ldots, 2) \times S^{1} \cong V(2,\ldots,2)_{{\text{reg}}}/\Gamma$, see Equation \[embeddingstiefel\].
\[unittangent\] An interesting characterization of $\mathscr{V}_{2}(\mathbb{R}^{N})$ which is worth mentioning, can be described as follows. Let $(M,g)$ be a Riemannian manifold, the [*tangent sphere bundle*]{} of radius $r_{0}$ over $(M,g)$ is the hypersurface defined by $$T_{r_{0}}M = \Big \{ \big (p;v \big ) \in TM \ \ \Big | \ \ g_{p}(v,v) = r_{0} \Big \}.$$ For the particular case that $r_{0} = 1$, $T_{1}M$ is called the [*unit tangent bundle*]{} of $(M,g)$. Also, by considering the natural projection map $\pi \colon T_{1}M \to (M,g)$, such that $\pi(p;v) = p$, $\forall (p;v) \in T_{1}M$, it follows that $\pi^{-1}(p) \cong S^{d-1}$, $\forall p \in M$, where $d = \dim(M)$. If one considers $(S^{N-1},g_{0})$, such that $g_{o} = \langle \cdot ,\cdot \rangle|_{S^{N-1}}$, where $\langle \cdot ,\cdot \rangle$ is the canonical inner product on $\mathbb{R}^{N}$, since
$TS^{N-1} = \Big \{ \big (p;v \big ) \in S^{N-1} \times \mathbb{R}^{N} \ \ \Big | \ \ \langle p,v \rangle = 0\Big\},$
it follows that $T_{1}S^{N-1} = \mathscr{V}_{2}(\mathbb{R}^{N})$. Notice that, for the both particular cases $N = 4$ and $N = 8$, since $S^{3}$ and $S^{7}$ are, respectively, parallelizable [@Bott2], it follows that $$T_{1}S^{3} = S^{3} \times S^{2} \ \ {\text{and}} \ \ T_{1}S^{7} = S^{7} \times S^{6},$$ thus we have $\mathscr{V}_{2}(\mathbb{R}^{4}) = S^{3} \times S^{2}$ and $\mathscr{V}_{2}(\mathbb{R}^{8}) = S^{7} \times S^{6}$. Moreover, from the previous comments, for the latter case we obtain
$S^{7} \times S^{6} \times S^{1} \cong V(2,\ldots,2)_{{\text{reg}}}/\Gamma$,
such that $V(2,\ldots,2) = \{(z_{1},\ldots,z_{8}) \in \mathbb{C}^{8} \ | \ z_{1}^{2} + \cdots + z_{8}^{2} = 0\}$. It is worth mentioning that by applying Theorem \[Theo2\], we recover the result introduced in [@OrneaPiccinni] on (locally Ricci-flat) l.c.K. structures on $\mathscr{V}_{2}(\mathbb{R}^{N}) \times S^{1}$ ($N>4$). The case $N = 4$ is presented in details below.
Now, by considering $N = 4$, we observe that from the characterization \[Stiefeldefinition\], it follows that $$S^{1} \hookrightarrow \mathscr{V}_{2}(\mathbb{R}^{4}) \to \mathcal{Q}_{3} \cong {\rm{Spin}}(4)/{\rm{Spin}}(2)\times {\rm{Spin}}(2).$$ Moreover, since ${\rm{Spin}}(4) = {\rm{SU}}(2) \times {\rm{SU}}(2)$, see Remark \[lowdimspin\], and ${\rm{Spin}}(2) = {\rm{U}}(1)$, we have $$\label{apexconofold}
\mathcal{Q}_{3} \cong \mathbb{C}{\rm{P}}^{1} \times \mathbb{C}{\rm{P}}^{1}.$$ the above description shows that this particular case is different from the previous ones, in fact, one can see that $b_{2}(\mathcal{Q}_{N-1}) = 1$, $\forall N > 4$, and $b_{2}(\mathcal{Q}_{3}) = 2$, notice that $\mathcal{Q}_{3}$ is a full flag manifold (cf. Example \[examplefullflag\]). Therefore, in order to describe in terms of elements of representation theory the projective embedding of \[apexconofold\], we need to proceed slightly different from the previous cases. In what follows, for the sake of simplicity, by following Remark \[lowdimspin\] we shall denote $$\label{splittingspin}
\mathfrak{spin}_{4}(\mathbb{C}) = \mathfrak{sl}(\alpha_{1}) \times \mathfrak{sl}(\alpha_{2}).$$ From the characterization above, the irreducible $\mathfrak{spin}_{4}(\mathbb{C})$-modules can be described by means of irredicible $\mathfrak{sl}(2,\mathbb{C})$-modules as follows: Given an integral weight $\mu_{\ell_{1},\ell_{2}} = \ell_{1}\omega_{\alpha_{1}} + \ell_{2}\omega_{\alpha_{2}}$, $\ell_{i} \in \mathbb{Z}_{\geq 0 }$, one can consider the irreducible $\mathfrak{spin}_{4}(\mathbb{C})$-module defined by the linear representation $$V(\mu_{\ell_{1},\ell_{2}}) = \bigg (V(\ell_{1}\omega_{\alpha_{1}}) \boxtimes V(\ell_{2}\omega_{\alpha_{2}}), \rho_{\ell_{1}} \boxtimes \rho_{\ell_{2}}\bigg),,$$ where $\rho_{\ell_{1}} \colon \mathfrak{sl}(\alpha_{j}) \to \mathfrak{gl}\big(V(\ell_{j}\omega_{\alpha_{j}})\big)$ is an irreducible $\mathfrak{sl}(\alpha_{j})$-module, $j = 1,2$, here the symbol “$\boxtimes$” stands for the tensor product of irreducible representations of two different Lie algebras (Deligne’s tensor product). Notice that the action of $\mathfrak{spin}_{4}(\mathbb{C})$ on $V(\mu_{\ell_{1},\ell_{2}})$ through the characterization \[splittingspin\] is given by
$\rho_{\ell_{1}} \boxtimes \rho_{\ell_{2}}(X,Y)(v_{1} \boxtimes v_{2}) = \big ( \rho_{\ell_{1}}(X)v_{1} \big ) \boxtimes Y + X \boxtimes \big ( \rho_{\ell_{2}}(Y)v_{2} \big ),$
$\forall (X,Y) \in \mathfrak{spin}_{4}(\mathbb{C}) = \mathfrak{sl}(\alpha_{1}) \times \mathfrak{sl}(\alpha_{2})$, $\forall v_{j} \in V(\ell_{j}\omega_{\alpha_{j}})$, $j = 1,2$. In what follows, on Lie group level, we also denote by $\rho_{\ell_{1}} \colon {\rm{SL}(\alpha_{j})} \to {\rm{GL}}\big(V(\ell_{j}\omega_{\alpha_{j}})\big)$ the induced irreducible ${\rm{SL}(\alpha_{j})}$-module, $j = 1,2$.
Now, back to the geometric setting, since $$\label{picarddeligne}
{\text{Pic}}(\mathcal{Q}_{3}) = \mathbb{Z}c_{1}\big (\mathscr{O}_{\alpha_{1}}(1)\big) \oplus \mathbb{Z}c_{1}\big (\mathscr{O}_{\alpha_{2}}(1)\big),$$ given a negative line bundle $L \in {\text{Pic}}(\mathcal{Q}_{3})$, such that $L = \mathscr{O}_{\alpha_{1}}(-\ell_{1}) \otimes \mathscr{O}_{\alpha_{2}}(-\ell_{2})$, $\ell_{j} \in \mathbb{Z}_{>0}$, $j =1,2$, from the Borel-Weil theorem it follows that
$H^{0}(\mathcal{Q}_{3},L^{-1})^{\ast} = V(\mu(L)) = V(\ell_{1}\omega_{\alpha_{1}}) \boxtimes V(\ell_{2}\omega_{\alpha_{2}}),$
notice that $\mu(L) = \ell_{1}\omega_{\alpha_{1}} + \ell_{2}\omega_{\alpha_{2}}$. By observing that $I(\mathcal{Q}_{3}) = 2$, i.e.
$\mu(K_{\mathcal{Q}_{3}}) = 2\varrho = 2\big ( \omega_{\alpha_{1}} + \omega_{\alpha_{2}}\big),$
see Equation \[sumweights\], we obtain $\mathscr{O}_{\mathcal{Q}_{3}}(-1) = \mathscr{O}_{\alpha_{1}}(-1) \otimes \mathscr{O}_{\alpha_{2}}(-1)$. The very ample line bundle $\mathscr{O}_{\mathcal{Q}_{3}}(1) \to \mathcal{Q}_{3}$ provides a projective embedding $$\label{projectiveembeddingQ3}
\iota \colon \mathcal{Q}_{3} \hookrightarrow {\text{Proj}}\big (H^{0}(X_{P},\mathscr{O}_{\mathcal{Q}_{3}}(1))^{\ast} \big),$$ whose the image is defined by the projective orbit $\iota(\mathcal{Q}_{3}) = {\rm{Spin}}_{4}(\mathbb{C})\cdot[v_{\omega_{\alpha_{1}}}^{+} \boxtimes v_{\omega_{\alpha_{2}}}^{+}]$. By considering $V(\omega_{\alpha_{j}}) = \mathbb{C}^{2}$, it follows that $v_{\omega_{\alpha_{j}}}^{+} = e_{1}$, $j = 1,2$, so we have
$H^{0}(X_{P},\mathscr{O}_{\mathcal{Q}_{3}}(1))^{\ast} = \mathbb{C}^{2} \boxtimes \mathbb{C}^{2}$.
Therefore, if we consider the tensor product $\mathbb{C}^{2} \boxtimes \mathbb{C}^{2} = \big({\rm{M}}(2,\mathbb{C}),\boxtimes \big)$, such that
$\boxtimes \colon \big ((u_{1},u_{2}),(v_{1},v_{2})\big) \mapsto \begin{pmatrix}
u_{1}v_{1} & u_{1}v_{2}\\
u_{2}v_{1} & u_{2}v_{2}
\end{pmatrix}$,
$\forall (u_{1},u_{2}),(v_{1},v_{2}) \in \mathbb{C}^{2}$, we obtain $$\label{segreembedding}
\mathcal{Q}_{3} \cong \Big \{ \big[\mathcal{W}\big] \in \mathbb{P}\big({\rm{M}}(2,\mathbb{C})\big ) \ \Big | \ \det(\mathcal{W}) = 0 \Big \}.$$
\[Segreremark\] It is worth pointing out that the above description recovers the Segre embedding $\mathbb{C}{\rm{P}}^{1} \times \mathbb{C}{\rm{P}}^{1} \hookrightarrow \mathbb{C}{\rm{P}}^{3}$. In fact, if one considers the identification $\mathbb{C}^{4} \cong {\rm{M}}(2,\mathbb{C})$ defined by
$ \big (X,U,V,Y\big) \longleftrightarrow \begin{pmatrix}
X & U\\
V & Y
\end{pmatrix}$,
from the Segre embedding $\mathbb{C}{\rm{P}}^{1} \times \mathbb{C}{\rm{P}}^{1} \hookrightarrow \mathbb{C}{\rm{P}}^{3}$, $([z_{1}:z_{2}],[w_{1}:w_{2}]) \mapsto [z_{1}w_{1}:z_{1}w_{2}:z_{2}w_{1}:z_{2}w_{2}]$, it follows that
$\mathbb{C}{\rm{P}}^{1} \times \mathbb{C}{\rm{P}}^{1} \cong \Big \{ \big [X:U:V:Y\big] \in \mathbb{C}{\rm{P}}^{3} \ \Big | \ XY - UV = 0\Big \}.$
Hence, the projective embedding defined by \[projectiveembeddingQ3\] is precisely the Segre embedding.
From the projective algebraic realization \[segreembedding\] we have the affine cone $K\mathcal{Q}_{3} \subset {\rm{M}}(2,\mathbb{C})$ defined by the HV-veriety $Y = \overline{{\rm{Spin}}_{4}(\mathbb{C})\cdot (e_{1} \boxtimes e_{1})}$, i.e., the [*conifold*]{}[^22] $$K\mathcal{Q}_{3} = \Big \{ \mathcal{W} \in {\rm{M}}(2,\mathbb{C}) \ \Big | \ \det(\mathcal{W}) = 0 \Big \},$$ notice that ${\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times}) \cong \big (K\mathcal{Q}_{3}\big)_{{\text{reg}}} = {\rm{Spin}}_{4}(\mathbb{C})\cdot (e_{1} \boxtimes e_{1})$.
\[matrixinnerporduct\] In what follows it will important for us to consider the inner product $\langle \cdot ,\cdot \rangle \colon {\rm{M}}(2,\mathbb{C}) \times {\rm{M}}(2,\mathbb{C}) \to \mathbb{R}$, such that $$\label{spininnerproduct}
\big \langle \mathcal{W}_{1}, \mathcal{W}_{2} \big \rangle = \Re\Big({\rm{Tr}}\big(\mathcal{W}_{1}\mathcal{W}_{2}^{\dagger}\big)\Big ),$$ $\forall \mathcal{W}_{1},\mathcal{W}_{2} \in {\rm{M}}(2,\mathbb{C})$. Notice that, from the identification $\mathbb{C}^{4} \cong {\rm{M}}(2,\mathbb{C})$ provided in Remark \[Segreremark\], it follows that the inner product described above coincides with the canonical inner product in $\mathbb{C}^{4}$.
As we have seen previously, we can realize $\mathscr{V}_{2}(\mathbb{R}^{4}) \subset S^{3} \times S^{3}$, so by considering
$S^{3} = \big \{ (z_{1},z_{2}) \in \mathbb{C}^{2} \ \big | \ |z_{1}|^{2} + |z_{2}|^{2} = 1\big \}$,
we obtain an identification $$\label{linkstiefelspin4}
\mathscr{V}_{2}(\mathbb{R}^{4}) \cong K\mathcal{Q}_{3} \cap S^{7}.$$ In fact, since
$S^{7} = \Big \{ \mathcal{W} \in {\rm{M}}(2,\mathbb{C}) \ \Big | \ {\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big) = 1 \Big \},$
and $S^{3} \cong {\rm{SU}}(2)$ through the correspondence
$(z_{1},z_{1}) \longleftrightarrow \begin{pmatrix}
z_{1} & -\overline{z}_{2}\\
z_{2} & \ \ \overline{z}_{1}
\end{pmatrix},$
we can define a smooth map $I \colon {\rm{SU}}(2) \times {\rm{SU}}(2) \to {\rm{M}}(2,\mathbb{C})$, such that
$I \colon \big ((u_{1},u_{2}),(v_{1},v_{2}) \big) \mapsto \Bigg (\begin{pmatrix}
u_{1} & -\overline{u}_{2}\\
u_{2} & \ \ \overline{u}_{1}
\end{pmatrix}\cdot e_{1} \Bigg )\boxtimes \Bigg ( \begin{pmatrix}
v_{1} & -\overline{v}_{2}\\
v_{2} & \ \ \overline{v}_{1}
\end{pmatrix}\cdot e_{1} \Bigg ) = \begin{pmatrix}
u_{1}v_{1} & u_{1}v_{2}\\
u_{2}v_{1} & u_{2}v_{2}
\end{pmatrix},$
$\forall \big ((u_{1},u_{2}),(v_{1},v_{2}) \big) \in {\rm{SU}}(2) \times {\rm{SU}}(2)$. It is straightforward to see that
$\det(I(u,v)) = 0$, and ${\rm{Tr}}\big(I(u,v)I(u,v)^{\dagger}\big) = 1$,
$\forall (u,v) \in {\rm{SU}}(2) \times {\rm{SU}}(2)$, so we have $I({\rm{Spin}}(4)) \subset K\mathcal{Q}_{3} \cap S^{7}$. Also, by considering the Iwasawa decomposition of ${\rm{SL}}(2,\mathbb{C})$, and the fact that ${\rm{Spin}}_{4}(\mathbb{C}) = {\rm{SL}}(2,\mathbb{C}) \times {\rm{SL}}(2,\mathbb{C})$, one can see that
$K\mathcal{Q}_{3} \cap S^{7} = {\rm{Spin}}_{4}(\mathbb{C})(e_{1}\boxtimes e_{1}) \cap S^{7} = {\rm{Spin}}(4) (e_{1}\boxtimes e_{1}) \cap S^{7}.$
Hence, considering the (proper) smooth action of ${\rm{Spin}}(4) = {\rm{SU}}(2) \times {\rm{SU}}(2)$ on ${\rm{M}}(2,\mathbb{C})$ (induced from the associated representation) it follows that
$I({\rm{Spin}}(4)) = {\rm{Spin}}(4) (e_{1}\boxtimes e_{1}) = K\mathcal{Q}_{3} \cap S^{7}.$
Therefore, since the isotropy subgroup of $e_{1}\boxtimes e_{1}$ with respect to the aforementioned action is given by ${\rm{U}}(1) = {\rm{Spin}}(2) \subset {\rm{Spin(4)}}$, the diffeomorphism \[linkstiefelspin4\] is obtained from the following commutative diagram:
& [[Spin]{}]{}(4) &\
[[Spin]{}]{}(4)/[[Spin]{}]{}(2) & & K\_[3]{} S\^[7]{}
Notice that $\widetilde{I} \colon \mathscr{V}_{2}(\mathbb{R}^{4}) \to K\mathcal{Q}_{3} \cap S^{7}$ satisfies $\widetilde{I} \circ \pi = I$, where $\pi \colon {\rm{Spin}}(4) \to {\rm{Spin}}(4)/{\rm{Spin}}(2)$ is the natural projection. From Remark \[unittangent\] and the above comments, we obtain
$K\mathcal{Q}_{3} \cap S^{7} \cong \mathscr{V}_{2}(\mathbb{R}^{4}) = S^{3} \times S^{2}$.
As we have seen, the manifold defined by $K\mathcal{Q}_{3} \cap S^{7}$ is also identified with the sphere bundle of $\mathscr{O}_{\mathcal{Q}_{3}}(-1)$ via Cartan-Remmert reduction, it is also denoted by $T^{1,1} = K\mathcal{Q}_{3} \cap S^{7}$. In general, from Equation \[picarddeligne\], for any holomophic line bundle
$L = \mathscr{O}_{\alpha_{1}}(-p)\otimes \mathscr{O}_{\alpha_{2}}(-q) \to \mathcal{Q}_{3}$,
such that $p > q$ are relatively prime, we have that the sphere bundle $Y^{p,q} = Q(L)$ is diffeomorphic to $S^{2}\times S^{3}$, e.g., [@WangZiller]. It is worth mentioning that the manifolds $Y^{p,q}$ admits Sasaki-Einstein metrics different from the homogeneous one, see for instance [@Gauntlett], due to their application in AdS/CFT correspondence they have been widely studied by mathematicians and physicists, see [@Martelli] and references therein.
Now, if one considers the manifold
$M = {\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times})/\Gamma, \ \ \ {\text{where}}$ $\Gamma = \bigg \{ \lambda^{n} \in \mathbb{C}^{\times} \ \bigg | \ n \in \mathbb{Z} \bigg \}$,
for some $\lambda \in \mathbb{C}^{\times}$, such that $|\lambda| <1$, it follows that
$M = {\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times})/\Gamma \cong \mathscr{V}_{2}(\mathbb{R}^{4}) \times S^{1} = S^{3} \times S^{2} \times S^{1}.$
Therefore, by applying Theorem \[Theo4\], one obtains an embedding $$\mathscr{R}^{(\Gamma)} \colon S^{3} \times S^{2} \times S^{1} \hookrightarrow {\rm{H}}_{\Gamma} = \Big ({\rm{M}}(2,\mathbb{C}) \backslash \{0\} \Big)/ \Gamma,$$ which can be explicitly described by $$\mathscr{R}^{(\Gamma)}\big (\wp_{1}([(g,h),w]) \big ) = \Bigg [\begin{pmatrix}
g_{11}h_{11} & g_{11}h_{21} \\
g_{21}h_{11} & g_{21}h_{21}
\end{pmatrix}w \Bigg ]_{\Gamma},$$ $\forall [(g,h),w] \in {\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times})$, such that $\wp_{1} \colon {\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times}) \to S^{3} \times S^{2} \times S^{1}$, notice that $(g,h) \in {\rm{Spin}}_{4}(\mathbb{C}) = {\rm{SL}}(2,\mathbb{C}) \times {\rm{SL}}(2,\mathbb{C})$.
\[HEWproductsphres\] Considering $\mathbb{C}{\rm{P}}^{1} \cong S^{2}$, the ideas developed in this last example provide a complete understanding, in terms of the representation theory of ${\rm{SL}}(2,\mathbb{C})$, for the the different types of geometric structures arising from the following diagram:
S\^[3]{}S\^[2]{} S\^[1]{} & S\^[3]{} S\^[2]{} & S\^[2]{}S\^[2]{} & S\^[2]{}\
S\^[3]{}S\^[1]{} & S\^[3]{} & S\^[2]{} & {}
The different types of geometries which we have in both upper and lower horizontal sequence of fibrations are, respectively, from the left to the right side of the diagram, Hermitian-Einstein-Weyl, Sasaki-Einstein and Kähler-Einstein cf. [@OrneaPiccinni].
As we have seen in Example \[exampleP1\], the geometric structures on the lower sequence of fibrations in the aforementioned diagram can be obtained from a suitable Kähler potential ${\rm{K}}_{H} \colon {\rm{Tot}}(\mathscr{O}_{\mathbb{C}{\rm{P}}^{1}}(-1)^{\times}) \to \mathbb{R}^{+}$. From the result provided in Theorem \[Theo2\], the same can be done when one considers the upper line of the fibrations. In fact, in this case one can consider ${\rm{K}}_{H} \colon {\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times}) \to \mathbb{R}^{+}$ such that $$\label{potentialhyperquadric}
{\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)v_{2\varrho}^{+}\big | \big |^{2} \Big)^{\frac{1}{I(\mathcal{Q}_{3})}} w\overline{w} = \Big (\big | \big |s_{U}(z)(e_{1}\boxtimes e_{1}) \big | \big |^{2} \Big)^{\frac{2}{I(\mathcal{Q}_{3})}} w\overline{w}$$ where $ s_{U} \colon U \subset \mathcal{Q}_{3} \to {\rm{Spin}}_{4}(\mathbb{C})$ is some local section and $||\cdot ||$ is the norm induced by the ${\rm{Spin}}(4)$-invariant inner product defined in \[spininnerproduct\]. Therefore, if one considers the opposite big cell $U = N^{-}x_{0} \subset \mathcal{Q}_{3}$, from the fact that ${\rm{Spin}}_{4}(\mathbb{C}) = {\rm{SL}}(2,\mathbb{C}) \times {\rm{SL}}(2,\mathbb{C})$, we obtain a parameterization
$z = (z_{1},z_{2}) \in \mathbb{C}^{2} \mapsto \big(n(z_{1}) \times n(z_{2})\big )x_{0} \in U$,
such that
$n(z_{j}) = \begin{pmatrix}
1 & 0 \\
z_{j} & 1
\end{pmatrix}, \ \ \forall z_{j} \in \mathbb{C}, \ \ j = 1,2.$
From this, if we consider the local section $s_{U} \colon U \subset \mathcal{Q}_{3} \to {\rm{Spin}}_{4}(\mathbb{C})$, defined by
$s_{U} \colon \big(n(z_{1}) \times n(z_{2})\big )x_{0} \mapsto n(z_{1}) \times n(z_{2}),$
since $I(\mathcal{Q}_{3}) = 2$, the Kähler potential \[potentialhyperquadric\] is given explicitly by $$\label{potentialspheres}
{\rm{K}}_{H}\big (z,w \big ) = {\rm{Tr}}\Bigg [\begin{pmatrix}
1 & z_{2} \\
z_{1} & z_{1}z_{2}
\end{pmatrix}\begin{pmatrix}
1 & z_{2} \\
z_{1} & z_{1}z_{2}
\end{pmatrix}^{\dagger}\Bigg]w\overline{w}.$$ As we have seen, from the potential above one can derive a Kähler-Einstein metric on $\mathcal{Q}_{3} \cong S^{2} \times S^{2}$, a Sasaki-Einstein metric on $\mathscr{V}_{2}(\mathbb{R}^{4}) = S^{3} \times S^{2}$ and a Hermitian-Einstein-Weyl structure on
$S^{3} \times S^{2} \times S^{1} \cong (K\mathcal{Q}_{3})_{{\text{reg}}}/\mathbb{Z}.$
In this latter case, from Theorem \[Theo2\], we have the homogeneous Hermitian-Einstein-Weyl metric $g$ on $S^{3} \times S^{2} \times S^{1}$ completely determined by the Lee form $$\theta_{g} = -d\log \Bigg\{ {\rm{Tr}}\Bigg [\begin{pmatrix}
1 & z_{2} \\
z_{1} & z_{1}z_{2}
\end{pmatrix}\begin{pmatrix}
1 & z_{2} \\
z_{1} & z_{1}z_{2}
\end{pmatrix}^{\dagger}\Bigg] \Bigg \} - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ Therefore, we have a complete description of different types of geometric structures on the manifolds of the latter diagram in terms of elements of representation theory of ${\rm{SL}}(2,\mathbb{C})$ (cf. [@OrneaPiccinni]).
In the setting of the example described above, consider the analytic complex function
$P \colon {\rm{M}}(2,\mathbb{C}) \to\mathbb{C}$, $\mathcal{W} \mapsto \det(\mathcal{W}).$
From the function above we obtain a family of analytic spaces $(X_{\epsilon})_{\epsilon \geq 0}$, such that
$X_{\epsilon} = P^{-1}(\epsilon^{2}) = \Big \{ \mathcal{W} \in {\rm{M}}(2,\mathbb{C}) \ \Big | \ \det(\mathcal{W}) = \epsilon^{2} \Big \}$
for every $\epsilon \geq 0$. For $\epsilon = 0$, we have $X_{0} = K\mathcal{Q}_{3}$, and for $\epsilon > 0$ we have a smooth analytic space $X_{\epsilon}$ which is in fact Calabi-Yau. Being more precise, by considering the smooth function
$r^{2} \colon X_{\epsilon} \to [\epsilon^{2},+\infty)$, $\mathcal{W} \mapsto {\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)$,
the (Stenzel’s [@Stenzel]) Calabi-Yau metric $\omega_{\epsilon} = \sqrt{-1}\partial \overline{\partial}{\rm{K}}_{\epsilon}$ is obtained by solving the homogeneous Monge-Ampère equation $$\label{mongeampere}
\big ( \sqrt{-1}\partial \overline{\partial}{\rm{K}}_{\epsilon}\big)^{3} = 0,$$ for the ansatz ${\rm{K}}_{\epsilon} = \mathcal{F}_{\epsilon}(r^{2})$, for some smooth function $\mathcal{F}_{\epsilon} \colon [\epsilon^{2},+\infty) \colon \mathbb{R}$. Since in this latter case we have $$\label{stenzelansatz}
\omega_{\epsilon} = \sqrt{-1}\Big [\mathcal{F}'_{\epsilon}{\rm{Tr}}\big(d\mathcal{W} \wedge d\mathcal{W}^{\dagger}\big) + \mathcal{F}''_{\epsilon}{\rm{Tr}}\big(d\mathcal{W}\big) \wedge {\rm{Tr}}\big(d\mathcal{W}^{\dagger}\big)\Big],$$ the problem of solving the aforementioned homogeneous Monge-Ampère equation reduces to the problem of solving the ordinary differential equation $$\label{odeansatz}
{\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)(\mathcal{F}'_{\epsilon})^{3} + \big({\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)^{2} - \epsilon^{4}\big)(\mathcal{F}'_{\epsilon})^{2}\mathcal{F}''_{\epsilon} = 1.$$ In order to solve the ordinary differential equation above, we consider the following: By taking the linear complex change of variables on ${\rm{M}}(2,\mathbb{C})$ defined by $$\label{linearrelations}
X = z_{1} + \sqrt{-1}z_{4}, \ \ U = \sqrt{-1}(z_{2} +\sqrt{-1}z_{3}), \ \ Y = z_{1} - \sqrt{-1}z_{4}, \ \ V = \sqrt{-1}(z_{2} -\sqrt{-1}z_{3}),$$ such that $(z_{1},\ldots, z_{4}) \in \mathbb{C}^{4}$, since
$T^{\ast}S^{3} = \Big \{ \big (p;v \big ) \in S^{3} \times \mathbb{R}^{4} \ \ \Big | \ \ \langle p,v \rangle = 0\Big\},$
from the previous change of variables we obtain a diffeomorphism $\mathcal{W}_{\epsilon } \colon T^{\ast}S^{3} \to X_{\epsilon}$, such that
$\displaystyle \big (p;v \big ) \mapsto \epsilon \Bigg( p \cosh(|v|) + \sqrt{-1}\frac{\sinh(|v|)}{|v|}v \Bigg),$
notice that, $\forall (p;v) \in T^{\ast}S^{3}$, we have that $\mathcal{W}_{\epsilon }(p;v \big )$ is a matrix whose the entries are determined by the functions $$z_{j}(p;v) = \epsilon \Bigg( p_{j} \cosh(|v|) + \sqrt{-1}\frac{\sinh(|v|)}{|v|}v_{j} \Bigg), \ \ j = 1,\ldots,4,$$ subjected to the linear relations \[linearrelations\]. From this, we have $$r^{2}(p;v) = {\rm{Tr}}\big((\mathcal{W}_{\epsilon }(p;v )\mathcal{W}_{\epsilon }(p;v)^{\dagger}\big) = \epsilon^{2}\cosh(2|v|),$$ notice that on the left side of the equation above we have $\mathcal{W}_{\epsilon }^{\ast}(r^{2}) = r^{2} \circ \mathcal{W}_{\epsilon }$. Therefore, by taking $t = 2|v|$, the pullback[^23] of Equation \[odeansatz\] by the diffeomorphism $\mathcal{W}_{\epsilon }$ provides $$\label{PbMA}
\epsilon^{2}\cosh(t)(\mathcal{F}'_{\epsilon})^{3} + \frac{\epsilon^{2}\sinh(t)}{3}\frac{d}{dt}(\mathcal{F}'_{\epsilon})^{3} = 1$$ such that $r^{2} = \epsilon^{2}\cosh(t)$. The ordinary differential equation above can be solved by means of the integrating factor given by $$\mathcal{F}'_{\epsilon}(t) = \Big(\frac{3}{4\epsilon^{2}} \Big)^{\frac{1}{3}}\frac{\sqrt[3]{\sinh(2t) - 2t}}{\sinh(t)},$$ $\forall t \in [0,\infty)$, such that $r^{2} = \epsilon^{2}\cosh(t)$. From the expression above, we can set $$\Upsilon'_{\epsilon}\big({\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)\big) = \mathcal{F}'_{\epsilon}\Big(\cosh^{-1}\Big(\frac{{\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)}{\epsilon^{2}}\Big)\Big) - 1, \ \ \forall \mathcal{W} \in X_{\epsilon}$$ and obtain a complete Calabi-Yau metric $\omega_{\epsilon}$ on $X_{\epsilon}$, $\forall \epsilon >0$, such that ${\rm{K}}_{\epsilon} = r^{2} + \Upsilon_{\epsilon}(r^{2})$, i.e. $$\omega_{\epsilon} = \sqrt{-1}\Big [{\rm{Tr}}\big(d\mathcal{W} \wedge d\mathcal{W}^{\dagger}\big) + \partial \overline{\partial} \big(\Upsilon_{\epsilon}({\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big))\big)\Big ],$$ see Equation \[stenzelansatz\]. For $\epsilon = 0$, by looking at the homogeneous Monge-Ampère equation associated to some ansatz ${\rm{K}}_{0} = \mathcal{F}_{0}(r^{2})$, we obtain the following ordinary differential equation $$\label{odeansatzsingular}
{\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)(\mathcal{F}'_{0})^{3} + {\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)^{2} (\mathcal{F}'_{0})^{2}\mathcal{F}''_{0} = 1.$$ By considering the initial condition $\mathcal{F}_{0}(0) = 0$, the solution for the equation above is
$\mathcal{F}_{0}({\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)) = \Big(\frac{3}{2}\Big)^{\frac{4}{3}}\sqrt[3]{{\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)^{2}}$,
so we have $${\rm{K}}_{0}(\mathcal{W}) = \Big(\frac{3}{2}\Big)^{\frac{4}{3}}\sqrt[3]{{\rm{Tr}}\big(\mathcal{W}\mathcal{W}^{\dagger}\big)^{2}},$$ and obtain a (singular) Calabi-Yau metric $\omega_{0}$ on $X_{0} = K\mathcal{Q}_{3}$ given by $$\omega_{0} = \sqrt{-1}\partial \overline{\partial}\big({\rm{K}}_{0}(\mathcal{W})\big).$$ In the setting of the construction above, the 1-parameter family of Calabi-Yau manifolds $(X_{\epsilon},\omega_{\epsilon})_{\epsilon >0}$ is called a deformation of the singular Calabi-Yau manifold $(X_{0},\omega_{0})$.
We observe that, by considering the biholomorphism $ {\mathscr{R}} \colon {\rm{Tot}}(\mathscr{O}_{\mathcal{Q}_{3}}(-1)^{\times}) \to \big (K\mathcal{Q}_{3}\big)_{{\text{reg}}}$, it follows that
${\rm{K}}_{H} = r^{2} \circ {\mathscr{R}} = \mathscr{R}^{\ast}(r^{2}),$
see Equation \[potentialreduction\] and Remark \[matrixinnerporduct\], so we have that $$\mathscr{R}^{\ast}\omega_{0} = \sqrt{-1}\partial \overline{\partial} \mathcal{F}_{0}({\rm{K}}_{H}).$$ Therefore, if we denote ${\rm{K}}_{St} = \mathcal{F}_{0}({\rm{K}}_{H})$, by considering the Higgs field $\theta_{g}$ on $S^{3} \times S^{2} \times S^{1}$ associated to the Hermian-Einstein-Weyl structure $([g],D,J)$ provided by Theorem \[Theo2\], from Remark \[HEWproductsphres\] we have (locally) the following relation $$\theta_{g} = -d\log \big( {\rm{K}}_{H}(z,w) \big) = - \frac{3}{2}d\log \big( {\rm{K}}_{St}(z,w) \big),$$ here we consider the local expression of ${\rm{K}}_{H}(z,w)$ given in Equation \[potentialspheres\], notice that $${\rm{K}}_{St}(z,w) = \Big(\frac{3}{2}\Big)^{\frac{4}{3}}\Bigg\{ {\rm{Tr}}\Bigg [\begin{pmatrix}
1 & z_{2} \\
z_{1} & z_{1}z_{2}
\end{pmatrix}\begin{pmatrix}
1 & z_{2} \\
z_{1} & z_{1}z_{2}
\end{pmatrix}^{\dagger}\Bigg]w\overline{w} \Bigg \}^{\frac{2}{3}}.$$ The ideas described above provide a constructive method in which the homgeneous Hermian-Einstein-Weyl geometry of $S^{3} \times S^{2} \times S^{1}$ is related to the Calabi-Yau geometry of the family $(X_{\epsilon},\omega_{\epsilon})_{\epsilon >0}$ at the limit $\epsilon \to 0$.
### Cones over Segre varieties
In this subsection, we will show how some ideas involved in the computation for the particular example of $\mathbb{C}{\rm{P}}^{1} \times \mathbb{C}{\rm{P}}^{1}$, provided in the previous section, can be naturally generalized for an arbitrary product $X_{P_{1}} \times X_{P_{2}}$, where $X_{P_{j}} = G_{j}^{\mathbb{C}}/P_{j}$ is a flag manifold associated to a pair $(G_{j}^{\mathbb{C}},P_{j})$, such that $G_{j}^{\mathbb{C}}$ is a simply connected complex simple Lie group, and $P_{j} \subset G_{j}^{\mathbb{C}}$ is some parabolic Lie subgroup, $j = 1,2$.
In order to generalize the ideas of the particular example $\mathbb{C}{\rm{P}}^{1} \times \mathbb{C}{\rm{P}}^{1}$ described previously, in the setting introduced above we consider the following facts:
1. ${\text{Rep}}_{\mathbb{C}}\big(G_{1}^{\mathbb{C}}\times G_{2}^{\mathbb{C}}\big) = {\text{Rep}}_{\mathbb{C}}\big(G_{1}^{\mathbb{C}}\big) \boxtimes {\text{Rep}}_{\mathbb{C}}\big(G_{2}^{\mathbb{C}}\big)$, where ${\text{Rep}}_{\mathbb{C}}\big(G_{j}^{\mathbb{C}})$ is the category of representations of $G_{j}^{\mathbb{C}}$ over $\mathbb{C}$, $j=1,2$, and the symbol “$\boxtimes$” stands for Deligne’s tensor product (e.g. [@Tensorcategory])[^24];
2. Given $L_{j} \in {\text{Pic}}(X_{P_{j}})$, and $\pi_{j} \colon X_{P_{1}} \times X_{P_{2}} \to X_{P_{j}}$, $j = 1,2$, denoting $L_{1} \boxtimes L_{2}:= \big (\pi_{1}^{\ast}L_{1} \big) \otimes \big (\pi_{2}^{\ast}L_{2} \big)$, from the Künneth formula[^25], we have
$H^{0}\big (X_{P_{1}} \times X_{P_{2}},L_{1} \boxtimes L_{2} \big) \cong H^{0}\big(X_{P_{1}} ,L_{1} \big) \boxtimes H^{0}\big(X_{P_{2}} ,L_{2}\big)$.
From the facts listed above, given $L_{1} \boxtimes L_{2} \in {\text{Pic}}(X_{P_{1}} \times X_{P_{2}})$, such that $c_{1}(L_{j}) < 0$, by identifying $X_{P_{1}} \cong G_{j}^{\mathbb{C}}\cdot [v_{\mu(L_{j})}^{+}] $, $j = 1,2$, one can consider the Segre embedding defined by
$X_{P_{1}} \times X_{P_{2}} \subset \mathbb{P}(V(\mu(L_{1}))) \times \mathbb{P}(V(\mu(L_{2}))) \hookrightarrow \mathbb{P}\big(V(\mu(L_{1})) \boxtimes V(\mu(L_{2}))\big)$.
For the sake of simplicity, let us denote $X = X_{P_{1}} \times X_{P_{2}}$. From the embedding defined above, we obtain
$X \cong G_{1}^{\mathbb{C}} \times G_{2}^{\mathbb{C}} \cdot \big [v_{\mu(L_{1})}^{+} \boxtimes v_{\mu(L_{2})}^{+}\big]$,
and $${\rm{Tot}}\big ( (L_{1} \boxtimes L_{2})^{\times}\big) \cong C\big(X_{P},L_{1} \boxtimes L_{2}\big)_{{\text{reg}}} = G_{1}^{\mathbb{C}} \times G_{2}^{\mathbb{C}} \cdot \big (v_{\mu(L_{1})}^{+} \boxtimes v_{\mu(L_{2})}^{+}\big),$$ where $C(X,L_{1} \boxtimes L_{2})$ denotes the affine cone over the image of the Segre embedding.
Now, if we consider a smooth section $s_{U} \colon U \subset X \to G_{1}^{\mathbb{C}} \times G_{2}^{\mathbb{C}}$, by fixing a $G_{1} \times G_{2}$-invariant inner product $\langle \cdot ,\cdot \rangle$ on $V(\mu(L_{1})) \boxtimes V(\mu(L_{2}))$, we can consider ${\rm{K}}_{H} \colon {\rm{Tot}}\big ( (L_{1} \boxtimes L_{2})^{\times}\big) \to \mathbb{R}^{+}$, such that (locally) $$\label{potentialdeligne}
{\rm{K}}_{H}\big (z,w \big ) = \Big (\big | \big |s_{U}(z)\big (v_{\mu(L_{1})}^{+} \boxtimes v_{\mu(L_{2})}^{+}\big)\big | \big |^{2} \Big) w\overline{w}.$$ The smooth function above defines a Kähler potential which allows us to equip the diagram below with geometric structures as follows:
([[Tot]{}]{}( (L\_[1]{} L\_[2]{})\^),\_,J\_ ) & ([[Tot]{}]{}( (L\_[1]{} L\_[2]{})\^)/,g,J ) & (Q(L\_[1]{} L\_[2]{}),g\_[Q(L\_[1]{} L\_[2]{})]{},,,)\
& (X,\_[X]{})
Similarly to the construction presented in Subsection \[negativeelliptic\], the geometric structures obtained from ${\rm{K}}_{H}$ are:
- $\big ({\rm{Tot}}\big ( (L_{1} \boxtimes L_{2})^{\times}\big),\omega_{\mathscr{C}},J_{\mathscr{C}} \big )$ and $\big (X,\omega_{X} \big)$ (Kähler structures);
- $\big ({\rm{Tot}}\big ( (L_{1} \boxtimes L_{2})^{\times}\big)/\Gamma,g,J \big )$ (l.c.K. structure);
- $\big (Q(L_{1} \boxtimes L_{2}),g_{Q(L_{1} \boxtimes L_{2})},\phi,\xi,\eta \big)$ (Sasaki structure).
In order to make more clear the expression of \[potentialdeligne\], we fix a $G_{j}$-invariant inner product $\langle \cdot ,\cdot \rangle_{j}$ on $V(\mu(L_{j}))$, so from this we can take an orthonormal basis $f_{1}^{(j)} = v_{\mu(L_{j})}^{+}, f_{1}^{(j)}, \ldots,f_{d_{j}}^{(j)}$ for $V(\mu(L_{j}))$, here we suppose $\dim_{\mathbb{C}}(V(\mu(L_{j}))) = d_{j}$, and obtain an isomorphism $V(\mu(L_{j})) \cong \mathbb{C}^{d_{j}}$, $j=1,2$. Now, we can write
$gP_{1} \in X_{P_{1}} \mapsto \big [ u_{1}(g) : \ldots : u_{d_{j}}(g) \big ] \in \mathbb{C}{\rm{P}}^{d_{1}-1}$ and $gP_{2} \in X_{P_{2}} \mapsto \big [ v_{1}(g) : \ldots : v_{d_{j}}(g) \big ] \in \mathbb{C}{\rm{P}}^{d_{2}-1}$.
Therefore, if we consider the (outer) tensor product $\mathbb{C}^{d_{1}} \boxtimes \mathbb{C}^{d_{2}}= \big({\rm{M}}_{d_{1}\times d_{2}}(\mathbb{C}),\boxtimes \big)$, such that
$\boxtimes \colon \big ((u_{1},\ldots, u_{d_{1}}),(v_{1},\ldots, v_{d_{2}})\big) \mapsto\begin{pmatrix}
u_{1}v_{1} & \cdots &u_{1}v_{d_{2}} \\
\vdots & \ddots & \vdots \\
u_{d_{1}}v_{1} & \cdots & u_{d_{1}}v_{d_{2}}
\end{pmatrix}$,
$\forall (u_{1},\ldots, u_{d_{1}}) \in \mathbb{C}^{d_{1}}$ and $\forall (v_{1},\ldots, v_{d_{2}}) \in \mathbb{C}^{d_{2}}$, we obtain
$\big (gP_{1},hP_{2} \big) \in X_{P_{1}} \times X_{P_{2}} \mapsto \big [ W(g,h) \big] \in \mathbb{P}\big ({\rm{M}}_{d_{1}\times d_{2}}(\mathbb{C}) \big).$
such that $W(g,h) = (W_{ij}(g,h))$, where $W_{ij}(g,h) = u_{i}(g)v_{j}(h)$, for $i = 1,\ldots,d_{1}$, and $j =1,\ldots,d_{2}$. From this, if we consider $U = U_{1}\times U_{2}$, such that $U_{j} = R_{u}(P_{j})^{-}P_{j} \subset X_{P_{j}}$ (opposite big cell), $j = 1,2$, we obtain a parameterization
$Z = (z_{1},z_{2}) \in \mathbb{C}^{m_{1}} \times \mathbb{C}^{m_{2}} \mapsto \big(n(z_{1})P_{1},n(z_{2})P_{2}\big )\in U$,
such that $\dim_{\mathbb{C}}(X_{P_{j}}) = m_{j}$, $j= 1,2$. Then, we consider the local section $s_{U} \colon U \subset X_{P_{1}} \times X_{P_{2}} \to G_{1}^{\mathbb{C}} \times G_{2}^{\mathbb{C}}$, defined by
$s_{U} \colon \big(n(z_{1})P_{1},n(z_{2})P_{2}\big ) \mapsto \big (n(z_{1}), n(z_{2}) \big).$
Now, as before, we can take the inner product $\langle \cdot ,\cdot \rangle \colon {\rm{M}}_{d_{1}\times d_{2}}(\mathbb{C}) \times {\rm{M}}_{d_{1}\times d_{2}}(\mathbb{C}) \to \mathbb{R}$, such that
$\big \langle W_{1}, W_{2} \big \rangle = \Re\Big({\rm{Tr}}\big(W_{1}W_{2}^{\dagger}\big)\Big ),$
$\forall W_{1},W_{2} \in {\rm{M}}_{d_{1}\times d_{2}}(\mathbb{C})$, here we consider $W^{\dagger} = \overline{W}^{T}$. Notice that
${\rm{Tr}}\big((u\boxtimes v)(u\boxtimes v)^{\dagger}\big) = \big|\big|u\big|\big|_{1}^{2}\big|\big|v\big|\big|_{2}^{2},$
so we have the the inner product above is $G_{1} \times G_{2}$-invariant. From the data above, the potential \[potentialdeligne\] can be written as $$\label{Delignepotential}
{\rm{K}}_{H}\big (Z,w \big ) = {\rm{Tr}}\big(W(z_{1},z_{2})W(z_{1},z_{2})^{\dagger}\big)w\overline{w} = \Big (\big|\big|s_{U_{1}}(z_{1})v_{\mu(L_{1})}^{+}\big|\big|_{1}^{2}\big|\big|s_{U_{2}}(z_{2})v_{\mu(L_{2})}^{+}\big|\big|_{2}^{2} \Big )w\overline{w},$$ such that $s_{U_{j}} \colon U_{j} \subset X_{P_{j}} \to G_{j}^{\mathbb{C}}$, $n(z_{j})P_{j} \mapsto n(z_{j})$, $j = 1,2$.
From the construction above, if one considers the particular case that $X = \mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}$, by taking
$\mathscr{O}_{\mathbb{C}{\rm{P}}^{m}}(-1) \boxtimes \mathscr{O}_{\mathbb{C}{\rm{P}}^{m}}(-1) \to \mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m},$
we obtain the Segre embedding
$\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m} \hookrightarrow \mathbb{P}\big(\mathbb{C}^{m+1} \boxtimes \mathbb{C}^{m+1}\big) \cong \mathbb{P}\big({\rm{M}}(m+1,\mathbb{C})\big).$
By identifying $\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}$ with its image through the Segre map above, we can consider the principal circle bundle
$S^{1} \hookrightarrow V_{m,m} \to \mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}$,
such that $$V_{m,m} = \Big ( {\rm{SL}}(m+1,\mathbb{C})\times{\rm{SL}}(m+1,\mathbb{C})\big(e_{1}\boxtimes e_{1}\big) \Big ) \cap S^{2(m+1)^{2}-1}.$$ Now, by considering the same Lie-theoretical data as in Example \[exampleP1\], in this particular, the potential given in Equation \[Delignepotential\] takes the following concrete expression: $$\displaystyle {\rm{K}}_{H}(Z,w) = \bigg (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \bigg ) \bigg (1+ \sum_{k = 1}^{m}|z'_{k}|^{2} \bigg )w\overline{w},$$ here we consider coordinates $Z=(z,z') \in \mathbb{C}^{m_{1}} \times \mathbb{C}^{m_{2}}\cong U \subset \mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}$. As we have mentioned, the potential above allows to obtain different types of geometric structures, particularly, we have from the potential above a Sasaki structure $(g,\phi,\xi,\eta)$ on $V_{m,m}$. In fact, it is not difficult to see that
${\rm{e}}\big (V_{m,m} \big) = c_{1}\big(\mathscr{O}_{\mathbb{C}{\rm{P}}^{m}}(-1) \boxtimes \mathscr{O}_{\mathbb{C}{\rm{P}}^{m}}(-1)\big),$
so by identifying $V_{m,m}$ with the sphere bundle of $\mathscr{O}_{\mathbb{C}{\rm{P}}^{m}}(-1) \boxtimes \mathscr{O}_{\mathbb{C}{\rm{P}}^{m}}(-1)$, we obtain the desired Sasaki structure $(g,\phi,\xi,\eta)$ as in Equation \[sasakistructure\]. Notice that the underlying contact structure $\eta$ on $V_{m,m}$ can be described locally as $$\eta = \frac{1}{2}\Bigg [d^{c}\log \bigg (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \bigg ) + d^{c}\log \bigg (1+ \sum_{k = 1}^{m}|z'_{k}|^{2} \bigg )\Bigg ] + d\sigma_{U},$$ so we have $d\eta = \pi^{\ast}\omega_{\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}}$, such that $\omega_{\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}}$ is a Kähler structure on $\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}$, satisfying $${\text{Ric}}\big (\omega_{\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}} \big) = (m+1)\omega_{\mathbb{C}{\rm{P}}^{m} \times \mathbb{C}{\rm{P}}^{m}}.$$ Therefore, after a suitable $\mathscr{D}$-homothetic transformation on $(g,\phi,\xi,\eta)$, such that
$\displaystyle a = \frac{m+1}{2m+1},$
see Remark \[homothetic\], we obtain an explicitly Sasaki-Einstein structure $(g_{a},\phi,\frac{1}{a}\xi,a\eta)$ on $V_{m,m}$, whose the associated Sasaki-Eisntein metric is given by $$g_{a} = \displaystyle \frac{m+1}{2m+1} \Bigg ( \frac{1}{2}d \eta ({\rm{id}} \otimes \phi) + \frac{m+1}{2m+1}\eta \otimes \eta \Bigg ).$$ Further, proceeding as in the proof of Theorem \[Theo2\], we obtain from the previous potential a (locally Ricci-flat) l.c.K. structure $(\Omega,J,\theta)$ on $V_{m,m} \times S^{1}$ completely determined by the Lee form $$\theta = -\Bigg [d\log \bigg (1+ \sum_{k = 1}^{m}|z_{k}|^{2} \bigg ) + d\log \bigg (1+ \sum_{k = 1}^{m}|z'_{k}|^{2} \bigg )\Bigg ] - \frac{\overline{w}dw + wd\overline{w}}{|w|^{2}}.$$ Therefore, the general construction described above for $X = X_{P_{1}} \times X_{P_{2}}$ provides a constructive explicit method to generalize the results introduced in [@OrneaPiccinni] on Sasaki-Einstein metrics on $V_{m,m}$ and (locally Ricci-flat) l.c.K. structures on $V_{m,m} \times S^{1}$.
[BGGSM]{}
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[^1]: Here we consider the fact that a holomorphic line bundle over a complex flag manifold is positive, in the sense of Kodaira, if and only if it is very ample, see [@BorelK Theorem 2] and [@PARABOLICTHEORY Theorem 3.2.8].
[^2]: Here we consider $\beta_{1} \odot \beta_{2} = \beta_{1} \otimes \beta_{2} + \beta_{2} \otimes \beta_{1}, \forall \beta_{1},\beta_{2} \in \Omega^{1}(M)$. Thus, we have $\beta \odot \overline{\beta} = 2\Re\big ( \beta \otimes \overline{\beta}\big )$, $\forall \beta \in \Omega^{1}(M) \otimes \mathbb{C}.$
[^3]: Notice that, from Remark \[realhomolomrphic\] it follows that $r\partial_{r}$ is real holomorphic, i.e. $\mathscr{L}_{r\partial_{r}}J_{\mathscr{C}} = 0$, which implies that that $\xi = J_{\mathscr{C}}(r\partial_{r})$ is real holomorphic, so we have that $\xi - \sqrt{-1}J_{\mathscr{C}}(\xi)$ is holomorphic, see for instance [@Moroianukahler].
[^4]: It is a consequence of the fact that $\xi$ and $r\partial_{r}$ are real holomorphic vector fields, so their associated flow are biholomorphism of $(\mathscr{C}(Q(L)),J_{\mathscr{C}})$. From this, it is not difficult to see that $\varphi_{\lambda} \colon \mathscr{C}(Q(L)) \to \mathscr{C}(Q(L))$ is obtained from the flow of $\xi$ and $r\partial_{r}$.
[^5]: We say that two Riemannian metrics $g$ and $g'$ are equivalent if and only if $g' = {\rm{e}}^{f}g$, where $f$ is a smooth function on $M$.
[^6]: For the sake of simplicity, we shall denote by $\nabla = \nabla^{g}$ when the choice of the Riemannian metric is implicit.
[^7]: Given a Hermitian manifold $(M,g,J)$, with fundamental $2$-form $\Omega = g(J\otimes{\text{id}})$, one can consider the [*Lefschetz*]{} operator $L \colon \bigwedge^{k}T^{\ast}M \to \bigwedge^{k+2}T^{\ast}M$, such that $L(\psi) = \psi \wedge \Omega$, $\forall \psi \in \bigwedge^{k}T^{\ast}M$, see for instance [@DANIEL]. From this operator we have a vector bundle isomorphism $L^{n-k} \colon \bigwedge^{k}T^{\ast}M \to \bigwedge^{2n-k}T^{\ast}M$, such that $L^{n-k}(\psi) = \psi \wedge \Omega^{n-k}$, $\forall \psi \in \bigwedge^{k}T^{\ast}M$, see for instance [@DANIEL $\S$ 1.3 and $\S$ 3.1], [@Voisin].
[^8]: Notice that the condition $||\nabla f||_{g} = 1$ implies that $f(\gamma(p,t)) = f(p) + t$, $\forall (p,t) \in \widetilde{M} \times \mathbb{R}$, where $t \mapsto \gamma(p,t)$ is the flow of $\nabla f$ through the point $p \in \widetilde{M}$.
[^9]: By considering the characterization $T\widetilde{Q} = \ker(df)$, one can define a Sasaki structure $(g_{\widetilde{Q}},\phi,\xi,\eta)$ by setting $g_{\widetilde{Q}} = g|_{\ker(df)}$, $\xi = J\nabla f |_{\widetilde{Q}}$, $\eta = -(df \circ J) |_{\ker(df)} $, and $\phi = J|_{\ker(df)} + \eta \otimes \xi$, see for instance [@Dragomir], [@VaismanI].
[^10]: Note that, $d\Omega_{p} = \int_{K}(k^{\ast}d\Omega)_{p}d\nu = \Big(\int_{K}(k^{\ast}\theta)_{p}d\nu\Big) \wedge \Omega_{p}$, $\forall p \in M$, here $d\nu$ denotes the Haar measure on $K$ normalized, i.e. $\int_{K} 1 \cdot d\nu = 1$. Thus, since $d\Big ( \int_{K}(k^{\ast}\theta)d\nu\Big) = 0$, we can suppose that $\theta = \int_{K}(k^{\ast}\theta)d\nu$.
[^11]: Observe that in the setting of Theorem \[LCKisVaisman\], it follows that $H^{1}(M,\mathbb{R}) \cong H_{K}^{1}(M,\mathbb{R})$, where $H_{K}^{\bullet}(M,\mathbb{R})$ denotes the invariant cohomology group of $(M,g,J)$ as a left $K$-manifold cf. [@Algmodels Theorem 1.28].
[^12]: See for instance [@Greub].
[^13]: It is worth pointing out that, the condition $||\nabla f||_{\widetilde{g}} = 1$ implies that $\nabla f$ is complete, see for instance [@Udriste Theorem 1.7].
[^14]: It is worthwhile to observe that $v_{2\varrho}^{+} = v_{\varrho}^{+} \otimes v_{\varrho}^{+}$, such that $v_{\varrho}^{+} \in V(\varrho)$, see for instance [@Procesi Page 345].
[^15]: Here we use the notation “" to stand for the complex algebraic variety underlying the complex analytic affine cone over a flag manifold.
[^16]: Here we consider the divisor ${\rm{div}}(f) = \{v \in C(X_{P},L)_{{\text{reg}}} \ | \ f(v) = 0\}$.
[^17]: Notice that the Cartan-Remmert reduction $\mathscr{R} \colon {\rm{Tot}}(L) \to Y$, in this particular case, is defined by $\mathscr{R}([g,z]) = zgv_{\mu(L)}^{+}$, $\forall [g,z] \in {\rm{Tot}}(L)$. Also, if we consider the zero section $\sigma_{0} \colon X_{P} \to L$, such that $\sigma_{0}(gP) = [g,0]$, $\forall gP \in X_{P}$, it follows that $\mathscr{R}(\sigma_{0}(X_{P})) = \{0\} \subset Y$.
[^18]: For more details on this subject we suggest [@Meinrenken], [@Lawson] and [@Onishchik].
[^19]: A morphism between two quadratic vector spaces $(V_{1},q_{1})$ and $(V_{2},q_{2})$ is a linear map $f \colon V_{1} \to V_{2}$, which satisfies $q_{2}(f(u),f(v)) = q_{1}(u,v)$, $\forall u,v \in V_{1}$. Given a morphism of quadratic vector spaces $f \colon (V_{1},q_{1}) \to (V_{2},q_{2})$, there is a unique morphism ${\rm{Cl}}(f) \colon {\rm{Cl}}(V_{1},q_{1}) \to {\rm{Cl}}(V_{2},q_{2})$, which extends $f$, see for instance [@Meinrenken], [@Lawson].
[^20]: For more details about resolutions of Kleinian singularities, in the general setting, we suggest [@BrieskornI], [@DuVal], [@Artin].
[^21]: Here we consider $\mathcal{W}^{\dagger} = \overline{\mathcal{W}}^{T}$, $\forall \mathcal{W} \in {\rm{M}}(2,\mathbb{C})$. It is worth mentioning that the Cartan-Killing form $\kappa$ of $\mathfrak{sl}(2,\mathbb{C})$ is defined by $\kappa(\mathcal{W}_{1},\mathcal{W}_{2}) = 4{\rm{Tr}}(\mathcal{W}_{1}\mathcal{W}_{2})$, $\forall \mathcal{W}_{1}, \mathcal{W}_{2} \in \mathfrak{sl}(2,\mathbb{C})$.
[^22]: The conifold has been playing an important role in the study of conformal field theory and string theory, e.g. [@Candelas], [@Klebanov].
[^23]: Despite the fact that $T^{\ast}S^{3}$ is not a complex submanifold of $\mathbb{R}^{4} \times \mathbb{R}^{4} \cong \mathbb{C}^{4}$, it can be endowed with a complex structure by pulling back the complex structure of $X_{\epsilon}$ through the diffeomorphism $\mathcal{W}_{\epsilon } \colon T^{\ast}S^{3} \to X_{\epsilon}$. From this, we can pullback the homogeneous Monge-Ampère equation \[mongeampere\], which leads to the ordinary differential equation \[PbMA\].
[^24]: Notice that $\mathscr{O}(G_{1}^{\mathbb{C}}\times G_{2}^{\mathbb{C}}) = \mathscr{O}(G_{1}^{\mathbb{C}}) \otimes \mathscr{O}(G_{2}^{\mathbb{C}})$, and $ \mathscr{O}(G_{j}^{\mathbb{C}}) = \bigoplus_{V \in {\text{Rep}}_{\mathbb{C}}\big(G_{j}^{\mathbb{C}}\big)}{\rm{End}}(V)$, $j = 1,2$, see for instance [@Procesi].
[^25]: See for instance [@Kempf].
|
Q:
All sub menu items are being displayed
I am trying to create a navigation menu similar to sample link
I have created a jquery and css based navigation menu from scratch, which should display the dropdown menu as shown in the above link.I want to achieve dropdown style as shown in the above menu.
I have created 4 depth-level menus and when I open all the menu-items once and when I click on the first depth level menu-item all the menu-items till depth level 4 are being displayed.
click on menu-item1>menu-item1.2>menu-item1.2.2>menu-item1.2.2.2
now click on menu-item1.2 twice (all sub-items are being display instead of only menu-item1.2.1 and menu-item1.2.2)
where I am doing mistake?
$(document).ready(function(){
$('.menu-item').click(function(event){
event.stopPropagation();
$(this).toggleClass('active');
$('.col ul li').removeClass('active');
});
$('.col ul li').click(function(event){
event.stopPropagation();
$(this).toggleClass('active');
$('.col ul li ul').removeClass('active');
});
$('.col ul li ul').click(function(event){
$(this).toggleClass('active');
});
$('.col ul ul li ul').click(function(event){
$(this).toggleClass('active');
});
});
.menu>li {
display: inline-block;
padding: 10px 20px;
}
.dropdown > .col {
display: none;
}
.visible {
display: block;
}
.col {
position: absolute;
top: 100px;
display: block;
}
.sub-col {
position: absolute;
left: 150px;
top: 0;
display: none;
}
.sub-sub-col {
position: absolute;
left: 150px;
top: 0;
display: none;
}
.sub-sub-sub-col {
position: absolute;
left: 150px;
top: 0;
display: none;
}
.active .col {
display: block;
}
.col .active .sub-col{
display: block;
}
.col .sub-col .active .sub-sub-col {
display: block;
}
.col .sub-col .sub-sub-col .active .sub-sub-sub-col{
display: block;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<ul class="menu">
<li class="menu-item"><a href="#">Menu-item-1</a>
<div class="dropdown">
<div class="col">
<ul>
<li>menu-item: 1.1</li>
<li>menu-item: 1.2
<div class="sub-col">
<ul>
<li>menuitem1.2.1</li>
<li>menuitem1.2.2
<div class="sub-sub-col">
<ul>
<li>menuitem1.2.2.1</li>
<li>menuitem1.2.2.2
<div class="sub-sub-sub-col">
<ul>
<li>menuitem1.2.2.2.1</li>
<li>menuitem1.2.2.2.2</li>
</ul>
</div>
</li>
</ul>
</div><!--sub-sub-col-->
</li>
</ul>
</div><!--sub-col-->
</li>
</ul>
</div> <!--col-->
</div><!--dropdown-->
</li>
<li class="menu-item"><a href="#">Menu-item-2</a></li>
</ul>
A:
try this one
change .col ul ul li to .col ul li ul and change .col ul ul ul li to .col ul ul li ul
$(document).ready(function() {
$('.menu-item').click(function(event) {
event.stopPropagation();
$(this).toggleClass('active');
$('.col ul li').removeClass('active');
});
$('.col ul li').click(function(event) {
event.stopPropagation();
$(this).toggleClass('active');
});
$('.col ul li ul').click(function(event) {
$(this).toggleClass('active');
});
$('.col ul ul li ul ').click(function(event) {
$(this).toggleClass('active');
});
});
.menu>li {
display: inline-block;
padding: 10px 20px;
}
.dropdown>.col {
display: none;
}
.visible {
display: block;
}
.col {
position: absolute;
top: 100px;
display: block;
}
.sub-col {
position: absolute;
left: 150px;
top: 0;
display: none;
}
.sub-sub-col {
position: absolute;
left: 150px;
top: 0;
display: none;
}
.sub-sub-sub-col {
position: absolute;
left: 150px;
top: 0;
display: none;
}
.active .col {
display: block;
}
.col .active .sub-col {
display: block;
}
.col .sub-col .active .sub-sub-col {
display: block;
}
.col .sub-col .sub-sub-col .active .sub-sub-sub-col {
display: block;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<ul class="menu">
<li class="menu-item"><a href="#">Menu-item-1</a>
<div class="dropdown">
<div class="col">
<ul>
<li>menu-item: 1.1</li>
<li>menu-item: 1.2
<div class="sub-col">
<ul>
<li>menuitem1.2.1</li>
<li>menuitem1.2.2
<div class="sub-sub-col">
<ul>
<li>menuitem1.2.2.1</li>
<li>menuitem1.2.2.2
<div class="sub-sub-sub-col">
<ul>
<li>menuitem1.2.2.2.1</li>
<li>menuitem1.2.2.2.2</li>
</ul>
</div>
</li>
</ul>
</div>
<!--sub-sub-col-->
</li>
</ul>
</div>
<!--sub-col-->
</li>
</ul>
</div>
<!--col-->
</div>
<!--dropdown-->
</li>
<li class="menu-item"><a href="#">Menu-item-2</a></li>
</ul>
|
Weather
HOT LUNCH PROGRAM
posted Sep 21, 2016, 9:28 AM by Gary Clayton
The cost of hot lunch this year is $2.70 and milk is .35 cents. If your child would like to purchase an extra milk/juice or a milk/juice for their cold lunch, please send cash. We are no longer allowing charges for extra milk or juice purchases. |
/* eslint-env jest */
import Bossbat from '../index';
const BossbatCJS = require('../index');
describe('index', () => {
it('exports for commonJS and ES modules', () => {
expect(Bossbat).toEqual(BossbatCJS);
});
});
|
NASCAR’s Bank of America 500 brings us a most confusing Tale of Two Kyles. One Kyle wants to pit. One Kyle might not want to pit. Both end up in a hit. It’s Kyle-on-Kyle action, but not the kind you’d expect from terrible NASCAR fanfiction. No, this was from today’s race at Charlotte Motor Speedway.
The Sprint Cup field was bunched up behind a safety car after Dale Earnhardt Jr.’s car scraped the wall. Kyle Busch in the pink number 18 car looked as if he was going to go into the pits. Further up the banking, Kyle Larson’s number red 42 car looked to be staying out on course.
Unfortunately, Kyle Larson decided that he wanted to make a last-minute cut back down into pit lane at about the same time Kyle Busch decided he’d like to turn back up onto the track.
Busch spun Larson’s car around, with the number 42 barely missing the pit lane wall in the process.
“What the [expletive] was he thinking?” Kyle Busch radioed in to his team, as quoted by NASCAR. “He was never going to make the [expletive] commitment cone anyway.”
Busch is a contender in the Chase for the Sprint Cup, so to say he’s a tad on edge out right now might be an understatement.
Associated Press NASCAR reporter Jenna Fryer said that it sounded like Larson was told that they would pit for 4 tires, but then the call was made to stay out if the rest of the field looked to be staying out. If that’s the case, it’s almost understandable Busch’s almost-entry would trigger Larson to move down into pit lane. Even then, Larson was still past the commitment cone for entering pit lane and thus, probably should have caught the pits he next lap around.
Larson was incredibly sorry for the screw-up.
“I’m sure it doesn’t matter, but please apologize as much as you can,” said Larson over his team radio, as quoted by NASCAR.
Fortunately, neither driver was hurt in the bizarre crash. After several repairs were made to his car, Busch rejoined in 23rd position, with Larson right behind him in 24th. They were the last two cars on the lead lap when they rejoined the field.
Contact the author at stef.schrader@jalopnik.com.
|
Clinical and biological significance of never in mitosis gene A-related kinase 6 (NEK6) expression in hepatic cell cancer.
Nek6 is a cell cycle regulatory gene, which can control cell proliferation and survival. Recent studies suggested that desregulation of Nek6 expression plays a key role in oncogenesis. This study was aimed to investigate the potential roles of Nek6 in hepatocellular carcinoma (HCC) development. Immunohistochemistry and Western blot analysis was performed for Nek6 in 80 hepatocellular carcinoma samples. The data were correlated with clinicopathological features. The univariate and multivariate survival analyses were performed to determine the prognostic significance of Nek6 in HCC. In addition, Nek6 expression vector was used to detect its role in cell cycle control. Nek6 was overexpressed in hepatocellular carcinoma as compared with the adjacent normal tissue. High expression of Nek6 was associated with histological grade and the level of alpha fetal protein, and Nek6 was positively correlated with proliferation marker Ki-67. Univariate analysis showed that Nek6 expression was associated with poor prognosis. Multivariate analysis indicated that Nek6 and Ki-67 protein expression was an independent prognostic marker for HCC. While in vitro, following release from serum starvation of HuH7 HCC cell, the expression of Nek6 was upregulated. Overexpression Nek6 in Huh7 cell could promote the cell cycle. In conclusion, Nek6 is involved in the pathogenesis of hepatocellular carcinoma. It may be a favorable independent poor prognostic parameter for hepatocellular carcinoma. |
Brandon Jennings is going to bypass his freshman year in college to play professionally in Europe. Jennings wasn't eligible for the NBA draft because the league has a rule stating players must be a year removed from high school.
By Steve Wieberg and Jim Halley, USA TODAY
TUCSON Is Brandon Jennings blazing a new trail to the NBA?
Forced to wait a year before entering the league's annual draft, the gifted young point guard says he'll spend that time playing professionally in Europe rather than in college, an unprecedented move that will be watched by other underage prospects and officials in the NBA and college basketball.
Succeed, and others are almost certain to follow.
Fail, and Jennings risks becoming a footnote.
"I think he'll do very well," says Louisville coach Rick Pitino, who spent Wednesday monitoring the Reebok All American Camp in Philadelphia. "But with other kids, if they don't last long, then their (basketball-playing) life is over. It could be a big gamble but, in his case, it's a worthwhile gamble."
Says a less optimistic Doug Gottlieb, the former Oklahoma State guard who played overseas and then turned ESPN analyst, "I'm not sure anybody is ready to go over there as an 18- or 19-year-old."
Jennings, a 6-2 lefty deemed the nation's best schoolboy point guard coming out of Oak Hill Academy in Mouth of Wilson, Va., confirmed his decision Tuesday night along with his attorney, Jeff Valle. Jennings had committed to attend and play at Arizona, but his eligibility was in doubt pending the results of his third attempt to earn a qualifying score on the SAT.
The NBA isn't an option until 2009 because of the league's rule holding players out of its draft until they're 19 and a year out of high school. Jennings, an all-USA selection, turns 19 on Sept. 23.
Traditionally, that year-in-waiting is spent in college — even though players might have little to no academic enthusiasm. Jennings' reroute through Europe uncovers an alternative.
"It gives him a chance to make some money and see a different side of basketball overseas," says Oderah Anosike, a Staten Island, N.Y., senior-to-be participating in the Reebok camp in Philadelphia. "(And) it's a good move for guys who are highly rated and who don't like school."
Donn Nelson, the Dallas Mavericks' president of basketball operations and general manager and one of the NBA's most internationally connected executives, likes the point guard's moxie.
"It's a brave undertaking," he says. "I don't know if it's about the grades or financial pressures or what have you, but a young man who's willing to do that … step out of the box and try something like that, I have to take my hat off to him. I think others in the league would look at it the same way."
Nelson predicts Jennings will benefit from the unrestricted practice time and higher level of competition in Europe. "Generally speaking, if you want to be good, go where the real competition is," Nelson says. "Playing in Italy, Spain, Greece, Turkey or France — in those really good A leagues — you're playing against pros. In college, you might only get a pro or two or maybe three a night."
Longtime shoe company executive Sonny Vaccaro, acting as an adviser to Jennings, says he has been in contact with "numerous" teams in Europe, Israel and Russia since Jennings' family contacted him a few weeks ago to gauge interest overseas. In the 24 hours after the player's decision was made known Tuesday night, Vaccaro said, shoe and soda company representatives also have been in touch regarding potential endorsement deals.
Vaccaro has neither reservations about Jennings' decision nor doubts about its impact.
"He's opened the floodgates," he says. "This is unmistakably a groundbreaking decision. Naysayers are going to say 10 years from now, 'That kid from Compton (Calif.) did this.' "
Says Jennings' Oak Hill coach, Steve Smith: "This is a business decision. That's the way he looks at it. He might be wiser than people think. … He's going to get a shoe deal by the end of the summer. People think he's not going to make much money. He'll make more on a shoe deal than on a contract. If he goes to college, he's not going to make anything. Brandon wants to play, and he wants to get paid to play. Who are we to say it's right or wrong?"
Others are more cautious.
There are questions and concerns about a teenager making the competitive and cultural leap to European pro basketball. "You have to understand," Pitino says. "You're going over to Europe, on foreign soil, where people don't speak your language. It becomes a job. It's not college."
ESPN's Jay Bilas, who played professionally in Italy, says even well-traveled elite high school players aren't necessarily prepared for another continent: "It was a totally different culture, and I was 22 years old and had a little bit of experience behind me. It's a lot harder when you're 18 years old ... and you're playing against men and expected to carry a professional load. I'm not saying the kid can't do it. But he's still a young man."
Cautions Gottlieb: "So much of basketball is being in just the right situation. Having the right coach who is willing to work with you. Are you playing for the right team that will understand you are a rookie point guard? Do you get along with other players? Do you get along with people in the town? Do you adjust culturally?
"I think it's interesting. It could be really, really bad for all parties or really, really good for European basketball and give people the understanding how hard it is to succeed as a young player."
NCAA President Myles Brand says this is evidence that athletes need to concentrate on academics.
"We wish Mr. Jennings well," Brand said in a statement. "He's made his decision, and college basketball at Arizona and elsewhere will move on. The most important point to be made about the NBA rule, which the NCAA had no role in making, is not where high school basketball athletes decide to play, but that they can't ignore their high school academic work and expect to play college ball."
To report corrections and clarifications, contact Reader Editor Brent Jones. For publication consideration in the newspaper, send comments to letters@usatoday.com. Include name, phone number, city and state for verification.
Guidelines: You share in the USA TODAY community, so please keep your comments smart and civil. Don't attack other readers personally, and keep your language decent. Use the "Report Abuse" button to make a difference. Read more. |
Q:
"Sign In" button of yahoo mail is not clicked by WebDriver using Java
I am using Selenium WebDriver(a.k.a. Selenium 2) with Java. For clicking "Sign In" button of Yahoo mail, I have written the following code:
driver.findElement(By.id(".save")).click();
But unfortunately, it is not working. What's the wrong with my code? Can anybody help me?
A:
If the browser is not maximized during execution of the test, please maximize the browser by using the following line of code:
driver.manage().window().maximize();
Then, "Sign In" button would be visible on the page and hopefully it should work.
|
The reason you stay vertical is because you have the "free leg" out in front of you or behind you to catch you from "falling" on your face. If you took the “free leg” out of the equation then you would “fall” flat on your face. The exercise to this action is to get used to the feel of “falling” on your face. The partner helps you so that you don’t actually reorganize your face.
Remember there is a big difference between what you feel, what you do and what you look like. Don’t get them mixed up as it will mix up your dancing and make everything take longer to understand and do.
Again this is one of those things that have to be experienced to be understood. There is really no other way of explaining it other then saying you should “fall flat on your face”. You understand the words as of right now and the words don’t make sense until you have experienced the action full out. You do have to get out of the “box” to get a full understanding of what is really meant.
My teachers told me to “fall on my face” for months and I totally understood the words but I was not doing it. It was not until I did the exercise and then implemented the exercise into my dancing that I really understood the words. I remember the day that I told my teachers (like I was teaching them ) that it should feel like you were “falling on your face” . They just smiled and said “you got it”.
So, to put it another way, you transfer your weight beyond the comfort zone? I've been working on something related with pro. The question is where does the moving foot go and when does it make contact. The answer, as I'm up to now is that it goes where the weight goes and makes contact approximately under the head. However, its definitely secondary to the movement of the body. In this the body (crotch up) acts as a different 'body' from the legs: the body maintains shape and poise while the legs are totally loose to swing.
So, to put it another way, you transfer your weight beyond the comfort zone?
YES, most defiantly yes.
Quote
The question is where does the moving foot go and when does it make contact. The answer, as I'm up to now is that it goes where the weight goes and makes contact approximately under the head. However, its definitely secondary to the movement of the body. In this the body (crotch up) acts as a different 'body' from the legs: the body maintains shape and poise while the legs are totally loose to swing.
This will be a little different depending on what School of Thought you follow.
Good dancers dance "on the edge" and to do that with no fear, your do need to learn to go to the edge and beyond/over the edge with no fear. Many people have a fear of falling and most people can't even remember when they fell the last time (except you, SW ). That means they are letting a fear control their action/life and they can't even remember when happened last time. Once they have learned to fall flat on the floor, they are not afraid of going to the edge and even beyond. This means they will be experiencing a sensation of free fall for a split of a second, helping them to move further and with less effort.
I absolutely agree with you on this one.
I think that is one of the main difference as well between good and awesome dancers.
The awesome ones always seems to push things further, harder but still at the same time looking like they are truling enjoying themselves otherwise they will just look like a try hard dancer.
One couple in my studio can't understand why they can't beat another couple in the same studio.
I watched both couple practising and realise that the other couple always seems to be so full of energy and pushing themself to the limit.
The first couple actually have much better technique but when they dance the energy seems to be 50 % of the second couple. It just seems so flat and passionless.
When you projects energy and passion from the inside this is what should drive you to dance on the edge for the thrill and love of it not merely to impress other people.
I'm not sure that is true - that it can't be taught. DSV has described 'falling classes' (she alludes to them above) where dancers learn to fall so that they leanr to lose their fear of being on the edge. I've no idea, however, how well it works...
I'm not sure that is true - that it can't be taught. DSV has described 'falling classes' (she alludes to them above) where dancers learn to fall so that they leanr to lose their fear of being on the edge. I've no idea, however, how well it works...
Speaking of fear of being on the edge, I crap my pants every time I think I'm going to hit the mirror of the studio or crash into the chairs in the corner when doing consecutive pivots.
I'm not sure that is true - that it can't be taught. DSV has described 'falling classes' (she alludes to them above) where dancers learn to fall so that they leanr to lose their fear of being on the edge. I've no idea, however, how well it works...
I meant the passion and energy ... not the falling bit LOL !
I bungee jumped once ... well let's just say I never felt scared falling down ever again since !
I think that is one of the main difference as well between good and awesome dancers.
The awesome ones always seems to push things further, harder but still at the same time looking like they are truling enjoying themselves otherwise they will just look like a try hard dancer.
One couple in my studio can't understand why they can't beat another couple in the same studio.
I watched both couple practising and realise that the other couple always seems to be so full of energy and pushing themself to the limit.
The first couple actually have much better technique but when they dance the energy seems to be 50 % of the second couple. It just seems so flat and passionless.
When you projects energy and passion from the inside this is what should drive you to dance on the edge for the thrill and love of it not merely to impress other people.
This, can't be taught.
I do think this can be taught but of cause the dancer has to want to change. Change can only happen if the dancer wants to change. I saw my teacher help students that were terrified of falling get to a point of total fearless falling just by learning to fall in a controlled environment for the first many times. Then after a couple of weeks they lost the fear and would fall with no fear. I have also myself taught this technique to many students with equal success. I would say it is totally possible to learn to fall with 100% commitment.
I would that is right that you can't teach anybody passion and energy because we all have both within us. What I have found is that great teachers are able to help you find the passion and energy within. They are then able to help you find a way to express both in your dancing. Not everybody is going to feel the same passion for the same thing. So it is important to find what you are passionate about and show that. Energy is again something that is different for everybody. What and how each dancer chooses to use their energy varies greatly. What the teacher needs to help with is to find a way where your passion and energy has the biggest effect and again that varies. That is what makes dancing so beautiful to me in that everybody can dance the same step and yet look totally different.
I do think this can be taught but of cause the dancer has to want to change. Change can only happen if the dancer wants to change. DSV
Sorry for the OT post, but I am so completely frustrated w/ a teacher at this very moment b/c he sees that what he is doing is good, but not great. He asks for my opinion/advice, then spends incredible amounts of energy telling me/trying to prove that I am wrong, and that what he is doing will be better once everyone else adapts to his way of thinking.
Sorry for the OT mini-vent, but I am at my last nerve w/ this bloke, but have to work w/ him for a wee while longer.
Logged
The most beautiful part of the dance is often found in between the steps... and in the movement within the stillness.
I do think this can be taught but of cause the dancer has to want to change. Change can only happen if the dancer wants to change. DSV
Sorry for the OT post, but I am so completely frustrated w/ a teacher at this very moment b/c he sees that what he is doing is good, but not great. He asks for my opinion/advice, then spends incredible amounts of energy telling me/trying to prove that I am wrong, and that what he is doing will be better once everyone else adapts to his way of thinking.
Sorry for the OT mini-vent, but I am at my last nerve w/ this bloke, but have to work w/ him for a wee while longer.
Sorry, when this happen I will normally tell the student that I was asked for an opinion and I gave it. I was not asking for his opinion and therefore don't want it. |
/* Copyright (c) 2008 The Board of Trustees of The Leland Stanford Junior University */
/* Copyright (c) 2011, 2012 Open Networking Foundation */
/* Copyright (c) 2012, 2013 Big Switch Networks, Inc. */
/* See the file LICENSE.loci which should have been included in the source distribution */
#ifdef __GNUC__
#ifdef __linux__
/* glibc */
#include <features.h>
#else
/* NetBSD etc */
#include <sys/cdefs.h>
#ifdef __GNUC_PREREQ__
#define __GNUC_PREREQ __GNUC_PREREQ__
#endif
#endif
#ifndef __GNUC_PREREQ
/* fallback */
#define __GNUC_PREREQ(maj, min) 0
#endif
#if __GNUC_PREREQ(4,6)
#pragma GCC diagnostic ignored "-Wunused-but-set-variable"
#endif
#endif
#include "loci_log.h"
#include "loci_int.h"
/**
* \defgroup of_list_bsn_gentable_entry_desc_stats_entry of_list_bsn_gentable_entry_desc_stats_entry
*/
/**
* Create a new of_list_bsn_gentable_entry_desc_stats_entry object
*
* @param version The wire version to use for the object
* @return Pointer to the newly create object or NULL on error
*
* Initializes the new object with it's default fixed length associating
* a new underlying wire buffer.
*
* Use new_from_message to bind an existing message to a message object,
* or a _get function for non-message objects.
*
* \ingroup of_list_bsn_gentable_entry_desc_stats_entry
*/
of_list_bsn_gentable_entry_desc_stats_entry_t *
of_list_bsn_gentable_entry_desc_stats_entry_new(of_version_t version)
{
of_list_bsn_gentable_entry_desc_stats_entry_t *obj;
int bytes;
bytes = of_object_fixed_len[version][OF_LIST_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY] + of_object_extra_len[version][OF_LIST_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY];
/* Allocate a maximum-length wire buffer assuming we'll be appending to it. */
if ((obj = (of_list_bsn_gentable_entry_desc_stats_entry_t *)of_object_new(OF_WIRE_BUFFER_MAX_LENGTH)) == NULL) {
return NULL;
}
of_list_bsn_gentable_entry_desc_stats_entry_init(obj, version, bytes, 0);
return obj;
}
/**
* Initialize an object of type of_list_bsn_gentable_entry_desc_stats_entry.
*
* @param obj Pointer to the object to initialize
* @param version The wire version to use for the object
* @param bytes How many bytes in the object
* @param clean_wire Boolean: If true, clear the wire object control struct
*
* If bytes < 0, then the default fixed length is used for the object
*
* This is a "coerce" function that sets up the pointers for the
* accessors properly.
*
* If anything other than 0 is passed in for the buffer size, the underlying
* wire buffer will have 'grow' called.
*/
void
of_list_bsn_gentable_entry_desc_stats_entry_init(of_list_bsn_gentable_entry_desc_stats_entry_t *obj,
of_version_t version, int bytes, int clean_wire)
{
LOCI_ASSERT(of_object_fixed_len[version][OF_LIST_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY] >= 0);
if (clean_wire) {
MEMSET(obj, 0, sizeof(*obj));
}
if (bytes < 0) {
bytes = of_object_fixed_len[version][OF_LIST_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY] + of_object_extra_len[version][OF_LIST_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY];
}
obj->version = version;
obj->length = bytes;
obj->object_id = OF_LIST_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY;
/* Set up the object's function pointers */
/* Grow the wire buffer */
if (obj->wire_object.wbuf != NULL) {
int tot_bytes;
tot_bytes = bytes + obj->wire_object.obj_offset;
of_wire_buffer_grow(obj->wire_object.wbuf, tot_bytes);
}
}
/**
* Associate an iterator with a list
* @param list The list to iterate over
* @param obj The list entry iteration pointer
* @return OF_ERROR_RANGE if the list is empty (end of list)
*
* The obj instance is completely initialized. The caller is responsible
* for cleaning up any wire buffers associated with obj before this call
*/
int
of_list_bsn_gentable_entry_desc_stats_entry_first(of_list_bsn_gentable_entry_desc_stats_entry_t *list,
of_bsn_gentable_entry_desc_stats_entry_t *obj)
{
int rv;
of_bsn_gentable_entry_desc_stats_entry_init(obj,
list->version, 0, 1);
if ((rv = of_list_first((of_object_t *)list, (of_object_t *)obj)) < 0) {
return rv;
}
of_object_wire_init((of_object_t *) obj, OF_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY,
list->length);
if (obj->length == 0) {
return OF_ERROR_PARSE;
}
return rv;
}
/**
* Advance an iterator to the next element in a list
* @param list The list being iterated
* @param obj The list entry iteration pointer
* @return OF_ERROR_RANGE if already at the last entry on the list
*
*/
int
of_list_bsn_gentable_entry_desc_stats_entry_next(of_list_bsn_gentable_entry_desc_stats_entry_t *list,
of_bsn_gentable_entry_desc_stats_entry_t *obj)
{
int rv;
if ((rv = of_list_next((of_object_t *)list, (of_object_t *)obj)) < 0) {
return rv;
}
rv = of_object_wire_init((of_object_t *) obj, OF_BSN_GENTABLE_ENTRY_DESC_STATS_ENTRY,
list->length);
if ((rv == OF_ERROR_NONE) && (obj->length == 0)) {
return OF_ERROR_PARSE;
}
return rv;
}
/**
* Set up to append an object of type of_bsn_gentable_entry_desc_stats_entry to an of_list_bsn_gentable_entry_desc_stats_entry.
* @param list The list that is prepared for append
* @param obj Pointer to object to hold data to append
*
* The obj instance is completely initialized. The caller is responsible
* for cleaning up any wire buffers associated with obj before this call.
*
* See the generic documentation for of_list_append_bind.
*/
int
of_list_bsn_gentable_entry_desc_stats_entry_append_bind(of_list_bsn_gentable_entry_desc_stats_entry_t *list,
of_bsn_gentable_entry_desc_stats_entry_t *obj)
{
return of_list_append_bind((of_object_t *)list, (of_object_t *)obj);
}
/**
* Append an item to a of_list_bsn_gentable_entry_desc_stats_entry list.
*
* This copies data from item and leaves item untouched.
*
* See the generic documentation for of_list_append.
*/
int
of_list_bsn_gentable_entry_desc_stats_entry_append(of_list_bsn_gentable_entry_desc_stats_entry_t *list,
of_bsn_gentable_entry_desc_stats_entry_t *item)
{
return of_list_append((of_object_t *)list, (of_object_t *)item);
}
|
Chloroplast heterogeneity and historical admixture within the genus Malus.
• The genus Malus represents a unique and complex evolutionary context in which to study domestication. Several Malus species have provided novel alleles and traits to the cultivars. The extent of admixture among wild Malus species has not been well described, due in part to limited sampling of individuals within a taxon.• Four chloroplast regions (1681 bp total) were sequenced and aligned for 412 Malus individuals from 30 species. Phylogenetic relationships were reconstructed using maximum parsimony. The distribution of chloroplast haplotypes among species was examined using statistical parsimony, phylogenetic trees, and a median-joining network.• Chloroplast haplotypes are shared among species within Malus. Three major haplotype-sharing networks were identified. One includes species native to China, Western North America, as well as Malus domestica Borkh, and its four primary progenitor species: M. sieversii (Ledeb.) M. Roem., M. orientalis Uglitzk., M. sylvestris (L.) Mill., and M. prunifolia (Willd.) Borkh; another includes five Chinese Malus species, and a third includes the three Malus species native to Eastern North America.• Chloroplast haplotypes found in M. domestica belong to a single, highly admixed network. Haplotypes shared between the domesticated apple and its progenitors may reflect historical introgression or the retention of ancestral polymorphisms. Multiple individuals should be sampled within Malus species to reveal haplotype heterogeneity, if complex maternal contributions to named species are to be recognized. |
t w.
420*w**2
Let m(o) be the third derivative of -3/8*o**4 - 1/120*o**6 + 0*o**5 + 0 + 0*o**3 + 15*o**2 + 0*o. Find the second derivative of m(w) wrt w.
-6*w
What is the third derivative of 23*v**2 + 42*v**2 - 149*v**2 + v**6 - 479*v**4 wrt v?
120*v**3 - 11496*v
Let h(x) = -x. Let o be h(-3). Suppose -11*g + 512 = 5*g. Differentiate g - 3*p**o - p**3 - 33 with respect to p.
-12*p**2
Let z be (3076/(-6) - 2)*-3. Find the third derivative of -9*k**2 - 1544*k + z*k - 18*k**3 wrt k.
-108
Let j(a) be the second derivative of 101*a**3/6 - 9*a**2 - 2*a - 87. Differentiate j(m) wrt m.
101
Suppose 4*s + 8 = -5*y + 23, -y = 5*s - 3. Let v(p) be the first derivative of -1/2*p**2 + s*p - 3 + 1/3*p**3. What is the second derivative of v(q) wrt q?
2
Let f(k) = -53*k**2 - 4*k + 38. Let t(n) = -35*n**2 - 3*n + 25. Suppose -31 = 8*g + 9. Let m(o) = g*f(o) + 8*t(o). Find the second derivative of m(j) wrt j.
-30
Differentiate -459 + 1270*w**3 + 721*w**3 - 2863*w**3 wrt w.
-2616*w**2
Let d(v) be the first derivative of 3*v**2 + 16. Let t(h) = 6*h - 1. Let f(q) = -6*d(q) + 4*t(q). What is the derivative of f(a) wrt a?
-12
Let b = -747 + 747. Let o(u) be the first derivative of u**3 - 3 + b*u - 1/2*u**4 + 0*u**2. What is the third derivative of o(q) wrt q?
-12
What is the second derivative of 4*r**3 + 103*r**4 - 48*r**4 - 49*r**4 + 146*r wrt r?
72*r**2 + 24*r
Suppose l - 6 = -a, 7*l - 12 = 4*l. What is the third derivative of -13*u**4 - 9*u**a + 16*u**2 - 3*u**2 wrt u?
-312*u
Let a(v) be the third derivative of -v**6/120 - 37*v**4/12 + 271*v**3/3 + 173*v**2. What is the derivative of a(s) wrt s?
-3*s**2 - 74
Let m(i) be the third derivative of -i**4/12 + 3*i**2. Let t be m(-1). What is the third derivative of 18*q**6 - 2*q + 2*q + q**t - 16*q**6 wrt q?
240*q**3
Let b(j) be the third derivative of -j**8/112 - 2*j**7/35 + 2*j**5/15 + j**4/12 + 18*j**2. What is the third derivative of b(d) wrt d?
-180*d**2 - 288*d
Let w = 39 - 27. What is the third derivative of -185*n + 185*n - 3*n**6 - w*n**2 wrt n?
-360*n**3
Let x be ((-1)/3)/((-9)/162). Let i(m) = -6*m**3 - 5*m**2 + 1. Let a(j) = 5*j**3 + 6*j**2. Let h(q) = x*i(q) + 5*a(q). Find the first derivative of h(g) wrt g.
-33*g**2
Let a = -22/17 + 61/34. Let c(q) be the second derivative of -a*q**4 + 2*q**2 + 0 + 0*q**3 + 7*q. Find the first derivative of c(y) wrt y.
-12*y
Suppose 0*z - 5*r = -z + 39, -2*z + 2*r = -38. Let l = z + -12. Find the second derivative of 5*j**3 - l*j**2 - 2*j**3 - 6*j + 2*j**2 wrt j.
18*j
Find the second derivative of 2*l**3 + 2*l**3 + 10*l**4 + 28*l**4 - 3*l + 4*l - 10 - 4*l wrt l.
456*l**2 + 24*l
Let d(j) be the second derivative of 1/3*j**4 - j + 1/15*j**6 + 0*j**3 + 0*j**2 + 0*j**5 + 0. What is the third derivative of d(n) wrt n?
48*n
Let c(z) be the third derivative of 13*z**7/210 + z**6/40 + 47*z**3/2 + 193*z**2. Differentiate c(o) wrt o.
52*o**3 + 9*o**2
Let g = 21 - 0. Suppose 0 = 2*a + 3*b - g, 5*a = 5*b - 9 - 1. Find the third derivative of -a*y**2 - 4*y**6 + 3*y**6 + 3*y**6 wrt y.
240*y**3
Suppose -8*z - 76 + 1108 = 0. Find the first derivative of -177*x + 349*x - z*x + 14 wrt x.
43
Let d(k) = -k**3 - 11*k**2 - 9. Let q = -40 - -36. Let s(u) = -4*u**2 - 3. Let j(w) = q*d(w) + 11*s(w). What is the derivative of j(p) wrt p?
12*p**2
Let v(r) = -r. Let p = -10 + 13. Let i(o) = -1 - 3*o + p*o + 5*o + 5. Let y(g) = i(g) + 2*v(g). What is the first derivative of y(l) wrt l?
3
Let m(k) be the first derivative of 28/3*k**3 + 0*k - 5 - 27/2*k**2. What is the second derivative of m(y) wrt y?
56
Suppose -3*t + 9 = -2*s, 3*s + 3*t = t + 19. Let q(p) be the second derivative of 2/3*p**4 + 0 + 0*p**s + 2*p**2 + 11*p. What is the derivative of q(f) wrt f?
16*f
Let i be ((-210)/36 + 6)*(-18)/(-2). Let n(r) be the first derivative of -5*r - 4 + i*r**2. Differentiate n(v) with respect to v.
3
Let u(n) = -12*n + 17. Let j(q) = 12*q - 18. Let d(h) = 2*j(h) + 3*u(h). Let o(t) = 6*t - 8. Let z(f) = -4*d(f) - 7*o(f). What is the derivative of z(r) wrt r?
6
Find the third derivative of -34 + 11*t**4 + 108*t**2 - 110*t**2 + t**3 - 14 - 36 wrt t.
264*t + 6
Let j(s) be the second derivative of -54*s**5/5 - 21*s**2 + 157*s. Differentiate j(o) wrt o.
-648*o**2
Let k(y) be the third derivative of 256*y**7/105 + 137*y**3/6 - 476*y**2. Differentiate k(b) with respect to b.
2048*b**3
Let f(y) be the first derivative of 22*y**2 + 15 - 1/2*y**4 + 0*y - 4/5*y**5 + 0*y**3. What is the second derivative of f(b) wrt b?
-48*b**2 - 12*b
Let k be (-2)/(2/(-1)) + 7 + -4. Let p(x) be the second derivative of 7*x + 3/10*x**5 + 0*x**3 + 0*x**k + 3/2*x**2 + 0. Differentiate p(o) wrt o.
18*o**2
Let t = -8 + 7. Let m be t + (0/(-2) - -4). What is the third derivative of k**2 - 4*k**2 + k**m + 0*k**2 + 6*k**2 wrt k?
6
Find the second derivative of -21*b - 55*b**2 - 738*b + 431*b wrt b.
-110
Let n(d) = 649*d - 116. Let t(q) = -217*q + 38. Let x(g) = -3*n(g) - 8*t(g). Differentiate x(m) wrt m.
-211
Let m(o) be the second derivative of 0*o**3 - 2*o + 0*o**5 + 0 + 3/10*o**6 - 1/4*o**4 + 0*o**2. Find the third derivative of m(w) wrt w.
216*w
Let p(x) = 461*x - 792. Let y(d) = -231*d + 393. Let s(b) = -6*p(b) - 13*y(b). Find the first derivative of s(o) wrt o.
237
Find the third derivative of -4*h**2 - 4*h**4 + 0*h**2 + 46*h**4 + 20*h**2 wrt h.
1008*h
Let g(a) = a**3 + 4*a**2 - 5*a + 2. Let x be g(-5). Find the second derivative of -4 + 1 + 3 + 4 + x*z + 6*z**3 wrt z.
36*z
Suppose 3*d - 4 = 8. Suppose -5*x = -d - 6. What is the third derivative of x*k**2 - k**2 - 4*k**2 + 3*k**5 - k**2 wrt k?
180*k**2
Let a(b) be the first derivative of 105*b**4/4 + 166*b - 4. Find the first derivative of a(g) wrt g.
315*g**2
Let u = 16 + -16. What is the second derivative of 0 + 20*r + 8*r**2 - 17*r**2 + u wrt r?
-18
Let p = 295 + -289. Let u(k) be the third derivative of 0*k**5 + 7/6*k**3 + 0 - k**2 - 1/120*k**p + 0*k + 0*k**4. Differentiate u(z) with respect to z.
-3*z**2
Let p(j) = 108*j + 100. Let g(q) = -108*q - 97. Let t(k) = 6*g(k) + 7*p(k). What is the derivative of t(l) wrt l?
108
Let o = 14 + 4. Suppose -6 = 5*i + 2*j - 37, 4*i + 5*j = o. Find the second derivative of 8*g**3 - 5*g**3 - 9*g**3 - i*g wrt g.
-36*g
Let i(a) be the first derivative of 0*a + 0*a**2 + 16/5*a**5 + 14/3*a**3 + 29 + 0*a**4. Find the third derivative of i(u) wrt u.
384*u
Let v(j) = j**3 + 5*j**2 - 7*j - 1. Let c be v(-6). Let q(n) be the first derivative of 4*n - c + 5 - 4*n**2 + 2*n**2 + 7. What is the derivative of q(p) wrt p?
-4
Let s(v) be the first derivative of -5*v**7/168 + 13*v**4/12 + 12*v**3 - 22. Let m(y) be the third derivative of s(y). Differentiate m(l) with respect to l.
-75*l**2
Let y = -31 + 29. Let f = y - -10. Differentiate -39*c + 7 + 39*c - f*c**4 wrt c.
-32*c**3
Let a(u) be the second derivative of 31*u**7/42 - u**5/20 - 71*u**4/6 + 4*u - 4. What is the third derivative of a(f) wrt f?
1860*f**2 - 6
What is the derivative of 2*b**2 - 31 - 6*b + 54*b - 404 wrt b?
4*b + 48
Let d(r) be the second derivative of -83*r**3/6 - 216*r**2 - 344*r. Find the first derivative of d(b) wrt b.
-83
Let p(f) = f**5 + 2*f**4 + 71*f**3 - 3*f**2 - 2*f + 7. Let r(a) = a**4 - a + 1. Let n(u) = p(u) - 2*r(u). Find the third derivative of n(j) wrt j.
60*j**2 + 426
Let z(a) be the first derivative of 309*a**2/2 + 896*a + 1001. Differentiate z(q) with respect to q.
309
Let j be 1 + -3*(-2)/6. Let r be (-1)/((-7)/j + 3). Differentiate 0*q**r + 2*q**3 - 8 + 3*q**3 + 0*q**2 wrt q.
15*q**2
Let r(p) = 710*p**2 + 5*p - 508. Let f(l) = -357*l**2 - 3*l + 253. Let d(q) = -5*f(q) - 3*r(q). What is the derivative of d(k) wrt k?
-690*k
Let s(b) = 551. Let g(k) = -k - 182. Let y(u) = -21*g(u) - 6*s(u). Differentiate y(p) wrt p.
21
Let d(v) be the first derivative of 2*v**7/35 + v**5/10 - v**2/2 - 6. Let g(p) be the second derivative of d(p). Find the third derivative of g(z) wrt z.
288*z
Let i = -3 - -15. Let c = 16 - i. What is the second derivative of -y**2 - 7*y**2 + c*y + 2*y + 0*y wrt y?
-16
Let q(v) be the first derivative of 831*v**4/4 - 176*v + 530. Find the first derivative of q(i) wrt i.
2493*i**2
Let y(s) = 4*s + 34. Let n be y(-8). What is the third derivative of 6*x**4 - 2192 + 2192 - 18*x**n wrt x?
144*x
Let m(g) = -35*g**2 + g + 1. Let u be m(-1). Let b be (-7)/(-2)*(-20 |
Q:
Duplicate SCSI HHD to SATA SSD
I got the Cavalry Dual Bay SATA Hard Drive Duplicator Docking Station and it works great when I Clone/Duplicate SATA Hard Drives but I have a couple of SCSI HHDs that I want to Clone/Duplicate but the connectors are not the same and I don't know what cable or adapter if any I can/should get to make it work, or if there is anything I can get for $150 or less if possible to get this done.
Can someone please help me out and let me know if there is anything out there that will get this done for me, I would like to use the Station VS a software solution.
Thank you so much for any help!
( The SCSI HHD I have is: 15k Ultra 320 SCSI 80 Pin )
A:
With 80 pins you probably have a SCA connector. Can you verify that it looks the same as in the picture below?
Image courtesy of Wikipedia
To connect such a drive you need a SCSI hostadapter (in sales droid terms: A SCSI controller). Once you have that you need a proper backplane or a caddy to mount it in (SCA drives are usually used in hot-swap backplanes) and then you can duplicate it with normal disk cloning software. (E.g. Clonezilla, Norton Ghost, Acronis trueimage, or just use plain single dd, cp or cat on a raw unix device).
|
Nothing last week I’m afraid. A bit of a domestic, doctor called. (Not for me some will be sorry to hear) Always on a bloody Sunday isn’t it? Still managed to cook a wild pigeon (You’d be wild too, if you were stuck into a hot oven and basted with your own juices) and four of five veg. This week it’s a simple pousin (My spell check doesn’t recognise the word, so I might well have spelled it wrong. Baby chicken I mean) poached in white wine with swede, carrot and onion, plus mashed potato and sprouts (frozen. But Marco Pierre White said they’re best). Read More |
Relationship between the Kihon Checklist and the clinical parameters in patients who participated in cardiac rehabilitation.
The Kihon Checklist is a useful screening tool for assessing frailty in older individuals. However, the clinical significance of the Kihon Checklist in cardiac rehabilitation patients remains unclear. The present study aimed to evaluate the relationship between the Kihon Checklist and the clinical parameters in patients who participated in cardiac rehabilitation. We enrolled 845 consecutive patients (584 men, mean age 71 years) who participated in cardiac rehabilitation at Juntendo University Hospital, Tokyo, Japan, between November 2015 and October 2017. The patients were divided into non-frailty (n = 287), pre-frailty (n = 270) and frailty (n = 288) groups according to their Kihon Checklist scores. Cardiopulmonary exercise testing was carried out in 302 patients. The frailty group was older and had a higher prevalence of history of heart failure than the non-frailty group, although left ventricular ejection fraction did not differ significantly between groups. Nutritional index, trunk and limb muscle mass, lean body weight, and grip strength were significantly lower in the frailty and pre-frailty groups than those in the non-frailty group. In the cardiopulmonary exercise test, a stepwise significant decrease in peak oxygen uptake was observed across the three groups (non-frailty 17.2 ± 3.6, pre-frailty 16.0 ± 3.4, frailty 14.4 ± 3.5 mL/kg/min, P < 0.01). Multivariate regression analyses showed that the Kihon Checklist score was significantly and independently associated with peak oxygen uptake (r = -0.34, P < 0.0001). The Kihon Checklist, which was associated with frailty and exercise tolerance, could be used as a clinical assessment method for patients who participated in cardiac rehabilitation. Geriatr Gerontol Int 2019; 19: 287-292. |
Whats all this 7900p45 business? I thought the 7900 was the old model and the 7910 was the new one? Also that filter is tiny unless the whole underside is all filter area? Whats the deal with the whole saw tell me more please.
Click to expand...
Dolmar renamed the saws from PS-7900 to PS-7910, Makita renamed them from DCS7900 to EA7900!
The PS-7910 and EA7900 are the current versions featuring a SLR muffler and a coil limited to 12.8k rpm WOT no load.
The SLR muffler is easily fixed and the coil doesn't truly impact performance.
The AF cartridge is filter surface all around and works surprisingly well.
His little brother is an a hole. Jax is the boss and even when zeke does this to him he doesn't back down and he doesn't try and hurt him back. It's the craziest thing how smart he is. He is a gentleman for sure.
Usually they get along, zeke is trying to be the boss but he won't ever be.
I agree, a bigger filter is gonna help. Hoping to get some K&n run time this weekend. I dont have anything to cut at my house, but I'm sure my neighbors would love it if I did.
I'm curious what roll tighter tater is gonna do
Click to expand...
Make sure the arm is straight and the blade sharp. And run at least a 3 foot blade. I bet it still won’t outcut my farm BOSS. Might want to invest in a big MAGNUM sticker too... bigger is better. Even the bishes agree...
Make sure the arm is straight and the blade sharp. And run at least a 3 foot blade. I bet it still won’t outcut my farm BOSS. Might want to invest in a big MAGNUM sticker too... bigger is better. Even the bishes agree...
7900 is much bigger than 044. Anyone tried bigger carbs?
I would still think a bigger air filter would be better for a saw that size.
Click to expand...
10%. I’ve run Zama in 046s and they run fine after Poleman modded them. 7900 with the HD filter can choke them down to the tune of 10% slower cut times on a few I did. I used a green weenie on one just pushing it over the filter neck and messy I g the cover down since the stock didn’t fit with the carb I used. It worked but wasn’t ideal.
10%. I’ve run Zama in 046s and they run fine after Poleman modded them. 7900 with the HD filter can choke them down to the tune of 10% slower cut times on a few I did. I used a green weenie on one just pushing it over the filter neck and messy I g the cover down since the stock didn’t fit with the carb I used. It worked but wasn’t ideal. |
Q
How long must a Texas inmate serving a 3(g) sentence serve before being eligible for parole?
A
In Texas an offender convicted of an offense with a deadly weapon or any other 3(g) offense must serve at least two years before they will become eligible for parole. On a 3(g) offense an TDCJ inmate must serve half of their sentence before being eligible for parole but they must serve two years before being eligible. Therefore, if an offender was sentenced to two years they will not become eligible for parole. If they are sentenced to three years they will eligible after two years. If they received a sentence of four years or more they will be eligible after serving half of their sentence.
As Texas parole lawyers we regularly represent and advocate for our clients at Parole Offices across the State of Texas. We do not shy away from difficult cases including 3(g) offenses. We are always honest with prospective clients about the strengths and negatives about a given case and will provide an honest evaluation of the case. However, it is also our belief that many offenders and their families want a voice and an advocate throughout the parole process independent of what the chances may be. We work diligently on every case to provide the parole board with a thorough picture of our client, their support systems and the reasons it is believed that they will be successful on parole. Please call us or fill out an online contact form for a free consultation with an experienced Texas Parole Lawyer. |
G-201 FUSED GLASS SUN CATCHERS & NIGHT LIGHTS (age 10-adult)
SKU:
$35.00
$35.00
Unavailable
Saturdays, 10:00 a.m. – 1:00 p.m.
G-201a: Saturday, Sept. 29
G-201b: Saturday, Oct. 13
G-201c: Saturday, Dec. 8
TUITION: $35 (materials included)
INSTRUCTOR: GEORGE AYARS
Make a colorful, charming fused glass sun catcher in this easy, 3 hour introduction to the basics of glass fusing! No previous glass cutting experience is required for this brief introduction to fused glass. Students will compose a design, approximately 4 x 6”, using small scraps of fusible glass, and fusible glass frit. Each piece is fired to over 1200 degrees, melting the glass into a finished sun catcher, ready to hang in a sunny window. Youths age 10 and up are welcome, if attending with a paying adult.
Students may choose to make two fused glass night lights instead of a sun catcher, for an additional fee of $9.00, to cover the cost of two fixtures, bulbs and clips, which are available in class. The night lights panels are 2 ½ x 3” each. Finished projects can be picked up at the School office after firing is completed, or mailed for a small additional fee. For ages 10 and up (and a great activity for a child to do with an older sibling, parent, or grandparent!) |
This invention relates to a process for preparing organic sulfides. In another aspect, this invention relates to a process for preparing unsymmetrical dihydrocarbyl sulfides. In still another aspect, this invention relates to the preparation of methyl alkyl sulfides and methyl alkenyl sulfides. In a further aspect, this invention relates to a process for the catalytic production of organic sulfides.
Processes for preparing organic sulfides such as symmetrical and unsymmetrical dialkyl sulfides and alkyl alkenyl sulfides, which are useful as solvents, surfactants and intermediates in organic syntheses, are known. One such prior art process is the reaction of an olefin and a mercaptan in the presence of a free radical initiator as catalyst. However, there is an ever present need to develop new processes for preparing sulfides, especially processes utilizing simpler catalyst systems and simpler reaction conditions than those previously known. |
Q:
Reference array value in PHP query result
I am trying to create a geoJSON array, but I am having difficulty access the values inside the query result. When I test my function by hardcoding the "coordinates" to the values '2' and '4' everything works fine.
However, when I then try to set the "coordinates" by referencing the $row['v2_lat'] and $row['v2_lng'], this causes an error. Here is the PHP code:
function getGeoJSON2(){
$m = $this->input->POST('municipality');
$l = $this->input->POST('saleslimit');
$q = $this->db->query("SELECT RollNum, Address, v2_lat, v2_lng FROM tblontario WHERE Municipality = '".$m."'"." LIMIT ".$l);
$res = $q->result();
if(!empty($res)){
$geoArr = Array(
"type" => "MultiPoint",
"coordinates" => Array()
);
foreach ($q->result() as $row) {
//$geoArr["coordinates"][] = Array(2,4);
$geoArr["coordinates"][] = Array($row['v2_lat'],$row['v2_lng']);
}
$geoJSON = json_encode($geoArr);
echo $geoJSON;
} else {
echo "{}";
}
}
And here is a sample of the query result and after applying json_encode:
[
{
"RollNum": "193601001000100",
"Address": "12 STEELES AVE E",
"v2_lat": "43.8561002",
"v2_lng": "-79.3370188"
},
{
"RollNum": "193601001000400",
"Address": "18 STEELES AVE E",
"v2_lat": "43.7986849",
"v2_lng": "-79.4178564"
},
{
"RollNum": "193601001002100",
"Address": "36 STEELES AVE E",
"v2_lat": "43.7987492",
"v2_lng": "-79.4169781"
}
]
I don't understand why I can't seem to be able to access the $row['v2_lat'] and $row['v2_lng'] values in my foreach loop. Any ideas? Thanks.
EDIT: I wanted to see the $res variable and this is the format of the array that comes back from the query:
Array
(
[0] => stdClass Object
(
[RollNum] => 180501000100100
[Address] => 2 HILEY AVE
[v2_lat] => 43.8526509
[v2_lng] => -79.0499877
)
[1] => stdClass Object
(
[RollNum] => 180501000100200
[Address] => 4 HILEY AVE
[v2_lat] => 43.8526678
[v2_lng] => -79.0501957
)
[2] => stdClass Object
(
[RollNum] => 180501000100300
[Address] => 6 HILEY AVE
[v2_lat] => 43.8526848
[v2_lng] => -79.0504037
)
)
What is the significance of the difference between these last two results?
A:
You access it like:
$lat = $row->v2_lat; //and not like $row['v2_lat']
Or if you want to access it like a simple array, you will need to change the following line:
//$res = $q->result(); //this produces objects
$res = $q->result_array(); //this produces array
This will produce an array for you of the results and you can use your earlier code as well.
|
Needlestick injuries among German medical students: time to take a different approach?
Medical students are at risk of occupational exposure to blood-borne viruses following needlestick injuries (NSIs) during medical school. The reporting of NSIs is an important step in the prevention of further injuries and in the initiation of early prophylaxis or treatment. The objective of this study was to describe the mechanisms whereby medical students experience occupational percutaneous blood exposure through NSIs and to discuss rational strategies for prevention. Incidents of exposure to blood-borne pathogens among medical students at a large German university were analysed. Year 6 medical students completed a written survey immediately before the clinical part of their training began, describing incidents that had occurred during the previous 5 years. In our study, 58.8% (183/311) of participating medical students recalled at least one NSI that had occurred during their studies. Overall, 284 NSIs were reported via an anonymous questionnaire. Occupational exposure to blood is a common problem among medical students. Efforts are required to ensure greater awareness of the risks associated with blood-borne pathogens among German medical students. Proper training in percutaneous procedures and how to act in the event of injury should be given in order to reduce the number of injuries. |
Abu Kharjah
Abu Kharjah (أقاب خارجهائئ) is a place on the Tigris River east of Kirkuk and south of Erbil in northern Iraq, at latitude 35.5917 north and longitude 44.0439 east.
The etymology of the name means I stand outside.
The climate exhibits extremely hot summers and cool wet winters.
References
Category:Kirkuk Governorate |
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https://dsoriib.wordpress.com/2012/08/06/hello-world/#commentsMon, 06 Aug 2012 11:18:14 +0000http://dsoriib.wordpress.com/?p=1続きを読む →]]>Welcome to WordPress.com! This is your very first post. Click the Edit link to modify or delete it, or start a new post. If you like, use this post to tell readers why you started this blog and what you plan to do with it. |
Q:
How to collect time periods, calculate the total, and present in ISO 8601 format?
I am trying to get few inputs of time periods from users (it can be by choosing in dropdown).
let's say a user will be able to input in one field: "1 hour and 5 minutes" or 01:05;
In second field the user will be able to input: "2 hours and 25 minutes" or 02:25;
Then, I want to calculate and add up the two time peirods for: "3 hours and 30 minutes" or 03:30;
It's important that all the date and time fields will be saved in ISO 8601 format, for google.
Someone can help, How can I do that?
Thanks!
A:
You're asking two completely unrelated questions here. To rephrase:
How can I provide Google with dates in a format that Google's scraper will recognize, as specified in Rich snippets - Reviews?
How can I accumulate time periods and present the result in a format suitable for humans?
Asking two unrelated question in one question is depreciated on SE, and will usually get your question closed, but I am going to answer them anyway.
First question: The ISO 8601 format requested by Google is YYYY-MM-DD with leading zeros if required, so July the 4th 2013 would be 2013-07-04.
To format the ISO date you can use the PHP function date like this:
$isodate = date('Y-m-d', $timestamp);
where $timestamp is an integer with the number of seconds that has elapsed since the Unix Epoch (January 1 1970 00:00:00 GMT).
Further, Google expects the $isodate string to be embedded in the page attached to the dtreviewed property using the syntax one of three metadata schemes (microdata, microformats, or RFDa). You do this in Drupal by overriding the template of page, node, etc. you want to embed the dtreviewed property in.
Second question: To accumulate time periods, you need to convert periods input by users (such as "1 hour and 5 minutes" or "01:05") to an integer format with seconds resolution. To convert to seconds you multiply the days with 86400, the hours with 3600 and the minutes with 60, so "1 hour and 5 minutes" becomes:
1 (hour) * 3600 + 5 (minutes) * 60 + 0 (seconds) = 3900 (seconds)
and "2 hours and 25 minutes" becomes:
2 (hour) * 3600 + 25 (minutes) * 60 + 0 (seconds) = 8700 (seconds)
To accumulate, you just add the seconds:
3900 + 8700 = 12600 (seconds)
To display in a suitable format, you can use format_interval.
$interval = 3900 + 8700;
$intervalstring = format_interval($interval, 2);
and $intervalstring will now be:
3 hours 30 minutes
(You will lose the "and" by using this standard function, if this "and" is important, you'll need to write your own.)
For avoidance of misunderstanding: Goggle is not interested in your time period data. A time period is typically an interval made up of hours and minutes, and not attached to a particular date. Google wants dates (as in year, month and day of month). Dates and time periods are different things, and there is no way you can format a time period as a date, or vice versa.
|
Herpetiform keratitis and palmoplantar hyperkeratosis: warning signs for Richner-Hanhart syndrome.
Richner-Hanhart syndrome (RHS, tyrosinemia type II) is a rare, autosomal recessive inborn error of tyrosine metabolism caused by tyrosine aminotransferase deficiency. It is characterized by photophobia due to keratitis, painful palmoplantar hyperkeratosis, variable mental retardation, and elevated serum tyrosine levels. Patients are often misdiagnosed with herpes simplex keratitis. We report on a a boy from Brazil who presented with bilateral keratitis secondary to RHS, which had earlier been misdiagnosed as herpes simplex keratitis. |
975 F.2d 710
UNITED STATES of America, Plaintiff-Appellee,v.Onofre R. GALLEGOS, Defendant-Appellant.
No. 91-2259.
United States Court of Appeals,Tenth Circuit.
Sept. 15, 1992.
Jana M. Miner, Asst. Federal Public Defender, Las Cruces, N.M., for defendant-appellant.
Robert J. Gorence, Asst. U.S. Atty. (Don J. Svet, U.S. Atty., with him on the brief), Albuquerque, N.M., for plaintiff-appellee.
Before BALDOCK, SETH and KELLY, Circuit Judges.
PAUL KELLY, JR., Circuit Judge.
1
Defendant-appellant Onofre R. Gallegos appeals his conviction of making a false statement in regard to a loan application. 18 U.S.C. § 1014. Mr. Gallegos was convicted by a jury and sentenced to twenty-four months imprisonment. On appeal, he argues that the district court erred by (1) not admitting expert testimony; (2) not questioning a juror who revealed that he knew a defense witness; and (3) not determining the actual loss to the bank in sentencing under U.S.S.G. § 2F1.1. He also argues that (4) he received ineffective assistance of counsel due to counsel's conflicts of interest.
2
In connection with obtaining a $1.25 million line of credit at the First National Bank of Albuquerque, Mr. Gallegos executed a borrower's certificate stating that financial statements furnished to the bank "were true and correct as of the date thereof" and "that no material adverse change in said financial condition reflected herein or the prospects of the same has changed since the date thereof." Mr. Gallegos had furnished the bank with unaudited interim financial statements which indicated that his company, Metro-Tec, had net income of $124,882 on revenues of $1.768 million. He claimed that more current statements were not available.
3
Contrary to his claim, however, the government's evidence tended to show that he knew that unaudited 1988 fiscal year-end financial statements were available and reflected a loss of $599,209. This loss resulted in negative shareholders' equity of $321,824. III R. 69-70. An accounts receivable clerk at Metro-Tec indicated that in order to comply with loan covenants imposed by Metro-Tec's prior bank, Mr. Gallegos instructed her to inflate the accounts receivable by overstating the amount actually billed to customers. III R. 45-46. The unaudited interim statements may have been a product of this overstatement. III R. 110. The bank's credit analyst testified that the change in financial status between the two statement dates was material: "No one would give a loan based on [the year-end financial statements.]" III R. 106.
I.
4
Mr. Gallegos attempted to introduce expert testimony that the financial condition reflected on the financial statements understated the true worth of existing and prospective government contracts Metro-Tec would complete. The district court declined to admit the testimony because it dealt with forecasted information and was not relevant to whether the financial condition had changed adversely between the interim and year-end dates.
5
The district court was correct. Section 1014 required the government to prove that Mr. Gallegos knowingly made a false material statement for the purpose of influencing the bank's action. See Williams v. United States, 458 U.S. 279, 284, 102 S.Ct. 3088, 3091, 73 L.Ed.2d 767 (1982); United States v. Smith, 838 F.2d 436, 439 (10th Cir.1988), cert. denied, 490 U.S. 1036, 109 S.Ct. 1935, 104 L.Ed.2d 407 (1989). The mere fact that the financial statements might be inaccurate under a different system of valuation is irrelevant. The issue is whether Mr. Gallegos falsely stated that the interim financial statements of MetroTec were correct and that no material adverse change in financial position had occurred. The crime is not that the interim financial statements were incorrect or that the financial position of Metro-Tec had changed; rather, it is the false statement to the contrary. See United States v. Kingston, 971 F.2d 481, 488 n. 3 (10th Cir.1992). The district court did not abuse its discretion in excluding this testimony.
II.
6
Mr. Gallegos next argues that the district court erred by not questioning a juror who notified the court before deliberations that he was acquainted with a defense witness (Mike Gallegos). After the witness testified, the juror sent the court a note stating: "I have personal knowledge of Mike Gallegos at UNM when he was student body president. Is it fair for me to bring up during deliberation?" The district court declined to question the juror, reasoning that defendant could have addressed the point during voir dire. Instead, the court sent back the juror's note with a "no" answer. When Mr. Gallegos subsequently moved to excuse the juror, the district court conducted voir dire of the juror, asking questions recommended by both counsel. VI R. 625. The juror indicated that he had not discussed the matter with the other jurors and that his personal knowledge would not prevent him from being fair and impartial. The district court then invited objections with the remark, "Satisfied, everybody?" VI R. 626. No objection followed.
7
Mr. Gallegos now contends that the district court should have asked whether the juror's knowledge formed a basis for an opinion, either positive or negative, about the defense witness so that Defendant could have explored a challenge for cause. On proper objection, our review of a district court's questioning concerning juror qualifications is deferential. See United States v. Berryhill, 880 F.2d 275, 278 (10th Cir.1989), cert. denied, 493 U.S. 1049, 110 S.Ct. 853, 107 L.Ed.2d 846 (1990). Given that this point was never raised below, our review is for plain error. Fed.R.Crim.P. 52(b); United States v. Young, 470 U.S. 1, 15-16, 105 S.Ct. 1038, 1046-47, 84 L.Ed.2d 1 (1985). Keeping in mind the juror's responses to the judge's questions and the juror's mere acquaintance with a defense witness, we are hard pressed to see how the district court's conduct constituted error, let alone plain error. See United States v. Bohle, 475 F.2d 872, 876 (2d Cir.1973).
III.
8
In computing the offense level under U.S.S.G. § 2F1.1(b)(1)(J), in effect at the time of the offense, see U.S.S.G. app. C at 70 n. 154 (Nov.1991), the district court enhanced the base offense level of six by nine levels, reasoning that the loss in this fraud case was $1.25 million or the entire line of credit. The presentence report indicates that $211,482 was repaid. In a civil action, the bank sought to recover on the note from various parties including Mr. Gallegos. The civil defendants filed a counterclaim predicated on a lender liability theory. Thereafter, the bank settled its claims for a total of $336,661 and the civil defendants dismissed their counterclaim. A balance of $312,340 remains on the settlement judgment with payments commencing on June 30, 1992.
9
We review a district court's determination of a U.S.S.G. § 2F1.1 loss under the clearly erroneous standard, but the factors a district court properly may consider is reviewed de novo. United States v. Levine, 970 F.2d 681, 689 (10th Cir.), cert. denied, --- U.S. ----, 113 S.Ct. 289, 121 L.Ed.2d 214 (1992). Mr. Gallegos argues that any loss under U.S.S.G. § 2F1.1(b) must be reduced by the amount repaid and the maximum loss to the bank is limited to the amount of the civil settlement agreement. In the district court, Mr. Gallegos argued that there has been no loss to the bank as a result of the civil settlement agreement. I R. doc. 47 at 5. He maintains that he intended to repay the loan.
10
Subsequent to sentencing, we decided United States v. Smith, 951 F.2d 1164 (10th Cir.1991). We acknowledged that the greater of actual or intended loss may be used to enhance, but "that actual loss should be measured by the net value, not the gross value, of what was taken." Id. at 1166. Accord U.S.S.G. § 2F1.1 comment. (n. 7(b)); United States v. Gennuso, 967 F.2d 1460, 1461-63 (10th Cir.1992). Smith involved mortgage loans extended in part on a false representation that $500 earnest money payments had been made by borrowers. The loans were secured by real property and none were in default. 951 F.2d at 1166. Under those circumstances, the government failed to prove actual loss. Id. at 1167.
11
Here, the bank has reduced its claim against Mr. Gallegos to the amount of the settlement agreement. It would be incongruous to hold that the actual loss to the bank was greater than the amount the bank now seeks to collect. The settlement agreement may be viewed as an offset; Mr. Gallegos has foregone his claims against the bank in exchange for a reduction of the debt owing the bank, a debt which is apparently to be paid in installments.
12
The district court did not specify whether the $1.25 million represented an actual or an intended loss. In light of Smith, we remand for this determination. See United States v. Haddock, 956 F.2d 1534, 1554, on reh'g, 961 F.2d 933 (10th Cir.), cert. denied, --- U.S. ----, 113 S.Ct. 88, 121 L.Ed.2d 50 (1992). Should the district court determine that the actual loss is appropriate, it should consider the likelihood of repayment of the settlement amount and sentence accordingly.
IV.
13
Mr. Gallegos next contends that he received ineffective assistance of counsel at trial because his counsel's daughter and law partner represented a key government witness, Mr. Littlefield, during Mr. Littlefield's immunity negotiations and subsequent testimony in this trial. Assuming without deciding that law partners should be considered as one lawyer, Burger v. Kemp, 483 U.S. 776, 783, 107 S.Ct. 3114, 3120, 97 L.Ed.2d 638 (1987); Martinez v. Sullivan, 881 F.2d 921, 930 (10th Cir.1989), cert. denied, 493 U.S. 1029, 110 S.Ct. 740, 107 L.Ed.2d 758 (1990), a defendant is entitled to a presumption of prejudice if he can prove that his lawyer " 'actively represented conflicting interests' and 'that an actual conflict of interest affected his lawyer's performance.' " Strickland v. Washington, 466 U.S. 668, 692, 104 S.Ct. 2052, 2067, 80 L.Ed.2d 674 (1984) (quoting Cuyler v. Sullivan, 446 U.S. 335, 350, 348, 100 S.Ct. 1708, 1719, 1718, 64 L.Ed.2d 333 (1980)). See also United States v. Bowie, 892 F.2d 1494, 1501-02 (10th Cir.1990); United States v. Winkle, 722 F.2d 605, 611-12 (10th Cir.1983). Of course, conflict-of-interest claims may be waived, Holloway v. Arkansas, 435 U.S. 475, 483 n. 5, 98 S.Ct. 1173, 1178 n. 5, 55 L.Ed.2d 426 (1978); Moore v. United States, 950 F.2d 656, 660 (10th Cir.1991), and we lack a record indicative of the professional relationship between trial counsel and his daughter. Normally, we do not consider ineffectiveness claims raised for the first time on appeal. Beaulieu v. United States, 930 F.2d 805 (10th Cir.1991). But because we are remanding this case, we deem it appropriate for the district court to consider defendant's ineffective assistance claim, notwithstanding that it was not raised below.
14
REMANDED.
|
1. Introduction {#sec1-polymers-12-00894}
===============
Aliphatic polyamides (PA) belong to the group of crystallizable polymers with many commercial applications. Their properties depend on the specific semi-crystalline morphology, which develops during melt-processing including, for example, injection molding. Important parameters of the semi-crystalline morphology are the crystal fraction, the crystal structure, habit, size, and orientation. In this study, an analysis of the injection-molding induced morphology of PA 66 was performed, by using polarized-light optical microscopy (POM), transmission electron microscopy (TEM), and X-ray scattering.
During injection molding, crystallization/solidification occurs under influence of shear- and temperature-gradients, which typically leads to the development of different semi-crystalline morphologies between the skin and core \[[@B1-polymers-12-00894],[@B2-polymers-12-00894],[@B3-polymers-12-00894]\]. Molded parts often appear transparent and featureless under an optical microscope in the surface-near regions, which is due to fast cooling of the oriented melt and solidification at rather high melt-supercooling. With increasing distance from the surface, a shear layer/zone is observed, which develops while the molecule segments are subjected to high shear forces during the mold-filling process along the skin layer which has already solidified. Next to the shear layer, towards the center of the cavity, the molecules typically exhibit lower orientation and are cooled more slowly, thus allowing spherulitic crystallization.
For a large number of aliphatic PA, different crystal polymorphs as a function of the conditions of melt-crystallization are reported \[[@B4-polymers-12-00894],[@B5-polymers-12-00894],[@B6-polymers-12-00894],[@B7-polymers-12-00894],[@B8-polymers-12-00894],[@B9-polymers-12-00894],[@B10-polymers-12-00894],[@B11-polymers-12-00894],[@B12-polymers-12-00894]\]. Concerning crystallization of the relaxed melt, a similar crystal/mesophase polymorphism is reported such as for isotactic polypropylene (iPP). Slow cooling results in the formation of α-crystals and spherulites, while fast cooling leads to development of non-lamellar mesophase domains \[[@B4-polymers-12-00894],[@B5-polymers-12-00894],[@B6-polymers-12-00894],[@B7-polymers-12-00894],[@B8-polymers-12-00894],[@B9-polymers-12-00894],[@B10-polymers-12-00894],[@B11-polymers-12-00894],[@B12-polymers-12-00894]\]. However, the critical cooling rate to suppress crystallization at high temperature and crystal perfection is different from iPP. It is worthwhile noting that the distinct high- and low-temperature crystallization processes in PA (6, 66, and 11) have been associated with different mechanisms of crystal nucleation \[[@B13-polymers-12-00894],[@B14-polymers-12-00894]\], ultimately yielding to qualitatively different semi-crystalline morphologies, and having impact on, for example, mechanical and optical properties \[[@B15-polymers-12-00894]\].
Prediction of structure formation of injection molded parts requires knowledge of the time- and position dependent temperature and shear-rate profiles and knowledge of the solidification/crystallization process at such conditions. Recent advances in this field have been achieved by application of fast scanning chip calorimetry (FSC). This method allows the mimicking of crystallization at cooling conditions that are present in injection molding, that is, at cooling rates of the order of magnitude of hundreds of K/s \[[@B16-polymers-12-00894]\]. In conjunction with X-ray analyses information about the formation of specific crystal polymorphs can even be obtained \[[@B9-polymers-12-00894],[@B11-polymers-12-00894],[@B17-polymers-12-00894]\]. Recently, the cooling-rate profile during low-shear injection molding of poly(butylene terephthalate) (PBT) was simulated in the skin and core regions and correlated with experimental quiescent-crystallization data obtained by FSC, yielding a very good prediction of the crystallinity difference in the various regions of the molded part \[[@B18-polymers-12-00894]\]. In the present work, we attempt to expand our efforts in characterization and prediction of structure formation in injection molding of polymorphic materials by using the example of PA 66.
It is the aim of this study to provide an experimental route to identify the process-induced morphology in terms of crystal shape and structure, orientation, crystallinity, and spherulitic superstructure in injection molded parts as a function of the distance from the surface and to correlate them with the microstructure assessed by optical microscopy. To achieve this aim a method was developed to investigate the process induced morphology layer by layer. With this research we follow prior and similar work performed for example on PA 6 and 66 \[[@B19-polymers-12-00894],[@B20-polymers-12-00894],[@B21-polymers-12-00894]\] but also iPP \[[@B1-polymers-12-00894],[@B22-polymers-12-00894]\]. However, we attempt to go beyond existing knowledge by interpretation of results in the context of our recent studies about crystallization at processing-relevant cooling conditions.
2. Materials and Methods {#sec2-polymers-12-00894}
========================
2.1. Material {#sec2dot1-polymers-12-00894}
-------------
A commercial injection-molding grade of PA 66 was used, as received from BASF (Ludwigshafen, Germany). The material had a melt flow index of 115 cm³/10 min (275 °C/5 kg, ISO 1133). It was non-nucleated, unfilled, and natural colored.
2.2. Processing {#sec2dot2-polymers-12-00894}
---------------
Before processing, the material was dried under vacuum at 80 °C for 12 h. Tensile test bars (DIN EN ISO 527, Type 1A) with a cross-section of 10 mm × 4 mm, shown in [Figure 1](#polymers-12-00894-f001){ref-type="fig"}, were injection molded. The barrel and mold temperatures were 300 and 40 °C, respectively, the injection speed was 40 mm/s, the holding pressure was 70 bar and the cycle time was 58 s.
2.3. Sample Preparation {#sec2dot3-polymers-12-00894}
-----------------------
After processing, the samples were prepared for morphological studies. As shown in [Figure 1](#polymers-12-00894-f001){ref-type="fig"}, a sample piece was removed from the tensile bar center. From this sample, thin sections corresponding to [Figure 1](#polymers-12-00894-f001){ref-type="fig"}a,b were removed using a rotary microtome at −30 °C, equipped with a tungsten-carbide knife. The thin sections were taken from different parts of the sample depending on the characterization method used. The marked eyes in [Figure 1](#polymers-12-00894-f001){ref-type="fig"} indicate the viewing direction from which the sample was analyzed with respect to the used methods. For POM investigations, 10 µm thick thin sections in flow direction were taken from the sample center ([Figure 1](#polymers-12-00894-f001){ref-type="fig"}a). For TEM, the cross-section of the sample was trimmed ([Figure 1](#polymers-12-00894-f001){ref-type="fig"}a) and contrasted with a formaldehyde/osmium tetroxide solution. For X-ray scattering, 50 µm thick thin sections were taken layer by layer from the skin (S) to the core (C) parallel to the flow direction ([Figure 1](#polymers-12-00894-f001){ref-type="fig"}b).
2.4. Polarized-Light Optical Microscopy (POM) and Transmission Electron Microscopy (TEM) {#sec2dot4-polymers-12-00894}
----------------------------------------------------------------------------------------
The POM images were taken by an Axio Imager from ZEISS (Oberkochen, Germany) with an objective of 50× magnification. Images of the skin and core areas of the cross-section were collected by using a TEM Tecnai G2 from FEI-Company (Hillsboro, OR, USA), which now belongs to Thermo Fisher Scientific (Waltham, MA, USA). An acceleration voltage of 200 kV was used.
2.5. X-ray Scattering {#sec2dot5-polymers-12-00894}
---------------------
X-ray analyses in different scattering ranges were performed on thin sections, which were cut layer by layer from the skin to the core, parallel to the direction of flow (see [Figure 1](#polymers-12-00894-f001){ref-type="fig"}b). Wide- and small-angle X-ray scattering (WAXS and SAXS, respectively) experiments were conducted consecutively by means of the multirange device Ganesha 300 XL+ (SAXSLAB ApS, Copenhagen, Denmark) equipped with a µ-focus X-ray tube and a Göbel mirror, providing monochromatic, parallel-beam Cu Kα radiation. Scattering intensities were measured by a two-dimensional (2D) Pilatus 300K detector (DECTRIS Ltd., Baden-Daettwil, Switzerland). The experiments were done in vacuum, in asymmetric-transmission mode, that is, the sample surface was oriented perpendicular to the primary beam. To connect WAXS and SAXS, the raw data were corrected for absorption (primary intensities in different measuring configurations were adjusted by using the sample transmission values). The results are presented as 2D scattering patterns and 1D line scans (radial scattering curves).
For calculation of the crystallinity, X~C,~ based on the intensity parts scattered by crystalline as well as amorphous amounts, a peak-fitting with pseudo-Voigt functions were executed by using the software package Analyze (part of the RayfleX software package for X-ray scattering devices; GE Sensing and Inspection Technologies GmbH, Ahrensburg, Germany). The minor anisotropies due to processing were neglected in the calculation of crystallinity. The obtained full widths at half maximum (FWHM) of selected scattering peaks served as basis to calculate average crystallite sizes 〈L〉~hkl~ perpendicular to the related lattice plane (hkl) by using the Scherrer equation \[[@B23-polymers-12-00894]\]. Long periods, D~L,~ were derived from the maximum position in the SAXS curve I(q) by using the Bragg equation \[[@B24-polymers-12-00894]\] (after background correction).
3. Results and Discussion {#sec3-polymers-12-00894}
=========================
During injection molding, the process parameters, especially the temperature profile, have a significant influence on the morphology of the moldings. It is known from the literature \[[@B25-polymers-12-00894],[@B26-polymers-12-00894]\], that the melt forms a fountain flow as it flows into the cavity. Within this fountain flow, the melt moves from the inner core out to the cold mold wall. When the melt gets into contact with the mold wall, the melt solidifies due to the high temperature difference between the melt and the mold, as shown in [Figure 2](#polymers-12-00894-f002){ref-type="fig"}.
The optical analysis of the PA 66 sample revealed the significant/notable influence of the temperature difference between the melt and the mold (220 K) on the morphology gradient from the skin to the core inside the molded part. As can be seen by the POM images of [Figure 3](#polymers-12-00894-f003){ref-type="fig"}a,b, a skin layer of about 50 µm thickness was observed, mainly due to the high temperature difference between the melt and the mold of 260 °C. Inside the skin layer, structural heterogeneities are not visible. At 50 µm distances from the mold, in the direction towards the core, spherulites are visible, increasing in their size with increasing distance from the skin. The visualization of crystalline structures is limited by the low resolution of POM, and therefore TEM was employed to gain further information about the crystal morphology. [Figure 3](#polymers-12-00894-f003){ref-type="fig"}c,d show TEM images obtained from the skin and core. At the skin, no heterogeneities are observed whereas in the core, lamellae with a thickness of 5.1 nm are visible (average of 10 lamellae). Therefore, [Figure 3](#polymers-12-00894-f003){ref-type="fig"} illustrates the strong influence of the thermal process history on the formation of the semi-crystalline structures during the injection-molding process. No visible crystalline structures could be detected in the skin layer. In contrast, behind the skin layer towards the core, spherulites could be observed increasing in size. The morphology in the core depends on the heat transfer through the solidified skin layer to the mold. Parameters with the greatest influence, are the thermal conductivity of the material (0.23 W/(K·m) \[[@B27-polymers-12-00894]\]), the temperature difference between the melt and the mold and the flow velocity of the melt, and therefore the resulting thickness of the skin layer. By considering the high temperature difference of 220 K between melt and mold and taking into account the literature \[[@B12-polymers-12-00894],[@B13-polymers-12-00894],[@B14-polymers-12-00894],[@B15-polymers-12-00894],[@B16-polymers-12-00894],[@B17-polymers-12-00894],[@B18-polymers-12-00894],[@B19-polymers-12-00894],[@B20-polymers-12-00894]\], it is assumed that cooling in the core is in the order of 400--600 K/s slower than at the skin. Accordingly, inside the core, the chains have sufficient time to form crystals and arrange in a spherulitic superstructure.
The results of X-ray scattering experiments are shown in [Figure 4](#polymers-12-00894-f004){ref-type="fig"}. Samples with a thickness of 50 μm were cut from the injection molded test specimen such that they are perpendicular to the samples which were analyzed using POM as is shown in [Figure 1](#polymers-12-00894-f001){ref-type="fig"}. The WAXS and SAXS patterns are shown such that their position corresponds to the appropriate position on the POM micrograph. The azimuthally integrated radial scattering curves are shown above and below, respectively. The WAXS curves feature peaks at spacing of 1.4, 0.66, 0.43, and 0.38 nm caused by scattering at the (001), (002), (100), and (010)/(110) lattice planes, indicating the presence of the triclinic α-modification with the unit cell (shown in [Figure 5](#polymers-12-00894-f005){ref-type="fig"}) parameters a = 0.49 nm, b = 0.54 nm, c = 1.72 nm (chain direction), α = 48.5°, β = 77°, and γ = 63.5° \[[@B28-polymers-12-00894]\]. The 2D-pattern revealed that the crystals are weakly oriented (negligible in the sense of discussion). Most important, however, is the observation that the maxima of the (100) and (010/110) peaks change their position as a function of the distance from the surface.
According to the literature \[[@B28-polymers-12-00894]\], the angular distance between the (100) and (010)/(110) diffraction maxima is proportional to the crystal perfection. A so-called crystal perfection index (CPI) can be calculated according to Equation (1) \[[@B29-polymers-12-00894]\]: $$CPI~\left\lbrack \% \right\rbrack = \frac{\left( {d_{100}/d_{010}} \right) - 1}{0.189} \times 100$$ where d~100~ is the interplanar spacing of α-(100) planes and d~010~ of α-(010)/(110) planes. The denominator correlates to the distance in a well-crystallized sample (with CPI = 100%), reported in \[[@B27-polymers-12-00894],[@B28-polymers-12-00894]\]. [Table 1](#polymers-12-00894-t001){ref-type="table"} shows the CPI values at different distances from the skin. As such, the crystal perfection increases from skin to core up to 90%. The WAXS data were also used to calculate the crystallinity of the core and the skin-layer. The former is about 37%, whereas the latter is, in contrast, much lower at 25%.
In PA 66, a lamellar morphology of alternating crystalline and amorphous sublayers can be assumed, which cause an interference maximum (long period D~L~) in the SAXS region. Long periods of 9.2 and 5.8 nm were estimated for the core and skin, respectively. Thus, a linear crystallinity model (two-phase model) can be applied. Additionally, supposing that the interface between the sublayers and the (001) net plane are nearly parallel to each other (due to the hydrogen bridges), the crystallite sizes (Scherrer) 〈L〉~001~ and 〈L〉~002~, respectively, can be compared with the thickness l~C~ of the crystalline sublayer. Consequently, the crystallized polymer chains continue tilted through its sublayers. Finally, the relation l~C~ = D~L~ × X~C~ (with X~C~ as crystallinity) can serve as unbiased estimate. The trend of the l~C~ values is given in [Table 1](#polymers-12-00894-t001){ref-type="table"}. Inside the error limits, the calculated l~C~ values follow the trend as shown of D~L~.
4. Conclusions {#sec4-polymers-12-00894}
==============
For understanding and predicting of the structure formation of melt-processed polymer materials, a detailed knowledge of phase transitions is necessary in well-resolved scales of position, time, and temperature. In this paper, the semi-crystalline morphology of injection molded PA 66 was investigated in terms of the influence of cooling conditions during the manufacturing of test specimens. Here, the results, achieved ex-situ after completion of manufacturing, were presented. The investigations have shown that the thermal history, especially during the filling phase, has a strong influence on the morphology in the skin and core region. With decreasing of the cooling rate from skin to core, the spherulite sizes increase. This was accompanied by an improved perfection of the crystalline phase inside the core, where a triclinic α-phase structure existed/formed. In comparison to the core, at the skin, the rapid cooling when the melt got into contact with the cold mold surface hindered the formation of spherulites. However, the results of X-ray analysis show that the skin layer contains a rather poor crystalline α-phase, which confirms that crystalline fractions can also form under process conditions which promote rapid cooling of the material. This knowledge can help to direct the processing of injection-molded construction materials to improve their application properties. A detailed study of structure--property relationships can be the basis for fabricating tailor-made materials.
Our sincere thanks go to Henning, Fraunhofer-Institut für Mikrostruktur von Werkstoffen und Systemen (Fraunhofer IMWS, Halle (Saale)), for his support.
Y.S.---Conceptualization, investigation, visualization and original draft; R.A.---Analysis, investigation, visualization and review and editing; D.J.---Investigation, methodology, visualization and original draft; I.K.---Conceptualization, supervision, visualization and review and editing. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
The authors declare no conflict of interest.
{#polymers-12-00894-f001}
{#polymers-12-00894-f002}
{#polymers-12-00894-f003}
{#polymers-12-00894-f004}
{#polymers-12-00894-f005}
polymers-12-00894-t001_Table 1
######
Determined values for selected reflections: position, perfection, crystallite size, crystallinity, long period, and crystalline sublayer (WAXS and SAXS results).
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
Sample Position Reflections Crystal Perfection Index Mean Crystallite Sizes (Scherrer) Crystallinity Long Period Crystalline Sublayer\
(l~C~ = D~L~ × X~C~)
----------------- ------------- -------------------------- ----------------------------------- --------------- ------------- ----------------------- ------- -------
S1 0.4335 0.3942 52.7 2.2 4.7 25.2 5.8 1.5
S2 0.4367 0.3852 70.7 2.9 6.4 36.6 7.3 2.7
S3 0.4377 0.3813 78.3 2.9 7.4 37.8 7.9 3.0
C 0.4397 0.3754 90.6 4.0 9.3 37.3 9.2 3.5
Errors ± Δx \~0.0005 \~0.0010 \~1.6 \~1.2 \~0.8 \~1.5 \~0.1 \~1.3
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
|
Prognostic significance of nuclear pSTAT3 in oral cancer.
Aberrant nuclear accumulation of proteins influences tumor development and may predict biologic aggressiveness and disease prognosis. This study determined the prognostic significance of pSTAT3 (phosphorylayed signal transducer and activator of transcription 3) in oral squamous cell carcinomas (OSCCs). Using immunohistochemistry, a significant increase in nuclear accumulation of pSTAT3 was observed in 49 of 90 leukoplakias (54.4%) and 63/94 OSCCs (67%) (p(trend) < .001). Increased pSTAT3 was associated with tumor stage (p = .01), nodal metastasis (p = .0018), and tobacco consumption (p = .004). Kaplan-Meier analysis demonstrated that OSCC with increased nuclear pSTAT3 showed significantly reduced disease-free survival (13 months), compared with the patients with no nuclear pSTAT3 expression (64 months, p = .019). Cox regression analysis revealed nuclear pSTAT3 as the most significant predictor of poor prognosis (p = .024, hazard ratio [HR] = 2.7). Increased nuclear accumulation of pSTAT3 occurs in early premalignant stages and is a marker for poor prognosis of OSCC. |
AP
The Panthers spent a boatload of money in free agency last year, and the bill has come due. While the team doesn’t have many key players to re-sign, Carolina needs to shed a lot of salary before the start of the new league year.
One candidate to be released, according to the Charlotte Observer, is Jimmy Clausen. The former quarterback of the future in Carolina was the No. 3 quarterback of the present behind Cam Newton and Derek Anderson in 2011.
He’s due a roster bonus of $923,000 in March, which is a lot of money for a No. 3 quarterback. It’s not crazy money if the Panthers are confident in Clausen’s ability to be a backup. Anderson is a free agent and the team could simply choose to elevate Clausen.
It will be a remarkably far fall from grace for Clausen if the Panthers let him go. Just two years ago, Clausen was viewed as a likely first round pick. He had a disastrous rookies season and was an afterthought last year. G.M. Marty Hurney may not want to admit his mistake on Clausen so soon.
If the Panthers do let Clausen go, perhaps the Broncos can sign him so he can say bad things about Tim Tebow to Mike Silver. |
Paul Begala dismissed Donald Trump's campaign reshuffling. | AP Clinton super PAC adviser dismisses Trump reshuffle: 'It's the candidate, stupid’
Former Bill Clinton adviser Paul Begala, who serves a senior adviser to the pro-Hillary Clinton Priorities USA super PAC, mocked the notion that Donald Trump’s campaign shakeup represented anything more than a rearranging of the deck chairs of the Titanic.
"It's the candidate, stupid," Begala told CNN Wednesday morning.
He was adapting a phrase from Bill Clinton's 1992 campaign.
Early Wednesday morning, Trump’s campaign announced the hiring of pollster Kellyanne Conway as campaign manager and Breitbart News Chairman Stephen Bannon as campaign chief executive.
Weekly Standard editor Bill Kristol echoed some of Begala's sentiment in an interview on MSNBC Wednesday morning, saying: "I don’t think it matters because the problem is Donald Trump. You know, his unfavorable rating has been consistently too high to win a presidential election. Hillary Clinton’s, you would normally say is too high, but it’s about 10 points lower than Trump’s. And that’s what it’s about." |
Fractures of the radial head and neck in children.
In thirty-three children with fractures of the radial head or neck, and with a minimum follow-up of two years, fractures of the radial neck were most common. The proximal radial epiphysis was usually closed by the age of fifteen years. The older children in the series had the worst prognosis, as did children with other associated upper-extremity injuries, usually on the medial aspect of the elbow. The results were best if treatment was initiated early, and closed reduction usually gave the most satisfactory results. Roentgenograms of the involved elbow were often abnormal even with a good clinical result, but the long-term result in such situations is uncertain. |
Army Chief General Datuk Seri Ahmad Hasbullah Mohd Nawawi said civilians wearing army fatigues is against the law. — Picture by Yusof Mat Isa
KENINGAU, Jan 26 — The act of civilians wearing army fatigues is against the law where they could be prosecuted for doing so, said Army Chief General Datuk Seri Ahmad Hasbullah Mohd Nawawi.
He said all army fatigues were gazetted and civilians were prohibited from having or using them.
Despite the issue been raised since a long time ago, he said, the wearing of army fatigues among civilians was still widespread.
He attributed it to lack of enforcement.
“On enforcement, it should be referred to the relevant authority because the army does not have that power,” he said when met by reporters at the Army Tactical Headquarters at Pos Keningau here.
Ahmad Hasbullah said suppliers of the army fatigues should also be aware of the matter and to not sell them to the public.
The authorities should do something to address the matter, he added. — Bernama |
Alphabet drops after earnings
The shares of the parent company of Google – Alphabet traded lower in after-hours trading on Monday. The online advertising giant reported revenue of $26.01 billon, which represents roughly a 21% increase year-over-year.
Affected by the EU anti-trust case, in which the company was fined $2.74 billion for prioritizing their own shopping service over the shopping services of the competitors, the reported profit fell sharply.
The main worry investors have are the rising traffic acquisition costs, which came in at $5.09 billion, higher than the estimated $4.75 billion. The CFO of Alphabet – Ruth Porat said that the company is focusing on “dollar growth” in revenue and operating income, “not margins”.
Reportedly, YouTube has been one of the main drivers of revenue growth this quarter, yet reports of the individual performance of the website were nowhere to be seen in the earnings release. |
Abortion Recovery
We are here for you.
If you’ve had an abortion, you already know it can be a traumatic experience. Maybe you felt like you had no other option. Whatever your circumstances, know this: it is not the end of your story. No, you can’t undo the past, but you are valuable, and your story is still being written.
The following list has been designed to help you identify symptoms in your life that may be related to a past abortion experience.
Some of these symptoms may pertain to you:
Sadness
Feelings of loss
Guilt
Regret
Recurring thoughts about the abortion(s)
Crying episodes
Anxiety
Inability to sustain an intimate relationship
Preoccupation with anniversaries, i.e., date of the abortion(s) or due date(s)
Obsession with children or childbearing issues
Avoidance of small children and babies
Increased alcohol use
Drug abuse
Repeat abortions
Multiple sexual relationships
Engaging in any of the following to excess: school, work, exercise, eating, or dieting
Difficulty sleeping
Feelings of numbness
Lack of self-esteem
Suicidal impulses
Desire for secrecy about the abortion
Disinterest in sex
If this list describes you, take heart. There is hope for you—you don’t have to stay where you are!
If you are experiencing these or similar symptoms and would like to talk with a trained, non-judgmental counselor in a confidential environment, please call Pathways of Pella at 641-628-4827 or 1-800-395-HELP or come in during our office hours (listed below). |
Role of gram-negative and gram-positive gastrointestinal flora in temperature regulation of mice.
An earlier study showed that the presence of gut flora elevates body temperature of mice and rats. In these experiments, we questioned whether the signal coming from the gut was endotoxin from gram-negative (Gm-) bacteria or some signal derived from gram-positive (Gm+) microorganisms. To test the idea that endotoxin is responsible for the effects of flora, we compared the temperature of the endotoxin-resistant mouse (C3H/HeJ) with that of endotoxin-sensitive strains of mice (C3H/SnJ and C3H/HeN). Temperature of C3H/HeJ was not different from that of C3H/SnJ or C3H/HeN during the light period but was significantly lower during the later hours of the dark period. We speculated that, if endotoxin leaking across the gut wall were responsible for elevating temperature, then reduction of gut flora with nonabsorbable antibiotics would depress the temperature of the endotoxin-sensitive mice more than that of the endotoxin-resistant mice. Because antibiotics lowered the temperature of both strains of mice to the same extent, the signal coming from the gut is unlikely to be endotoxin. To test whether Gm+ flora can be responsible for elevating temperature, we inoculated one group of germfree mice with Gm+ organisms. Their mean temperature was significantly higher than that of mice that remained germfree. Cecectomy had no effect on temperature, indicating that the special properties of the germfree cecum were not involved in lowering the temperature of germfree mice. These data support the hypotheses that Gm+ organisms are a major source of the stimulatory effect of flora on normal body temperature and that the presence of Gm- organisms is unnecessary. |
Custom Garage Door
Repair & Installation
Garage Door Repair Berwyn
6/12/2009
I just got off the phone with Robbie. He called to apologize and find out what he can do to make things right. It took alot of stones to do that and I respect him for it.
9.9 times out of 10 business don’t give a damn when they F up, but clearly Robbie does. He admitted fault, apologized, and wanted to make things right. Heck, he even offered the door for free but services were rendered, and I didn’t write the review looking for a handout.
So I upped my rating to 4 stars… Maybe the 2 was a little harsh, but I was damn sick of manually opening my garage door with my scrawny arms.
Kudos to Robbie for having the stones to be a good business man. Hopefully I won’t need him in the future, but I will reco him to friends. |
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Beer and I played it and as with early access it has a ways to go BUT, it was fun! You are either a pirate or British and you attack each other by either ramming/boarding or cannon fire.
The cannon were pretty straight forward to use but, being alone on a ship it gets to be a pain to load and run them out as you try to steer etc..
I hope they fix it so you can actual aim them up and down basically you just wait until they appear in your line of fire.Character development is thin and not very well defined but, maybe they will improve soon. Water was good graphic wise as were most items.Hopefully, they will add more to it $20.00 bucks.
Hey guys, we just want to give you all an update on the most recent patch. 3.0 required a lot of code to be rewritten and we have essentially back tracked on our progress bug wise. After a year in early access you'd expect us to be near complete by now, but this is not the case. To see why visit the news post below. For a TL;DR: We were not happy with the direction we were headed design wise and decided to overhaul some fundamentals of the game, ignoring any existing bugs because we saw no sense in patching things that would be changed.
This genre is still young and unprecedented, with little reference to go off of we were not surprised when halfway through Early Access we would completely change course. By that time we had seen enough public play to know where we wanted to go.
One could argue we were putting content before bugs, which is half true. We were in a constant tug of war with retaining players long enough to have incentive to fix bugs. According to negative reviews from 2.0 and previous, players stopped playing due to lack of content, not because of a buggy experience. Because of this we shifted our focus onto content and only prioritized gamebreaking bugs that could be recreated almost all of the time. We would much prefer a content rich game that has some bugs to fix than a polished game that doesn't hold you.
Before going live 3.0 was tested as extensively as it could have been for those who volunteered to join the testing branch. As far as bugs go we have many on our list and are working as quickly as we can. We have been asked by server owners to actually slow the rate of our patches because it's causing them to restart too often.
It is worthing noting that many bugs can be fixed by simply reconnecting. In some cases a single error during connect will botch the state of the game and cause a multitude of seemingly random bugs.
Obvious signs that this has happened are invisible ships, being unable to jump, or seeing players on deck with the swimming animation. We are working to at least warn a user that this error has occurred while we investigate more into the issue. It doesn't appear to be consistent at all (making it difficult for us to get logs of what really went wrong).
As for the design and forts, this is being tweaked constantly. We also have analytics in place. A lot of things are currently happening that aren't exactly bugs, but oversights in design that we would never have known until the patch was public. As expected we are now patching reactively to these "tactics".
Please keep in mind 3.0 has only been out for 5 days and there are only two of us (minus any outsourcing to speed things up). Only one being a programmer. We're knocking bugs out as quickly as we can and we definitely won't stop until things are cleared up.
Overall the amount of positive feedback we've gotten has been overwhelming. We're much happier to see the game head in this direction even if it means going through a rough patch from such a large overhaul. Bugs have never been more crucial to us than they are now.
Dakota & Tyler
Side note
There seems to be a misunderstanding about our plans to leave early access being mistaken for planning to stop updating the game, this is not true. Also this initial bare bones 3.0 patch is not all of 3.0, we just released it in its rawest form to make bug fixing easier. Other features are done and not turned on yet. (See our 3.0 preview post in news history) |
Q:
Logical explanation why exponentiation operation is not commutative and associative
Considering Peano axioms we'll define addition, multiplication and exponentiation operations.
We can then prove that addition and multiplication operations are commutative and associative. The proof for addition operation is pretty straightforward but for for multiplication it a bit more difficult. For example, to prove associativity, we have to prove distributivity first.
Moreover, for the definition itself it is not obvious that commutativity should take place, because the definition is not symmetric.
$a \cdot 0 := 0$
$a \cdot S(b) := a \cdot b + a$
Here $S(b)$ is the "next" function from Peano axioms
This definition is not symmetric with respect to $a$ and $b$, so commutivity is not obvious.
For exponentiation operation neither commutivity nor associativity take place. And I am very curious about the logical explanation why that happens
NOTE: The explanation I am looking for is somehow similar to a Abel–Ruffini theorem which explains WHY there is no an algebraic solution for a general polynomial equation
A:
I'm not sure how to explain it using the Peano axioms other than that it's not, that this follows from the axioms and the definition and there's nothing more to it.
On the other hand, if you need some intuitions, you can find them by interpreting number $n$ as $\big|\{0,1,2,...\ldots,n-1\}\big|$ and observing that $a = |A|$ and $b = |B|$ implies $a \cdot b = |A \times B|$.
No wonder multiplication is commutative, because the roles of sets $A$ and $B$ are symmetric. Similarly for $(a\cdot b)\cdot c = a \cdot (b \cdot c)$ which follows from $(A \times B) \times C \equiv A \times (B \times C)$
(I'm assuming you have the intuition why the last one is true).
Yet, for exponentation we have $a^b = |B \to A|$, i.e., the size of the set of functions from $B$ to $A$ (it is sometimes also written as $A^B$). However, note that function of the form $B \to A$ is much different than a function $A \to B$, for example if $B$ is empty and $A$ is not, then the latter cannot exist. The two operands on both sides of the arrow have different roles, so no wonder exponentiation is not commutative.
Also $(A \to B) \to C$ differs a lot from $A \to (B \to C)$. To see this, observe that the first one expects as input a rather huge amount of information compared to the second which is equivalent to $(A \times B) \to C$ (the formula $(a^b)^c = a^{b\cdot c}$ is not a coincidence). Thus, it's not surprising that the exponent is not associative.
I hope this helps $\ddot\smile$
|
Q:
Oracle SQL Delete Record without affecting child tables
I've got many tables in my database, but my problem is concerned to this two tables:
.
The definiton of those is:
Employee (EmployeeID_PK, FirstName, LastName, Email, ...)
Order (OrderID_PK, EmployeeID_FK, OrderDate, MenuID, ...)
*_PK = Primary Key
*_FK = Foreign Key
Now I want to delete a Employee so that he's not able to login anymore, but I still need to have his orders and his attributes (like FirstName, LastName, ...).
So the data should stay in the tables, but the employee shouldn't be able to login anymore.
I know that adding a column 'isactive' to Employee would be a solution, but I don't think that it's the best way to do this. Please tell me anyway, if it's the only way to do it.
Thanks in advance!
A:
As you said one option is to have a status column - 1 for active, 0 for inactive, 2 for login blocked may be.
If you don't want to take that approach, you can create a copy of both primary and secondary tables ( Employee_Archive and Order_Archive with equivalent pk and fk constraints) and move both the sets of data there. That way the user can no longer login, no data is lost and the integrity is retained.
If there is however too much data in Order table though, you may still only have the first approach left - to add a status column.
|
Q:
subtract part of text
I have this code
public void descargarURL() {
try{
URL url = new URL("https://www.amazon.es/MSI-Titan-GT73EVR-7RD-1027XES-Ordenador/dp/B078ZYX4R5/ref=sr_1_1?ie=UTF8&qid=1524239679&sr=8-1");
BufferedReader lectura = new BufferedReader(new InputStreamReader(url.openStream()));
File archivo = new File("descarga2.txt");
BufferedWriter escritura = new BufferedWriter(new FileWriter(archivo));
BufferedWriter ficheroNuevo = new BufferedWriter(new FileWriter("nuevoFichero.txt"));
String texto;
while ((texto = lectura.readLine()) != null) {
escritura.write(texto);
}
lectura.close();
escritura.close();
ficheroNuevo.close();
System.out.println("Archivo creado!");
//}
}
catch(Exception ex) {
ex.printStackTrace();
}
}
public static void main(String[] args) throws FileNotFoundException, IOException {
Paginaweb2 pg = new Paginaweb2();
pg.descargarURL();
}
}
And I want to remove from the URL the part of the reference that is B078ZYX4R5, and this entity /
After the html that is saved in the text file there is a part of the code that has *"<div id =" cerberus-data-metrics "style =" display: none; "data-asin =" B078ZYX4R5 "data-as-price = "1479.00" data-asin-shipping = "0" data-asin-currency-code = "EUR" data-substitute-count = "0" data-device-type = "WEB" data-display-code = "Asin is not eligible because it has a retail offer "> </ div>"*, and I want to only get the price from there that is 1479.00, it is included among the tags "data-as-price = "
I dont want to use external libraries, I know that it can be done with split, index of, and substring
Thanks!!!!
A:
You could solve both tasks by using regular expressions. Yet for the second task (extraction of the price from the HTML) you could use JSOUP which is much better suited to extract content from HTML.
Here are some possible solutions based on regular expressions for your tasks:
1. Change URL
private static String modifyUrl(String str) {
return str.replaceFirst("/[^/]+(?=/ref)", "");
}
This is just a replacement using a regular expression using a positive look-ahead (?=/ref) (see https://docs.oracle.com/javase/7/docs/api/java/util/regex/Pattern.html)
Extract Price
private static Optional<String> extractPrice(String html) {
Pattern pat = Pattern.compile("data-as-price\\s*=\\s*[\"'](?<price>.+?)[\"']", Pattern.MULTILINE);
Matcher m = pat.matcher(html);
if(m.find()) {
String price = m.group("price");
return Optional.of(price);
}
return Optional.empty();
}
Here you can use also a regular expression (data-as-price\s*=\s*["'](?<price>.+?)["']) to locate the price. With a named group ((?<price>.+?)) you can then extract the price.
I am returning an Optional here so that you can deal with the case that the price was not found.
Here is a simple test case for the two methods:
public static void main(String[] args) throws IOException {
String str = "https://www.amazon.es/MSI-Titan-GT73EVR-7RD-1027XES-Ordenador/dp/B078ZYX4R5/ref=sr_1_1?ie=UTF8&qid=1524239679&sr=8-1";
System.out.println(modifyUrl(str));
String html = "<div id =\" cerberus-data-metrics \"style =\" display: none; \"data-asin =\" B078ZYX4R5 \"data-as-price = \"1479.00\" data-asin-shipping = \"0\" data-asin-currency-code = \"EUR\" data-substitute-count = \"0\" data-device-type = \"WEB\" data-display-code = \"Asin is not eligible because it has a retail offer \"> </ div>";
extractPrice(html).ifPresent(System.out::println);
}
If you run this simple test case you will see on the console this output:
https://www.amazon.es/MSI-Titan-GT73EVR-7RD-1027XES-Ordenador/dp/ref=sr_1_1?ie=UTF8&qid=1524239679&sr=8-1
1479.00
Update
If you want to extract the reference from the URL, you can do it using similar code to the one used to extract the price. Here is a method which extract a specific named group from a pattern:
private static Optional<String> extractNamedGroup(String str, Pattern pat, String reference) {
Matcher m = pat.matcher(str);
if (m.find()) {
return Optional.of(m.group(reference));
}
return Optional.empty();
}
Then you can use this method for extracting the reference and price:
private static Optional<String> extractReference(String str) {
Pattern pat = Pattern.compile("/(?<reference>[^/]+)(?=/ref)");
return extractNamedGroup(str, pat, "reference");
}
private static Optional<String> extractPrice(String html) {
Pattern pat = Pattern.compile("data-as-price\\s*=\\s*[\"'](?<price>.+?)[\"']", Pattern.MULTILINE);
return extractNamedGroup(html, pat, "price");
}
You can test the above methods with:
public static void main(String[] args) throws IOException {
String str = "https://www.amazon.es/MSI-Titan-GT73EVR-7RD-1027XES-Ordenador/dp/B078ZYX4R5/ref=sr_1_1?ie=UTF8&qid=1524239679&sr=8-1";
extractReference(str).ifPresent(System.out::println);
String html = "<div id =\" cerberus-data-metrics \"style =\" display: none; \"data-asin =\" B078ZYX4R5 \"data-as-price = \"1479.00\" data-asin-shipping = \"0\" data-asin-currency-code = \"EUR\" data-substitute-count = \"0\" data-device-type = \"WEB\" data-display-code = \"Asin is not eligible because it has a retail offer \"> </ div>";
extractPrice(html).ifPresent(System.out::println);
}
This will print:
B078ZYX4R5
1479.00
Update 2: Using URL
If you want to use the java.net.URL class to help you narrow down the search scope you can do it. But you cannot use this class to do the full extraction.
Since the token you want to extract is in the URL path you can extract the path and then apply the regular expression explained above to do the extraction.
Here is the sample code you can use to narrow down the search scope:
public static void main(String[] args) throws IOException {
String str = "https://www.amazon.es/MSI-Titan-GT73EVR-7RD-1027XES-Ordenador/dp/B078ZYX4R5/ref=sr_1_1?ie=UTF8&qid=1524239679&sr=8-1";
URL url = new URL(str);
extractReference(url.getPath() /* narrowing the search scope here */).ifPresent(System.out::println);
String html = "<div id =\" cerberus-data-metrics \"style =\" display: none; \"data-asin =\" B078ZYX4R5 \"data-as-price = \"1479.00\" data-asin-shipping = \"0\" data-asin-currency-code = \"EUR\" data-substitute-count = \"0\" data-device-type = \"WEB\" data-display-code = \"Asin is not eligible because it has a retail offer \"> </ div>";
extractPrice(html).ifPresent(System.out::println);
}
|
Universal Orlando announces expansion of ‘Wizarding World of Harry Potter’
This is an archived article and the information in the article may be outdated. Please look at the time stamp on the story to see when it was last updated.
Universal Orlando announces expansion of ‘Wizarding World of Harry Potter’
ORLANDO, Fla. — Universal Orlando Resort and Warner Bros. Entertainment today announced an expansion of historic proportion with the entirely new themed environment, The Wizarding World of Harry Potter – Diagon Alley.
Scheduled to open in 2014, the world’s first centrally themed, multi-park experience expands The Wizarding World of Harry Potter across both Universal Orlando theme parks and allows Universal’s creative team to bring an unparalleled vision to this unique project.
The new area will bring to life some of the experiences and places found in and around London in the Harry Potter books and films, offering brand-new adventures for fans and theme park guests from around the world.
Diagon Alley and ‘London’ will be located within the Universal Studios Florida theme park, which is adjacent to Universal’s Islands of Adventure theme park, where guests now experience Hogwarts and Hogsmeade. The new area within Universal Studios will be just as expansive, immersive and authentic as the existing themed environment.
And – just like in the books and films – guests will be able to travel between ‘London’ and Hogsmeade aboard the Hogwarts Express.
Work on the new area is already underway. When complete, it will feature shops, a restaurant and an innovative, marquee attraction based on Gringotts bank – all directly inspired by the fiction and films. |
Daily variation in USA mortality.
Daily variation in each of several major causes of death in the United States during the period 1962-1966 is described by annual graphs showing deaths chronologically and by frequency distribution of deaths per day. The most significant sporadic factors affecting mortality in the 5 years studied appeared to be the influenza epidemic in 1963 and the unusually hot weather occurring in mid-1966. Deaths were also studied by day of the week and by holiday. Significant and consistent variations were found for violent deaths and those due to coronary heart disease. |
Persons born on 11 October 1945with first names starting with G
Providing for free what some websites charge money for.
This is a privately owned genealogy website based on public information which everybody
has a right to know. If your name is on a voter registration entry, click the link to go to a page with information on
removal requests. Social Security information on deceased persons is public under the Freedom of Information Act and
will not be deleted. We are always looking for good transcription works: Email webmaster@gedcomindex.com
Entries are sorted by given names first, to help you find a remarried aunt. These are
in strict alphabetical order, not word-by-word, so that Rose Mary and Rosemary are
treated the same, but it also puts the Josephine names between the Joseph I
and Joseph J names. Surnames are considered next, so that Daniel Zimmerman
would come before Danielle Abbott.
We have begun adding death residence and death benefit ZIP Codes to some Death Master File entries.A person may be listed twice.
Gail B Woodworth was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 3901 Se St Lucie Blvd, K22, Stuart, Florida 34997. Florida voter ID number 115197678. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478621
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GAIL H YONOV was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 35 Watergate Dr, ##904, Sarasota, Florida 34236. Florida voter ID number 100328068. This is the most recent information, from the Florida voter list as of 31 May 2012. 29478623
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GAIL MARIE BEDNAR was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 18549 Lakeside Gardens Dr, Jupiter, Florida 33458. Florida voter ID number 109117340. Her telephone number is 1-305-965-9625. . This is the most recent information, from the Florida voter list as of 30 September 2018. Previous information:31 May 2012 voter list: Gail Marie Bednar, 19645 E Saint Andrews Dr, Hialeah, FL 33015 Republican Party of Florida. 29478624
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Gail S. Eldridge was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 6456 Cedar St, Milton, Florida 32570. Florida voter ID number 107598172. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478625
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Garry Edward Kennedy was born 11 October 1945, is male, registered as No Party Affiliation, residing at 2317 Emerald Lake Dr, Sun City Center, Florida 33573. Florida voter ID number 121789953. This is the most recent information, from the Florida voter list as of 30 June 2018. 29478627
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Gary Alan Lee was born 11 October 1945, is male, registered as Republican Party of Florida, residing at 165 4Th St, Bonita Springs, Florida 34134. Florida voter ID number 103058716. His telephone number is 1-239-272-5680. . His email address is gator165@gmail.com. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478633
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Gary Don Smith was born 11 October 1945, is male, registered as Republican Party of Florida, residing at 11841 Mandarin Rd, Jacksonville, Florida 32223. Florida voter ID number 103558275. His telephone number is 1-904-268-0785. . This is the most recent information, from the Florida voter list as of 30 September 2018. 29478637
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GARY M SCHAEFER was born 11 October 1945, registered as Republican Party of Florida, residing at 335 Amesbury Ct, Longwood, Florida 32779-4647. Florida voter ID number 107668653. This is the most recent information, from the Florida voter list as of 30 November 2016. 29478642
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GARY R WOOD wasborn 11 October 1945, received Social Security number 114-38-4052 (indicating New York) and, Death Master File says, died 18 July 2003. Research in ZIP Code 33809.29478643
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE MAILE wasborn 11 October 1945, received Social Security number 319-38-4569 (indicating Illinois) and, Death Master File says, died June 1980. Research in ZIP Code 61554.29478650
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE PATTERSON wasborn 11 October 1945, received Social Security number 527-66-1148 (indicating Arizona) and, Death Master File says, died 15 May 1998. Research in ZIP Code 85706.29478651
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE SHOEMAKER wasborn 11 October 1945, received Social Security number 393-44-8381 (indicating Wisconsin) and, Death Master File says, died April 1981. Research in ZIP Code 53963.29478652
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE H COOK wasborn 11 October 1945, received Social Security number 435-66-4802 (indicating Louisiana) and, Death Master File says, died 09 May 2007. Research in ZIP Code 71268.29478655
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE L REID wasborn 11 October 1945, received Social Security number 319-38-0137 (indicating Illinois) and, Death Master File says, died 28 July 2006. Research in ZIP Code 62914.29478656
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE R NANCE wasborn 11 October 1945, received Social Security number 510-44-4143 (indicating Kansas) and, Death Master File says, died 25 November 2005. Research in ZIP Code 66061.29478657
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE T BURTON wasborn 11 October 1945, received Social Security number 405-64-0891 and, Death Master File says, died 08 November 1997. Research in ZIP Code 42501.29478658
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGE W MURPHY wasborn 11 October 1945, received Social Security number 235-70-4710 (indicating West Virginia) and, Death Master File says, died 09 June 1989. Research in ZIP Code 15120.29478661
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
GEORGIA T ISAACSON was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 1793 N 60Th Ave, Pensacola, Florida 32506. Florida voter ID number 117682883. Her telephone number is 1-850-456-0960. . This is the most recent information, from the Florida voter list as of 30 September 2018. 29478664
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GERALDINE CYRUS wasborn 11 October 1945, received Social Security number 251-84-1860 (indicating South Carolina) and, Death Master File says, died 11 February 1998. Research in ZIP Code 29556.29478674
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
Geraldine Ellen Wainscott was born 11 October 1945, is female, registered as No Party Affiliation, residing at 815 7Th Ave S, Jacksonville Beach, Florida 32250. Florida voter ID number 119677472. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478675
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GERALD P. McDunnah was born on 11 October 1945 and heregistered to vote, living at 95 Quarry Road R in Granby, Connecticut 06035-112429478677For more information, click here (free), then check
Archives.com (fee-based)
Gerald Page Langlykke was born 11 October 1945, is male, registered as No Party Affiliation, residing at 3711 Royal Palm Dr, Bradenton, Florida 34210. Florida voter ID number 121162642. This is the most recent information, from the Florida voter list as of 30 September 2018. Previous information:31 December 2014 voter list: Gerald Page Langlykke, 906 24th AVE W, Palmetto, FL 34221 No Party Affiliation. 29478678
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Geraldyne Marie Hatch Carlton was born 11 October 1945, is female, registered as No Party Affiliation, residing at 9230 Hall Rd, Lakeland, Florida 33809. Florida voter ID number 105419635. Her telephone number is 1-863-712-5627. . This is the most recent information, from the Florida voter list as of 30 September 2018. Previous information:30 June 2018 voter list: Geraldyne H. Carlton, 324 Tarpon St, Anna Maria, FL 34216 No Party Affiliation. 29478679
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GERARD Nucera was born on 11 October 1945 and heregistered to vote, living at 67 Sherman Court in Fairfield, Connecticut 06824-582729478680For more information, click here (free), then check
Archives.com (fee-based)
German A Pardo was born 11 October 1945, is male, registered as Republican Party of Florida, residing at 5404 Sw 140Th Pl, Miami, Florida 33175. Florida voter ID number 109829061. His email address is fpardo2011@att.net. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478681
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GIORA Y SINGER was born 11 October 1945, is male, registered as No Party Affiliation, residing at 205 Spring Lake Hills Dr, Altamonte Springs, Florida 32714. Florida voter ID number 107858633. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478684
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Gladys Alvarado was born 11 October 1945, is female, registered as No Party Affiliation, residing at 9508 Larkbunting Dr, Tampa, Florida 33647. Florida voter ID number 111258046. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478686
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GLENDA E PHILLIPS wasborn 11 October 1945, received Social Security number 242-68-4400 (indicating North Carolina) and, Death Master File says, died 25 November 1994. Research in ZIP Code 27358.29478687
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
Glenna Marx was born 11 October 1945, is female, registered as Florida Democratic Party, residing at 13007 Titleist Dr, Hudson, Florida 34669. Florida voter ID number 106430707. Her telephone number is 1-727-857-0065. . This is the most recent information, from the Florida voter list as of 30 September 2018. 29478689
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GLENN R. Bell was born on 11 October 1945 and heregistered to vote, living at 209 Mariners Walk in Milford, Connecticut 06460-636929478691For more information, click here (free), then check
Archives.com (fee-based)
Glenny R. Maneritch was born 11 October 1945 and registered to vote, giving the address 207 SE 1st in Russell, White County, Arkansas 72139, United States of America. 29478692For more information, click here (free), then check
Archives.com (fee-based)
GLENVILLE C PAYNE wasborn 11 October 1945, received Social Security number 232-70-9610 (indicating West Virginia or North Carolina) and, Death Master File says, died 03 October 2000. Research in ZIP Code 25071.29478693
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
Gloria A. Greeno was born 11 October 1945 and registered to vote, giving the address 107 Wood Street in Weiner, Poinsett County, Arkansas 72479, United States of America. 29478697For more information, click here (free), then check
Archives.com (fee-based)
GLORIA A HARRIS wasborn 11 October 1945, received Social Security number 515-48-2343 (indicating Kansas) and, Death Master File says, died 06 July 2009. Research in ZIP Code 67203.29478698
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
Gloria J Rosenthal was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 1904 Hammocks Ave, Lutz, Florida 33549. Florida voter ID number 118677548. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478700
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
Gloria Jean McClendon was born 11 October 1945, is female, registered as Florida Democratic Party, residing at 2168 Hagood Loop, Crestview, Florida 32536-5457. Florida voter ID number 120095491. The voter lists a mailing address and probably prefers you use it: 1200 Dorsey Ave LOT 14 Morgantown WV 26501-7023. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478701
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GLORIA JEAN WOUDENBERG was born 11 October 1945, is female, registered as Republican Party of Florida, residing at 9509 Sw Nuova Way, Pt St Lucie, Florida 34986. Florida voter ID number 118518337. Her telephone number is 1-201-888-9345. . This is the most recent information, from the Florida voter list as of 30 September 2018. 29478702
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GORRELL ROBERT STINSON was born 11 October 1945, is male, registered as No Party Affiliation, residing at 1309 Bando Ln, The Villages, Florida 32162-0115. Florida voter ID number 108318834. His telephone number is 1-352-217-0759. . His email address is SCRAPIRON15@GMAIL.COM. This is the most recent information, from the Florida voter list as of 30 September 2018. Previous information:31 May 2013 voter list: GORRELL ROBERT STINSON, 1309 BANDO LN, THE VILLAGES, FL 32162 No Party Affiliation. 29478703
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based)
GREGORY V FLACCAVENTO wasborn 11 October 1945, received Social Security number 077-34-4664 (indicating New York) and, Death Master File says, died 13 January 2001. Research in ZIP Code 13905.29478708
Source: Social Security Death Master File (public domain). Check
Archives.com (fee-based)
G Roger Prado-Flores was born 11 October 1945, is male, registered as Republican Party of Florida, residing at 3630 Sw 18Th Ter, Miami, Florida 33145. Florida voter ID number 109404034. This is the most recent information, from the Florida voter list as of 30 September 2018. 29478709
Source: Florida voter lists from Florida Dept. of State (public domain). Check
Archives.com (fee-based) |
Q:
Using pointers as template non-type arguments
I would like to use an array pointer (with array arithmetic) as a non-type argument. I understand that the argument should be known at compile-time, but isn't it the case for a fixed size global array?
This example can print the first 2 lines, but not the third one. Is there any workaround for this?
EDIT: I am looking for an answer for not only aa+1, but all aa+is where i is less than the size of aa
#include <iostream>
void print (int n) {
printf("the value is: %d\n", n);
}
template <int *n>
void myWrapper() {
print(*n);
}
void myCall(void (*CALLBACK)(void)) {
CALLBACK();
}
int a = 1; int aa[4] = {2,3,4,5};
int main()
{
myCall(myWrapper<&a>); // prints 1
myCall(myWrapper<aa>); // prints 2
/* the following line gives error: no matches converting function 'myWrapper' to type 'void (*)()'
note: candidate is: template<int* n> void myWrapper()
*/
myCall(myWrapper<aa+1>);
}
A:
This is excluded by a note to [temp.arg.nontype]:
3 - [ Note: Addresses of array elements and names or addresses of non-static class members are not acceptable
template-arguments. [...]
A workaround could be to supply the array index as another template parameter:
template <int *n, unsigned N = 0>
void myWrapper() {
print(n[N]);
}
// ...
myCall(myWrapper<aa, 1>);
|
Fire hero climbs up Highbury flats building to help pensioner
A NEIGHBOUR climbed into a pensioner’s flat during a fire at a housing block in Highbury to reassure her until firefighters arrived.
Residents in Aberdeen Park told how the man scaled the outside of the building to get to the woman who was unable to leave her home.
The woman’s flat was above a second-floor property where the London Fire Brigade tackled a blaze in the early hours of Tuesday.
Four fire engines and 21 firefighters had been called to the scene.
An elderly man whose flat was at the centre of the blaze had left before rescue crews arrived and was later treated in hospital.
Fernando Collado-Lopez, who lives on the ground floor and was there at the time of the fire, told the Tribune: “There was one lady that was above the flat that had the fire. She couldn’t get out because she said there was smoke in the lobby so she said she couldn’t breathe.”
He added: “The flat below was burning, the smoke was coming up, I think people have the Grenfell tower in their heads. She was panicking so she opened the window which meant the smoke was coming in.
The boyfriend of someone who lives on the second floor climbed up the façade of the other side of the building and went into the flat to reassure her until the firemen came.”
Emergency crews rescued a man and a woman from the second-floor flat on arrival and gained control of the fire around an hour later. Three other adults were led to safety by the internal staircase.
Mr Collado-Lopez, who lives with his wife and 17-month-old child, said: “Everyone came out of the building and neighbours brought out blankets and jumpers to help each other out.”
The Brigade said the fire was believed to have started in the bedroom and been caused by candles left unattended.
“These items should always be held firmly in heat-resistant holders and placed on a stable surface where they won’t be knocked over,” it said in a statement. “Be aware that tea lights get very hot and without proper holders can melt through plastic surfaces like a TV or bath.” |
Assessment of thoracic aortic conformational changes by four-dimensional computed tomography angiography in patients with chronic aortic dissection type b.
To characterize the heartbeat-related distension of dissected and non-dissected thoracic aortic segments in chronic aortic dissection type b (CADB) ECG-gated computed tomography angiography was performed in ten CADB patients. For 20 time points of the R-R interval, multiplanar reformations were taken at non-dissected (A, B) and dissected (C) aorta: ascending aorta (A), aortic vertex (B), 10 cm distal to left subclavian (Ct, true channel; Cf, false channel). Relative amplitudes of aortic area and major and minor axis diameter changes were quantified. Area amplitudes were 12.9 +/- 3.7%, 11.4 +/- 1.8%, 16.5 +/- 5.9% and 10.5 +/- 5.7% at A, B, Ct and Cf, respectively. Area amplitudes were significantly greater at Ct than at Cf and B (p < 0.05). Major axis diameter amplitudes were 7.7 +/- 1.9%, 6.2 +/- 1.3%, 5.9 +/- 2.0% and 6.1 +/- 3.6% at A, B, Ct and Cf, respectively. There were no differences in major axis diameter amplitudes. Minor axis diameter amplitudes were 6.7 +/- 2.1%, 8.4 +/- 1.9%, 12.7 +/- 6.3% and 6.0 +/- 2.2% at A, B, Ct and Cf, respectively. Minor axis diameter amplitudes were significantly the greatest at Ct (p < 0.05). In CADB, the heartbeat-related distension of aortic area and diameter is evenly distributed over the non-dissected aortic arch. As a result from different blood flow properties, there are significantly greater conformational changes in the true channel of the dissected aorta. |
Parafoveal processing of inflectional morphology in Russian: A within-word boundary-change paradigm.
The present study examined whether the inflectional morphology on Russian nouns is processed parafoveally in words longer than five characters while the eyes are fixated on the word. A modified boundary-change paradigm was used to examine parafoveal processing of nominal case markings within a currently fixated word n. The results elicited identical preview benefit for both first and second-pass measures on the post boundary and whole word regions. The morphologically related preview benefit (vs. nonword) was observed for first and second-pass measures as early as pre-boundary, post-boundary, and whole word regions. Additionally the morphologically related preview elicited cost (vs. identical) for first-pass measures on the post-boundary region, total time for the whole word, and regressions into the pre-boundary region. The contribution of the study is two-fold. First, this is the first study to use within-word boundary changes to study the parafoveal processing of inflectional morphology in Russian. Second, we provide additional evidence that inflectional morphology can be integrated parafoveally while reading a language with linear concatenative morphology. |
Bitcoin to paypal blockchain
Blockchain Technology, Bitcoin, Ethereum, and the
Find all you need to know and get started with Bitcoin on bitcoin.org.Add https://blockchain.info/api/api_receive api to my website anopong.com...X-Crypto is online crypto-currency exchange where you can sell bitcoins instantly and receive money to Paypal.
Taking the stage at a dinner organized by bitcoin and blockchain advocacy group.
Buy Bitcoins With Paypal & Credit Cards Www.Virwox.com
7 opciones para intercambiar entre bitcoins y PayPal
Why Bitcoin Is Important For Your Business. needing a bank or having to pay fees to Paypal or another.Start accepting bitcoin, store and spend bitcoin securely, or get the BitPay Card.
Might the Blockchain Outlive Bitcoin? - InfoQ
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1. Introduction {#sec1-materials-13-00286}
===============
Generally used in the building industry as a filling of windows or glass facades, insulating glass units (IGUs) consist of two or more component glass panes, connected at the edges with a glass spacer. The space between the component glass panes forms a tight gap filled with gas. In order to improve the thermal performance of the building partition constructed this way, the gap is filled with gas with lower thermal conductivity than air, most often argon. Further improvement of the performance is achieved by the use of component glass panes with a low-E coating---such a coating must be located on the side of the gap because it corrodes quickly when exposed to weather conditions. The tightness of the gap in insulating glass units is therefore a necessary factor to maintain good thermal insulation of transparent glazing \[[@B1-materials-13-00286],[@B2-materials-13-00286]\].
Tight gaps, however, determine some specific properties of IGUs in the context of environmental loads transfer and the associated deformation of structural elements. The gap is filled with gas in the production process of the unit, therefore the gas in the gap has some initial parameters of pressure, temperature, and volume. Under operating conditions, an insulating glass unit is exposed to climatic loads which generate loads and deflections of the component glass panes due to the pressure difference between the gap and the environment. For example, an increase in atmospheric pressure or a decrease in the gas temperature in the gap results in a concave form of deflection of the panes ([Figure 1](#materials-13-00286-f001){ref-type="fig"}a) and the opposite changes of these parameters in a convex form ([Figure 1](#materials-13-00286-f001){ref-type="fig"}b). The magnitude of the under or overpressure in the gap depends not only on the value of climatic loads, but also on the structure of IGU. In general, it increases with reduced IGU dimensions (width × length), increased thickness of the gas gap, and increased thickness of the component glass plates. How the pressure difference affects the deflections in IGU will be presented later in this article.
In the case of wind exposure ([Figure 1](#materials-13-00286-f001){ref-type="fig"}c), the tightness of the gap has a positive effect on the load distribution in an IGU. Due to changes in gas pressure in the gaps, the external load is partly transferred to the other panes of the unit.
The deflections of the glass described above result in deformation of the image viewed in the light reflected from the glass in windows or on glass facades ([Figure 2](#materials-13-00286-f002){ref-type="fig"}). It is important that under conditions of low air temperatures, i.e., during the heating season, insulating glass units tend to take the concave form of deflection. The result is a reduction in the thickness of the gas space---especially in the central part of the glazing, where the component glass panes are closest to each other---which makes it possible to reduce the thermal insulation of the IGU.
The aim of the analysis carried out in the paper was to determine the effect of taking into account the reduction in thickness of gas-filled gaps in insulating glass units in winter conditions on the calculated heat losses through these partitions. The analysis was made for example for double- and triple-glazed IGUs. A detailed numerical quantification of this phenomenon was carried out for various IGU constructions.
In the literature, studies describing previous research in this area can be found: Barnier and Bourret \[[@B3-materials-13-00286]\] analyzed the effect of plate curvature in IGUs on the thermal transmittance (*U*~g~-value). The authors determined the *U*~g~-value for IGUs with variable gap thickness (limited by the surfaces of deflected panes), considering the average gap thickness in the loaded IGU as reliable. The authors stated that this assumption becomes reasonable when plate curvature is small and it is certainly acceptable in the conduction regime, where the convective movement is not significant. This article provides the results of sample calculations for double- and triple-glazed units under winter conditions. It was found that taking into account the plate curvature increases the calculated *U*~g~-value from 4.4% to 5.8%. Calculations were also made to account for changes in weather conditions (typical meteorological year) for Montreal and Toulouse. The results indicate that *U*~g~ may vary up to 5% above and 10% below the yearly average.
Hart et al. \[[@B4-materials-13-00286]\] analyzed the *U*~g~-value calculated from real deflections of double and triple-glazed units, measured in summer and winter at several locations in the USA. It was found that a 20 °C temperature difference reduces thermal performance by 4.6% for double-glazed IGUs and by 3.6% for triple-glazed IGUs.
Penkova et al. \[[@B5-materials-13-00286]\] presented examples of numerical analysis and experimental research regarding both parameters related to heat flow and climate loads. However, no detailed analysis of the change in thermal transmittance related to the deflections in the IGUs was carried out.
Thermal imaging photographs illustrating a decrease in thermal insulation in the central part of the glazing were published as well \[[@B6-materials-13-00286]\]. Examples are presented where the temperature in the central part of the glazing is 1--3 °C higher than the average on its surface (in images from the outside). An example of an IGU in which the component panes came into contact due to climatic loads is also presented.
2. Methodology for the Calculation of Static Quantities in IGUs {#sec2-materials-13-00286}
===============================================================
The methods of calculation of static quantities in double-glazed IGUs loaded with climatic factors are described in the literature. Mention may be made here of analytical models presented in papers \[[@B7-materials-13-00286],[@B8-materials-13-00286],[@B9-materials-13-00286],[@B10-materials-13-00286]\] and numerical models allowing for consideration of the possibility of non-linear deflections of component glass \[[@B11-materials-13-00286],[@B12-materials-13-00286]\]. The results of calculations presented in this paper were obtained using the author's analytical model proposed in the article \[[@B13-materials-13-00286]\], which allows to calculate the load and deflection of component glass panes in units with any number of tight gaps.
The basis for the calculation of static quantities in IGUs is the assumption that the gas in the gaps meets the ideal gas equation:$$\frac{p_{0} \cdot v_{0}}{T_{0}} = \frac{p_{op} \cdot v_{op}}{T_{op}} = {const},$$ where:*p*~0~, *T*~0~, *v*~0~---initial gap gas parameters: pressure \[kPa\], temperature \[K\], volume \[m^3^\], obtained in the production process,*p*~op~, *T*~op~, *v*~op~---operating parameters---analogously.
It is also assumed that the glass panes are simply supported at the edges and that the linear dependence of deflection *w* \[m\] of the component glass pane on its resultant surface load *q* \[kN/m^2^\] is assumed. The latter assumption is considered to be sufficiently accurate if the deflection is not greater than the thickness of the glass \[[@B14-materials-13-00286]\]. The deflection function of a simply supported single pane of the *a* \[m\] width and *b* \[m\] length, subjected to the *q* \[kN/m^2^\] load, placed centrally in the x-y coordinate system, can be recorded as \[[@B15-materials-13-00286]\]:$$w\left( {x,y} \right) = \frac{4qa^{4}}{\mathsf{\pi}^{5}D}{\sum\limits_{i = 1,3,5\ldots}{\frac{\left( - \left. 1 \right) \right.^{{(i} - {1)/2}}}{i^{5}}\cos\frac{i\mathsf{\pi}x}{a}}} \cdot \left( {1 - \frac{\beta_{i} \cdot {th}\beta_{i} + 2}{2 \cdot {ch}\beta_{i}} \cdot {ch}\frac{i\mathsf{\pi}y}{a} + \frac{1}{2 \cdot {ch}\beta_{i}} \cdot \frac{i\mathsf{\pi}y}{a} \cdot {sh}\frac{i\mathsf{\pi}y}{a}} \right),$$ with $$\beta_{i} = \frac{i\mathsf{\pi}b}{2a},$$
*D* \[kNm\] is the flexural rigidity of glass pane:$$D = \frac{E \cdot d^{3}}{12 \cdot \left( {1 - \mu^{2}} \right)},$$ where:*d*---is the glass pane thickness \[m\],*E*---is the Young's modulus of glass \[kPa\],*μ*---is the Poisson's ratio \[-\].
Change in gap volume ∆*v* \[m^3^\] resulting from the deflection of one of the limiting glass pane may be determined by integration of Equation (2):$$\mathsf{\Delta}v = {\int\limits_{- b/2}^{b/2}{\int\limits_{- a/2}^{a/2}{w\left( x,y \right)dxdy}}},$$ $$\mathsf{\Delta}v = \frac{4qa^{6}}{\mathsf{\pi}^{7}D}{\sum\limits_{i = 1,3,5\ldots}{\frac{{( - 1)}^{(i - 1)/2}}{i^{7}} \cdot \frac{\sin\frac{i \cdot \mathsf{\pi}}{2}}{\left( {{ch}\beta_{i}} \right)^{2}}}} \cdot \left( {4 \cdot \beta_{i} + 2 \cdot \beta_{i} \cdot {ch}\left( {2 \cdot \beta_{i}} \right) - 3 \cdot {sh}\left( {2 \cdot \beta_{i}} \right)} \right),$$
After the relevant calculations have been made:$$\mathsf{\Delta}v = {\alpha_{\prime}}_{v} \cdot \frac{q \cdot a^{6}}{D} = \alpha_{v} \cdot q,$$ where:
1. *α*′~v~---is the dimensionless coefficient dependent on the *b*/*a* ratio ([Table 1](#materials-13-00286-t001){ref-type="table"}) \[-\],
2. *α*~v~---is the proportionality factor, \[m^5^/kN\].
Any change in climatic conditions (atmospheric pressure, temperature, wind) results in a change in the gas pressure in the gaps which affects the resultant operating load of each of the component glass panes. For each gap of an IGU it is possible to formulate the equation of state:$$p_{0} \cdot v_{0} \cdot T_{op} = p_{op} \cdot \left( {v_{0} + {\sum{\mathsf{\Delta}v}}} \right) \cdot T_{0},$$ where:
Σ∆*v*---is the change in gap volume caused by deflection of both panes limiting it \[m^3^\].
As already mentioned in the article, double- and triple-glazed IGUs were analyzed. In the remaining part, the parameters of the individual component glass panes and gaps were marked with appropriate indices ([Figure 3](#materials-13-00286-f003){ref-type="fig"}). It is also assumed that loads and deflections are positive if they face the interior, i.e., from left to right as in [Figure 3](#materials-13-00286-f003){ref-type="fig"}.
Taking into account the adopted markings and conventions, Equation (8) for a double-glazed IGU can be presented in the form:$$\frac{p_{0} \cdot v_{01} \cdot T_{{op}1}}{T_{0}} = p_{{op}1} \cdot \left\lbrack {\left( {p_{{op}1} - c_{ex}} \right) \cdot \alpha_{v,{ex}} + \left( {p_{{op}1} - c_{in}} \right) \cdot \alpha_{v,{in}}} \right\rbrack,$$ with $${c_{ex} = p_{a} + q_{z,{ex}}}{,\ c_{in} = p_{a} - q_{z,{in}},}$$ where:p~a~---current atmospheric pressure \[kPa\],*q*~z,ex~, *q*~z,in~---load per area from outer factors, primarily wind \[kN/m^2^\], almost always *q*~z,in~ = 0.
After the relevant transitions have been made:$$B \cdot p_{{op}1}^{2} - A \cdot p_{{op}1} - \frac{p_{0} \cdot v_{01} \cdot T_{{op}1}}{T_{0}} = 0,$$ with $$A = c_{ex} \cdot \alpha_{v,{ex}} + c_{in} \cdot \alpha_{v,{in}} - v_{01},$$ $$B = \alpha_{v,{ex}} + \alpha_{v,{in}}.$$
Equation (11) has one solution giving non-negative results:$$p_{{op}1} = \frac{A}{2 \cdot B} + \sqrt{\left( \frac{A}{2 \cdot B} \right)^{2} + \frac{p_{0} \cdot v_{01} \cdot T_{{op}1}}{B \cdot T_{0}}}.$$
In the case of a triple-glazed unit, a system of quadratic equations should be solved:$$\left\{ \begin{matrix}
{p_{{op}1} \cdot \left\lbrack {v_{01} + \left( {p_{{op}1} - c_{ex}} \right) \cdot \alpha_{v,{ex}} + \left( {p_{{op}1} - p_{{op}2}} \right) \cdot \alpha_{v,1 - 2}} \right\rbrack - \frac{p_{0} \cdot v_{01} \cdot T_{{op}1}}{T_{0}} = 0} \\
{p_{{op}2} \cdot \left\lbrack {v_{02} + \left( {p_{{op}2} - p_{{op}1}} \right) \cdot \alpha_{v,1 - 2} + \left( {p_{{op}2} - c_{in}} \right) \cdot \alpha_{v,{in}}} \right\rbrack - \frac{p_{0} \cdot v_{02} \cdot T_{{op}2}}{T_{0}} = 0} \\
\end{matrix} \right.$$
This system has no analytical solution, but it can be solved numerically by iteration.
After calculating the operating pressure *p*~op~ for each of the gaps, the resultant loading *q* for each of the component glass panes can be determined: for a double-glazed IGU $${q_{ex} = c_{ex} - p_{{op}1}}{,\ q_{in} = p_{{op}1} - c_{in},}$$for a triple-glazed IGU $${q_{ex} = c_{ex} - p_{{op}1}}{,\ q_{1–2} = p_{{op}1} - p_{{op}2}}{,\ q_{in} = p_{{op}2} - c_{in}}$$
Deflection *w*~c~ \[mm\] in the center of the glass pane can be determined by the formula:$$w_{c} = {\alpha^{\prime}}_{w} \cdot \frac{q \cdot a^{4}}{D} \cdot 1000,$$ where:
*α*′~w~---is the dimensionless coefficient dependent on the *b*/*a* ratio ([Table 1](#materials-13-00286-t001){ref-type="table"}) \[-\].
However, the average deflection of the component glass panes *w*~m~ \[mm\] was determined from the formula:$$w_{m} = \frac{\mathsf{\Delta}v}{a \cdot b} \cdot 1000.$$
3. Materials and Methods {#sec3-materials-13-00286}
========================
Thermal transmittance *U*~g~ \[W/(m^2^·K)\] of IGUs was calculated on the basis of the methodology described in standard \[[@B16-materials-13-00286]\], and heat losses were expressed by density of heat-flow rate *Φ* \[W/m^2^\] from the formula:$$\Phi = U_{g} \cdot \left( {t_{i} - t_{e}} \right),$$ where:
*t*~i~, *t*~e~---are the internal and external air temperature \[°C\].
The heat flow through an insulating glass unit is complex---through conduction, convection, and radiation. The thermal resistance of gas-filled gaps *R*~s~ \[(m^2^·K)/W\] has the greatest influence on the *U*-value. For each gap:$$R_{s} = \frac{1}{h_{g} + h_{r}},$$ with $$h_{g} = \frac{\lambda_{{}_{g}} \cdot Nu}{s},$$ $$h_{r} = \frac{4 \cdot \mathsf{\sigma} \cdot T_{m}^{3}}{\frac{1}{\varepsilon_{{sur}1}} + \frac{1}{\varepsilon_{s{{ur}2}}} - 1},$$ where: *h*~r~---is the thermal conductance by radiation \[W/(m^2^·K)\],*h*~g~---is the thermal conductance of gas (by conduction and convection) \[W/(m^2^·K)\].*λ*~g~---is the thermal conductivity of gas \[W/(m·K)\],*s*---is the gas gap thickness \[m\],*Nu*---is the Nusselt number \[-\],σ---is the Boltzmann constant 5.6693 × 10^−8^ W/(m^2^·K^4^)*T*~m~---is the average temperature of both surfaces delimiting the gap \[K\],*ε*~sur1~, *ε*~sur2~---are the emissivity of surfaces delimiting the gap \[-\].
It is particularly important whether convection occurs in the gaps. In the case of narrow gaps (*Nu* \< 1) it is assumed that convection does not occur---thermal insulation of the gap increases linearly with its thickness. If a certain gap thickness limit (for *Nu* = 1) is exceeded, the effect of convection is taken into account. In this non-linear range (for *Nu* \> 1) thermal insulation of the IGU does not improve. The value of this thickness limit depends on many factors (see also \[[@B17-materials-13-00286],[@B18-materials-13-00286]\]), first of all on: the type of gas; the calculation was based on the use of argon,location in the structure; the calculations assume a horizontal position, in units situated horizontally or diagonally convection increases.increasing the temperature difference on the surfaces of the glass panes limiting the gap affects the increase in convection,convection also increases when the average gas temperature in the gap increases.
The thermal resistance of the gaps is primarily influenced by the use of low-emission glass. Glass without coating has a standard coefficient of emission of *ε* = 0.837. Application of low-emission coating reduces the emissivity of the plate surface, which results in a significant reduction of heat transfer by radiation. Currently, in Central and Northern Europe, IGUs are most often produced, in which each gap is adjacent to one coated surface and one without coating ([Figure 3](#materials-13-00286-f003){ref-type="fig"}). This solution is most often used in units currently produced in Central and Northern Europe. The values *ε*~sur1~ = 0.837 and *ε*~sur2~ = 0.04 were used in the calculations.
Glass conducts heat well, therefore the thickness of the glass panes has no significant effect on the *U*~g~-value. Physical parameters of argon were adopted on the basis of the standard \[[@B17-materials-13-00286]\].
Of course, thermal insulation is also affected by thermal surface resistance at the external side (*R*~e~ \[(m^2^·K)/W\]) and at the internal side of a window (*R*~i~ \[(m^2^·K)/W\]). They depend primarily on the positioning of the window in the structure and the velocity of air (a short analysis on this subject is presented in Chapter 5). The calculations assumed *R*~i~ = 0.13 (m^2^·K)/W (vertical position) and *R*~e~ = 0.04 (m^2^·K)/W (for wind velocity *V* = 4 m/s). These are often accepted comparative values, also in the standard \[[@B16-materials-13-00286]\].
The calculations according to the adopted model require the use of numerical methods, because we encounter several interdependent values here. For example, the temperature values of gas and glass surfaces depend on the temperature distribution in the cross-section of the IGU. This distribution depends on the resulting thermal resistance values. The results of calculations were obtained by iteration after building the appropriate spreadsheet, assuming the steady state of heat transfer.
[Figure 4](#materials-13-00286-f004){ref-type="fig"} shows the effect of gap thickness *s* \[mm\] on the design *U*~g~-value for double- and triple-glazed IGUs, with glass thickness *d* = 4 mm, assuming *t*~i~ = 20 °C and in two variants of the outside air temperature *t*~e~ = 0 °C and *t*~e~ = −20 °C. The dashed line was used to determine the limits of gap thickness at which *Nu* = 1.
[Figure 4](#materials-13-00286-f004){ref-type="fig"} shows that at low temperatures *t*~e~ the limit thickness decreases. It can also be stated that in the case of triple-glazed IGUs, the difference in temperature in the gap is smaller, and the thickness of the boundary increases.
4. IGUs Under Pressure and Temperature Changes---Presentation and Discussion of Test Results {#sec4-materials-13-00286}
============================================================================================
An analysis of the influence of climate loads on heat loss through IGUs under winter conditions was carried out for sample units with dimensions 0.7 × 1.4 m. Glass material parameters were adopted according to the standard \[[@B19-materials-13-00286]\]: *E* = 70 GPa, *μ* = 0.2.
It was also assumed that the following initial parameters were obtained in the argon-filled gaps during the production process *T*~0~ = 20 °C = 293.15 K, *p*~0~ = 100 kPa. In these conditions, the component glass panes are flat.
Two variants of the temperature drop load were used.
*Variant 1.* Reduced temperature conditions: *t*~i~ = 20 °C, *t*~e~ = −20 °C; the gas temperature in each gap was calculated for each case based on the temperature distribution in the particular IGU: for a double-glazed IGU *T*~op1~ = −2.37 to −2.25 °C, for triple-glazed IGU *T*~op1~ = −10.09 to −9.66 °C, *T*~op2~ = 7.60 to 7.97 °C.
*Variant 2*. Conditions for a "mild winter": *t*~i~ = 20 °C, *t*~e~ = 0 °C; gas temperature: for a double-glazed IGU *T*~op1~ = 8.80 to 9.01 °C, for triple-glazed IGU *T*~op1~ = 4.97 to 5.08 °C, *T*~op2~ = 13.66 to 14.10 °C.
First, the effect of varying glass thickness on the gap width in the loaded set was investigated. IGUs with 16 and 12 mm gap thickness were analyzed in various combinations of 3, 4, and 6 mm thick panes. It was assumed that IGUs are only loaded with the temperature drop, as in variant 1, i.e., the current atmospheric pressure *p*~a~ = *p*~0~ = 100 kPa. The results of the calculations are presented in [Table 2](#materials-13-00286-t002){ref-type="table"}.
The resultant loading *q*~ex~ and *q*~in~ (absolute value) illustrates the underpressure in the gaps in relation to atmospheric pressure. The parameter *q*~1-2~ is the difference in operating pressure between the gaps in a triple-glazed IGU. From Equations (18) and (19) the extreme deflection (in the center of the pane) *w*~c~ and the average deflection *w*~m~ (the *w*~m~ values are given between parentheses) were calculated for each pane.
On the basis of these deflections, the minimum gap thickness in the center of the IGU *s*~c~ \[mm\] and the average gap thickness *s*~m~ \[mm\] were calculated.
On the basis of the calculations presented in [Table 2](#materials-13-00286-t002){ref-type="table"}, it was found that the calculated values of *s*~c~ and *s*~m~ for analyzed IGUs with gaps of the same nominal thickness do not differ much from each other. This is despite the fact that the deflection of component glass panes varies considerably. The effect of gas interactions in tight gaps can be seen here. Rigid panes are less susceptible to deflection, but the external load is less compensated for by the gas pressure in the gap. After changing the thickness of all the panes in a unit, the load changes, but the deflections are similar. Therefore, when one of the glass panes changes to a stiffer one, the absolute load values of component glass panes increase, although their algebraic sum for each IGU is equal to 0. After such conversion, the less rigid panes deflect more because they are exposed to a higher loading---for this reason, a loaded IGU has approximately constant volume of gaps, despite the change in thickness of the component panes.
To identify the extent of the phenomenon described above, another example was solved. [Figure 5](#materials-13-00286-f005){ref-type="fig"} shows the influence of IGU width (at a constant ratio *b*/*a* = 2) on the maximum deflection of component panes *w*~c~ in double-glazed units at 3, 4 i 6 mm thick panes and 16 mm thick gap. It was assumed that IGU is loaded only by a change in atmospheric pressure by ∆*p* = *p*~a~ − *p*~o~ = 3 kPa. This means that the current atmospheric pressure is *p*~a~ = 103 kPa. It can be added here that the results of calculations of static quantities are not very sensitive to the value of *p*~o~, and to a significant extent to ∆*p*. This means that if we assumed, for example, *p*~o~ = 950 kPa i *p*~a~ = 980 kPa, the results would be almost identical.
[Figure 5](#materials-13-00286-f005){ref-type="fig"} shows that that greater diversity of deflections for units of different component panes thickness occurs in the case of smaller IGU sizes. Then, however, the deflection values are smaller and it can be expected that changes in the gap thickness are also small.
In the context of the above, further analysis was carried out for IGUs with the same thickness of component glass panes *d* = 4 mm and different gap thicknesses *s* were assumed. [Table 3](#materials-13-00286-t003){ref-type="table"} presents calculations of *w*~c~, *w*~m~, *s*~c~, and *s*~m~ values for units loaded with temperature change as in variant 1 and simultaneously operating loading with external atmospheric pressure increase of ∆*p* = 3 kPa. These are particularly unfavorable operating conditions in the context of reducing the thickness of the gaps. An analogous calculation was carried out for variant 2 ([Table 4](#materials-13-00286-t004){ref-type="table"}).
Based on the above data, [Table 5](#materials-13-00286-t005){ref-type="table"} (for variant 1) and [Table 6](#materials-13-00286-t006){ref-type="table"} (for variant 2) present the results of calculations of the thermal transmittance *U* and the density of heat-flow rate *Φ* \[W/m^2^\]:*U*~g~, *Φ*~g~---describe heat loss without taking into account the curvature of the panes, calculated for the nominal thickness of the gaps,*U*~c~, *Φ*~c~---describe possible local heat loss near the IGU center, i.e., where the distance between the panes is the smallest, calculated for the thickness of the gaps *s*~c~,*U*~m~, *Φ*~m~---describe the average heat loss through the IGU, calculated for the thickness of the gaps *s*~m~.
Finally, the percentage increase in the calculated quantities is also presented (∆*Φ*~c~*,* ∆*Φ*~m~*)* for units of nominal gap thickness.
The data presented in [Table 5](#materials-13-00286-t005){ref-type="table"} and [Table 6](#materials-13-00286-t006){ref-type="table"} and [Figure 4](#materials-13-00286-f004){ref-type="fig"} indicate that the reduction in the thickness of the gaps of insulating glass units due to their deflection under a drop in gas temperature and a rise in atmospheric pressure may result in an increase in design heat losses in relation to the calculations without taking into account the curvature of the panes. The increase in heat loss occurs in the linear range of the *U*~g~-value change, i.e., when the conditions inside the gap lead to *Nu* \< 1. It is different when the *U*~g~-value changes in the non-linear range (*Nu* \> 1). Heat losses do not increase. Then, the reduction of gap thickness can lead to a slight decrease in the calculated *Nu* value, which translates into a slight reduction in the calculated heat losses.
In this context, it is preferable to design IGUs such that it has *Nu* \> 1 with a certain margin based on glazing deflections. However, this task should be approached with great caution, taking into account local climate conditions. It is necessary to check if the thickening of the gaps between the panes will not lead to excessive overpressure during the summer, due to the heating of gas in the gaps.
One more feature of the described phenomenon should be noted. In the linear range of changes in the *U*~g~-value, the indices ∆*Φ~c~* and ∆*Φ*~m~ almost do not depend on the thickness of the gaps. This is due to the fact, as additional calculations have shown, that the relationship between the thickness of IGU gaps and static quantities (resultant loading of component glass and their deflections) is also linear.
For many years, double-glazed IGUs dominated the market. Currently, due to the need to save energy, in Central and Northern Europe, triple-glazed IGU 4-16-4-16-4 is the most commonly produced and sold glazing for windows. [Figure 6](#materials-13-00286-f006){ref-type="fig"} presents an analysis illustrating the dependence of the percentage change in the calculated heat loss ∆*Φ*~m~ for these units on their width (at a constant ratio *b/a* = 2), under different external temperature conditions *t*~e~. Simultaneous pressure increase ∆*p* = 3 kPa was assumed. Other data was used as in previous examples.
It was found that the described effect is important for the currently sold glazing in "mild winter" conditions, i.e., when the outside temperatures fluctuate within between −5 °C and 5 °C. For IGU width above 0.7m, the ratio ∆*Φ*~m~ changes from 3.9% to 5.0%. These values are characteristic of the average temperature during the winter months in many places around the world.
5. Notes on IGUs Wind Load {#sec5-materials-13-00286}
==========================
Wind pressure or suction are also factors that cause deflection of the component glass panes in an IGU. As already mentioned, the wind velocity pressure acts directly only on the outer pane, but due to the change in the gas pressure in the gaps, the resultant load is distributed over all the panes of the unit. [Table 7](#materials-13-00286-t007){ref-type="table"} shows the resultant loads and deflections in sample unit's surface loaded with 0.3 kN/m^2^, which approximately corresponds \[[@B20-materials-13-00286]\] to a pressure of wind with velocity *V* of approx. 80 km/h (22.2 m/s).
[Table 7](#materials-13-00286-t007){ref-type="table"} demonstrates that in the majority of units the deflections of component glass have similar values. Greater variations may occur when thicker panes are used, but the deflection values are small. It can therefore be concluded that the change in the thickness of the gaps due to wind load is small and has no noticeable effect on heat loss by IGUs.
Wind velocity has an indirect effect on heat loss. It is a factor influencing external thermal surface resistance on the outside, which translates into the *U*~g~-value. Graphic illustration of this effect is shown in [Figure 7](#materials-13-00286-f007){ref-type="fig"}. The calculations were made for units with gap thickness of 16 mm. It can be noted that in the case of triple-glazed IGUs, the effect of wind velocity is negligible.
6. Conclusions {#sec6-materials-13-00286}
==============
One of the factors influencing thermal transmittance *U*~g~ of insulating glass units is the thickness of gas-filled tight gaps. It is assumed in the calculation procedures that this thickness is not dependent on temporary changes in climatic factors. The thickness is variable under real operating conditions. In winter conditions in particular, IGU component glass panes take a concave form of deflection, which reduces the thickness of the gaps. This effect increases if the atmospheric pressure increases at the same time.
Based on the example calculations carried out, it has been shown that the increase in the calculated heat losses associated with the reduction of the gap thickness occurs when the conditions in the gap lead to *Nu* \< 1, i.e., when the thermal transmittance of the gas layer is linearly dependent on its thickness. Heat losses can then increase to about 4.6% for double-glazed IGUs and to about 5% for triple-glazed ones, for external air temperature *t*~e~ = 0 °C. These values almost do not depend on the nominal thickness of the gaps, which results from the linear dependence of static quantities in an IGU on this thickness. Under certain conditions, heat losses calculated according to standard procedures may therefore be underestimated.
It is different in the non-linear range of the *U~g~*-value change (*Nu* \> 1), i.e., when the outside temperature drops significantly or the gaps are thick enough. The thermal performance of glazing does not deteriorate. It is therefore advantageous to design IGUs so that *Nu* \> 1, but it is necessary to take into account local climatic conditions and analyze loads that may also occur during the summer period.
In the case of the most commonly sold triple-glazed units 4-16-4-16-4 heat losses may be underestimated when the outside temperatures fluctuate between −5 °C and 5 °C. For large IGU dimensions, the ∆*Φ*~m~ index totals then from 3.9% to 5.0%.
It was also shown that the effect of wind load on gap thickness change is negligible in the context of heat loss estimation.
This research received no external funding.
The author declares no conflict of interest.
A
auxiliary parameter \[m
3
\]
a
width (of glass pane) \[m\]
B
auxiliary parameter \[m
5
/kN\]
b
length (of glass pane) \[m\]
c
auxiliary parameter \[kPa\]
D
flexural rigidity (of glass pane) \[kNm\]
d
thickness (of glass pane) \[m\] or \[mm\]
E
Young's modulus \[kPa\] or \[GPa\]
h
thermal conductance \[W/(m
2
·K)\]
i
consecutive natural number
Nu
Nusselt number
p
pressure \[kPa\]
q
load per area, \[kN/m
2
\]
R
thermal resistance \[(m
2
·K)/W\]
s
thickness (of gas gap) \[mm\]
T
temperature (of gas in the gap) \[K\]
t
temperature (of air) \[K\] or \[°C\]
U
thermal transmittance \[W/(m
2
·K)\]
V
wind velocity \[km/h\]
v
volume (of the gap) \[m
3
\]
w
deflection \[mm\]
w(x,y)
function of deflection, \[m\]
x-y
coordinate system
Greek letters
α
proportionality factor, \[m
5
/kN\]
α'
dimensionless coefficient \[-\]
β
i
auxiliary parameter \[-\]
∆
p
pressure change \[kPa\]
∆
T
temperature difference \[K\]
∆
v
volume change \[m
3
\]
∆
Φ
percentage increase in density of heat-flow rate \[%\]
ε
surface emissivity \[-\]
λ
thermal conductivity \[W/(m·K)\]
μ
Poisson's ratio \[-\]
π
number "pi"
σ
Boltzmann constant \[W/(m
2
·K
4
)\]
Φ
density of heat-flow rate \[W/m
2
\]
Subscripts and markings
0
initial gas parameters
1, 2
specific gas-filled gap
1-2
glass pane (between gaps)
c
center (of glass pane)
a
atmospheric
e
external
ex
exterior glass pane
g
regarding gas or regarding IGU (at
U
g
,
Φ
g
and ∆
Φ
g
)
i
internal
in
interior glass pane
m
mean, average
op
operating gas parameters
r
radiative
s
regarding gas gap
sur1, sur2
regarding surfaces
v
regarding volume
w
regarding deflection
z
outside
{#materials-13-00286-f001}
{#materials-13-00286-f002}
{#materials-13-00286-f003}
{#materials-13-00286-f004}
{#materials-13-00286-f005}
{#materials-13-00286-f006}
{#materials-13-00286-f007}
materials-13-00286-t001_Table 1
######
Coefficients for calculating volume change and deflection for simply supported glass pane.
------------- ---------- ---------- ---------- ---------- ---------- ----------
***b/a*** 1.0 1.1 1.2 1.3 1.4 1.5
***α*′~v~** 0.001703 0.002246 0.002848 0.003499 0.004189 0.004912
***α*′~w~** 0.004062 0.004869 0.005651 0.006392 0.007085 0.007724
***b/a*** 1.6 1.7 1.8 1.9 2.0 3.0
***α*′~v~** 0.005659 0.006427 0.00721 0.008004 0.008808 0.017055
***α*′~w~** 0.008308 0.008838 0.009316 0.009745 0.010129 0.012233
------------- ---------- ---------- ---------- ---------- ---------- ----------
materials-13-00286-t002_Table 2
######
Static quantities and gap thicknesses in IGUs---under reduced temperature conditions (Variant 1).
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Structure of IGU \[mm\] Resultant Loading *q* \[kN/m^2^\] Deflection *w*~c~ (*w*~m~) \[mm\] Resultant Thickness of Gap *s* \[mm\]
-------------------------------------------- ----------------------------------- ----------------------------------- --------------------------------------- -------- --------- --------- ------- ------- ------- -------
***d*~ex~-*s*~1~-*d*~in~** **Double-glazed units**
4-16-4 0.218 \- −0.218 1.36\ \- −1.36\ 13.28 14.82 \- \-
(0.59) (−0.59)
6-16-4 0.330 \- −0.330 0.61\ \- −2.08\ 13.31 14.83 \- \-
(0.27) (−0.90)
4-12-4 0.164 \- −0.164 1.03\ \- −1.03\ 9.94 11.10 \- \-
(0.45) (−0.45)
6-12-4 0.251 \- −0.251 0.47\ \- −1.57\ 9.96 11.12 \- \-
(0.20) (−0.68)
3-12-3 0.070 \- −0.070 1.04\ \- −1.04\ 9.92 11.10 \- \-
(0.45) (−0.45)
***d*~ex~-*s*~1~-*d*~1-2~-*s*~2~-*d*~in~** **Triple-glazed units**
4-16-4-16-4 0.463 −0.117 −0.346 2.90\ −0.73\ −2.16\ 12.37 14.42 14.57 15.38
(1.26) (−0.32) (−0.94)
6-16-4-16-4 0.840 −0.311 −0.529 1.56\ −1.94\ −3.34\ 12.50 14.47 14.63 15.41
(0.68) (−0.85) (−1.44)
4-12-4-12-4 0.345 −0.087 −0.258 2.16\ −0.54\ −1.61\ 9.30 10.77 10.93 11.54
(0.99) (−0.24) (−0.70)
6-12-4-12-4 0.631 −0.233 −0.398 1.17\ −1.46\ −2.49\ 9.37 10.86 10.97 11.55
(0.51) (−0.63) (−1.08)
6-12-3-12-4 0.540 −0.112 −0.429 1.00\ −1.66\ −2.68\ 9.34 10.85 10.98 11.55
(0.43) (−0.72) (−1.17)
3-12-3-12-3 0.149 −0.037 −0.112 2.21\ −0.55\ −1.66\ 9.24 10.80 10.89 11.52
(0.96) (−0.24) (−0.72)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
materials-13-00286-t003_Table 3
######
Static quantities and gap thicknesses in IGUs under reduced temperature conditions (Variant 1) and atmospheric pressure increase by ∆*p* = 3 kPa.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Structure of IGU \[mm\] Resultant Loading *q* \[kN/m^2^\] Deflection *w*~c~ (*w*~m~) \[mm\] Resultant Thickness of Gap *s* \[mm\]
-------------------------------------------- ----------------------------------- ----------------------------------- --------------------------------------- -------- --------- --------- ------- ------- ------- -------
***d*~ex~-*s*~1~-*d*~in~** **Double-glazed units**
4-16-4 0.295 \- −0.295 1.84\ \- −1.84\ 12.32 14.38 \- \-
(0.81) (−0.81)
4-14-4 0.259 \- −0.259 1.62\ \- −1.62\ 10.76 12.60 \- \-
(0.70) (−0.70)
4-12-4 0.233 \- −0.233 1.39\ \- −1.39\ 9.22 10.78 \- \-
(0.61) (−0.61)
4-10-4 0.187 \- −0.187 1.17\ \- −1.17\ 7.66 8.98 \- \-
(0.51) (−0.51)
***d*~ex~-*s*~1~-*d*~1-2~-*s*~2~-*d*~in~** **Triple-glazed units**
4-16-4-16-4 0.613 −0.114 −0.500 3.83\ −0.71\ −3.13\ 11.46 14.02 13.58 14.95
(1.67) (−0.31) (−1.36)
4-14-4-14-4 0.540 −0.100 −0.440 3.38\ −0.63\ −2.75\ 9.99 12.26 11.88 13.07
(1.47) (−0.27) (−1.20)
4-12-4-12-4 0.460 −0.085 −0.375 2.87\ −0.53\ −2.35\ 8.60 10.52 10.18 11.21
(1.25) (−0.23) (−1.02)
4-10-4-10-4 0.387 −0.070 −0.317 2.42\ −0.44\ −1.98\ 7.14 8.76 8.46 9.33
(1.05) (−0.19) (−0.86)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
materials-13-00286-t004_Table 4
######
Static quantities and gap thicknesses in IGUs under conditions for a "mild winter" (Variant 2) and atmospheric pressure increase by ∆*p* = 3 kPa.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Structure of IGU \[mm\] Resultant Loading *q* \[kN/m^2^\] Deflection *w*~c~ (*w*~m~) \[mm\] Resultant Thickness of Gap *s* \[mm\]
-------------------------------------------- ----------------------------------- ----------------------------------- --------------------------------------- -------- --------- --------- ------- ------- ------- -------
***d*~ex~-*s*~1~-*d*~in~** **Double-glazed units**
4-16-4 0.188 \- −0.188 1.18\ \- −1.18\ 13.64 14.98 \- \-
(0.51) (−0.51)
4-14-4 0.165 \- −0.165 1.03\ \- −1.03\ 11.94 13.10 \- \-
(0.45) (−0.45)
4-12-4 0.143 \- −0.143 0.89\ \- −0.89\ 10.22 11.20 \- \-
(0.40) (−0.40)
4-10-4 0.120 \- −0.120 0.75\ \- −0.75\ 8.50 9.34 \- \-
(0.33) (−0.33)
***d*~ex~-*s*~1~-*d*~1-2~-*s*~2~-*d*~in~** **Triple-glazed units**
4-16-4-16-4 0.384 −0.058 −0.326 2.41\ −0.36\ −2.04\ 13.23 14.80 14.32 15.27
(1.04) (−0.16) (−0.89)
4-14-4-14-4 0.339 −0.050 −0.289 2.12\ −0.31\ −1.81\ 11.57 12.94 12.50 13.35
(0.92) (−0.14) (−0.79)
4-12-4-12-4 0.293 −0.042 −0.251 1.83\ −0.26\ −1.56\ 9.91 11.09 10.70 11.43
(0.80) (−0.11) (−0.68)
4-10-4-10-4 0.246 −0.035 −0.212 1.54\ −0.22\ −1.32\ 8.24 9.24 8.90 9.52
(0.67) (−0.09) (−0.57)
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
materials-13-00286-t005_Table 5
######
Quantities describing heat losses by IGUs under conditions of reduced temperature (Variant 1) and atmospheric pressure increase by ∆*p* = 3 kPa.
Gas Gap Thickness \[mm\] Thermal Transmittance \[W/(m^2^·K)\] Density of Heat-Flow Rate *Φ* \[W/m^2^\] ∆*Φ~c~* \[%\] ∆*Φ*~m~ \[%\]
-------------------------- -------------------------------------- ------------------------------------------ --------------- --------------- ------- ------- ------ ------
**Double-glazed units**
16 1.330 1.299 1.317 53.20 51.96 52.68 −2.3 −1.0
14 1.314 1.298 1.302 52.56 51.92 52.08 −1.2 −0.9
12 1.296 1.441 1.297 51.84 57.64 51.88 11.2 −0.1
10 1.364 1.630 1.467 54.56 65.20 58.68 19.5 7.6
**Triple-glazed units**
16 0.643 0.653 0.636 25.72 26.12 25.44 1.6 −1.1
14 0.634 0.727 0.648 25.36 29.08 25.92 14.7 2.2
12 0.675 0.817 0.730 27.00 32.68 29.20 21.0 8.1
10 0.776 0.941 0.839 31.04 37.64 33.56 21.3 8.1
materials-13-00286-t006_Table 6
######
Quantities describing heat losses by IGUs under conditions for a "mild winter" (Variant 2) and atmospheric pressure increase by ∆*p* = 3 kPa.
Gas Gap Thickness \[mm\] Thermal Transmittance \[W/(m^2^·K)\] Density of Heat-Flow Rate *Φ* \[W/m^2^\] ∆*Φ~c~* \[%\] ∆*Φ*~m~ \[%\]
-------------------------- -------------------------------------- ------------------------------------------ --------------- --------------- ------- ------- ------ ------
**Double-glazed units**
16 1.113 1.142 1.107 22.26 22.84 22.14 2.6 −0.5
14 1.122 1.250 1.174 22.44 25.00 23.48 11.4 4.6
12 1.246 1.388 1.305 24.92 27.76 26.10 11.4 4.7
10 1.408 1.567 1.473 28.16 31.34 29.46 11.3 4.6
**Triple-glazed units**
16 0.563 0.631 0.590 11.26 12.62 11.80 12.1 4.8
14 0.623 0.699 0.653 12.46 13.98 13.06 12.2 4.8
12 0.700 0.786 0.735 14.00 15.72 14.70 12.3 5.0
10 0.804 0.903 0.844 16.08 18.06 16.88 12.3 5.0
materials-13-00286-t007_Table 7
######
Static quantities in IGUs loaded with wind pressure of 0.3 kN/m^2.^
Structure of IGU \[mm\] Resultant Loading *q* \[kN/m^2^\] Deflection *w*~c~ \[mm\]
-------------------------------------------- ----------------------------------- -------------------------- ------- ------ ------ ------
***d*~ex~-*s*~1~-*d*~in~** **Double-glazed units**
4-16-4 0.154 \- 0.146 0.96 \- 0.91
8-16-4 0.268 \- 0.032 0.21 \- 0.20
4-16-8 0.047 \- 0.253 0.29 \- 0.20
***d*~ex~-*s*~1~-*d*~1-2~-*s*~2~-*d*~in~** **Triple-glazed units**
4-16-4-16-4 0.109 0.098 0.092 0.68 0.61 0.58
8-14-4-14-4 0.247 0.028 0.026 0.19 0.17 0.16
4-12-4-12-8 0.053 0.039 0.208 0.33 0.24 0.16
|
When a hyperbolic metamaterial is corrugated, its reflectance is greatly reduced and it becomes "darker than black." Image credit: E. E. Narimanov, et al.
(PhysOrg.com) -- If typical black paint absorbs about 85% of incoming light, then a newly designed metamaterial that absorbs up to 99% of incoming light may be considered darker than black." By taking advantage of the unique light-scattering properties of metamaterials, researchers have discovered that a hyperbolic metamaterial with a corrugated surface can have a very low reflectance, which could make it promising for high-efficiency solar cells, photodetectors, and radar stealth technology.
The researchers, E. Narimanov, et al., from Purdue University and Norfolk State University, have posted their study on the radiation-absorbing metamaterial at arXiv.org.
In their study, the researchers fabricated a hyperbolic metamaterial out of arrays of silver nanowires grown in alumina membranes. They found that this material absorbed about 80% of incoming light. Then, they ground the surface of the metamaterial to produce corrugations and defects, which they predicted would dramatically reduce the light reflection, increasing the absorption. Their measurements showed that the corrugated metamaterial absorbed up to 99% of incoming light, and that the radiation-absorbing capability is applicable to all parts of the electromagnetic spectrum.
As the scientists explained, the metamaterials very low reflectivity results from one of its hyperbolic properties: an infinite density of photonic states. This super singularity greatly increases the amount of light scattering from surface defects and corrugations in the metamaterial. The defects and corrugations scatter light primarily inside the material, basically sucking photons inside the hyperbolic medium.
The researchers predict that the new metamaterial will provide a new route toward designing radiation-absorbing materials. As light absorption plays a key role in solar cells and many other applications, the researchers plan to investigate these possibilities in the near future.
Explore further Bending light with better precision
© 2011 PhysOrg.com |
Q:
Python Line_profiler and Cython function
So I'm trying to profile a function within a python script of my own using line_profiler, because I want line-by-line timings. The only problem is that the function is a Cython one, and line_profiler isn't working correctly. On the first runs it was just crashing with an error. I then added
!python
cython: profile=True
cython: linetrace=True
cython: binding=True
at the top of my script and now it runs fine, except the timings and statistics are blank!
Is there a way to use line_profiler with a Cythonized function?
I could profile the non-Cythonized function, but it's so much slower than the Cythonized one that I could not use the information coming from the profiling - the slowness of the pure python one would make it impossible how I could improve the Cython one.
Here is the code of the function I'd want to profile:
class motif_hit(object):
__slots__ = ['position', 'strand']
def __init__(self, int position=0, int strand=0):
self.position = position
self.strand = strand
#the decorator for line_profiler
@profile
def find_motifs_cython(list bed_list, list matrices=None, int limit=0, int mut=0):
cdef int q = 3
cdef list bg = [0.25, 0.25, 0.25, 0.25]
cdef int matrices_length = len(matrices)
cdef int results_length = 0
cdef int results_length_shuffled = 0
cdef np.ndarray upper_adjust_list = np.zeros(matrices_length, np.int)
cdef np.ndarray lower_adjust_list = np.zeros(matrices_length, np.int)
#this one need to be a list for MOODS
cdef list threshold_list = [None for _ in xrange(matrices_length)]
cdef list matrix_list = [None for _ in xrange(matrices_length)]
cdef np.ndarray results_list = np.zeros(matrices_length, np.object)
cdef int count_seq = len(bed_list)
cdef int mat
cdef int i, j, k
cdef int position, strand
cdef list result, results, results_shuffled
cdef dict result_temp
cdef int length
if count_seq > 0:
for mat in xrange(matrices_length):
matrix_list[mat] = matrices[mat]['matrix'].tolist()
#change that for a class
results_list[mat] = {'kmer': matrices[mat]['kmer'],
'motif_count': 0,
'pos_seq_count': 0,
'motif_count_shuffled': 0,
'pos_seq_count_shuffled': 0,
'ratio': 0,
'sequence_positions': np.empty(count_seq, np.object)}
length = len(matrices[mat]['kmer'])
#wrong with imbalanced matrices
upper_adjust_list[mat] = int(ceil(length / 2.0))
lower_adjust_list[mat] = int(floor(length / 2.0))
#upper_adjust_list[mat] = 0
#lower_adjust_list[mat] = 0
#-0.1 to adjust for a division floating point bug (4.99999 !< 5, but is < 4.9!)
threshold_list[mat] = MOODS.max_score(matrix_list[mat]) - float(mut) - 0.1
#for each sequence
for i in xrange(count_seq):
item = bed_list[i]
#TODO: remove the Ns, but it might unbalance
results = MOODS.search(str(item.sequence[limit:item.total_length - limit]), matrix_list, threshold_list, q=q, bg=bg, absolute_threshold=True, both_strands=True)
results_shuffled = MOODS.search(str(item.sequence_shuffled[limit:item.total_length - limit]), matrix_list, threshold_list, q=q, bg=bg, absolute_threshold=True, both_strands=True)
results = results[0:len(matrix_list)]
results_shuffled = results_shuffled[0:len(matrix_list)]
results_length = len(results)
#for each matrix
for j in xrange(results_length):
result = results[j]
result_shuffled = results_shuffled[j]
upper_adjust = upper_adjust_list[j]
lower_adjust = lower_adjust_list[j]
result_length = len(result)
result_length_shuffled = len(result_shuffled)
if result_length > 0:
results_list[j]['pos_seq_count'] += 1
results_list[j]['sequence_positions'][i] = np.empty(result_length, np.object)
#for each motif
for k in xrange(result_length):
position = result[k][0]
strand = result[k][1]
if position >= 0:
strand = 0
adjust = upper_adjust
else:
position = -position
strand = 1
adjust = lower_adjust
results_list[j]['motif_count'] += 1
results_list[j]['sequence_positions'][i][k] = motif_hit(position + adjust + limit, strand)
if result_length_shuffled > 0:
results_list[j]['pos_seq_count_shuffled'] += 1
#for each motif
for k in xrange(result_length_shuffled):
results_list[j]['motif_count_shuffled'] += 1
#j = j + 1
#i = i + 1
for i in xrange(results_length):
result_temp = results_list[i]
result_temp['ratio'] = float(result_temp['pos_seq_count']) / float(count_seq)
return results_list
I'm pretty sure the triple nested loop is the main slow part - it's job is just to rearrange the results coming from MOODS, the C module doing the main work.
A:
Till Hoffmann has useful information on using line_profiler with Cython here: How to profile cython functions line-by-line.
I quote his solution:
Robert Bradshaw helped me to get Robert Kern's line_profiler tool working for cdef functions and I thought I'd share the results on stackoverflow.
In short, set up a regular .pyx file and build script and pass to cythonize the linetrace compiler directive to enable both profiling and line tracing:
from Cython.Build import cythonize
cythonize('hello.pyx', compiler_directives={'linetrace': True})
You may also want to set the (undocumented) directive binding to True.
Also, you should define the C macro CYTHON_TRACE=1 by modifying your extensions setup such that
extensions = [
Extension('test', ['test.pyx'], define_macros=[('CYTHON_TRACE', '1')])
]
A working example using the %%cython magic in the iPython notebook is here:
http://nbviewer.ipython.org/gist/tillahoffmann/296501acea231cbdf5e7
A:
Api was changed. Now:
from Cython.Compiler.Options import get_directive_defaults
directive_defaults = get_directive_defaults()
directive_defaults['linetrace'] = True
directive_defaults['binding'] = True
|
Q:
Lua language context/scope implementation
I am implementing a Lua interpreter in C# and I've stumbled across a performance issue, which I assume it is caused by incorrect design:
in my implementation, scopes are arranged in a hierarchical manner, that is, each scope has a parent scope which may or may not be null; each time a variable is being set or requested the scope checks whether the specified variable is contained in it. If not, asks its parent to do the same or, if there is no parent, it creates it / returns nil.
The problem is that scopes use Dictionary<string, LuaObject> underneath, and Get / Set functions are recursive. Is there a better way to do it? I've been thinking about different approaches, and I've found this one, which is not very elegant nor sounds efficient (inspired by both V8 and C# closures):
Use types and fields as variables.
Everytime you request a variable which was not previously declared, create a new type using Reflection that inherits the previous one.
If a function is declared, check whether it uses a variable declared in a outer scope. If so, re-create a field in that type and make the outer variable point to the new one. (I know this point is explained really awfully, but it is nothing but the standard way C# closures are implemented. You can go and read that if you didn't understand)
Are there simpler / more elegant solutions? How is Lua internally implemented?
A:
The way it's done in the reference implementation, and in many other implementations of similar languages: At compile time,
Figure out at compile time which name refers to which scope.
For each scope (except global), enumerate the variables and assign an index to each.
Instead of mentioning the variable names in the bytecode, use the indices from the previous step.
Then during execution,
Store the values of the locals in an array (per activation record). Read and write opcodes carry the necessary index.
Implement globals like a Lua table (and get nil for missing variables for free).
This is much simpler, easier, and portable than mucking around with dynamically created types. It may even be faster, but certainly won't be much slower.
Closure support is mostly orthogonal. So-called upvalues refer to (a slot in) the locals array while it's still alive, and adopt the closed-over value when the locals array dies. You could do something similar with your original design. For correctness, you must take care to only create one upvalue per closed-over variable and to flatten closures. As an example, g in the below code must also close over x so that it still exists by the time the h closure is created:
function f()
local x = 1
function g()
function h()
return x
end
return h
end
return g
end
print(f()()())
|
Q:
How to get json file local and repeat data?
I am new in angular5 and I am trying to load json file and repeat it but I can't repeat data
( i can get and show data form jason file but can't repeat data )
in src/assets/data.json
{ "status": "S","messeage": "error","Card":[
{
"ID": "01",
"Price": "30,000",
"Color": "Black"
},
{
"ID": "02",
"Price": "32,000",
"Color": "Red"
}]}
in src/app/app.component.ts
data;
constructor(private http:Http) {
this.http.get('../assets/data.json')
.subscribe(res => this.data = res.json());
}
in src/app/app.component.html
{{data?.Card[0].Color}} // => black
{{data?.Card[0].Price}} // => 30,000
But I need to know how to repeat it.
Sorry for my english
Thank you for help
Edit
last 2 answer it work
but it have error in console
how to fix it ?
thank you so much
enter image description here
A:
Now you get your json .
You can repeat it like this
In js code .
for(i=0;i<data.Card.length;i++){
console.log(data.card[i].Color);
}
In angular5 html template
<li *ngFor="let card of data.Card">
{{ card.Color}}
</li>
|
Introduction
============
In species with a complex social life, affiliative interactions represent an important component of the behavioral repertoire. Affiliative patterns vary across species but some behaviors are common to most mammals, including mother--infant contact, grooming, social play, and mating-related affiliative patterns. Presumably, the adaptive value of affiliative relationships has evolved into a set of physiological mechanisms that predispose social animals to relate to conspecifics and to experience pleasure and positive affect from social contact and relatedness. In human beings, social hedonic capacity, or the ability to experience pleasure and reward from social affiliation, is considered a personality characteristic that is normally distributed in non-clinical populations.^[@bib1],\ [@bib2]^ A few individuals (notably those with schizoid personality or autistic spectrum disorders) seem to lie at the lower extreme of this continuum, experiencing little or no positive feelings during affiliative interactions.^[@bib3],\ [@bib4]^
Over the last decades, animal models have achieved enormous insights into how neurobiological factors contribute to the regulation of social reward. Recent animal studies have shown that the neuropeptides oxytocin and ariginine vasopressin could have a specific impact on social approach behavior, social affiliation and social attachment.^[@bib5],\ [@bib6]^ There is preliminary evidence that both neuropeptides may exert similar effects in human subjects.^[@bib7],\ [@bib8]^
Another neurobiological system that is likely to be implicated in the modulation of social reward is opioid activity. Originally formulated by Panksepp,^[@bib9]^ the brain opioid hypothesis of social attachment posits that reductions in opioid activity should increase desire for social companionship, and increases in this system should reduce the need for affiliation.^[@bib10]^ Observations consistent with this hypothesis have been collected in a large variety of species using several distinct behavioral measures.^[@bib11]^ Animals treated with moderate doses of opiates tend to socially isolate themselves; conversely, the opioid antagonists naloxone and naltrexone (NTRX) have opposite effects, increasing affiliation.
In a previous study, we demonstrated that a genetic-induced deficit in the μ-opioid system was associated with a defective infant--mother bond in mouse pups.^[@bib12]^ Compared with intact controls (wild-type; WT), μ-opioid receptor knockout pups (μ-KO) emitted fewer ultrasonic vocalizations (USVs) when isolated from dam and littermates. Such a reduction in USVs was specific to social isolation and not seen when pups were exposed to cold temperature or male mice odors. In addition, these KO pups showed decreased preference/ability to discriminate their mother\'s scent and no potentiation of callings after repeated separation from the mother. Taken together, these data suggest that endogenous opioids binding to μ-opioid receptors are important for the formation of infants\' attachment bond in mice. In line with these findings, Roth and Sullivan^[@bib13],\ [@bib14]^ reported similar results in neonatal rats, showing that NTRX, an opioid antagonist, was able to prevent acquisition, consolidation and expression of odor preference during the sensitive period.
The aim of the present study was to confirm and extend our previous findings showing an opioid-related early attachment deficit and ascertain whether it translates into long-lasting alterations of social behavior. In the first part of the study, we compared pups and young μ-KO with aged matched WT animals on a battery of behavioral tests designed to evaluate attachment behavior and to assess affiliation and social reward. We decided to focus on juveniles because human clinical syndromes characterized by pervasive social deficits (for example, autism and schizophrenia spectrum disorders) generally manifest during adolescence and respond better to early interventions.^[@bib15],\ [@bib16]^
In the second part of the study, we aimed to ascertain whether the behavioral alterations observed in young μ-KO mice were the consequence of abnormal mother--infant interactions or whether they reflected a permanent malfunctioning of the opioid system. To test these alternative hypotheses, we induced a reversible pharmacological blocking of the opioid system with NTRX,^[@bib17]^ during the first postnatal days, to prevent the formation of mother--infant bond in outbred mice. The specificity of such a bond is based on mutual recognizing capability.^[@bib12],\ [@bib18]^ The rationale for using NTRX requires an explanation. In fact, whereas NTRX in adult animals induces a transient increase in social affiliation, including the display of attachment-related behaviors,^[@bib19]^ administration of NTRX in pups that have not yet formed an attachment bond with their mothers interferes with the neurochemical opioid mechanisms of social reward.^[@bib20]^ We hypothesized that such experimental manipulation would impede the formation of the infant--mother attachment and cause social behavioral modifications later in life.
Materials and methods
=====================
Animals, housing conditions and breeding procedures
---------------------------------------------------
*Orpm*+/+ (WT) and *Orpm*−/− (μ-KO) mice were used to confirm and investigate the role of a genetic deletion of μ-opioid receptors on sociability in youth (experiment 1); NMRI outbred mice (Harlan, Harlan, Italy) were used to investigate the role of a pharmacological-induced deficit in attachment during early life on subsequent sociability (experiment 2). The generation of mice lacking μ-opioid receptors was previously described.^[@bib21]^ WT and μ-KO subjects of these experiments derived from homozygous breeding pairs. NMRI mice used in this study arrived in our lab when 5--6 weeks old and were housed in groups of four in transparent high-temperature polysulfone cages (27 × 21 × 14 cm^3^) with water and food available *ad libitum*. Room temperature (21±1 °C) and a 12:12 h light--dark cycle (lights on at 1900 hours) were kept constant.
In both experiments the mating protocol consisted in housing one male with two females for 15 days. Pregnant females were isolated around delivery and cages were inspected twice a day for pups: the day of birth was considered postnatal day 0 (PND0). Weaning was performed between PND28 and PND30: animals of similar age/sex and genotype were housed in groups of 2--5 *per* cage.
Another group of NMRI mice born and reared under standard conditions in our animal facility were used as stimulus partners. All animal used procedures were in strict accordance with standard ethical guidelines (European Community Guidelines on the Care and Use of Laboratory Animals 86/609/EEC) and the Italian legislation on animal experimentation (Decreto L.vo 116/92).
Experimental design
-------------------
In the first experiment USVs and preference for mother\'s cues were measured in WT and μ-KO pups, to replicate previous data. The relevance and the response to social stimuli were evaluated in juvenile mice in the social approach-avoidance test^[@bib22],\ [@bib23]^ or in a social place-conditioning paradigm.^[@bib24]^
In the second experiment a total of 22 NMRI litters were used. On PND1 litters were culled to eight pups (four males and four females). From PND1 to PND4 half of each litter (two males and two females) was injected twice a day (0800 and 1800 hours) with NTRX (1 mg kg^−1^, sc), the other half was saline (SAL) injected. After the first injection all pups (NTRX and SAL) were tattooed. USVs, preference for mother\'s cues, and response to social stimuli were measured in NTRX- and SAL-treated NMR1 mice by the social approach-avoidance and place preference conditioning test.
Behavioral measures
-------------------
### Pups\' attachment behavior
Attachment behavior was measured in pups by USVs during separation from the mother, and by homing behavior, that is, measuring capability/motivation to orient towards their mother\'s cues. USVs were measured on PND8. Vocalizations uttered by isolated pups in either (i) a beaker containing home-cage bedding, or (ii) a beaker containing clean bedding, were recorded for 5 min using Avisoft technology ([Supplementary Information](#sup1){ref-type="supplementary-material"}). No more than four pups *per* litter were tested. After the test, pup\'s body weight and sex (and tattoo for NMR1 mice) were noted. Homing behavior was measured in PND10 pups. The capability of pup to orient towards familiar odorous cues was evaluated in a small apparatus (5 × 33 × 10 cm^3^) with a central Plexiglas part (5 × 5 cm^2^, starting point) that separated, with sliding doors, two differently scented chambers, one covered with pups\' home-cage bedding, the other covered with bedding from the cage of un unknown mother with its litter (same postnatal day and genetic background). After 1 min of habituation in the central part of the maze, the lateral doors were opened and the pup could move freely in the apparatus for 5 min. The test was video-recorded and the time spent in the compartments was evaluated thereafter. No more than four pups *per* litter were tested.
### Social approach-avoidance test
Animals were tested immediately before weaning in a gray Plexiglas rectangular box (60 × 40 × 24 cm^3^) consisting of three interconnected chambers.^[@bib22],\ [@bib23]^ Two identical clear Plexiglas cylinders (8 cm in diameter) with multiple small holes were placed, one in each end chamber of the apparatus. During the habituation session (5 min) the mouse was placed in the central chamber and allowed to freely explore the whole apparatus. A stimulus NMRI mouse, age/sex matched, was then introduced into one cylinder (pseudo-randomly chosen) for 5 min, whereas a white object was introduced into the other cylinder. Both sessions were recorded and the time the subject mouse spent in each chamber was measured by a video-tracking system (Smart 1.1). After each test, the entire apparatus was carefully cleaned with 10% ethanol. Habituation preference scores were measured to evaluate a priori discrimination between lateral compartments ([Supplementary Information](#sup1){ref-type="supplementary-material"}) and animals were scored for sociability in the test session (sociability index: 100 × time in social chamber/(time in social chamber + time in non-social chamber).
### Rewarding value of social interactions
Behavioral testing was conducted as previously described^[@bib24]^ according to a Conditioned Social Place Preference protocol (CSPP). Briefly for the first 24 h following weaning (PND28-30) two males and two females from different litters, but same genotype, were housed together in a cage that contained a set of novel environmental cues (social housing condition). For the next 24 h mice were socially isolated within a second, distinct home-cage environment (isolated housing condition). The two conditions differed for the bedding used. Following the completion of 10 conditioning sessions the CSPP response of each mouse was evaluated ([Supplementary Information](#sup1){ref-type="supplementary-material"}). The preference score was evaluated as follow: CSPP index=100 × time social-cued chamber/(time social-cued chamber + time-isolated cued chamber).
Statistical analysis
--------------------
USVs parameters were evaluated by a three-way analysis of variance, the factors being the genotype/treatment, the sex of the pups and the experimental condition (clean or mother/nest bedding). Two-way analysis of variances, factors being genotype/treatment and sex, were used for homing, approach-avoidance and place-preference tests.
Results
=======
Effects of genetic deletion of the μ-opioid receptor gene on social anhedonia
-----------------------------------------------------------------------------
Confirming the results of our previous study, μ-KO pups showed a deficit in their attachment behavior^[@bib12]^ as shown by USVs data indicating that knockout pups did not discriminate between clean and mother/nest bedding during isolation ([Figure 1a](#fig1){ref-type="fig"}). By contrast, WT pups strongly reduced callings in the presence of their mother/nest odor, suggesting a calming effect of mother\'s cues (Tukey *post-hoc*: *P*\<0.05). This result is based not only on the number and mean duration of calls, but also on frequency modulations ([Table 1](#tbl1){ref-type="table"}). As for their approach/preference towards their mother/nest bedding, contrary to WT, μ-KO pups showed lower preference, both in terms of time and number of entries into the familiar arm (%time: F(1/64)=4.82, *P*\<0.05; %entries: F(1/64)=4.51, *P*\<0.05, [Figure 1b](#fig1){ref-type="fig"}). Both measures of infant--mother bond were not affected by sex of the pup and no genotype × bedding interaction reached significance.
Compared with control animals, adolescent μ-KO mice showed reduced interest in social partners and diminished conditioned social place preference ([Figure 2](#fig2){ref-type="fig"}). In the approach-avoidance apparatus, WT mice showed higher sociability in comparisons with μ-KO animals, spending more time in the social compartment ([Figure 2a](#fig2){ref-type="fig"}: genotype: F(1/41)=9.92, *P*\<0.01). No other significant effects emerged. In the social place preference test ([Figure 2b](#fig2){ref-type="fig"}), μ-KO mice avoided the compartment containing cues previously associated with social housing condition whereas the WT mice showed no preference between compartments (genotype: F(1/46)=7.86, *P*\<0.01). No other effect reached the significance level.
Effects of postnatal treatment with an opioid antagonist on social anhedonia in NMR1 mice
-----------------------------------------------------------------------------------------
The analysis of USVs revealed that overall, NTRX pups did not differ for the number of calls from SAL-treated pups, but only the latter were able to discriminate between clean and mother/nest beddings (treatment: F(1/40)=0.45, NS; bedding: F(1/40)=6.25, *P*\<0.05; treatment × bedding: F(1/40)=4.42, *P*\<0.05) emitting more calls in the absence of familiar cues (Tukey *post-hoc*: *P*\<0.05, [Figure 3a](#fig3){ref-type="fig"}). Exposure to mother/nest bedding was responsible for shorter duration of USVs, independently from the pharmacological treatment (treatment: F(1/40)=0.30, NS; bedding: F(1/40)=6.92, *P*\<0.05; treatment × bedding: F(1/40)=0.90, NS). No sex differences emerged for these parameters. The temporary block of opioid receptors also determined reduced preference for mother/nest bedding ([Figure 3b](#fig3){ref-type="fig"}: %time mother: treatment: F(1/34)=6.16, *P*\<0.05; %entries mother: treatment: F(1/34)=7.80, *P*\<0.01). No Sex and no sex × treatment effect reached the statistical significance. Overall, these results indicate that NTRX treatment disrupted pup\'s attachment bond preventing ultrasonic emotional response discrimination between home-cage and familiar bedding during isolation and abolishing selective preference orientation towards mother/nest stimuli.
SAL mice tested for sociability at weaning ([Figure 4a](#fig4){ref-type="fig"}) showed higher scores than NTRX mice, suggesting a greater interest towards conspecifics (treatment: F(1/30)=8,81, *P*\<0.01). No sex and no sex × treatment effect reached the significant level. Similar results were obtained in the CSPP test: juvenile NTRX mice showed lower preference for the social-cued chamber in comparison with SAL-treated mice ([Figure 4b](#fig4){ref-type="fig"}: Treatment: F(1/20)=23.22, *P*\<0.001). No sex and no sex × treatment effect reached the significant level.
Discussion
==========
The attachment behavior system, that is, the behavioral system that promotes proximity to the caretaker maximizing infant survival, influences and organizes motivational, cognitive, emotional and memory processes.^[@bib25]^ These processes are organized during early infancy with respect to significant caregiving figures and extend into adulthood including attachment to intimate adult partners in humans. Different brain substrates of infant--mother attachment have been identified, the opioid system being one of the possible mediators.^[@bib11]^ We have already demonstrated^[@bib12]^ that mice lacking μ-opioid receptors show deficits in two independent measures of attachment behavior: knockout pups were not able to selectively approach their mothers, and did not vocalize during maternal separation. These μ-KO pups seemed to be specifically less sensitive to the absence of maternal cues. The etiology of this insensitivity is likely due to a diminished capacity to experience pleasure and reward related to maternal stimuli. The mesocorticolimbic reward system has been implicated in social attachment and Roth and Sullivan\'s experimental studies^[@bib13],\ [@bib14],\ [@bib20]^ support the hypothesis that the μ-opioid system is a crucial component of the social reward processes modulating the positive affective states associated with maternal stimuli.^[@bib26],\ [@bib27]^
The aim of this study was to investigate whether, according to Panksepp\'s suggestion,^[@bib28]^ deficits in infant--mother attachment could affect the development of later social interactions. In line with our hypothesis, we found that juveniles\' social motivation and emotional profiles are altered in mice showing dysfunctional attachment in infancy. This finding was confirmed by manipulating the μ-opioid system functioning in two different ways: permanent genetic knocking-out or temporary pharmacological blocking during the first postnatal days.
Compared with our previous experiment,^[@bib12]^ in the present study we used more sophisticated equipment for the analysis of ultrasounds. μ-KO mice showed higher frequency modulation than WT pups and this could explain the greater number of USVs detected in this study. The mean number, duration and frequency modulation of USVs were strongly affected by experimental condition in WT pups, whereas μ-KO infants did not modify their calls according to environmental cues. In addition, no differences between male and female pups in both measures of attachment behavior were detected.
To ensure comparability with our previous experiment,^[@bib12]^ we used knockout and WT mice derived from homozygous lines. Such a strategy of subject selection implied that not only pups but also their mothers had different genetic profiles. Thus, to exclude maternal effects on pups\' attachment behavior, we carried out a detailed analysis of mothers\' behavior during the first week of life ([Supplementary Figure 1](#sup1){ref-type="supplementary-material"}) indicating that μ-KO and WT dams did not differ as for maternal behaviors directed towards their pups.
The results of the first experiment support Panksepp\'s hypothesis that both motivation for, and reward from, social stimuli are reduced in μ-KO animals. Juvenile μ-KO mice showed not only no preference for the social chamber and reduced sociability in comparison with WT peers, but also social place-conditioning avoidance. Importantly, based on data collected during the habituation session, mice did not differ for their general exploratory performance, but they reacted differently to social stimuli.
The second experiment was conducted to ascertain if altered social motivation and reward in juveniles depended on dysfunctional infant--mother attachment, or alternatively on abnormal opioid neurotransmission affecting knockout animals during their entire life. Blocking the opioid receptors, NTRX treatment prevents the formation of infant--mother bond. NTRX-treated pups did not increase vocalization rate in the absence of their mothers\' cues and did not discriminate between their mothers\' and an unknown dam\'s bedding. These effects are unlikely to depend on the acute effects of NTRX. NTRX is a short half-life drug^[@bib29]^ and there was a 4--6-day interval between the last injection (PND4) and behavioral tests (PND8-10). Thus, these effects are likely to result from a deficit in early bond formation due to the NTRX block of the μ-opioid component of the reward system.
As for the long-term effects on social behavior of this early sub-chronic treatment, we did find permanent deficits in juveniles. In particular, we found a reduced interest in social partners by NTRX animals. Confirming this, NTRX mice also showed no social conditioned place preference, contrary to SAL mice that spent more than 60% of their time in the chamber containing cues previously associated to social housing.
Taken together, the results of these experiments suggest that sociability in juvenile mice is highly dependent on the establishment of a positive affective relationship with their mothers during infancy. Both permanent and temporary disruption of the opioid reward circuit during infancy reduced affiliative behaviors during adolescence. Our findings suggest the existence of a very narrow time window, from PND1 to PND4 (NTRX treatment) that has a crucial role in terms of setting the neurobiological basis for social affiliation. It is worth noting that the sub-chronic NTRX treatment did not disrupt social behavior in term of competence or skills, but mainly affected the affiliative motivation and the affective component of social interactions. These alterations are more evident during adolescence and tend to disappear during adulthood. Adult reproductive behaviors as maternal behavior in μ-KO mice and exploratory courtship behavior (both in μ-KO/WT and NTRX/SAL mice, ([Supplementary Table 2](#sup1){ref-type="supplementary-material"}) were unaffected by the genetic and pharmacological treatments. Similarly, reproductive behavior is maintained in BTBR T1tf/J, a genetic mouse model of autism recently developed, even if these mice show altered social behaviors during dyadic interactions.^[@bib30]^
Conflicting results concerning the effects of age on sociability are reported in the literature.^[@bib23],\ [@bib31]^ These contrasting findings might depend on the difficulty to ascertain if social approach reflects affiliative or aggressive motivation. In fact, behavioral tests for assessing sociability are based on spatial proximity, but generally prevent physical contact that could reveal the real nature of the social interaction. A few studies have allowed a free interaction session after the approach-avoidance test, and sometimes aggression occurred within 3--5 min.^[@bib31],\ [@bib32]^ To minimize the likelihood that social approach was motivated by aggression, in this study we used young/not-already-weaned animals of the same sex/line/treatment of the experimental subject. The evaluation of sociability and affiliation in adult mice, a species characterized by male territoriality and aggression, is somewhat difficult.^[@bib33]^ In this study, adult animals were observed in a reproductive context (that is, mating and caregiving), and no abnormal behaviors were observed.
The present study has some limitations. First, human variability in opioid functioning does not reach the extreme dysfunctionality induced by genetic knocking-out. In this regard, animal models using subjects with genetic polymorphisms are closer to human variability.^[@bib34],\ [@bib35],\ [@bib36]^ Second, s.c. injections represent stressful events *per s*e for mouse pups and, even if we used SAL-injected controls to minimize this problem, changes in physiological and behavioral parameters could have occurred in our subjects.
The results of the present study, together with those of previous reports,^[@bib9],\ [@bib11],\ [@bib12],\ [@bib19]^ may be useful to build an animal model of human psychiatric disorders implying defective social motivation and reduced sensitivity to social reward. Psychiatric conditions with these clinical features include some forms of infantile autism and schizophrenia spectrum disorders.^[@bib4],\ [@bib37]^ There are several characteristics of the animal model that support its validity as a useful tool to investigate the pathogenesis and treatment of defective social motivation and reduced sensitivity to social reward. First, the model involves dysfunctional attachment processes, and there is evidence that early experience and relationships with caregivers have a role in the vulnerability to social anhedonia and dismissing attachment in human subjects.^[@bib38]^ Second, a role for altered opioid neurotransmission has been demonstrated in some patients suffering from psychiatric disorders associated with social anhedonia.^[@bib39]^ Third, natural genetic variation in human subjects (resembling that experimentally-induced in animal models^[@bib12],\ [@bib34],\ [@bib35],\ [@bib36]^) has been shown to influence social motivation and the capacity to experience pleasure from affiliative interactions.^[@bib40],\ [@bib41]^ Finally, the emergence of social deficits during childhood (and years before the onset of the clinical symptoms) is a peculiar feature of psychiatric conditions with high levels of social disability.^[@bib42]^ If the validity of the animal model will be confirmed by future research, translational studies focusing on the interaction between early experience and opioid neurotransmission could provide useful insights for identifying endophenotypes of human psychiatric disorders associated with social anhedonia and for developing effective intervention strategies to be implemented early in life.
Thanks to Brigitte L Kieffer for comments on the manuscript, scientific support and providing animals. This study has been granted by Telethon, Italy (Grant no. GGP05220) and also partially supported by funds from Regione Lazio for 'Sviluppo della Ricerca sul Cervello\'.
[Supplementary Information](#sup1){ref-type="supplementary-material"} accompanies the paper on the Translational Psychiatry website (http://www.nature.com/tp)
The authors declare no conflict of interest.
Supplementary Material {#sup1}
======================
######
Click here for additional data file.
{#fig1}
{#fig2}
{#fig3}
{#fig4}
###### ANOVA results for mean number, duration and frequency modulation of ultrasonic vocalizations emitted by μ-KO and WT 8-day old pups during 5 min of isolation in clean or home-cage bedding
*df* *Mean number* *Men duration* *Mean frequency modulation*[a](#t1-fn2){ref-type="fn"}
-------------- ------ --------------- ---------------- --------------------------------------------------------
Genotype (G) 1/58 F=1.19 F=2.99 F=1.84
Bedding (B) 1/58 F=1.29 F=2.43 F=3.18
G × B 1/58 F=6.58\* F=5.08\* F=9.90\*\*
Sex (S) 1/58 F=0.49 F=0.11 F=2.66
G × S 1/58 F=0.05 F=0.02 F=0.13
B × S 1/58 F=0.42 F=0.02 F=0.32
G × B × S 1/58 F=0.56 F=0.03 F=0.02
Abbreviations: ANOVA, analysis of variance; KO, knockout pups; WT, wild-type
Frequency Modulation: difference between the highest and the lowest peak frequency within each element.
\**P*\<0.05, \*\**P*\<0.01.
|
Functional reconstitution of a crenarchaeal splicing endonuclease in vitro.
Sulfolobus tokodaii strain 7 is one of Crenarchaea whose entire genome has been sequenced. The genome sequence revealed that it possesses two open reading frames (ORFs) that are homologous to EndA, a protein responsible for splicing endonuclease activity in Archaea. Interestingly, one of the two ORFs lacks a putative catalytic amino acid residue for the nuclease activity. To investigate their functions, the two ORF products were individually expressed in Escherichia coli, partially purified, and tested for their nuclease activities in vitro. Using in vitro transcribed tRNA precursor as a substrate, we found that the two ORF products are concurrently required to cleave exon-intron junctions. Our finding implies that the splicing endonuclease for the organism is a multi-subunit complex composed of the two endA gene products. |
There have been two big theme parks in gaming news recently. First, there’s Frontier’s Planet Coaster which doesn’t contain any dinosaurs at all, and then there’s Jurassic World, which contains all of the dinosaurs. I don’t think Parkitect [official site] has any dinosaurs but it does have gorgeous isometric rollercoasters and looks like the closest thing to Rollercoaster Tycoon 2 since Rollercoaster Tycoon 2. Kickstarted last year, it’s due to release in August and I want it now.
I don’t have a new video for you but the development blogs are updated weekly and packed with insights about building a game about building a theme park.
Slopes!
“We finally added the ¼ height slopes to the track builder. We can also put separate limits on up/down slope heights for tracked ride types now (notice how the log flume can have steep down slopes, but not steep up slopes). And we can put minimum size restrictions on slope transitions – the high and medium slopes in the back are the smallest possible sizes you can build, but the high slope transition is a bit longer.”
Rating rides!
“A good amount of time this week has been spent on calculating a bunch of statistical numbers for tracked rides. They’ll be used for determining how much guests enjoyed your ride. The balancing is still entirely off and it’ll most likely take lots of tweaking over a long amount of time to get it right, but it’s a start.”
Camera details!
“You can free-rotate the camera with the mouse currently, and we just added the ability to tilt it up and down freely as well. This should help with seeing rides and details, and helping to position new objects. You can use the keyboard to snap the camera to certain angles, which is usually the angles we take screenshots in.”
There are also some videos on the official Parkitect YouTube channel showing the developers building models for rides and… other things.
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DINOSAURS CONFIRMED |
WARSAW (Reuters) - An independent panel aimed at documenting cases of paedophilia in Poland’s Catholic Church will be set up early next year to help victims speak up and claim damages, opposition parliamentarians said on Monday.
The Roman Catholic Church worldwide is reeling from crises involving sexual abuse of minors, deeply damaging confidence in the Church in Chile, the United States, Australia and Ireland among other countries.
In stark contrast, in Poland, a deeply Catholic country, debate has only just begun while activists close to the nascent anti-paedophilia project, called “Nie lekajcie sie” (Don’t Be Afraid), say that around 600 priests in Poland may have inclinations towards paedophilia.
Around 100 people claiming to have been sexually molested by Polish priests phoned to tell their stories in the first 24 hours after organisers of the panel posted an interactive paedophilia map on the Internet on Sunday.
“The map of paedophilia was viewed by 500,000 people since yesterday. Public discussion has started and we want to set up a panel early next year to tackle the issue,” Joanna Scheuring-Wielgus, an opposition MP, told Reuters TV.
The map documents 280 cases of paedophilia committed by 60 priests convicted by Polish courts, but according to activists the real numbers are much higher since victims are often afraid to speak up for psychological and social reasons.
“One-hundred people have called us since yesterday to tell us about their cases, and we need to research and verify all these cases,” Scheuring-Wielgus said.
Several left-wing and liberal MPs, along with rights activists, declared their support in creating or running the investigative panel. The ruling nationalist (PiS) party, whose core support comes from devout Catholics and the Church, is not expected to be involved.
A PiS spokeswoman was not immediately available for comment, nor was a spokesman for Poland’s Church.
Last week a Polish court upheld a landmark ruling granting a one million zlotys ($266,084.83) in compensation and an annuity to a victim of sexual abuse by a Catholic priest, accepting that the church ore responsibility for the crimes of its cleric.
($1 = 3.7582 zlotys) |
Los Angeles police on Saturday named a suspect in the kidnapping and sexual assault of a young Northridge girl this week as a 30-year-old transient with an extensive criminal history.
Tobias Dustin Summers, who was last released from jail on Jan. 19 after serving a brief stint for a probation violation, is the primary focus of the police investigation, LAPD Deputy Chief Kirk Albanese said at an afternoon news conference at LAPD headquarters downtown.
Summers has a criminal history dating back to 2002 that includes arrests for kidnapping, robbery, explosives possession and petty theft, Albanese said. He is not a registered sex offender.
The girl told investigators that two men were involved in the incident and that she was taken to multiple locations in different vehicles, according to law enforcement sources. She was found bruised and scratched Wednesday near a Starbucks about six miles from her home.
But police said that Summers was the primary focus of their investigation.
“We’re trying to go through this slowly and methodically because we don’t want to overwhelm her. She’s a 10-year-old girl,” LAPD Capt. William P. Hayes said after the news conference.
The girl’s mother told authorities she last saw her daughter in her room about 1 a.m. Wednesday. About 3:40 a.m., police said, the mother heard a noise. When she went to check on her daughter, the girl was gone.
Authorities combed the area house by house, and the FBI joined the search. Shortly before 3 p.m., a man spotted the girl in a parking lot about six miles from her home and pointed her in the direction of nearby police.
LAPD officials said they believe the girl was dropped off at a Kaiser Permanente Medical Center in Woodland Hills. She then walked toward the Starbucks.
Law enforcement sources have said that detectives were trying to determine whether there was any connection between this case and a high-profile international child abduction in 2008.
Public records and court documents indicated one of the children kidnapped in the 2008 case was a relative of the Northridge girl.
Police said Saturday that so far there is no clear connection.
More than 20 detectives and the FBI are working on the Northridge case.
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Twitter: @RosannaXia |
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10 Dragon Ball Z Theories That Could Change Everything
Dragon Ball has been on the scene since 1984, and since then fans have fallen in love with the series. From its humble beginnings as a manga, to its transition to anime, and to its multiple theatrical film releases, the Dragon Ball series has truly stood the test of time. Loyal fans have stayed with the series for decades, and with the continued release of content, it seems that we will be enjoying Dragon Ball for years to come. We have video games like Dragon Ball Xenoverse 2 for PC, Playstation 4, Xbox One, and Nintendo Switch, Dragon Ball Fusions for the Nintendo 3Ds, and Dragon Ball Z Dokkan Battle for android and iOS. Of course, we are also getting new content from Akira Toriyama himself with his latest series Dragon Ball called Super. Dragon Ball Z has become an iconic anime and a must see for anyone new to anime or fans of shows like Fullmetal Alchemist, Trigun, and Attack on Titan. Between Goku, Vegeta, Gohan, Trunks, Piccolo, and all of the other protagonists, there’s no shortage of people to attach to. There are plenty of awesome villains like Frieza, Cell, and Majin Buu to keep things interesting, too. Plus with all of that content comes even more content from the fans as well. With all of the fan fiction, fan art, and especially fan theories, there is no shortage of DBZ goodness to keep you occupied. Some theories are even so impactful, they could change everything about the series past, present, and future. |
Q:
SQL - filter duplicate rows based on a value in a different column
I have an SQL table:
+-------------+-----------+---------+
| ID | position | user |
+-------------+-----------+---------+
| 1 | 1 | 0 |
| 2 | 2 | 0 |
| 3 | 3 | 0 |
| 4 | 4 | 0 |
| 5 | 5 | 0 |
| 6 | 6 | 0 |
| 7 | 7 | 0 |
| 8 | 7 | 1 |
+-------------+-----------+---------+
I would like to filter the duplicate row based on position column and the distinct value of user column, for the first query I need to have the following result:
+-------------+-----------+---------+
| ID | position | user |
+-------------+-----------+---------+
| 1 | 1 | 0 |
| 2 | 2 | 0 |
| 3 | 3 | 0 |
| 4 | 4 | 0 |
| 5 | 5 | 0 |
| 6 | 6 | 0 |
| 8 | 7 | 1 |
+-------------+-----------+---------+
For the second query I need the following:
+-------------+-----------+---------+
| ID | position | user |
+-------------+-----------+---------+
| 1 | 1 | 0 |
| 2 | 2 | 0 |
| 3 | 3 | 0 |
| 4 | 4 | 0 |
| 5 | 5 | 0 |
| 6 | 6 | 0 |
| 7 | 7 | 0 |
+-------------+-----------+---------+
What queries do I need to achieve this?
Thanks.
A:
In the absence of further information, the two queries below assume that you want to resolve duplicate positions by taking either the larger (maximum) user value, in the first case, or the smaller (minimum) user value in the second case.
First query:
SELECT t1.*
FROM yourTable t1
INNER JOIN
(
SELECT position, MAX(user) AS max_user
FROM yourTable
GROUP BY position
) t2
ON t1.position = t2.position AND
t1.user = t2.max_user
Second query:
SELECT t1.*
FROM yourTable t1
INNER JOIN
(
SELECT position, MIN(user) AS min_user
FROM yourTable
GROUP BY position
) t2
ON t1.position = t2.position AND
t1.user = t2.min_user
|
But the Dragon mission was anything but routine. It was a nail-biting, exhilarating, exciting, nerve-wracking, but ultimately historic voyage to low-Earth orbit, offering the world a glimpse of a possible commercial future for the exploration of space. As SpaceX CEO would say, this is also the first step in making mankind “multi-planetary.”
So, to mark this historic mission, SpaceX has compiled the Dragon launch, berthing and splashdown highlights and edited them to music. For me, the most emotional parts of the whole video are when you hear and see the SpaceX employees’ reaction to launch and unfurling of the Dragon solar panels — it’s a heartwarming reminder that space exploration isn’t a cold, robotic endeavor; it’s a very human experience. |
Molly Scott Cato
Green MEP for the South West of England and member of the Economics and Monetary Affairs Committee in the European Parliament
THE BLOG
Angela Merkel Visit: Lessons From Germany for David Cameron
323
25/02/2014 17:52 GMT
|
Updated
27/04/2014 10:59 BST
German chancellor Angela Merkel is being treated like political royalty, a consequence of her country's economic power as well as prime minister David Cameron's desperate need for friends in Europe. Few would argue about the position of Germany as the economic powerhouse of the European Union but what can Britain learn from the German economic model? Does Germany's economy suggest that the idealisation of competition and flexibility, touted by chancellor George Osborne and his elite friends, is the route to success?
We have a lot to learn from the German model, particularly in terms of the way the government frames the two important sources of economic dynamism: energy and money.
Germany's Energiewende or energy transition is one of the most dramatic and underreported developments taking place in Europe today. It has hugely ambitious targets for the reduction of carbon dioxide emissions, which are to fall by a minimum of 80% by 2050 with a staging-post of 55% reductions by 2030, as well as pledging to phase out nuclear energy by the end of 2022. The rejection of nuclear after the Fukushima accident was famously an example of Merkel's ability to listen, learn and change her mind, which we might also welcome being shared with our own government.
Not only has Germany turned its back on Europe's dirty fossil and nuclear past, it has also questioned ownership of energy and responded in a way that would be anathema to Britain's Conservative politicians. The energy revolution is being driven by communities and by local politicians; it would be quite impossible without a muscular role being played by the state, the same state that in Britain is being devastated by austerity cuts. As a result, local communities and local governments across Germany are benefiting from the energy transition: the 928 inhabitants of the village of Grossbardorf, in Bavaria, have united to develop photovoltaic roof systems, solar power plants, a biogas plant with a combined heat and power (ChP) unit and a district heating network; Jühnde in Göttingen began its journey to becoming a 'bio-energy village' in 2001 and by 2004 70% of the population of the village were members of the co-operative and use locally generated bio-heat, relying on the methane produced by fermenting agricultural waste products.
These sorts of developments would be impossible without a wholly different approach to finance exemplified by the state-owned development bank - the KfW (Kreditanstalt für Wiederaufbau or Reconstruction Credit Institute). Established under the Marshall Plan and originally focused on the reconstruction of a war-torn economy, the KfW has been able to provide the finance to enable Germany's development as political priorites have changed, through Reunification and now the Energiewende. The contrast with the situation in the UK is made clear through evidence given to the Environmental Audit Committee's Inquiry into Green Finance, which will report shortly. It will demonstrate a tussle over power and profits that has held back the energy transition in Britain, where high-risk activities are always more attractive to private finance than investment in vital sustainable and resilient infrastructure.
The KfW is just one example of a banking system that is oriented entirely differently from that of the UK. In Germany banks play a role in lending to businesses first, rather than extracting money from local economies to invest it in the global casino economy as is the case with the city of London. Germany's Mittelstand - the layer of medium-sized businesses that are the bedrock of the country's economic success - could not have succeeded without their close working relationships with local banks and local bank managers, now an extinct species in Britain. As finance speaker for the Green Party I have repeatedly called for RBS to be broken up into a network banks on this sort of model. The call has fallen on deaf ears because it would undermine the power of the City and decentralise banking profits.
When I was in Berlin in the autumn I was again impressed by the extremely high standard of living that the Germans enjoy. In many ways the country represents the highest level of economic development that we can achieve within our existing paradigm. Yet, while the CO2 emissions ambitions are laudable, the levels of energy per capita required to sustain this lifestyle are quite incompatible with a sustainable future for the planet, and this realisation is underlined if we take the only morally defensible position that all the world's people deserve an equivalent quality of life. Some of Germany's Green politicians such as Hermann Ott have recognised this and are seeking to explore what the economy is for and what is meant by quality of life. Ott's Enquete-Kommission of the Bundestag is taking the next step, exploring the limits of energy and resource efficiency and questioning how to achieve real prosperity with lower material impact.
Germany demonstrates what can be achieved by a functional polity that brings in new political movements. Because of its proportional electoral system the Greens have been represented in the parliament in significant numbers and have been involved in both national and regional governments. This has revolutionised understanding of how energy should be produced and has encouraged the development of pro-business and pro-society banking. By contrast, in Britain, visionaries are blocked in a way that prevents progress and entrenches existing power structures, undermining both our economic and our environmental performance. |
Three groups are vying to lead the designs behind the Defense Advanced Research Projects Agency’s (DARPA’s) XS-1 Program, which aims to make a craft that can go to space and launch satellites 10 times in 10 days.
On Monday the agency set the deadline for July 22, at which point it will pick between the designs of three groups, Northrop Grumman, partnered with Virgin Galactic; Boeing, partnered with Blue Origin; and Masten Space Systems, partnered with XCOR Aerospace. The winner of the public-private partnership with be awarded $140 million in DARPA funding to build the submitted designs for the reusable rocket.
The designs of this craft must meet four goals laid out by DARPA.
Fly 10 times in a 10-day period (barring weather) to demonstrate aircraft-like access to space. Achieve flight velocity sufficiently high to enable use of a small (and therefore low-cost) expendable upper stage. Launch a 900- to 1,500-pound payload and demonstrate the ability to eventually launch 3,000+-pound payloads during future missions. Reduce the cost of flight to just $5 million per flight.
A DARPA concept video shows a concept XS-1 craft launching a satellite. YouTube
Jeff Bezos’ Blue Origin has been the company at the forefront of this sort of launch so far. The Amazon and Washington Post owner has already demonstrated that his space company’s crafts are capable of launching and landing the same craft three times in a row. While these launches are lower in altitude than similar events from Elon Musk’s SpaceX, it seems DARPA is only looking to launch satellites, not deliver payloads to the International Space Station. For those purposes, Blue Origin seems like the player to beat here, especially when assisted by the folks at Boeing.
Virgin Galactic hasn’t proven itself a major player in the commercial space flight game yet, but maybe Richard Branson can take this opportunity to prove himself in a major way. Veterans from the company recently started a new company called Vector Space Systems, which aims to do many of the same things.
The third partnership is by far the least well known, but could be a formidable underdog. Masten Space Systems focuses on entry, descent, and landing technologies (EDL) while XCOR Aerospace is developing some pretty cool looking reusable rockets. Maybe these two specialties combined can win out to beat the big dogs. |
1. Field of the Invention
The present invention relates to a method and device for feeding a recording sheet at a certain speed without variation in feeding speed of the recording sheet, which is occurred in transferring the recording sheet from feeding means to another one.
2. Background Arts
In a printer processor, for instance, recording light beams are projected onto a sheet-type photosensitive recording material (hereinafter referred to xe2x80x9crecording sheetxe2x80x9d) in a main scan direction while the recording sheet is fed in a sub-scan direction perpendicular to the main scan direction, to record an image onto the recording sheet. Along a passage of the recording sheet are provided a plurality of feeding roller pairs, each of which transfers the recording sheet to the next one. The feeding roller pairs are controlled to be equal in the feeding speed.
In order to improve the image quality, it is needed to apply the recording light beams onto the recording sheet with high precision. The recording light beams can be projected in the main scan direction with high accuracy by controlling an optical system in an exposure unit. With regard to the sub-scan direction, the feeding roller pairs provided near an exposure position are designed to feed the recording sheet with high accuracy. However, other feeding roller pairs to provided at upstream and downstream positions of the exposure position are not able to feed the recording sheet with high accuracy. Under the influence of the upstream and downstream feeding roller pairs, the feeding speed of the recording sheet is changed. Thereby, density unevenness in a print picture could be occurred.
In view of the foregoing, an object of the present invention is to provide a method and device for feeding a recording sheet capable of preventing deviation in feeding speed of the recording sheet.
To achieve the above objects, a method for feeding a recording sheet includes the following steps: feeding the recording sheet at a speed V1 by use of first feeding means, transferring the recording sheet to second feeding means from the first feeding means, transferring the recording sheet to third feeding means from the second feeding means, and feeding the recording sheet at a speed V3 by use of the third feeding means. The feeding speed V2 of the second feeding means is set, to be faster than a maximum feeding speed of the first feeding means, and slower than a minimum feeding speed of the third feeding means.
In the preferred embodiment, the first feeding means includes at least one transfer roller pair including a one-way clutch. The second feeding means includes first and second feeding roller pairs arranged sequentially along a feeding passage. The first and second feeding roller pairs rotates at a same speed by a steel belt stretched therebetween. The third feeding means includes a belt conveyor to feed the exposed recording sheet toward a processor section. When the recording sheet is nipped by the first feeding roller pair, the one-way clutch is actuated to make said transfer roller pair free. Moreover, before the trailing end of the recording sheet passes the second feeding roller pair, the recording sheet is fed with the leading end portion thereof being slipped on the belt conveyor.
According to the present invention, since the recording sheet is conveyed at the same feeding speed V2 during transfer, it is possible to prevent deviation in feeding speed of the recording sheet. Thus, image quality of a print image could be improved. |
As we were sitting at Easter brunch, a friend of mine recounted how her family had recently grilled her about her future. Given that she has had a job since the beginning of the school year, this surprised me.
After returning to Grounds and seeing friends for the first time in months, I find myself frequently saying, “Want to get coffee?” or “We should grab lunch!” I’ve enjoyed the coffee dates I’ve had so far — and look forward to more in the future — but I’ve started to wonder why these encounters always involve some sort of food or beverage.
What does our need for food in social settings say about our relationships? |
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<meta http-equiv="content-type" content="text/html; charset=utf-8" />
<title>RiTa Reference</title>
<link rel="stylesheet" href="../../../css/bootstrap.css" type="text/css" />
<link rel="stylesheet" href="../../../css/syntax.css" type="text/css" />
<link rel="stylesheet" href="../../../css/style.css" type="text/css" />
<link rel="shortcut icon" type="image/x-icon" href="http://rednoise.org/rita/rita.ico"/>
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" href="../../../css/normalize.css">
<link rel="stylesheet" href="../../../css/main.css">
<script src="../../../js/vendor/modernizr-2.6.2.min.js"></script>
<script language="javascript" src="../../../js/highlight.js"></script>
</head>
<body>
<?php include("../../../header.php"); ?>
<div class="gd-section pad-large">
<div class="gd-center pad-large">
<div class="row">
<div class="col1"></div>
<div class="col10">
<h3>Reference</h3>
<div class="page row">
<div class="refbar span3">
<div id="index">
<!-- begin publish.classesIndex -->
<h3></h3>
<ul class="classList" >
<br />
<li style="top:60px;left:50px">
<a href="../../index.php">Back to index</a>
</li>
</ul>
<hr />
<!-- end publish.classesIndex -->
</div>
</div>
<div class="span11">
<table cellpadding="0" cellspacing="0" border="0" class="ref-item">
<tr class="name-row">
<th scope="row">Class</th>
<!-- ------------ METHODS PROPERTIES HERE ------------ -->
<!-- ClASS -->
<td><h3><a href="../../RiWordNet.php">RiWordNet</a></h3></td>
</tr>
<tr class="name-row">
<th scope="row">Name</th>
<!-- METHOD NAME -->
<td><h3>randomizeResults</h3></td>
</tr>
<tr class="">
<th scope="row">Description</th>
<!-- DESCRIPTION -->
<td>Sets/gets whether results from RiWordNet methods will be randomized (default=true)</td>
</tr>
<tr class='Syntax'>
<th scope="row">Syntax</th>
<!-- SYNTAX -->
<td><pre>randomizeResults(value) // as setter <br/>randomizeResults(); // as getter</pre></td>
</tr>
<tr class='Parameters'>
<th scope="row">Parameters</th>
<td>
<!-- PARAMETERS -->
<table cellpadding="0" cellspacing="0" border="0" class="sub-table">
<tr class=''><th width='25%' scope='row' class=nobold>boolean (optional)</th><td width='75%'>the desired value</td></tr>
</table></td>
</tr>
<tr class='Returns'>
<th scope="row">Returns</th>
<td>
<!-- RETURNS/TYPE (for variables) -->
<table cellpadding="0" cellspacing="0" border="0" class="sub-table">
<tr class=''><th width='25%' scope='row' class=nobold>boolean </th><td width='75%'>return true or false (as getter), or this RiWordNet object (as setter)</td></tr>
</table></td>
</tr>
<tr class='Related' style='display:none'>
<th scope="row">Related</th>
<!-- RELATED -->
<td>tmp_related</td>
</tr>
<tr class='Note'>
<th scope="row">Note</th>
<!-- NOTE -->
<td>Can be used as a setter or getter</td>
</tr>
<tr class='Example'>
<th scope='row'>Example</th>
<td>
<div class="example">
<!-- EXAMPLE -->
<!--img src="../../../img/RiTa-logo4.png" alt="example pic" /-->
<pre class="margin">myWordNet.randomizeResults(false); // as setter<br/>boolean val = myWordNet.randomizeResults(); // as getter</pre>
</div></td>
</tr>
<tr class="">
<th scope="row">Platform</th>
<!-- PLATFORM -->
<td>Java only</td>
</tr>
<tr class="">
<th scope="row"></th>
<td></td>
</tr>
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</div>
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Viscount Hewart
Viscount Hewart, of Bury in the County Palatine of Lancaster, was a title in the Peerage of the United Kingdom. It was created in 1940 for Gordon Hewart, 1st Baron Hewart, on his retirement as Lord Chief Justice. He had already been created Baron Hewart, of Bury in the County of Lancaster, in 1922, also in the Peerage of the United Kingdom. He was educated at Bury Grammar School. The titles became extinct on the death of his son, the second Viscount, in 1964.
Viscounts Hewart (1940)
Gordon Hewart, 1st Viscount Hewart (1870–1943)
Hugh Vaughan Hewart, 2nd Viscount Hewart (1896–1964)
References
Category:Extinct viscountcies in the Peerage of the United Kingdom |
Mississippi Highway 8
Mississippi Highway 8 (MS 8) runs east–west from U.S. Highway 278 (US 278) northeast of Aberdeen, to MS 1 in Rosedale.
Points of interest
Delta State University
Great River Road State Park
Grenada Lake
Hugh White State Park
Natchez Trace Parkway
Locales on route
From east to west
Aberdeen
Houston
Vardaman
Derma
Calhoun City
Grenada
Ruleville
Cleveland
Pace
Rosedale
Major intersections
See also
Mississippi Highway 335
List of Mississippi state highways
References
008
Category:Transportation in Bolivar County, Mississippi
Category:Transportation in Sunflower County, Mississippi
Category:Transportation in Leflore County, Mississippi
Category:Transportation in Grenada County, Mississippi
Category:Transportation in Calhoun County, Mississippi
Category:Transportation in Chickasaw County, Mississippi
Category:Transportation in Monroe County, Mississippi |
/*
Derby - Class org.apache.derby.client.net.NetDatabaseMetaData
Licensed to the Apache Software Foundation (ASF) under one or more
contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
package org.apache.derby.client.net;
import org.apache.derby.client.am.Configuration;
import org.apache.derby.client.am.ClientDatabaseMetaData;
import org.apache.derby.client.am.ProductLevel;
import org.apache.derby.client.am.SqlException;
class NetDatabaseMetaData extends ClientDatabaseMetaData {
NetDatabaseMetaData(NetAgent netAgent, NetConnection netConnection) {
// Consider setting product level during parse
super(netAgent, netConnection, new ProductLevel(netConnection.productID_,
netConnection.targetSrvclsnm_,
netConnection.targetSrvrlslv_));
}
//---------------------------call-down methods--------------------------------
public String getURL_() throws SqlException {
String urlProtocol;
urlProtocol = Configuration.jdbcDerbyNETProtocol;
return
urlProtocol +
connection_.serverNameIP_ +
":" +
connection_.portNumber_ +
"/" +
connection_.databaseName_;
}
}
|
This post has been contributed by a third party. The opinions, facts and any media content here are presented solely by the author, and The Times of Israel assumes no responsibility for them. In case of abuse, report this post.
Blogs Editor
More in this blog
So today started late due to being up until 3:30am for the opening ceremonies. I wasn’t up partying it was simply transportations issues getting back to Tel Aviv. So the morning track workout time was out the window. We were first told we could have the track from 6-8pm tonight but as this is Friday evening and thus Shabbat that was not going happen. So I planned to do the workout my coach Johnny Marino had for me as a road interval workout earlier in the day. The workout was 6 x 400m with 400 recovery at 5K pace with a good warm up and cool down. Terry Robinson, my new friend and teammate joined me for the workout which was GREAT!
I decided to program the workout in my Garminso I would not have to look at the GPS. BAD IDEA! The watch basically locked up so after the first interval I had to reset the watch and run the rest simply looking down to see when we ran 0.25 miles. That seemed to work. When we started the first interval we saw our (the USA) open team up ahead. Needless to say we chased them down so the first few intervals were way too fast. The final few were more on pace and felt VERY easy. Great sign!!
After the workout and a quick clean up, I joined a few members of our team for a traditional Shabbat service and dinner. It was really nice.
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Thermal destabilization of transmembrane proteins by local anaesthetics.
Local anaesthetics, in addition to anaesthesia, induce the synthesis of heat shock proteins (HSPs), sensitize cells to hyperthermia, and increase the aggregation of nuclear proteins during heat shock. Anaesthetics are membrane active agents, and anaesthesia appears to be due to altered ion channel activity; however, the direct effect of heat shock is protein denaturation. These observations suggest that local anaesthetics may sensitize cells to hyperthermia by interacting with and destabilizing membrane proteins such that protein denaturation is increased. It is shown, using differential scanning calorimetry (DSC), that the local anaesthetics procaine, lidocaine, tetracaine and dibucaine destabilize the transmembrane domains of the Ca2+ -ATPase of sarcoplasmic reticulum and the band III anion transporter of red blood cells. The transmembrane domain of the Ca2+ -ATPase has a transition temperature (Tm) of denaturation of 61 degrees C which is decreased, for example, to 53 degrees C by 15 mM lidocaine. The degree of destabilization (deltaTm) by each anaesthetic is proportional to the lipid to water partition coefficient, and the increased sensitization by anaesthetics with larger partition coefficients and at higher pH suggests that the uncharged forms of the anaesthetics are responsible for destabilization. A Hill analysis of deltaTm for the Ca2+ -ATPase as a function of the concentration of anaesthetic in water gives dissociation constants (Kd) on the order of 10(-4) M, if binding occurs directly from the aqueous phase. This demonstrates moderate affinity binding. However, dissociation constants of 1-3 M are obtained, if binding occurs through the lipid phase, which demonstrates low affinity binding. Thus, the interaction of local anaesthetics with the Ca2+ -ATPase may be moderately specific or non-specific depending on the mechanism of interaction. The observation that local anaesthetics also destabilize the transmembrane domain of the band III protein of erythrocytes suggests that destabilization of transmembrane proteins is a general property of anaesthetics, which is at least in part a mechanism of sensitization to hyperthermia. |
Difference of caveolin-1 expression pattern in human lung neoplastic tissue. Atypical adenomatous hyperplasia, adenocarcinoma and squamous cell carcinoma.
Caveolin-1 has been implicated in cellular transformation and tumorigenesis. We assessed lung cancer specimens for caveolin-1 expression immunohistochemistry. A majority of the cell types in the lung and the bronchial epithelium normally exhibited positive staining for caveolin-1. In adenocarcinomas (ADs) of positive staining for caveolin-1, pT1 tumors exhibited significantly higher staining than pT2-pT4 tumors (P=0.0240). In squamous cell carcinomas (SCCs), pT1-pT2 tumors expressed significantly lower expression levels than pT3-pT4 tumors (P=0.0175). In AD, loss of caveolin-1 may be essential for tumor extension and dedifferentiation. In contrast, caveolin-1 overexpression may be correlated with tumor extension in SCC. |
package net.osmand.plus.measurementtool.command;
import android.util.Pair;
import net.osmand.GPXUtilities.WptPt;
import net.osmand.plus.measurementtool.MeasurementEditingContext;
import net.osmand.plus.measurementtool.MeasurementEditingContext.RoadSegmentData;
import net.osmand.plus.measurementtool.MeasurementToolLayer;
import net.osmand.plus.settings.backend.ApplicationMode;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.Map;
import static net.osmand.plus.measurementtool.MeasurementEditingContext.DEFAULT_APP_MODE;
public class ReversePointsCommand extends MeasurementModeCommand {
private List<WptPt> oldPoints;
private List<WptPt> newPoints;
private Map<Pair<WptPt, WptPt>, RoadSegmentData> oldRoadSegmentData;
private ApplicationMode oldMode;
public ReversePointsCommand(MeasurementToolLayer measurementLayer) {
super(measurementLayer);
this.oldMode = getEditingCtx().getAppMode();
}
@Override
public boolean execute() {
MeasurementEditingContext editingCtx = getEditingCtx();
oldPoints = new ArrayList<>(editingCtx.getPoints());
oldRoadSegmentData = editingCtx.getRoadSegmentData();
newPoints = new ArrayList<>(oldPoints);
Collections.reverse(newPoints);
executeCommand();
return true;
}
private void executeCommand() {
MeasurementEditingContext editingCtx = getEditingCtx();
editingCtx.clearSnappedToRoadPoints();
editingCtx.getPoints().clear();
editingCtx.addPoints(newPoints);
if (!newPoints.isEmpty()) {
WptPt lastPoint = newPoints.get(newPoints.size() - 1);
editingCtx.setAppMode(ApplicationMode.valueOfStringKey(lastPoint.getProfileType(), DEFAULT_APP_MODE));
}
editingCtx.updateCacheForSnap();
}
@Override
public void undo() {
MeasurementEditingContext editingCtx = getEditingCtx();
editingCtx.getPoints().clear();
editingCtx.addPoints(oldPoints);
editingCtx.setAppMode(oldMode);
editingCtx.setRoadSegmentData(oldRoadSegmentData);
editingCtx.updateCacheForSnap();
}
@Override
public void redo() {
executeCommand();
}
@Override
public MeasurementCommandType getType() {
return MeasurementCommandType.REVERSE_POINTS;
}
}
|
Q:
XSLT character-escaping query
Lets say I have the following code snippet below, how do I also apply the disable-output-escaping to the {name} in the title attribute?
<a title="{name}"><xsl:value-of select="name" disable-output-escaping="yes" /></a>
This has really got me stumped.
Thanks guys.
A:
This cannot be done with XSLT. The spec says:
It is an error for output escaping to
be disabled for a text node that is
used for something other than a text
node in the result tree.
Thus it makes no difference if you use Attribute Value Templates or xsl:attribute with xsl:value-of, because you're generating an attribute node, not a text node. It's a limitation in the language.
|
from datetime import date
# data in the form of: ID,FullName,AssetPackageName,AsciiLabel,PackID,Quality,UnlockMethod
dump_file = input('Data dump (input) file name?->')
data = open(dump_file, 'r')
Bodies, Wheels, Boosts, Antennas, Decals, Toppers, Trails, Goal_Explosions, Paints, Banners, Engine_Audio = [
], [], [], [], [], [], [], [], [], [], []
car_dict = {'backfire': 'Backfire', 'force': 'Breakout', 'octane': 'Octane', 'orion': 'Paladin',
'rhino': 'Road Hog', 'spark': 'Gizmo', 'torch': 'X-Devil', 'torment': 'Hotshot', 'vanquish': 'Merc',
'venom': 'Venom', 'import': 'Takumi', 'musclecar': 'Dominus', 'zippy': 'Zippy', 'scarab': 'Scarab',
'interceptor': 'Ripper', 'wastelandtruck': 'Grog', 'musclecar2': 'Dominus GT', 'torch2': 'X-Devil Mk2',
'neocar': 'Masamune', 'rhino2': 'Road Hog XL', 'takumi': 'Takumi RX-T', 'aftershock': 'Aftershock',
'neobike': 'Esper', 'marauder': 'Marauder', 'carcar': 'Breakout Type-S', 'number6': 'Proteus',
'cannonboy': 'Triton', 'o2': 'Octane ZSR', 'gilliam': 'Vulcan', 'bone': 'Bone Shaker',
'scallop': 'Twin Mill III', 'endo': 'Endo', 'charged': 'Ice Charger', 'flatbread': 'Mantis',
'pumpernickel': 'Centio V17', 'focaccia': 'Animus GP', 'challah': "'70 Dodge Charger R/T",
'melonpan': "'99 Nissan Skyline GT-R R34", 'levain': 'Werewolf', 'greycar': 'DeLorean Time Machine',
'darkcar': "'16 Batmobile", 'takumi_ii': 'Takumi RX-T', 'sourdough': 'Jäger 619 RS',
'multigrain': 'Imperator DT5', 'berry': 'The Dark Knight Rises Tumbler', 'eggplant': "'89 Batmobile",
'Universal Decal': 'Universal Decal'}
categories = {
"Bodies": [Bodies, ['body']],
"Decals": [Decals, ['skin']],
"Wheels": [Wheels, ['wheel']],
"Boosts": [Boosts, ['boost']],
"Antennas": [Antennas, ['flag', 'antenna', 'pennant', '.at']],
"Toppers": [Toppers, ['hat', 'crown']],
"Trails": [Trails, ['ss']],
"Goal_Explosions": [Goal_Explosions, ['explosion']],
"Paints": [Paints, ['paintfinish']],
"Banners": [Banners, ['playerbanner']],
"Engine_Audio": [Engine_Audio, ['engineaudio']]
}
for line in data:
line = line.lower()
line_parts = line.split(',')
ID = line_parts[0]
FullName = line_parts[1]
AssetPackageName = line_parts[2]
AsciiLabel = line_parts[3]
PackID = line_parts[4]
Quality = line_parts[5]
UnlockMethod = line_parts[6]
# if it is platform specific, logo for season play, crate, bots, keys, packs
if (ID == '0' or AssetPackageName == 'seasonlogos' or AssetPackageName == 'itemcontainer' or
AssetPackageName == 'bots' or 'key' in AssetPackageName or 'pack' in AssetPackageName):
continue
else:
filtered = False
for cat in categories:
filters = categories[cat][1]
for filter in filters:
if filter in FullName and not filtered:
name = ('{:4} - {}'.format(ID, AsciiLabel))
categories[cat][0].append(name)
filtered = True
if not filtered:
print('unassigned:\n', line_parts)
def write_file(categories):
file_name = input('Clean file (output) name(with "".txt")->')
clean_file = open(file_name, 'w')
day = date.today()
today = day.strftime("%d/%m/%y")
title = "# List of Item ID's in game as of " + today + "\n\n"
clean_file.write(title)
clean_file.write('### ID - AsciiLabel')
for cat in categories:
write = '\n\n### ' + cat + '\n'
clean_file.write(write)
clean_file.write('```\n')
for item in categories[cat][0]:
clean_file.write(item)
clean_file.write('\n')
clean_file.write('```')
clean_file.close()
write_file(categories)
|
#!/bin/bash
# T&M Hansson IT AB © - 2020, https://www.hanssonit.se/
# shellcheck disable=2034,2059
true
SCRIPT_NAME="Locate Mirror"
# shellcheck source=lib.sh
. <(curl -sL https://raw.githubusercontent.com/nextcloud/vm/master/lib.sh)
# Must be root
root_check
# Use another method if the new one doesn't work
if [ -z "$REPO" ]
then
REPO=$(apt update -q4 && apt-cache policy | grep http | tail -1 | awk '{print $2}')
fi
# Check where the best mirrors are and update
msg_box "To make downloads as fast as possible when updating Ubuntu you should have download mirrors that are as close to you as possible.
Please note that there are no gurantees that the download mirrors this script will find are staying up for the lifetime of this server.
Because of this, we don't recommend to change the mirror, except you live far away from the default mirror.
This is the method used: https://github.com/jblakeman/apt-select"
msg_box "Your current server repository is: $REPO"
if ! yesno_box_no "Do you want to try to find a better mirror?"
then
print_text_in_color "$ICyan" "Keeping $REPO as mirror..."
sleep 1
else
if [[ "$KEYBOARD_LAYOUT" =~ ,|/|_ ]]
then
msg_box "Your keymap contains more than one language, or a special character. ($KEYBOARD_LAYOUT)\nThis script can only handle one keymap at the time.\nThe default mirror ($REPO) will be kept."
exit 1
fi
print_text_in_color "$ICyan" "Locating the best mirrors..."
curl_to_dir https://bootstrap.pypa.io get-pip.py /tmp
install_if_not python3
install_if_not python3-testresources
install_if_not python3-distutils
cd /tmp && python3 get-pip.py
pip install \
--upgrade pip \
apt-select
check_command apt-select -m up-to-date -t 4 -c -C "$KEYBOARD_LAYOUT"
sudo cp /etc/apt/sources.list /etc/apt/sources.list.backup && \
if [ -f sources.list ]
then
sudo mv sources.list /etc/apt/
fi
msg_box "The apt-mirror was successfully changed."
fi
clear
|
Q:
Java Web Application - Deployment Strategy Alternative to WAR - Managing UI Changes Separately from Full Code Base Patches
I've heavily edited this question because responses indicated I wasn't being clear
problem: UI changes to a Java web project can be tedious and time consuming because every web-app file is contained within the WAR
my proposed solution: Manage the JSP's, CSS, JS and Tags separately from the application code base which for the purpose of this question I'm defining as:
All Java Source Code
Custom Tag Libraries With Compiled Java (extending TagSupport)
Spring Configuration Files
Web.xml
Jar's
+ Source
+ WEB
- WEB-INF
- JSP
- Tags
- lib
- HTML
- CSS
- JS
What would be nice is if after the major initial release and maintenance cycle, changes to view files could be treated as a different kind of release than a change to the Source. Source changes would be committed normally, and the application version would change. However, a change to a CSS/JS/HTML and even a JSP could be made in a test environment that is internally viewable to test new looks, add links, and so on. Technically, a JSP could even be added and as long as the controllers (like mine do) can be configured to show new JSP's without any Source modification, pages could be added without any deployments.
USE CASE - Owner of the site is running a promotion, he has a fancy graphic to link to an informational pure HTML page and wants it added to the home page.
Now imagine this work-flow:
UI dev opens dreamweaver and can FTP into Staging (staging may be a bad name, but basically a live test server). He can see:
- JSP
- Tags
- HTML
- CSS
- JS
Now he brings in the HTML file with information on it, adds it to the HTML directory. He then goes into JSP/home.jsp and finds the component that renders an advertisement on the right column, directly below it he adds his nifty image, saves his changes, opens his browser and goes to the live Test URL. He sees his image, but the component no longer renders the advertisement. Oops, he calls a developer and the developer says no problem
$ staging - > ./rollbackView -mostRecentBackup
The UI guy checks the site, everything is back like he never touched it.. now he more carefully adds his graphic and HTML, realizing that he cut off a JSP custom tag before. Now, QA, whatever that is for the project looks at the site, runs selenium, whatever. It all looks great. The developer gets the ok to release the changes
$ production - > ./updateProductionView
the script checks the application versions, ensuring they are identical, then copies over the view files. It's now 8:45 and the website owner (for us internal) is very happy that his new idea was implemented in the first 15 minutes of the day
Now the developer wants to create a patch that allows something cool, he updates his project, and the new view files are present. Maybe this isn't possible, but he could run a script, or use a second source repository like Mercurial to manage the views (ideas?) and he has the project and view files he needs. He makes his changes to the source, and views, whatever he needs. Now that is complete he can check in his changes and bring the WAR to a directory on Staging
$ staging - > ./deployStaging -overwriteView
The full war is deployed, and the JSP's are now what he had in his project. If the UI guys had made changes to staging, they will be overwritten (backed up maybe?). He could leave off the '-overwriteView' flag and the view files would remain untouched. At this point a full QA regression, integration and unit tests have been run, it's time to patch the main application
$ staging - > ./deployProduction
A full deploy is there, the application version is now V1.1 and everyone is happy
My Questions:
First, has anyone done something like this? If so, are there any good recommendations you can make? Development is done on Windows, but the production and staging servers are running Unix. All servers run the same version of Tomcat.
I'm looking for ideas for scripts that would allow Staging web files to be backed up, and hopefully even committed to the main project, also scripts that could take
What has been overlooked? Can I keep the project structured the same? Will this cause problems with CVS?
Is there anything that isn't possible or technically feasible here?
A:
Can you point your UI resources to other folders? This way you set up the symlinks once on the test server(s) and allow the UI developers to manipulate the 'live' files. If these folders are source controlled then the UI devs could rollback their own changes if necessary.
http://www.isocra.com/2008/01/following-symbolic-links-in-tomcat/
|
from collections import defaultdict
import logging
import os
import string
import random
from google.protobuf import json_format
from job import Job
from peloton_client.pbgen.peloton.api.v0.job import job_pb2 as job
from peloton_client.pbgen.peloton.api.v0.task import task_pb2 as task
from peloton_client.pbgen.mesos.v1 import mesos_pb2 as mesos
from util import load_test_config
#########################################
# Constants
#########################################
# general configs.
CONFIG_FILES = ["test_job.yaml", "test_task.yaml"]
MESOS_MASTER = ["peloton-mesos-master"]
MESOS_AGENTS = [
"peloton-mesos-agent0",
"peloton-mesos-agent1",
"peloton-mesos-agent2",
]
JOB_MGRS = ["peloton-jobmgr0"]
RES_MGRS = ["peloton-resmgr0"]
HOST_MGRS = ["peloton-hostmgr0"]
AURORA_BRIDGE = ["peloton-aurorabridge0"]
PLACEMENT_ENGINES = ["peloton-placement0", "peloton-placement1"]
# job_query related constants.
NUM_JOBS_PER_STATE = 1
TERMINAL_JOB_STATES = ["SUCCEEDED", "RUNNING", "FAILED"]
ACTIVE_JOB_STATES = ["PENDING", "RUNNING", "INITIALIZED"]
# task_query related constants.
TASK_STATES = ["SUCCEEDED", "FAILED", "RUNNING"]
DEFAUILT_TASKS_COUNT = len(TASK_STATES)
HOSTPOOL_DEFAULT = "default"
HOSTPOOL_BATCH_RESERVED = "batch_reserved"
HOSTPOOL_SHARED = "shared"
HOSTPOOL_STATELESS = "stateless"
log = logging.getLogger(__name__)
"""
Creates a JobConfig object.
Arg:
file_name: Load base config from this file,
(either 'test_task.yaml' or 'test_job.yaml').
job_name: string type. The Job name.
job_owner: string type. The Job owner.
job_state: string type. Has the value of 'SUCCEEDED', 'RUNNING', or 'FAILED'.
task_states: a list of tuples, e.g. [<task_state>, <task_count>].
Needed for 'task_query' config.
Returns:
job_pb2.JobConfig object is returned.
"""
def generate_job_config(
file_name, job_name=None, job_owner=None, job_state=None, task_states=None
):
job_config = _load_job_cfg_proto(file_name)
if file_name == "test_job.yaml":
# Create job_config for `job_query`.
job_config.name = job_name
job_config.owningTeam = job_owner
task_cfg = create_task_cfg(job_state)
job_config.defaultConfig.MergeFrom(task_cfg)
else:
assert task_states
# Create config for `task_query`.
is_state_mixed = True if len(task_states) > 1 else False
tasks_count = sum(i[1] for i in task_states)
if is_state_mixed:
mixed_tasks_cfg = create_task_configs_by_state(task_states)
job_config.instanceConfig.MergeFrom(mixed_tasks_cfg)
else:
default_cfg = create_task_cfg(task_states[0][0])
job_config.defaultConfig.MergeFrom(default_cfg)
job_config.instanceCount = tasks_count
sla_config = _create_sla_cfg(job_config.sla, tasks_count)
job_config.sla.MergeFrom(sla_config)
return job_config
"""
Creates a TaskConfig object.
Arg:
task_state: a string value of 'SUCCEEDED', 'RUNNING', or 'FAILED'.
Returns:
a TaskConfig object is returned.
"""
def create_task_cfg(task_state="SUCCEEDED", task_name=None):
assert task_state in TASK_STATES
commands = {
"SUCCEEDED": "echo 'succeeded instance task' & sleep 1",
"RUNNING": "echo 'running instance task' & sleep 100",
"FAILED": "echo 'failed instance task' & exit(2)",
}
return task.TaskConfig(
command=mesos.CommandInfo(shell=True, value=commands.get(task_state)),
name=task_name,
)
#########################################
# Helper Functions.
#########################################
"""
Loads and returns the JobConfig object based on the input yaml file.
"""
def _load_job_cfg_proto(job_file):
job_config_dump = load_test_config(job_file)
job_config = job.JobConfig()
json_format.ParseDict(job_config_dump, job_config)
return job_config
"""
Get a map of job objects by state and their common identifier salt
Args:
_num_jobs_per_state: number of job objects per state.
Returns:
dict of jobs list per state is returned
"""
def create_job_config_by_state(_num_jobs_per_state=NUM_JOBS_PER_STATE):
salt = "".join(
random.choice(string.ascii_uppercase + string.digits) for _ in range(6)
)
name = "TestJob-" + salt
owner = "compute-" + salt
jobs_by_state = defaultdict(list)
# create three jobs per state with this owner and name
for state in TERMINAL_JOB_STATES:
for i in xrange(_num_jobs_per_state):
job = Job(
job_config=generate_job_config(
file_name="test_job.yaml",
# job name will be in format: TestJob-<salt>-<inst-id>-<state>
job_name=name + "-" + str(i) + "-" + state,
job_owner=owner,
job_state=state,
)
)
jobs_by_state[state].append(job)
return salt, jobs_by_state
"""
Creates an InstanceConfig object from given task_states
for `task query` tests.
Arg:
task_states: a list of tuples: [(`task_state`, `task_count`)].
For example, [('SUCCEEDED', 2), ('FAILED', 1)]
Returns:
a map of task configs by state in a job.
"""
def create_task_configs_by_state(task_states):
for i in task_states:
assert isinstance(i, tuple)
tasks, index = {}, 0
for state, num in task_states:
for i in range(index, index + num):
tasks[i] = create_task_cfg(state, task_name="task-" + str(i))
index += num
return tasks
"""
Updates `maximumRunningInstances` with number of tasks created in
SLAConfig object, which is needed for `task query` tests.
Arg:
curr: Current SlaConfig object read from JobConfig file.
tasks_count: total number of tasks to be created.
Returns:
a structured SlaConfig object.
"""
def _create_sla_cfg(curr, tasks_count=DEFAUILT_TASKS_COUNT):
assert isinstance(curr, job.SlaConfig)
updated_sla = job.SlaConfig(
priority=curr.priority,
preemptible=curr.preemptible,
maximumRunningInstances=tasks_count,
)
return updated_sla
# Returns the type of the minicluster. This will either be "k8s",
# "mesos" or "".
def minicluster_type():
return os.getenv("MINICLUSTER_TYPE")
# Returns whether host-pools should be used for placement decisions.
def use_host_pool():
return os.getenv("USE_HOST_POOL", "false").lower() in ["true", "1"]
|
Migrants of a “non-Western origin” are massively overrepresented in Denmark’s benefits system, according to new figures from the Ministry of Employment.
Of all totally dependent families in Denmark, married couples where both partners are on social assistance — state benefits — some 84 per cent are “non-Western origin” migrants. In total, a third of all cash paid out in benefits every month goes to these non-Western migrants in Denmark, according to the latest figures obtained by Ekstra Bladet.
These figures might be considered especially high, as among Denmark’s working age population non-Western migrants make up just eight per cent of residents. The paper reports experts agree the phenomenon of a minority group of eight per cent of the population making up such a significant part of the claimant count, and concede it is a “large and especially expensive problem”.
It is estimated Denmark’s migrants cost the government some 11 billion crowns (£1.1 billion) a year reports BT, a significant sum for a country of just five and a half million people.
Offering an explanation for this apparent predilection towards worklessness among non-Western migrants in Denmark, Ekstra Bladet links the latest news to the comments of a government troubleshooter tasked with getting migrants into work made in 2015. Specialist job centre chief Eskild Dahl, who remarked at the end of his tender at the facility that he had spent “an awful lot of money to virtually no effect” in trying to get migrants into work.
Those migrants he came into contact with on a daily basis saw government cash assistance as a right, and the so-called “refugees” generally saw work as “punishment” to be avoided at all costs. The fundamental problem was that the Danish welfare state was built on the Protestant work ethic and was incompatible with the new arrivals he said, reports the Berlingske.
It is the unusually frank bookkeeping practised by the Danish government that allows these statistics about worklessness in migrant communities to be reported. Unlike many Western countries, Denmark recognises, even among full Danish citizens who are legitimately ethnic Danish, those migrants from Western nations, and those from further afield.
With an eye on this count of the number of actual Danes in the country, the government expresses concern about demographic change and population levels. As Breitbart London reported despite the overall Danish population rising because of immigration the actual population is falling. According to government statistics, just 89 per cent of people in the country were Danish in 2014. |
The present invention relates to recording apparatuses for intermittently recording picture data of a monitor camera or the like at a constant interval and, more particularly, to time lapse recorders, which comprise a motion correcting circuit, typically MPEG2 (xe2x80x9cGeneric coding of moving pictures and associated audio information: videoxe2x80x9d, ISO/IEC IS 13818-2), provided in a picture coding part and an HDD (hard disc drive) or a VCR (Video Cassette Recorder) provided in a recording unit part.
Time lapse VCRs (video tape recorders) using VTRs are well known in the art as an apparatus for intermittently recording an extracted part of a picture signal from a camera at a constant interval.
With the recent development of digital signal processing techniques, time lapse recorders are also used, in which image signal is digitized and data compressed by using a still image coding system, typically JPEG (xe2x80x9cDigital Compression Coding of Continuous-Tone Still Imagesxe2x80x9d, ISO/IEC IS 10918-1) for recording of the compressed data in a recording medium, such as HDD or an optical disc.
The time lapse recorder currently in practical use, has a problem that in the case of an analog VTR it is difficult to record high quality picture signal.
In the digital system case, it is possible to obtain high quality picture signal recording owing to the use of a still picture cording system. However, for executing abnormality detection and automatically reducing the picture signal recording interval when the abnormality is detected, an exclusive abnormality detector is necessary, so that the apparatus is increased in scale, complicated and increased in cost.
The present invention was made in view of the above problem, and it has an object of providing a time lapse recorder, which uses a motion picture coding system having a motion compensating circuit typically MPEG2 and thus can detect abnormality without exclusive abnormality detector, and in which when the abnormality is detected the recording interval is adaptively changed or the recording picture quality is improved to permit accurate recording of the abnormal phenomenon.
According to an aspect of the present invention, there is provided a time lapse recorder comprising: a dividing means for dividing an input picture signal into signals corresponding to small area blocks each consisting of a plurality of pixels; a motion estimate means for retrieving motion vectors of blocks most similar to each other in picture frames of different times; a motion compensating means for obtaining a difference between a coding subject frame and a motion vector computation frame for each of the blocks on the basis of the motion vectors; an orthogonal transform means for executing orthogonal transform of the blocks; a quantizing means for quantizing transform coefficient values as the output of the orthogonal transform means; a code quantity control means for controlling picture quality and quantity of codes generated per frame in accordance with the complexity of the input picture; a variable length coding means for variable length coding quantized coefficient data outputted from the quantizing means; an abnormality detecting means for detecting abnormality of input picture; a recording interval control means for changing recording frame number at a constant interval in the normal state and in unit times in the abnormality detection state; and a recording means for recording the variable length coded data in a recording medium under control of the recording interval control means, wherein the abnormality detecting means computes the coding bit number of the variable length coding means for each frame, compares the difference of this coding bit number from the coding bit number of the immediately preceding frame having the same predicted structure and executes abnormal detection when the compared difference is above a predetermined threshold value.
According to another aspect of the present invention, there is provided a time lapse recorder comprising: a dividing means for dividing an input picture signal into signals corresponding to small area blocks each consisting of a plurality of pixels; a motion estimate means for retrieving motion vectors of blocks most similar to each other in picture frames of different times; a motion compensating means for obtaining a difference between a coding subject frame and a motion vector computation frame for each of the blocks on the basis of the motion vectors; an orthogonal transform means for executing orthogonal transform of the blocks; a quantizing means for quantizing transform coefficient values as the output of the orthogonal transform means; a code quantity control means for controlling picture quality and quantity of codes generated per frame in accordance with the complexity of the input picture; a variable length coding means for variable length coding quantized coefficient data outputted from the quantizing means; an abnormality detecting means for detecting abnormality of input picture; a recording interval control means for changing recording frame number at a constant interval in the normal state and in unit times in the abnormality detection state; and a recording means for recording the variable length coded data in a recording medium under control of the recording interval control means, wherein the abnormality detecting means computes statistical value of the distribution or amplitude values of the motion vectors for each frame, compares the statistical value of the motion vectors with a predetermined threshold value, and executes abnormality detection when the statistical value is above the threshold value.
The recording interval control means adaptively controls the recording frame number according to a multiple-valued output of the abnormality detecting means. When the output of the abnormality detecting means is abnormal, the recording interval control means controls the recording frame number such as to permit recording of all frames. The abnormality detecting means provides a multiple-valued output, and the recording interval control means decodes the value of the output of the abnormality detecting means, selects a recording time interval corresponding to the output value among a plurality of different recording times shorter than a recording time in a normal state and causes the recording means to record frames at the selected recording time interval. The code quantity control means executes code quantity control on the basis of the output of the abnormality detecting means such as to provide an improved recording picture quality when the abnormality is detected.
According to other aspect of the present invention, there is provided in an image processing apparatus of moving picture coding type for detecting motion vectors from an input picture, variable length coding a motion corrected signal, and recording the coded signal in recording means, a time lapse recorder comprising: means for comparing a unit time mean coding bit number of past frames and the coding bit number of the present frame and executing abnormality detection on the basis of the difference between the compared coding bit numbers; and means for controlling the number of frames recorded in the recording means such that the recording interval per unit time is reduced when the abnormality is detected.
According to still other aspect of the present invention, there is provided inn an image recording apparatus for detecting motion vectors from an input picture, generating an inter-frame difference value between the input picture and a predicted frame from the motion vectors, executing orthogonal transform of the inter-frame difference, quantizing coefficients obtained by the orthogonal transformation, coding the quantized coefficients and recording the quantized coefficients in a recording means, a time lapse recorder comprising: an abnormality detecting mean for comparing a unit time mean coding bit number of past frames and the coding bit number of the present frame and executing abnormality detection on the basis of the difference between the compared coding bit numbers; and a recording interval control means for controlling the number of frames recorded in the recording means such that the recording interval per unit time is reduced when the abnormality is detected.
According to further aspect of the present invention, there is provided in an image recording apparatus for detecting motion vectors from an input picture, generating an inter-frame difference value between the input picture and a predicted frame from the motion vectors, executing orthogonal transform of the inter-frame difference, quantizing coefficients obtained by the orthogonal transformation, coding the quantized coefficients and recording the quantized coefficients in a recording means, a time lapse recorder comprising: an abnormality detecting means including means for computing the unit time least mean coding bit number on the basis of the frame coding bit number of each frame, and means for comparing the coding bit number of the present frame and the least mean coding bit number and determines that the frame is abnormal when the difference is greater than a predetermined threshold value; and a recording interval control means for controlling the number of frames recorded in the recording means such that the recording interval per unit time is reduced when the abnormality is detected.
According to still further aspect of the present invention, there is provided in an image recording apparatus for detecting motion vectors from an input picture, generating an inter-frame difference value between the input picture and a predicted frame from the motion vectors, executing orthogonal transform of the inter-frame difference, quantizing coefficients obtained by the orthogonal transformation, coding the quantized coefficients and recording the quantized coefficients in a recording means, a time lapse recorder comprising: an abnormality detecting means including means for computing the unit time least mean coding bit number on the basis of the frame coding bit number of each frame, and means for comparing the coding bit number of the present frame and the least mean coding bit number and determines that the frame is abnormal when the difference is greater than a predetermined threshold value; and a recording interval control means for controlling the number of frames recorded in the recording means such that the recording interval per unit time is reduced when the abnormality is detected, wherein a plurality of threshold values are provided, and abnormality detection is executed by a multiple-valued state detection as a result of comparison between the difference and the plurality of threshold values.
The abnormality detecting means includes: means for selecting a frame having a predetermined predicted structure among input frames; means for receiving frame coding bit number of the selected frame and computing the unit time least mean coding bit number; and means for executing abnormality detection and providing an abnormality detection signal when the quotient of division of the mean coding bit number by the least mean coding bit number is greater than a predetermined threshold value and also when the quotient of division of the coding bit number of the selected present input frame by the unit time least mean coding bit number is greater than a predetermined threshold value. The frame coding bit number is inputted when P picture frames alone or P and B picture frames are inputted. The abnormality detecting means includes: means for analyzing the motion vector over one frame; and means for comparing the motion vector value distribution with a predetermined value, executing abnormality detection when it is larger than the predetermined value, and generating an abnormal detection signal. The abnormality detecting means includes: means for converting the motion vector amplitudes to corresponding scalar values; means for accumulating the scalar values for each frame; and means for comparing the accumulation value with a predetermined threshold value and outputting an abnormality detection signal when the accumulation value is greater than the threshold value. The recording interval control means normally executes a control for recording frames at a predetermined recording time interval in the recording means; and when the abnormality detecting means detects abnormality, the variable length coding means is controlled on the basis of the value of the output of the abnormality detecting means such that all frames are recorded continuously or at a recording time interval shorter than the recording time interval in the normal state. An abnormal detection signal is outputted when it is determined from a motion vector value and coordinate data of a base block of the motion vector that motion coordinates of the motion vector are in a predetermined monitoring area. When the abnormality detecting means detects abnormality, the code quantity control means controls the quantizing means and the variable length coding means for recording frames in the recording means at an increased coding rate and a reduced compression factor compared to the normal state.
Other objects and features will be clarified from the following description with reference to attached drawings. |
Protect Receiver From Damage When Clamping
Heavy-duty blocks allow you to firmly clamp AR-15/M16 receiver halves in a vise without risk of crushing, twisting, or otherwise distorting them when applying vise pressure or torquing the barrel nut. Keeps your rifle secure for safe, effective cleaning, assembly, and repair. Unbreakable urethane vise block takes all the clamping force, so your expensive lower receiver won’t be damaged or distorted. Insert block into the magazine housing from the top or bottom, then clamp the other end in your bench vise. The magazine catch locks to the block, safely and securely holding the rifle for assembly, disassembly, cleaning, or repair. Will not mar or scratch the finish. Fits tightly in most receivers and may require filing to fit others. |
1. Field
Exemplary embodiments of the invention relate to a display device and a method of manufacturing the display device.
2. Description of the Related Art
Display devices may be classified into liquid crystal display (“LCD”) devices, organic light emitting diode (“OLED”) display devices, plasma display panel (“PDP”) devices, electrophoretic display devices, and the like based on a light emitting scheme thereof.
Among those, an LCD device includes two substrates including electrodes formed thereon and a liquid crystal layer interposed between the two substrates. Upon applying voltage to the electrodes, liquid crystal molecules of the liquid crystal layer are rearranged such that an amount of transmitted light is controlled in the LCD device. |
Alias Julius Caesar (Charles Ray Productions, 1922) Lobby Card
Description
Alias Julius Caesar (Charles Ray Productions, 1922)
Billy Barnes gets into trouble over a practical joke on a golf course and ends up in jail, despite his high social standing. But he manages to redeem himself with the help of a jewel thief he befriends in jail. |
"Defendants deny that Plaintiffs are entitled to any relief, whether monetary, compensatory, declarative, equitable, costs, and/or fees relating to this matter, or in any other form sought by Plaintiffs," the 10-page response filed Thursday by the firm Aaron Marks of Kasowitz, Benson, Torres and Friedman reads. |
---
date: February 2016
display_date: February 2016
meetups:
- name: 'StatServer-Samza: Near Real-time Analytics'
host: LinkedIn
image:
presenters:
- name: Tomy Tsai
website:
image:
affiliation: LinkedIn
abstract: StatServer is a near real-time analytics service popularly used in LinkedIn and is in the process of being migrated to the Samza platform.
video:
url: https://youtu.be/eUaPd8t7Pac
image: https://img.youtube.com/vi/eUaPd8t7Pac/0.jpg
---
<!--
Licensed to the Apache Software Foundation (ASF) under one or more
contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
--> |
High-sulfur protein gene expression in a transgenic mouse.
We analyzed the effect of minoxidil on hair follicles isolated from transgenic mice. These transgenic animals synthesize the reporter enzyme CAT in their hair follicles only during the active phases of hair growth. The recombinant gene used to generate these mice contained the bacterial enzyme CAT under the control of the promoter from the gene of UHS protein. Studies using in situ hybridization showed that UHS proteins are expressed specifically in the matrix cells of the hair follicle during the terminal stages of hair differentiation. Hence the expression of the UHS proteins is a clear sign of active hair growth. With other in situ hybridization studies we demonstrated that CAT mRNA is expressed in differentiating matrix cells of the hair shaft in a location similar to that in which mRNA encodes UHS proteins. Thus we can use the levels of CAT activity as a measure of hair growth. We have confirmed that expression of the transgene is found in hair that is high in anagen and low in catagen follicles. The usefulness of our model was further demonstrated by showing that minoxidil, a drug that stimulates hair growth, increased the expression of CAT in cultured hair follicles. Thus we have demonstrated that expression of this reporter gene is sensitive, hair specific, and also useful for monitoring effects in cultured hair follicles. Hence these transgenic mice provide a model system for studying the biology of hair growth. |
Cytology and hormonal receptors in breast cancer.
Even in advanced tumors of the breast, it may be interesting to know the steroid receptor status (RS) for therapeutic and prognostic purposes. When, for clinical reasons, surgical biopsy is not advisable, a morphologic technique may be attempted on cytological material. In the past few years, we have employed the cytochemical method described by Lee (RSf) on cytological material obtained from 31 primary and 34 secondary tumors; adequate material and follow-up were available in 42 cases. Patients with positive results (strong and diffuse fluorescence of neoplastic cells) usually had better prognosis and longer disease-free interval and total duration of disease. The majority of patients responded to endocrine manipulation. In contrast, weak positivity or negative results were associated with a poorer prognosis. If these results are confirmed in larger series of cases and for longer periods of observation, the morphological evaluation of RS on cytological material could significantly contribute to the management of breast cancer patients. |
[The brain-gut axis: insights from the obese pig model].
The pig, which shares several similarities with humans, is increasingly used for biomedical research, particularly in nutrition and neurosciences. Recent studies in minipigs have shown that a deleterious nutritional environment (e.g. a high-fat and high-sugar diet) induces obesity which, as in humans, is associated with increased adiposity, insulin resistance, modified eating behaviour, and altered gastric function and intestinal sensitivity. These changes are accompanied by differences in the activation matrices and metabolic activity of several brain areas. Using this animal model, we have revisited the concept of dual hedonic and homeostatic control of food intake. We have thus developed a minimally invasive and potentially reversible surgical approach to the control of food intake, as an alternative to bariatric surgery, based on chronic vagal stimulation at the abdominal level. |
Q:
JavaScript Standard Library for V8
In my application, I allow users to write plugins using JavaScript. I embed V8 for that purpose. The problem is that developers can't use things like HTTP, Sockets, Streams, Timers, Threading, Crypotography, Unit tests, et cetra.
I searched Stack Overflow and I found node.js. The problem with it is that you can actually create HTTP servers, and start processes and more things that I do not want to allow. In addition, node.js has its own environment (./node script.js) and you can't embed it. And it doesn't support Windows - I need it to be fully cross platform. If those problems can be solved, it will be awesome :) But I'm open to other frameworks too.
Any ideas?
Thank you!
A:
There is CommonJS, which defines a "standard" and a few implementations available of that standard - one of which is node.js.
But from what I can see, it's still fairly immature and there aren't many "complete" implementations.
A:
In the end, I built my own library.
|
Latino Cafe & Restaurant - Out n' About
By Shaza Ossama
If you're looking for a place with a breathtaking view and reasonable prices to spend your day, then Latino is the way to go. Latino is a cafe/restaurant which has this mesmerizing Nile view which looks just perfect at both day and night and in my opinion it will be a good place for a first date.
I have tried most of their dishes so I can give you an overall opinion of whats going-on there. Let's start with the least favorite thing that I have tried; the burger. Burgers there are very straight to the point, classic with a medium quality beef.
Platters however are a whole different story; they serve a reasonable quantity for a good price. I've tried the paradise chicken and the monterrey chicken. Both tasted very good and chicken was juicy and perfectly cooked. All platters there whether beef or chicken are served with two side dishes of your choice (sautéed veggies, fries, mashed potatoes, penne pasta, Alfredo, rice, and the list goes on).
cocktails and milkshakes taste so good as well, and they do serve" shesha". When it comes to desserts they are so good to be true, one of their popular desserts there is "zalabya" covered with chocolate, and let me tell you if you went there and didn't order one it's your loss! brownies and cheese-cakes are also amazing.
Service wise, it`s not that good. The place is generally busy so with a very big place and a small staff, they always look stressful, and it may take them while to take your order.
Final Review:
Taste: 4\5
Price: 4.5\5
Atmosphere: 5\5
service: 3.5\5
whole experience: 4\5
Branches:
106 Nile street, Agouza |
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