text stringlengths 54 548k | label stringclasses 4
values | id_ stringlengths 32 32 |
|---|---|---|
Individual spectral studies of the most interesting Chandra sources will also be presented in follow-up work. Here, we have simply mentioned a few highlights. The most luminous ULX in the sample (a source in IC 3322A with {{formula:d21fc5b2-0101-4f51-a130-61724191bd0d}} erg s{{formula:06a0a4d3-bbd2-4f1f-a192-8d87bc5c7c8d}} ) may not be an accreting compact object, as its X-ray spectrum is more consistent with that of a young SN (which could have been missed by optical/IR searches). Apart from that mysterious source, the three most luminous ULXs (in NGC 4254, NGC 4496A and NGC 4579) all reached a luminosity {{formula:891fe0e5-e97e-4d0a-9558-2f8b35c1f303}} erg s{{formula:8bee92d0-9d31-4c6b-9b31-a3d0bdcdb480}} , which is the well-known characteristic threshold above which ULXs become much rarer {{cite:465a9c58a7f22a10478be6dd775f55fcdba17373}}, {{cite:2b4e83a4e3d9c023a2753fcfe71affd7774dc029}}, {{cite:bade5d716fba51d3963bd1b252acb5a68f3cfe78}}, {{cite:5b6b4e53f23bce3769da0687be579444aafde0e9}}. At least three Chandra sources seen inside the {{formula:8b68d20b-4e3f-44dd-8df6-a7f3a747941b}} of sample galaxies are likely to be the nuclear sources of smaller galaxies: further investigation is needed to determine whether those galaxies are also in Virgo (possibly satellites of their larger companions) or are a more distant background.
| d | ad54aea0b59c73db292ce85a07a41621 |
The proposed analysis also corroborates existing explanations for techniques that successfully improve generalization, such as data augmentation and contrastive learning.
Both were indeed shown to depend on the injection of additional knowledge, respectively in the design of the augmentations {{cite:9699115e4038223af08bc23fa383398f9debfa15}}, {{cite:65afefd61bb7ec8d5e576360e4339f335696855b}}, {{cite:e4d01550b10c8359a384bde10a5df6d6e500a884}} and pair selection strategy {{cite:d6141eff8ba6fdce39636ef115db3478228afebe}}.
And this extra knowledge is often task-specific {{cite:8a6a85221bec078091803f7e95c543542c9f0d75}}. For example, augmenting images with rotations may help in identifying flowers but not traffic signs.
Injecting task-specific knowledge is sometimes vilified in a “data-driven” culture.
This study suggests that we would rather benefit from highlighting this practice and making assumptions more explicit, thus helping one to identify the limits of applicability of various methods.
| d | 6ac3e7201e2153d9620f025110f96fcf |
Two of the most common strategies to solve OC problems are the Pontryagin Maximum Principle (PMP) {{cite:b12bdd9f45ca49ee026b778450a8a8473ea68064}} and Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) {{cite:d21a05d14838282dc0283399da8a933c651310e1}}. The PMP is often suitable for high-dimensional problems (Sec. REF ). A local solution method, the PMP finds the optimal policy for a single initial state, so deviations of the system from the optimal trajectory require re-computation of the solution.
In contrast, the HJB approach is a global solution method suitable for real-time applications.
It is based on solving the HJB PDE to obtain the value function (Sec. REF ). However, state-of-the-art HJB solvers, e.g., ENO/WENO {{cite:16311dbc0b7afb803d8d9d4e62162966812be9b4}}, are grid-based and can suffer from the curse of dimensionality (CoD) {{cite:d21a05d14838282dc0283399da8a933c651310e1}}, meaning that costs increase exponentially with dimension.
For OC problems with a state-space dimension exceeding four, the CoD renders using grid-based HJB solvers infeasible.
| i | af838390c106a3080982f5974c41f041 |
CVAE with VADE {{cite:49b231f1c72136c42710ad5344ac1df7249d9b0e}}, introducing a Gaussian mixture model in the latent space, to obtain a better representation of the text through the clustering task.
{{figure:c46b9dcb-b250-43a9-b638-63e2411e973b}} | m | acf41939a436150425844bd1de26aa93 |
We also perform the evaluation on UCF101, which is a bigger dataset than HDMB51. A comparison between our method with state-of-the-art results is shown in REF . Similar conclusions as with UCF-101 can be extracted. Our model gets 82.69%, only using RGB frames as input, outperforming the models using just spatial information, and is just slightly below the two stream model {{cite:285044620b2074a69658f85f543ad8a5e73d535e}} which achieves 86.2%.
| r | b0ef0725504cda5a36f9666eaff5eb78 |
The MPP has been simulated by means of the Whizard event generator {{cite:a07ee8daafa8f68c11648241705d9fbbb3a38034}}, {{cite:c2a3504c8a7805b51342f167aabb10943eb6dc32}}, run in such a way to take into account the incoming beams features. The muon beam transverse normalized emittance for all the schemes is also reported in Table REF and the number of muons per second divided by the emittance, crucial figure of merit as suggested in {{cite:0e50e1c0b2a95394e876de9198c553fb49876d2c}}, has been calculated. The best calculated value for muon beam normalized transverse emittance is {{formula:0f029d6f-cfc0-41b0-96be-48f619120702}} nm rad: this value compares to the analytical prediction of Eq. REF giving {{formula:785eaafc-f957-4a6e-ba85-f60375d42e5a}} nm rad.
The emitted muon beams features for the FCC-twin Linac, the CLIC parasitic and the FCC-ee parasitic schemes are reported in Fig. REF . The energy, angle and transverse momentum distrubutions are shown.
The Twin Linac option has been analysed in detail. The incoming electron beam, the FEL photon pulse and the MPP features are displyed in Fig. REF .
{{figure:47c322ac-12a3-4793-9e20-73f82cbe7f67}} | r | d407736f265e9ac542793ca6e5404b08 |
At the crux of our analysis lies a new restricted isometry property of the sub-differentials, which we call Sign-RIP.
Under Sign-RIP, the sub-differentials of the {{formula:a04868ca-8c47-44a3-8087-84923ed90564}} -loss are {{formula:8d9e0010-8889-47cf-889d-794fb1e9d921}} -away from the sub-differentials of an ideal, expected loss function (see Section REF for precise definitions). We will show that the classical notions of {{formula:681a29ce-2e70-484c-88b3-20199d88df11}} -RIP {{cite:2e064861a243dd1626f546946f70672abb77d3b5}} and {{formula:1574dccf-1cc4-4b22-ac4d-dba988061f19}} -RIP {{cite:913ed4d2834d654fd1cdd65fc7383e19963ef7eb}} face major breakdowns in the presence of noise. In contrast, Sign-RIP provides a much better robustness against noisy measurements, while being no more restrictive than its classical counterparts. We will show that, with Gaussian measurements, the Sign-RIP holds with an overwhelming probability under two popular noise models, namely outlier noise model and Gaussian noise model.
| r | d7012deef46bd748f42a1c352d77e0c9 |
Generally, these methods simply model the result consistency on observed transitions, with the transition {{formula:16f27604-4aa5-4e8d-af63-24b331825043}} implicitly conveyed within the training data.
For example,
Pix2Pix {{cite:75f9eb39582083e33f932608bc7ffb993f2d16f5}} use {{formula:ecca6bcf-35a6-4a16-9148-15fefbace0dc}} as the constraint for result consistency, with {{formula:c53d4b3c-86fd-44ad-89aa-830573cda82a}} implied within each data pair {{formula:846f5a51-6157-4761-b6d0-2b46b8312715}} , while CycleGAN {{cite:abd095c54e737ea867094577da25c35e6e444fe5}} tackles the problem with cycle-consistency {{cite:6da1327fca22c79bc89208d58cee8e26dccdca6c}}, i.e., {{formula:29726a77-5604-4ba2-92b0-5b69420ba30a}}.
Other methods, e.g. AttGAN {{cite:66463a6bcdcb67538fdbe969badfdd8c61ed4fd5}} and StarGAN {{cite:ece865f8445928549e9d0e130bcfd6af13cc447f}}, regularize result consistency via attribute prediction on the translated image i.e.{{formula:6582e423-5391-4c44-8cf7-3bab220464d9}}, with {{formula:f0848708-ee28-4796-85ba-ff6519685881}} implied within {{formula:d91edb8d-5297-488c-ae27-6bfc5c002e22}} .
The self-reconstruction constraint in {{cite:a52c2c255406678303b408e5795e245a6299a3f0}} can be explained with transition as {{formula:4988de72-af04-45de-8ecf-bd87b5f00079}}.
Without explicitly model {{formula:ebf05b0b-f572-4cef-b2a4-1d2935997a5f}} , these methods can only regularize consistency on the observed transitions, limiting their generation capacity when translating with unseen transitions.
| d | 99f6aae8d7898df26d31f7489047c5b0 |
Recall that {{formula:f141633e-36e4-4592-9b1a-90185aa147ca}} is topologically graded with conditional expectation {{formula:dba8a96a-2c99-4e87-a8fb-453fda4efa04}} . Since {{formula:bb9149ef-7042-45f0-93d3-7e80717e0a21}} strongly reduces to an amenable group, then by Lemma REF , the gauge action {{formula:6331ac2e-d429-40ca-9eaf-ab049420bfc2}} is normal and thus the conditional expectation {{formula:db83a40d-0eb2-45ba-a86c-da8f119595d8}} is faithful. But by {{cite:f8d843e14c51201f8c2ea14e6c59ccf7aa829e1d}}, the kernel of the regular representation {{formula:6f859e8d-0f83-4c4b-98c8-40e5b53b8f32}} is given by
{{formula:9ca371e1-631e-42fc-8e97-7189d3733610}}
| r | 58041bfe5f996b207be9d04a52fb8674 |
DIME and D{{formula:1054c8ec-0f8f-4c5f-bf23-ea5bdc700ade}} both reliably detect both types of OOD-examples, and are to our knowledge the the first methods achieving PR-AUC {{formula:cdabf392-b34a-4b10-b180-eff6b94d4a04}} 80 % on the Tweets dataset in this task (see Figure REF D).
Surprisingly, distance-based metrics not using a modelled embedding perform worse on both datasets and fail completely on tweets.
Both MC-Dropout and softmax confidence show fair results, and we observe better performance for softmax confidence on tweets compared to the original authors (66 % compared to the original 41 %) {{cite:144ee758c3434598d9b14b7f898cbb9db562f44e}}.
One difference that may explain this discrepancy is that they only trained their classifier on the Wall Street Journal-subset of Penn Treebank whereas we used the complete dataset.
That model calibration is improved by training on more diverse datasets is consistent with literature {{cite:cc107421f35023e67047ac11ce3b14bf305b56c4}}.
| m | d767e0433feecb9032061785c798d88a |
By studying only the quotient structure of certain spacetimes, it is possible to learn much about dynamics and quantum corrections on those backgrounds. A principal tool to accomplish this is the Selberg zeta function, a cousin of the Riemann zeta function in which prime numbers are replaced by prime geodesics on a hyperbolic spacetime {{formula:62dbccdb-dd75-4b9a-84d4-445cca800ad2}} , where {{formula:e77daefc-75c0-4b3f-bf7d-9a025d95970c}} is a discrete subgroup of SL(2,{{formula:364824e7-503e-42cd-bff2-a159ad01c286}} ) {{cite:71f644d098e69129710505df8a76c351b34ee0d5}}, {{cite:234c383f50540957ccd8ebcd2c3bcca5dc141106}}, {{cite:ff6b0bf556617d32d70e29d2382694a68c10faf3}}. For example, for {{formula:e38bf4c7-9b00-4daa-b24e-8a0d106f737f}} the Selberg zeta function is of the form
{{formula:048f94bf-208f-41e6-a8c3-927f224db4f5}}
| i | c7cd71af03323d5905fdd905e4f1382d |
The CelebA dataset consists of more than {{formula:79455d6d-923d-4642-9ee3-5791d19e39b1}} face images of celebrities, where each input image is cropped to a {{formula:e93fbe87-c8bb-4fba-b17e-ddb28e7b1734}} RGB image with {{formula:27dbfa1d-f1e4-4c5e-8b8e-1cc1ac66a59f}} . The generative model {{formula:57fe14e5-5e37-428a-b016-865c8dd4167e}} is set to be a
pre-trained Deep Convolutional Generative Adversarial Networks (DCGAN) model with latent dimension {{formula:618eda10-e488-4cab-8bda-a14b9aa87e35}} . We use the DCGAN model trained by the authors of {{cite:36045ec9916210b865e73d63887a5032f1bf15fb}} directly. We select the best estimate among 2 random restarts.
The Adam optimizer with 100 steps and a learning rate of {{formula:3dedb303-e70a-412e-81c8-d3820a0243bd}}
is used for the projection operator.
| r | 028d64933cef5403c1c6850f73164dee |
We expect to detect more flaring stars as well as extragalactic transients in a future systematic search for transients in the ACT data. This will permit an assessment of the associated event rates. The large area mm-wave coverage of ACT and the upcoming Simons Observatory {{cite:dbdce8f469cfadac1eaf383b31f3c862eca36abd}} and CMB-S4 {{cite:f0840caaf303bbc235809c7ec4d5284428bfe18b}} will nicely complement the Vera Rubin Observatory's detection of transients in the optical {{cite:a400e3d4a6106e824b7b182118dfe80f91c31feb}}.
The lack of time coverage in our detections highlights the need for a regular and frequent cadence to characterize such events well.
| d | 9fd30ed971bce36964f54a8ff3b25383 |
In {{cite:8f3ed1d4632293f084c71d057bc9e40cbaca6982}}, Gu et al. described a backdoor attack which, instead of forcing the network to classify any stamped image with the target label, only alters the label if the original image has a specific ground truth label {{formula:7f2d6dcb-57a3-494e-a05a-38dd28eaa188}} (e.g., Bob with the trigger will activate the backdoor and be classified as Alice the manager). Our verification approach can be easily adapted to verify the absence of this attack by focusing on images with label {{formula:9791f5f5-46d7-4286-b707-0e6aeaa95883}} in Algorithm REF and Algorithm REF .
| d | ed0bc6008eee7df731a89a85720ce2b0 |
Although exact EF1 and PO are not compatible, we prove that for an arbitrary FISP instance, there always exists a {{formula:b26bc88a-2b93-4996-8df3-6b20bf87050e}} -approximate EF1 and PO schedule, which coincides with {{cite:e0a239385883814ed00c879c1513abf69c06bbee}}.
In the setting of {{cite:e0a239385883814ed00c879c1513abf69c06bbee}}, each job has a value and a weight but there is no release time, processing time and processing time.
Every agent has a budget and every subset of jobs that the total weight does not exceed the agent's budget can be assigned to the agent.
If all jobs have unit processing time, a {{formula:acd1a8a6-495b-4373-9649-9387e6162714}} -approximate EF1 and PO schedule exists.
To prove this result, we consider Nash social welfare – the geometric mean of all machines' utilities.
We show that a Nash social welfare maximizing schedule satisfies the desired approximation ratio.
This result is in contrast to the corresponding one in {{cite:9aa061cee6986f7e7bec0950e53de817179bd8e7}}, which shows that without any feasibility constraints, such an allocation is EF1 and PO.
We also show that both approximations are tight.
| r | 4c685914cfdf32c5d10acc48ed49f4b2 |
Lastly, it is worth noting that shock propagation within impacting drops and along impacted surfaces has been investigated theoretically for compressible drops with the liquid Mach number {{formula:30f1bec0-a3ee-461c-a2a5-a2560e1600ad}} {{cite:2fa7c21c24f817c35990c940bb40e089f53d81f1}}. Nevertheless, the shock process of incompressible drops with {{formula:81914f79-3fcc-477d-a263-22c2b10a0a92}} that are relevant to most natural and industrial processes {{cite:881154a21054b5433470f1284a7cfc5510f91f88}}, {{cite:69b60ce01a5d2956cd5f434e0ddd19c836893f2f}}, {{cite:31caf05a7fcf08fc29d38da22c99d314ef70b2fd}} has not been discussed heretofore. The application of high-speed microscopy in low-speed drop impact in our study demonstrates its great potential to measure the impact stress of liquid drops in more diverse situations such as drop impact on patterned substrates, at low ambient pressures and with non-Newtonian drops {{cite:8d4d96cff8ee2ad5ab0dc110371fd712913300de}}, {{cite:5be496420a4972e86255b976437190a8b9566457}}.
| d | 3d1a44aebc7510a569cef83bf3d9426b |
•
We propose PokeConv, a binary convolutional block that can substantially improve BNN accuracy. We replace most of the convolutions in ResNet {{cite:df79c5a121877b3d910e161f7d0b3301943567ea}} with PokeConv.
•
We introduce a novel PokeInit block to replace ResNet's initial convolutional layer that is hard to binarize. PokeInit significantly reduces the network's cost. Together with PokeConv they form the foundation of the PokeBNN family.
•
We optimize an under-explored clipping bound hyper-parameter in BNNs that controls the binarization gradient approximation. Ablation in sec:ablation shows we gain more than 3% in top-1 accuracy through this parameter.
•
We motivate and define a novel hardware and energy inspired cost metric called ACE, which is informed by inference costs on hardware yet at the same time it is agnostic to the existing hardware platforms. With ACE we are aiming to improve
alignment of the research on energy-efficient neural networks with future ML hardware.
We use ACE to quantify the inference cost of PokeBNN.
•
We empirically show that on ImageNet {{cite:1ca4e7ed1b023bb86e9d48986779cd5d6fed18bb}} PokeBNN establishes the Pareto-SOTA of top-1 together with cost metrics — CPU64, ACE, and network size. We improve over the SOTA ReActNet-Adam by 5.1% top-1 at the same ACE cost (fig:energy-pareto).
| i | 309f00afa4e783a08408a3a06b8f31cd |
Here, we investigate behavioral cascades that spread socially through animal groups. Specifically, we study escape waves in schools of juvenile fish (golden shiners, Notemigonus crysoleucas) {{cite:bce0d3f6ac49681a5af7b89a9fdd6203706f9424}}, {{cite:49e6d9890c6548e98f1f416b3e6b7bc415f7f95f}}, {{cite:983fc5f077dc8bbd6b87e39f919640ec98ecd38e}}, a system that allows us to explore core aspects of previous studies of criticality in biological systems, namely: 1) quantifying where in parameter space a particular system operates with respect to a critical transition, including identification of an aggregate variable (called an order parameter in physics) that is best suited to identify the transition in question (e.g. the average marching direction in case of the locust {{cite:584f3a900cc0aa53a7afaa7f7253d5ad5379de84}}, {{cite:ceae1426883383261abd05477299092536faf494}}); 2) identifying functional benefits of operating near criticality, e.g. in terms of collective computation and information processing {{cite:2a2afc8930a4a559f16d452984c84a5b7c1d23dc}}, {{cite:f24378fe34dc0e57daef82d345aabf3b9288ee9a}}, {{cite:c935cf8f4a9ff8181905086e6a31295ef61ab01b}}, {{cite:92d1375b7ba3a8f215cdca23e6a570ee1fd5d91e}} and 3) revealing the mechanisms that enable biological systems to control their critical behavior, and to adapt in order to function properly {{cite:3270b72902bb77f6c6d5d3014e5ff72208a86f16}}, {{cite:6f915b95878bef9d47446d59b38137c3990293be}}, {{cite:2a329580982715341c20491503b8d7e1824079c4}}, {{cite:7d0f2b0fa6c4be3e8498386077067e992e76d3a5}}.
| i | 3b28e4cb5c20c228ec62ae8c89701adb |
In this paper, we provide a bit thread interpretation for the OSED tensor network (which is also named as “surface growth scheme” intuitively) proposed in {{cite:1b9c694575cec402f0dcb09859d56c85687852ee}} and later generalized in {{cite:466075b76868349e62f2ed5879da4ecc09dc82d8}}. More specifically, based on the locking theorem of bit thread proved by the bulk-cell gluing method in {{cite:93d59ea81949626a226928082c987fafdcfc682d}}, we match a class of locking thread configurations satisfying a set of conditions with the OSED tensor network, and argue that this class of locking thread configurations describe the entanglement details of the OSED tensor network. In this way, we show the connection between bit thread and entanglement distillation and obtain the explicit bit thread interpretation of the {{formula:f01feac7-45bf-47bb-b926-911b010e70a3}} tensor in the OSED tensor network. On the one hand, the locking bit thread configuration can provide a detailed description for the discretization of spacetime. On the other hand, the OSED tensor network provides a picture of reconstructing the spacetime with the surface growth scheme, i.e., the emergence of spacetime can be regarded as the reorganization of the boundary degree of freedom through the entanglement distillation. Therefore, the bit thread perspective will provide some new insights into the bulk reconstruction scheme in the framework of the holographic principle.
| d | ebf3552bf483fc1956bd036b25bce174 |
In this section, we first revisit the definition of vanilla KD {{cite:5d83f85d846cd04ffc611c2182489e17fe911c69}}, which transfers knowledge from the high-capacity teacher network to student network with soft labels. Then we describe the proposed similarity transfer for knowledge distillation. Fig. REF compares the vanilla KD with STKD and describes the whole framework of our approach.
| m | 1eef6e434697bdae0b3c56bac3e19665 |
Recently, with the unprecedented advances in computer vision and natural language processing, we have seen a considerable effort in developing artificial intelligence (AI) agents that can jointly understand visual and language information. Visual-language tasks, such as image captioning {{cite:638fed46f2349d557160685b6dfd5c9b8a5dec64}} and visual question-answering (VQA) {{cite:36bb73a1787cfbbb89a26c0e8084537c2da9ca7d}}, have achieved inspiring progress over the past few years. However, the applications of these agents in real-life are still quite limited, since they cannot handle the situation when continuous information exchange with a human is necessary, such as in visual-language navigation {{cite:d3a5ba1b551427c23b104f9a08a0d117b4856ebe}} and visual dialog {{cite:1fb0fabb083a0b4ce154d282f6bee894128591ae}}.
{{figure:0f85d674-0f65-4b0a-a657-567392e11d21}} | i | de6f2346d9d0250d101d146a588eae7f |
Moreover, the proposed framework can be also applied to high-order Markov decision process (high-order MDP). MDP has been commonly used as a baseline in control theory and reinforcement learning {{cite:914b0465164c3c4d9c53c873a8a6d48128a92086}}, {{cite:8f9ea8142bd51fbac7b0e1fd4c0daef63285ce68}}, {{cite:015f733022d08cd76f9f4a8d72e439b237d26c6f}}, {{cite:11b9ec61bee9e85df5cd6a1ee888fcf1f1aad2f5}}. Despite the wide applications of MDPs, most of the existing work focus on the first-order Markov processes. However, the high-order effects often appear, i.e., the transition probability at the current time depends not only on current, but also the past {{formula:dbe9c7ed-3c67-4107-a542-dcf2f430cb2d}} states and actions. See Figure REF for an example. Since the number of free parameters in such MDPs can be huge, a sufficient dimension reduction for the state and action space can be a crucial first step.
Similarly to the example of high-order Markov process in Section , the TTOI can be applied to achieve better dimension reduction and state aggregation for the high-order Markov decision processes.
{{figure:5925ab3e-71c8-40c8-820a-06b6f9cd34a5}} | d | 99adbf9a031049ffdbbb2994bead5590 |
The most important consequence of magnetic fields is that they increase the phase space for neutral mesons;
at large {{formula:6e5d8f69-b88c-431b-bf82-882556cdafba}} , the phase space enhancement of a factor {{formula:fff189bc-8da6-407f-ac2d-626fd884e113}} takes place.
At low temperature where the lightest neutral mesons dominate,
the entropy density is significantly larger at finite {{formula:f3c19177-9e98-47d4-b43f-622adfbf912f}} than the {{formula:c8675a4d-b290-4233-9826-d9104fa21e83}} case.
This tendency is very different from the PDG based HRG at finite {{formula:8f8cac22-c262-492f-bd50-3c8d63883adc}} , where neutral states are treated as elementary and do not depend on {{formula:79fd176c-6478-4cea-a188-3131fc47afab}} ;
the resulting entropy density is much smaller than ours and lattice results for {{formula:f0f20f89-3b62-444e-a681-231aa1372f79}} , see Fig.10 in Ref.{{cite:c5e6ec71f3517d58eda41dee7473ee8e76b1b721}}.
| r | 1bf8e4b1bc9c70e75bbf987919ddbe72 |
In contrast, there are several GAN-inspired methods for generative SSL.
{{cite:5a97521bafe232a577c528d77a862ff5026014a8}} devised the “bi-directional GAN" method, where an encoder maps the image distribution to the GAN latent space. Further works {{cite:97b20f7eee79e3cf60f70ea5d303babe5f8b4d96}}, {{cite:87e0a4e96a481726458ca6a4a3e38163aa784b95}} use the latent space as the classification feature space. These works generate a compact representation of the data distribution of the images {{formula:b6b22fe7-613c-4989-8897-d919ef98916c}} and use it to predict the image label in a supervised manner.
Other authors employed both generative modelling and adversarial training to learn the image label. Using architectures derived from TripleGAN {{cite:3e59bd8822cdf57c5f86d1199f3935fcbcb1fa83}}, {{cite:aad8695ed7d711878a2d51906a2b5421d2d0664a}} and {{cite:c09e138cd00dd4935fdfc10815b80bbed0f454ec}} propose to learn {{formula:580cb97b-2119-4909-8686-a863e513b3f4}} through a GAN generator. Samples from the GAN are then concatenated to labels discriminatively generated by a classifier and fed to a discriminator for adversarial training. To the best of our knowledge, no GAN-based method other than {{cite:9accf92ff4115d160420707d3a59601b9dbaab6d}} learns the joint distribution {{formula:31af6c1b-d7a2-416e-acf0-d4588b85d8f8}} . For this reason, our work focuses on the latter.
| m | 1ba2f590bcdcc7adb64dc53c3fefd908 |
The method presented in this paper also holds potential for other settings where one estimates entanglement of the easily accessed probes with precision advantage and reveals quantumness of a macroscopic mediating object.
In particular, this includes an extension of Refs. {{cite:bfa533373b3cf23b79e26cc0af67e6add0c1f942}}, {{cite:e14ebdc177260899cc8b4226262eaa860a467255}}, {{cite:2cc5bda2926cf376367e4683ed44e399ae102f5e}}, {{cite:9e1b4a8640232df25f444b92aeb1b5a077a3e02b}} towards showing quantum properties of photosynthetic bacteria {{cite:b44d0e1eeb8b8b6f80090c92311eb2f2a2daba66}} or that of a macroscopic mechanical membrane in the membrane-in-the-middle optomechanics setting {{cite:97070697a8d672b096ffdb86ec9972ef88ddf5a9}}, {{cite:627ff22f88047f94456a8f6070e86b64adf1dfbc}}, {{cite:a6de92a635fb0dc40b7a090eddffdd5bb6166f06}}.
Additionally, we note that our scheme can work not only for CV or discrete systems, but also hybrid configurations such as discrete systems as input and CV systems as the QN or vice versa (see Appendix B).
| d | baab560a78ba77ad57ad9a2b82911f48 |
To compare the effectiveness of leveraging monolingual data between backtranslation {{cite:ae500603a76bc42887ebe840491e300065eb8984}}, {{cite:72f1ad11c016f0360dc65ec6813128e779cd4925}} and our model, we train the document transformer {{cite:443f4544f07ad28a776f352cd1c2e66a6eb33aa0}} using additional synthetic parallel documents generated by backtranslation {{formula:7abb8153-ec57-46d5-81bf-fbb15ff34d8b}} . For fair comparison we use the same monolingual data for both models. As shown in Table REF that while both techniques improve translation backtranslation is less effective than our model. Since we have a new model {{formula:39fe99c6-90eb-4152-8556-01b48b077716}} , we can use it as a proposal model for our doc-reranker—effectively using the monolingual data twice. We find that this improves results even further, indicating that the effect of both approaches is additive.
{{table:7c82c27c-40ea-4e2c-a39c-2da7f0c63a59}}{{table:3d135f44-e039-4617-95d4-b490b1ef81a2}}{{table:78b3c616-3d03-44b4-9c2e-1d8377725550}}{{figure:b330c9e2-c9d4-4198-b39f-3ce961c601b9}} | r | 2d6d71c7023b057d94ba91f2c41c08a9 |
In this section we briefly present the two main components of the methodology adopted to classify and explain the ISIC dataset.
Details can be found in {{cite:1512d4364ad1a6065bfb7775417b4d123ef0454e}}, {{cite:fa89c9e026e21134015e9d6217e9f4ab6e717895}}, {{cite:84b4b608b75dbbd068e8c62c1520f8bfef636dd6}}.
| m | 83d1c800ae13dd08ad2e5ad2033ba345 |
To provide a complete comparison, the learning curves of the compared state-of-the-art unsupervised embedding learning methods plus ours at different learning epochs on CIFAR-10 are plotted in Fig. REF . As we can see, for the first 50 epochs, our performance is slightly lower than ISIF {{cite:7735a587a1b9909f25506a5712052a16e60e788f}} since the uncertainty of samples in the initial distribution learning phase is pretty large which requires more learning epochs to stabilize the distribution learning. While our method outperforms all the other rivals by a large margin after 50 epochs which demonstrates the effectiveness of modeling the uncertainty of samples via distribution learning.
{{figure:da61836e-87ab-4be3-9a5e-a35809ff449c}}{{figure:c91229a4-660e-4bad-bedd-6b76ade8705a}}{{figure:7c563587-e8fa-4b46-9c9a-24e431dba8c2}}{{figure:3927d6d5-df60-4e50-9964-e39d12b0d151}} | r | b23a3117b7af68dab0147d8f49c82dad |
From the defense perspective, existing studies focus on model-specific defending strategies for whit-box attacks. Other defending strategies are also worth exploring. For example, adversarial training is a widely-used strategy to improve model robustness against adversarial attacks by incorporating adversarial samples in the training set {{cite:c9d93f79a5243564d018744487fbde428e2fcd0c}}.
However, under the settings of unsupervised or one-class learning, we do not have anomalies. In such cases, how to craft adversarial samples, especially the anomalies in the purpose of evading detection, is challenging.
It is also interesting to develop new anomaly detection approaches to detect adversarial samples that are crafted for anomaly detection models, e.g., based on the idea of using out-of-distribution detection models to detect adversarial samples from a polluted training set {{cite:f279f38e22fe90fe4395962f1e6756c02e1334f0}}.
| d | 769341d4f60d36c1b27ab514932e5a9a |
The main difference between our multi-fold masking and previous work multi-crop {{cite:8446ba5b10a85ac9fd39f11b541ab3b70bbb54f8}} is discussed as follows.
The multi-crop strategy needs to train multiple random views with different sizes without concern of complexity.
By contrast, multi-fold masking just rearranges the mask tokens and does not create new views. Multi-fold masking can also save training complexity by calculating self-attention on only a small group of tokens.
| d | a1da8f31070f93da48866a1428deb6ad |
The phenomenon of {{formula:d5d8d9d1-e455-4625-b231-f8b1cc059055}} oscillation is analogous to that of neutron-antineutron ({{formula:630c6fd0-6f8d-4fd5-8707-82e04eb99fbd}} )
oscillation {{cite:830c86697840ba0e0589039f005d3471c9f92571}}
(for a review, see {{cite:d90437a9b14c092f89c25d33074b4c9a81099748}}), and in fact both phenomena
can be related to the same new physics {{cite:9e9c9714833d556e1a692d9eac186736b559f2d2}}, {{cite:492b3688a697816f069b69d77f90c3c4c6eaac5c}}.
However, {{formula:7b511e61-9265-49cb-a968-b75837f01f4a}} oscillation is strongly restricted by experiment.
Namely, the direct experimental limit on {{formula:b8323745-7e99-4de4-b8d2-df335e93142e}} oscillation is
{{formula:31775615-ae7c-4ad6-901a-5e0629df2e3d}} eV while the nuclear stability bounds
are yet stronger yielding {{formula:49776de7-24a0-4185-9c58-2148bd2fdcd3}} eV {{cite:d90437a9b14c092f89c25d33074b4c9a81099748}}.
As for {{formula:e456c71a-c0cb-4dab-8a59-843d32216a1f}} oscillation, it is kinematically forbidden for neutrons bound in nuclei,
simply by energy conservation {{cite:44c1766afe7afd7198ebe1abc1589c1176fc109f}}, and so nuclear stability gives no
limit on {{formula:c5c5bc0b-256a-45bb-b9c2-0fd33de72c41}} mixing.
| i | 0958e334c917a370d5e2376de42ef79b |
Quantum computing provides a fundamentally new approach to computation by exploiting the laws of quantum mechanics that govern our universe.
In this computational model, a quantum circuit consists of a sequence of operations each of which is either a quantum gate, characterized by a unitary matrix, or a quantum measurement, characterized by a Hermitian matrix (i.e., an observable) {{cite:732263f8fff94896b081420bfa58013486c59c6c}}.
So, a universal quantum computer must be capable of implementing arbitrary unitary operations and measuring any Hermitian operator on a given set of {{formula:f50fa473-6aea-4747-acf7-06bf7b94efdf}} qubits.
In 1999, Gottesman and Chuang demonstrated that such universal quantum computing can be performed just by using the quantum teleportation protocol if one has access to certain standard resources — Bell-state preparation, Bell-basis measurements, and arbitrary single-qubit rotations {{cite:2cbf2ff62a0d6f3de663a2e9ad1c56c77cdca7c0}}.
They defined the Clifford hierarchy as part of their proof, and this has proven to be a useful characterization of a large set of unitary operations, both in theory and practice.
In fact, in their teleportation model of computation, the level of a unitary in the hierarchy can be interpreted as a measure of complexity of implementing it.
Furthermore, this model is closely related to the currently widespread scheme of distilling “magic” states and injecting them via teleportation-like methods in order to fault-tolerantly execute unitary operations on qubits encoded in a quantum error-correcting code {{cite:c40fc68d66094fb1b64c219da976a20dcef27b08}}, {{cite:d65fb723e0f7f6756c629a4a83dbd9c87d125fdf}}.
Hence, it is very important to understand the structure of this hierarchy since it has important implications for fault-tolerant quantum computing.
| i | 808e16ba2f029c32a299814a60afd789 |
With respect to {{formula:cd594586-8178-46d8-8c95-ca5be5c7182b}} channel, the amplitude in Eq. (REF ) is calculated using the effective Lagrangian as Eq. (6) in Ref. {{cite:495482c2e4aada1e0d1c5a9051b9a0918becb9ff}} to describe the {{formula:f58e1b3f-5222-4080-a778-920bb1c886ad}} process and the interaction coefficients {{formula:d67fd50c-e5d8-4da6-88f0-6653d37383dc}} are listed in Tables REF and REF . Since the production ratio {{formula:724f8e28-0393-4f2c-91b2-9d4e4e47340e}} {{cite:f914a2f7033f6de30ac7ded88b48be9ae1a757d3}}, {{formula:5284af12-e941-4a16-84a3-b3d39b7433b6}} can be used to describe the invariant amplitude in Eq. (REF ) when ignoring the resonance structure in the {{formula:61d1bb88-a7f4-46a5-a6d6-dc26adc1c80d}} decay. Analogously, {{formula:5a601158-6ff3-4e9e-822e-49d59d177a26}} , {{formula:3f07ae29-3512-432b-a5f1-620f2bcfd79d}} can also be introduced as the invariant amplitude in Eq. (REF ) for {{formula:b3ab5870-0e57-4f73-9316-d3738d7cc25d}} and {{formula:22d08e9a-ff9a-4d9a-bd7f-3abb720d3428}} channels for simplicity. Besides, in the analyses below, it is noticed that the decay ratios of these channels account for only a tiny share. Hence, it is feasible to employ {{formula:ef8be623-a9f4-42fe-9403-197f614ae857}} , {{formula:622303e7-6947-45be-9f8b-66ea86d93e33}} and {{formula:818d662e-df3a-4092-9205-2863a7ec7c08}} for simplicity.
| d | 929cf7bfb8673874b660740bfb495c98 |
To achieve better performance, some recent methods exploit unlabeled or synthetic data for training. Based on the generated synthetic data {{cite:414df9faac8996c676dd60b069d96ab9925ae075}}, the supervised learning and domain adaptation strategies are proposed to improve the counting accuracy significantly. In {{cite:ee62a531b111139079d350a1fa089651da5b71cc}}, the ranked image sets are generated from unlabeled data for counting applications suffering from a shortage of labeled data. Sam et al. {{cite:d8e28ad13f0b123e005ca7b7a53d2dfdea371597}} propose the Grid Winner-Take-All autoencoder to learn features from unlabeled images such that weight update of neurons in convolutional output maps is restricted to the maximally activated neuron in a fixed spatial cell.
| m | 2c2ffba0658a0090165286580e24a74f |
Though we discussed the potential model extension with rolling horizon and incident duration uncertainty, we did not implement these extensions in the case study due to the complexity in updating inputs at the simulation environment. Incorporating real-time information as an adaptive RO would generally increase the model's performance {{cite:e0f9d974d218dcd92343aee77f59a615b87f8995}}. This can be done for future related models with easily adjustable inputs.
| d | 576cec494131dff9474f58b208c0a90a |
Such restrictions define certain sets of continuants.
The problem arises is to find the maximal and the minimal value of continuant
over these sets.
Some results related to this problem were obtained in
{{cite:2eef041777418bae296d88784c15a5331d5e93c7}}, {{cite:09eb837711db331c54156f9ee983886192951fe1}}, {{cite:6b59ab8c71fd8fa9eb315b61bd3cc60527b8ebd4}}.
In the present paper we obtain further reuslts.
| i | 879d279c7a129719ff0c1ed23953c494 |
The existing methods for self-supervised monocular depth estimation could be generally categorized into two groups according to the types of training data: the methods which are trained with monocular video sequences {{cite:1531303826149e21787df8c73b0d2f68643fc309}}, {{cite:7e6fc30016c3a6ef62e70c0e7966147510828858}}, {{cite:0df900955a9d8c5ce638256987182d95f1d0c4a5}} and the methods which are trained with stereo pairs {{cite:afa8c721333b0ed726224db8bacfd327d56649ed}}, {{cite:f0dde0695286e85f6e7eea83d9e00f0969f6a0e8}}, {{cite:0f789b686de352d519fba20aca5d656da129454f}}.
Regardless of the types of training data, many existing methods {{cite:bce8c2f82741c09677e23c4ab9acb1f766290f79}}, {{cite:0f789b686de352d519fba20aca5d656da129454f}}, {{cite:b986d1f1f3b3edd30eee1c4b8d68843e7a3048fc}}, {{cite:14fa65f800a83ad526197c1e0fc838d8ac690102}}, {{cite:337251548fa73e566fa544721d996bfd0197d7ee}}, {{cite:b4e2d2e81a9b86d87737757b24345ec2028625dc}}, {{cite:308c81a92a031b0090c341b0c68fd82df92a0a08}} employ various encoder-decoder architectures for depth prediction, and the estimation processes could be considered as a general process that sequentially learns multi-scale features and predicts scene depths.
In most of these works, their encoders extract multi-scale features from input images, and their decoders gradually aggregate the extracted multi-scale features via either straightforward concatenation or element-wise addition, however, although such feature aggregation operations have demonstrated their effectiveness to some extent in these existing works, they generally neglect the contextual consistency between the multi-scale features, i.e., the corresponding regions of the features from different scales should contain the contextual information for similar scenes.
This problem might harm a further performance improvement of these works.
| i | 3ec70985c196dcde05c8a27f43e2e9eb |
The Burrows-Wheeler Transform (BWT) {{cite:eae4432d3b51c6f4b52bb393030a1d806675e951}} is the basis of the popular compression method bzip2, yielding, on many types of possible input files, better compression than gzip and other competitors. As a matter of fact, BWT itself is not a compression method: its output is a permutation of its input, which has obviously the same size. The usefulness of the transformation is that it has a tendency to reorganize the data into what seems to be a more coherent form, grouping many, though not all, identical characters together. The output is therefore usually more compressible, by applying as simple methods as run-length coding and move-to-front.
| i | 43c6c85a7b53d45839ef07fb468078a3 |
We observe that KGIRNet outperforms other approaches for both goal (in-car) and non-goal oriented (soccer) dialogues except for BLEU and METEOR scores on the soccer dataset. The effect of knowledge groundedness is particularly visible in the case of soccer dialogues, where most models (except GLMP) produces feeble knowledge grounded responses. The Entity F1 score used here is adapted from {{cite:d61bc41b3dfd1a49b8b98da86e81b7669ee46d36}} and is defined as the average of F1-scores of the set of predicted objects, for all the questions in the test set.
| r | 54caf636a8da23bc7e9c2dfd0db0303a |
We see further applications needing more investigation, such as exploring the potential in consistency regularization and multi-modal datasets. For example, finding directions corresponding to adding or removing contrast in scans. Further, the approach we use has been shown to be effective in unsupervised saliency detection and segmentation on natural images {{cite:6ba8d1d7480b61e9dd7c3a2d6fadfa0676d18dbf}}, {{cite:878b8591435dd7462468a0f9afb5150b74c27ced}}, {{cite:ad2858310f85b85812d425626d50f70f2c2739b5}}.
| d | 07e5107f9ea0e9ff0feb75adc8884122 |
The LEFT (low-energy effective field theory) describes the physics
below the {{formula:fb446635-5019-4e13-9441-571c5e35b30c}} mass, and is produced when the heavy SM particles ({{formula:71663135-11d8-456a-9f55-d71b18fb7937}} ,
{{formula:29e205e1-af70-4d6b-b7bd-799eff176a1b}} , {{formula:4b29c45b-35f6-4d41-8e60-0aae722b3180}} , {{formula:84141718-9941-4c81-8f9a-a30b1d5e036f}} ) are also integrated out. (This is also called the WET
(weak effective field theory).) In Ref. {{cite:6bd3c6f81c5613226e62cba1f073479d58d2042c}},
Jenkins, Manohar and Stoffer (JMS) present a complete and
non-redundant basis of LEFT operators up to dimension 6, including
those that violate {{formula:557f9149-e431-4c62-830d-db08d2d6a816}} and {{formula:97edeea0-eb15-467f-a51d-d5c9285a957a}} . The matching to dimension-6 SMEFT
operators at tree level is also given. The one-loop contributions of
dimension-6 SMEFT operators can be taken into account through the
renormalization-group running of the coefficients of the LEFT. This is
computed in Refs. {{cite:0f284eb69fbd1fb4bd3daa4ccb4f483cdecaa637}}, {{cite:68eab83b6582f35df0ef1d5f0a6eb3bf1074b7d1}}, {{cite:19d3a524a582f5e0bec6a62b3fb8804401914251}}, {{cite:7353e5a379775bdf13cde455adeba858000184d1}}. With this information, if a discrepancy with the SM
is observed in a process that uses a particular LEFT operator, we will
know which dimension-6 SMEFT operators are involved.
| i | 4b7bf9a35a72fffe7523c7d7d43dc3a3 |
To further justify the effectiveness of the model, we also evaluate our model on the recently released DexYCB dataset and compare the results with {{cite:785f8483af8db7fe438c07feb276f991d089a4ab}}, {{cite:8ec68f50b9c40ebcccca7e7a30238990b91d25b5}}, {{cite:f94d4a5c6ce309960282679ecc046df10a48fff8}} in Table REF . Note that while {{cite:8ec68f50b9c40ebcccca7e7a30238990b91d25b5}} has the same setting with us, {{cite:785f8483af8db7fe438c07feb276f991d089a4ab}}, {{cite:f94d4a5c6ce309960282679ecc046df10a48fff8}} do not assume known object CAD models, therefore tackle a more challenging task and can perform worse in estimating accurate object meshes. Hence we only compare with them in the hand metric. The results show that our method consistently outperforms baseline methods in all comparable metrics.
{{table:fc07c15a-487e-4251-af36-1d44be8c5e1f}} | r | 700faeab75b53a8a420ec68bd427b20e |
Our understanding of the universe is intimately related to our description of many-body systems. The knowledge of the fundamental particles and forces of nature is as important as our ability to understand how these building blocks are organized to form complex systems. Remarkably, the emergence of simple and regular patterns are common features observed in strongly correlated many-body systems {{cite:1c334a29dd61f6ce5dbe6196dd4d7ff2be2c3d56}}, {{cite:5d58676e2f77824aeda82aade82da993f39659cc}}, {{cite:e54ec6048ed102186f0c3f930eff705fcb910055}}, {{cite:cb52bc79be9b090dafe84ae2a1a760d11d1a9725}}, {{cite:3d0fa9e88800dac77aed7ca82c13e0cde49a947e}}, {{cite:72559ba9e27135f211001f3f871bb3f67e372bda}}, {{cite:3dacd647689c6408c6b313b66b728d7759509346}}, {{cite:9ae83396a6be3af78397e21509f11e8c9d40c3fe}}, {{cite:a4c2bb2d829aec8168f8d21e78b0e866165a54c2}}. At the microscopic level, the individual parts of different physical systems can be described by fundamentally different interactions, however, their collective behaviour can exhibit similar patterns. These seemingly simple regularities of certain properties of a physical system tends to suggest the existence of underlying symmetries and allows simple models to provide a good description of the observed data {{cite:67fcfeba23beadf0943fe3c2c4ac09cdd5c33f68}}, {{cite:6200e78e75b5f2ab2dc31472d6541b5331c42d67}}, {{cite:bce913e3211bc416f8c0cb81fade88444dbb013e}}, {{cite:a4c2bb2d829aec8168f8d21e78b0e866165a54c2}}, {{cite:288fec49a72c46315efc3d7b78ad903599a6e670}}. However, the link between these models and their microscopic interactions is an open question in many fields of physics.
| i | 9a2aad1ce9975825756b86bdc5272a8c |
[leftmargin=*]
RBR:
A rule-based model from the Cubist R package {{cite:6eeb7ca82cffeaf9f553881d3697b858d86b82a7}}, which is a variant of the Model Tree {{cite:28b131dd4d561a663b7b9374247bfa99361d79dd}};
RF
A Random Forest method, which is an ensemble of decision trees {{cite:0a9ea0e2b9119b2dd5f628c70ef4acc18b6812ff}}. We use the implementation from the ranger R package {{cite:6817660679d7d0224b3358f25709514a7cb90b92}};
GP:
Gaussian Process regression. We use the implementation available in the kernlab R package {{cite:8a7c6bfededf2429a36c955ec48c91a960109a6e}};
MARS:
The multivariate adaptive regression splines {{cite:085d17110300a33963e03e7eb30ecc547b42c025}} method, using the earth R package implementation {{cite:c80d066eff34c3a95fa695004402097d215da563}};
GLM:
Generalized linear model {{cite:3104ac27a255d7693b218390c24e715c16cd06e1}} regression with a Gaussian distribution and a different penalty mixing. This model is implemented using the glmnet R package {{cite:0f7d72b1bbd109206001abd35908a8165778fefc}}.
| m | e2250116be6b490d9074112f9108429d |
We close our result discussion with a note of caution. While the density of active matter varies, the overall fluid is incompressible: in other words, the total density of active matter and underlying solvent is constant. Incompressibility is important especially for high activity and intermediate {{formula:20e7b69e-6608-4a05-9f0f-541b6a8fc576}} — i.e., in the bottom right region in the phase diagram in Fig. REF , which is challenging to characterise accurately. There we typically find active turbulent patterns with vortex correlation length similar to the system size. We found that these are replaced by an apparent phase separation into high and low-density patches if the Navier–Stokes solver allows for fluid compressibility and the Mach number is not sufficiently small, which occurs when using, for instance, hybrid Lattice Boltzmann simulations at {{formula:93c6c6cb-12a0-417b-89c4-2649c6a33685}} . These states are artifacts in the current model, and using a purely incompressible and 2D fluid they are not found, in line with simulations with tracers in turbulent flows which only show aggregation in compressible fluids, or due to inertial effects {{cite:9907aaa946e10125f8bfb53d44830b0170629c7b}}, {{cite:c3e26571508d7fa35261fbfe1aad338dd8a14235}}. It would however be of interest to see whether these patters may be recovered in thin active matter films, which can behave effectively as compressible fluids {{cite:5f15dc433520649b7889c4379bbd2eccf52626cb}}.
| d | 2b13f4ced19a813d0fbe094aecec7d0e |
The expression of the numerical method is given again by (REF )-(REF ) with the difference that the second-order reconstructions are used now to compute
the numerical fluxes. The TVDRK2 method {{cite:a52a7efc2a3f0668cec8217f689ded3bc75b76de}} is used in order to discretize the equations in time.
| m | 923a1debcb8fdda542415a34bdcdbe52 |
Notice that the above theorem shows that an {{formula:116a116b-13e6-4397-b659-b78485841446}} -approximation algorithm for MCNwRS immediately gives an {{formula:73ecdaf1-c5c1-4af9-82cb-508ac120246b}} -approximation algorithm for the Minimum Crossing Number problem. In fact, even much weaker guarantees for MCNwRS suffice: if there is an algorithm that, given an instance {{formula:47d3f983-a10e-42a1-b1cb-8ce40c749847}} of MCNwRS, computes a solution of value at most {{formula:c47b23ac-9c05-46de-aad2-471b2c1dd210}} , then there is an {{formula:31e5c14f-8b47-4a75-946c-e8889b1ae4ad}} -approximation algorithm for Minimum Crossing Number. Recall that {{cite:5764504d3a80bf4e593baff8e8b2600df17f1c0a}}, {{cite:ee28b463c7b7f1ce375db90a5724d5178fb9762e}} provide an algorithm for the Minimum Crossing Number problem that draws an input graph {{formula:3f2cc9e2-9378-4294-bb97-22b3248ad75a}} with {{formula:4b8066db-fa04-4a3a-99a4-4ad084676be8}} crossings. While it is conceivable that this algorithm can be adapted to the MCNwRS problem, it only gives meaningful guarantees when the maximum vertex degree in {{formula:62cf2019-3836-4815-833e-c387f95eeb62}} is low, while in the instances of MCNwRS produced by our reduction this may not be the case, even if the initial instance of Minimum Crossing Number had bounded vertex degrees. Our next result provides an algorithm for the MCNwRS problem.
| r | c09e681ca3c060d1c9114ca126ce8935 |
The origin and repercussions of the observed anomaly deserve careful consideration. Concerning the origin of the anomaly, the least interesting, and a highly unlikely possibility is instrumental systematics as source. This is considering the twin facts that the results are consistent across data releases and different processing pipelines, and that the Euler characteristic computed from the CMB maps obtained by Planck's predecessor Wilkinson Microwave Anisotropy Probe (WMAP) satellite also exhibits mildly significant deviations between observations and simulations {{cite:5e19a1730cb310b1ecf7b610a763812eb43f0577}}. Disregarding systematics, a more interesting possibility is the source of the anomaly being a genuine astrophysical signal, perhaps truly primordial in nature, or due to a yet unresolved foreground effect. If the signals are primordial, it opens up the possibility of the Universe admitting a non-trivial global topology {{cite:66ed9c4b7e24a2f47b30a31e633c557bbde7747e}}, {{cite:cc3d063040d6e79925079eebaaede5bae20f23d9}}, {{cite:e3457760d8b871a06f4147d114d1bd27b6564c12}}, possibly induced by large-scale topological defects {{cite:b932ab7b0647e9afe504c85eef54cd946461520d}}, {{cite:e50e8fc73e435481d476995a739a9831c7009be2}}, {{cite:5c7df35693d749fd5d7a2fb97ed669cca9b2b560}}, {{cite:1ec1b86e07c78275511e9e69f871edceeef9d9de}}, as well as that of primordial non-Gaussianity {{cite:f7f76186bab0632aeac42cb093df8c1a796a2ae2}}, {{cite:554e23be95992f14876041fe0cc43496ceca83fe}}. The latter scenario has justification based on the nature of the method used to detect anomalies. Methods based on comparing the power spectrum of different patches of the sky, as example hemispheres, inform about homogeneity properties {{cite:f647ffc202ad2edf1ac39eb022f695c1b8f4cec9}}, {{cite:da34ec83929736403a4f74dae59c8a33bd060c40}}, where as investigating the alignment of multipoles informs about isotropy properties {{cite:3d211a7b4313649b21da34dd2864a429c42e8d46}}. Results based on these methods do not encode information about higher orders, and hence cannot shed light on non-Gaussianities. Computing higher order correlations is expensive, however 3
-point correlation functions show the observations and simulations to be consistent (c.f. {{cite:4e39347338cd21df38d7c4beb559ee95b4dd6a17}}, {{cite:cb1d7622854893158bf594931b68dc88bb2d49be}}. Alternative methods, potentially encoding information of all orders emerge from geometry and topology. Principal such tools from integral geometry are the Minkowski functionals {{cite:4035d49b0ea467a83d1d871331c5b58335d69202}}, {{cite:13af7bcf158f85986e19a7e2c1d0de11db3afe97}}, {{cite:9f2da6b9c2cc6f00ac10c3b57f724971f35b84b1}}, {{cite:798fbe3ca368ea81797415a1c2732362cf025af5}}, that also show consistent behavior between simulations and observations of CMB {{cite:4e39347338cd21df38d7c4beb559ee95b4dd6a17}}, {{cite:cb1d7622854893158bf594931b68dc88bb2d49be}}. In this scenario, the purely topological tools employed in this paper present, for the first time, an anomaly that may have contributing influence from higher orders of correlation, thereby potentially pointing to a non-Gaussian signal. Future research will involve a coordinated effort in all the aforementioned directions to establish a deeper understanding of the origin and nature of the anomaly.
| d | 2db1ae835d8b5b0730c979276cc9a53f |
It is especially intriguing that the model of the QCD axion we discuss is consistent with the astrophysical hints suggested both by anomalous TeV-transparency of the Universe {{cite:2da4332d86ccb94552f3a1e9ab1b91fc6e3118f3}}, {{cite:5402c1e0787212de94b1fbdc32000f8556be167a}} and by excessive cooling of horizontal branch stars in globular clusters {{cite:3e183c4c15be037cf2d14141481f19541df8e5bd}}, see Fig. REF . Moreover, Fig. REF shows that the parameter space of our model is to be probed in the future by many experiments and astrophysical observatories, namely ALPS II {{cite:3e649835edee5c44ffbfb7646737ce80a3054268}}, BabyIAXO {{cite:252bdeca05e09386eedd3ae7721f08d351f72aed}}, IAXO {{cite:8511567107881bc730d755d0650e2ad2c600f63d}} and Fermi-LAT {{cite:4e64f85a4a820963848a4a84fd6768e4c7745ad9}}. Meanwhile, advanced LIGO {{cite:c48b4f06a23f4f33a0501668cad7155648d04bb7}}, KLASH {{cite:af25d44c5cbad82864c2fa05603f8793d5288274}} and ABRACADABRA {{cite:f47211ee6a90b31c24e4b6ca3066953f786b2072}} experiments have all the chances to discover the cosmic abundance of such axions. As to the experiments which probe the interactions of axion with neutrons, one can see in Fig. REF that although the projected reach of the CASPEr Wind experiment is not enough to probe the QCD axion model we propose, the gap between theory and experiment is way smaller than in the case of DFSZ axions.
| d | ed1c8f7900ba137e85d4faba035a2c58 |
With the small stellar radius, Proxima d has a transit probability just
above 2%. Its equilibrium temperature may reach 360K, assuming a Bond albedo
of 0.3 {{cite:df3b13337c37606910331ac39c5318a46d26579e}}. From the planetary properties and stellar
parameters, we can estimate a planetary radius of {{formula:0a608611-feb5-44cc-93ed-f6295e83b7e6}} {{formula:d0c08dfc-9dd9-4722-91fa-8583c73a720d}} using
the random forest model from {{cite:579a812ce9ff818283e18054e3befda2949182cf}}, leading to a transit
depth of about 0.3% (approximately half that of Proxima b;
{{cite:df14ce541225be62a0557f0536c4d5b6a8ecbe2c}}). A transit detection would allow a precise
measurement of the planetary radius and could place constraints on the planet
bulk density and possible atmosphere. However, a transit is unlikely given
that Proxima b has not been found to transit {{cite:105324eff9967694a50770202f52d0a52757f235}} and that transit events at periods below 5 days and
depths above 3 mmag have been ruled out {{cite:023d20baa86a6762ae0d6ece16fb91d7ad18118c}}, {{cite:228f76fc05fe88108adef56c4501f41e5189ee33}}.
| d | c6844a6478456242ffd0db4c0bde5d4f |
The HF calculations for nuclei starting from realistic nuclear forces have been successful {{cite:2444588a35be92bf57780d4a8f0a9a975056aa78}}. The {{formula:24204159-20b3-4afa-b4d8-613dbd9d4be9}} -body intrinsic nuclear Hamiltonian {{formula:b2f0748f-fb2c-457a-9f80-7582cb03b457}} is
{{formula:038134ea-4458-4907-aa2e-7537922c1a2f}}
| m | 75aeb00ab7b6ee8a57c2e53ef921159a |
Due to the fact that {{formula:d08ac7b8-714e-4a7b-a00a-e3fec828ce96}} is infinite dimensional, {{formula:21f00eb2-8cbf-4396-bbd8-f5728a75e63f}} cannot be obtained directly by solving (REF ). However, based on the representer theorem {{cite:9f51bcad955ec5ab2c62c4e3ade90b06af8edc57}}, (REF ) admits a solution in form of
{{formula:5edb2d2d-dcd3-4a0c-bd37-201b0c2d9f82}}
| m | 484c24ab8d8ed98c3225a4ae0f0ce7f1 |
We proceed to analyse the trajectory estimation performance of different implementations. The LP trajectory metric error versus time for all the implementations that estimate trajectories is shown in Fig. REF , and the numerical values of the average LP trajectory metric error are presented in Table REF . On the whole, BS-V-PMB has the best estimation performance averaged over different time steps, followed by T-PMBM. In particular, BS-V-PMB has the best performance when objects are in close proximity, whereas T-PMBM has the best performance on initiating and terminating trajectories. In principle, T-PMBM will produce optimal trajectory estimates if it is implemented without approximation, and we apply an optimal estimator (e.g. in the sense of minimising the mean trajectory metric error). However, in practice, we use pruning and a suboptimal estimator. When objects are well-spaced, only hypotheses with negligible weight are pruned, so the performance of T-PMBM is effectively optimal. With closely-spaced objects, there are many feasible hypotheses which cannot be effectively enumerated, and the additional ability of BS-V-PMB to reason over the entire sequence is clearly evident. BS-TO-PMB outperforms T-PMB, with the latter being an efficient approximation of T-PMBM. GLMB is less accurate than the other implementations using Poisson birth model due to their less efficient representation of the multi-object posterior {{cite:78872444e83564903820f5b798a8b5675667e10f}}, even though for the considered scenario where at most one object is born at a time it is more beneficial to use a Bernoulli birth model. M-GLMB has better performance than GLMB by improving the GLMB estimates using a multi-scan Gibbs sampler with batch smoothing.
| r | ca2316bfb8daa412e39bb6d9ade75ffb |
The mapping from the general non-convex optimization landscape of deep neural network training to a convex layer-wise objective has its parallels in continuous optimization strategies {{cite:ef4023a9f0e6c18c0dde29fca11183d1dd67cda1}}, {{cite:bcbfe40ebb2522c708ecb4a1a8d59592f4867b86}}. Convexification and layerwise optimization are themselves highly interesting and relevant areas of research that benefit both discrete and continuous optimization strategies.
| d | 0936b360a75e82e571bbd5eaa4fce855 |
Another challenge in style transfer is when the style and the content images have different types or different numbers of objects (Fig. REF ). One strategy would be to merge objects with similar classes (e.g., grass and tree) to minimize the content mismatch {{cite:21702dd039853dc943ae2c5a06892935b4ec8ec4}}. However, the strategy above would not be useful when the semantic objects are of different classes (Sec. ).
| i | 72db04c1aec18443143989d3e8942cd6 |
For {{formula:abffddf9-d3e9-4bdd-af4c-81fe0aa3308d}} points of the same {{formula:4b4c1dd8-cae6-4bf5-963f-2c4254219e98}} but
different {{formula:95ecf68b-16fb-4142-9e3c-df1ddbdee0e6}} 's correspond in general to different {{formula:882d2781-73b7-4af0-8c09-21f383c854de}} 's. Therefore one has
to interpolate {{formula:6690c6d8-1362-4984-8226-9bb56c150c85}} to {{formula:cce9f871-1d75-412a-aeca-36a1c418a4b4}} such that {{formula:23a84aca-67bd-424e-973f-101ebd1481a3}} .
This procedure is described in detail in
Refs. {{cite:5041a6b1dda37b5b662a9099a964f662597a644b}}.
| m | fb66a50ef7c1a936963ea49eb7ee3575 |
Recent methods that rely on deep Convolutional Neural Networks (CNNs) have shown a higher level of robustness against recompression {{cite:fb08aaac4204a6d03b425defa7c8aa893c66cc51}}, {{cite:08e2a61009ef8c4ed7626e74f9b1791800a19878}}.
However, most of them remain intrinsically weak because they can only detect forgeries that were seen during training. Therefore, a promising strategy for forgery detection is using one-class deep-learning methods that are only trained using pristine data. As such, they are not limited to specific types of forgery. For example, the Noiseprint algorithm learns to extract a fingerprint from the camera model using deep learning {{cite:516d15db7ff99454b7679f1485f8b6f2965ac5f2}}. Then, inconsistencies in this fingerprint reveal tampering. This is similar to traditional methods using PRNU fingerprints, but does not require images from the camera of the image under investigation. It is trained on images coming from many different cameras, hence being able to detect different sources, but no specific design was made during training to take into account the different JPEG history of such patches.
| i | e40869dc32df3441d6d61eece8752542 |
Finally, in Fig. REF , we compare the calculated
isothermal elastic constants as a function of temperature, with experimental
results. In Ref. {{cite:f49a79f48271afdc322b35d49c0fb3feb6f2ee35}}, experimental results
for isothermal elastic constants {{formula:5e38cd18-c447-4589-98b8-242492cb308e}} are given
in the temperature range 0-300K. In Ref. {{cite:e4215ff8b32b13d04123b95a4fa93d2207f9b563}},
experimental results for adiabatic elastic constants are given in
the range 300-800K. Since the isothermal and adiabatic elastic constants
are identical for {{formula:f2a9a9f9-2481-47d4-9d25-f7a2213adaee}} , we can use these experimental results
directly. On the other hand, the isothermal and adiabatic elastic
constants {{formula:7a3e5f2a-73b0-4860-8fa0-e8dc6f360479}} are not identical {{cite:652cf686cce83eb0417d30ed4a8ba6ffc8271fe2}}, {{cite:43d8b54731ef4bfc31e48b06f6863d0608e009cb}}, {{cite:e52278a329d8a736d926d4a4a37813e28a2b83e3}}{{cite:652cf686cce83eb0417d30ed4a8ba6ffc8271fe2}}, {{cite:43d8b54731ef4bfc31e48b06f6863d0608e009cb}}, {{cite:e52278a329d8a736d926d4a4a37813e28a2b83e3}},
and were not compared here. It is evident that the agreement between
the calculated and experimental values is relatively good, especially
given the fact that the Copper potential we used {{cite:57093ab19a5138b8a8465bfa07bcada04893c2da}}
was calibrated to slightly different values at {{formula:6ad1a08c-4ba1-4805-9169-9ec98a3895c5}} K (the experimental
values of Ref. {{cite:f49a79f48271afdc322b35d49c0fb3feb6f2ee35}} are {{formula:04d5e37c-2b1b-45d6-90a7-0e630172a384}} ,
{{formula:57cee097-1b95-4498-8e7e-48662f85e3fd}} and {{formula:f67ac67d-754f-4cd0-8fa9-699d5a988126}} ). The values
that were used to calibrate the tabulated potential (at {{formula:2d38ffab-2393-4971-be7a-550e035073ac}} , are
reproduced by our calculations to 5 significant digits. In addition, in Ref. {{cite:6c50260c1e11e2d31d75b20e02e847469380c67b}} adiabatic elastic constants are calculated by molecular dynamics simulations (using LAMMPS MD code {{cite:5fe5223af3fa78d3c30c7a2e810bb65eeaab4d6b}}) at {{formula:31c21962-fec5-40ba-af2b-ccc67498b6a2}} K, using the same EAM potential and employing the explicit deformation method. This result for {{formula:5b4f2567-f2f2-4ef5-9567-8dd7a047c82e}} is also shown in Fig. REF , showing a very good agreement with our calculation (less than 0.5%).
| r | ac5c0fb78dd6ece87475426acff05732 |
We show the formulation for our proposed method in Equation REF , where {{formula:93aa23e9-051c-4dd9-924a-5ddd59c81482}} is the training iteration and {{formula:061839c9-4910-41b7-a725-4d5f2b81bde0}} is the weight term in calculating the task loss EMA {{formula:a7e5ddeb-b065-41fa-ae8c-a37e42fd3cd2}} – the reciprocal of which is the loss coefficient {{formula:a79c5646-40a4-4c54-9058-7415bc60b285}} for task {{formula:77a64427-7f3f-4b56-a870-ee5957f088a2}} . {{cite:119415d7bd261e45d7094461e5181e5dd6d295be}} notes that the Uncertainty Weighting technique {{cite:ac49e2b13c0d24df2e78b66aaae88a4a3acee814}}, a successful loss balancing technique which learns {{formula:d59b7d1f-fc69-44fd-90bf-b540979551b6}} to optimize the overall loss through gradient descent, often learns {{formula:af30248c-812d-4cd4-80f9-633966e8bd55}} {{formula:f65749f3-c9a9-4eaa-8a28-5a4a97217fc1}} {{formula:eeba1603-a92c-4fad-8948-9a1a77f5f439}} . This validates our idea to directly scale task losses by their moving average. However, they also note Uncertainty Weighting (UW) can lead to volatile spikes in the {{formula:8b6db78c-c7d8-4be8-b3f4-2413e8b55435}} , which leads to performance deterioration – in our case, {{formula:c9621112-0557-493c-8469-20518b64eedf}} is a hyperparameter to help prevent such issues by tuning how fast to adapt the task loss EMA and thus the loss coefficients themselves.
| m | 53090fbd960422cc47646cd9e4830376 |
However, the most energetic phenomena in the jet, leading to the production of {{formula:e23b0f76-64ee-4edb-84dc-286cec68effb}} -rays up to the TeV regime, are known to occur at smaller distances from the BH, on sub-parsec and parsec scales {{cite:d6127237620e0331e065b845e19a96644533f743}}. Therefore, it is the jet composition closer to the central engine that may be most relevant in relation to neutrino production. Contrary to the scenario described above for the larger scales, modeling of the jet broadband emission {{cite:ab9c7d5eb0558ec6fd0aad4de68fdfa412f3c120}}, {{cite:2142bfa6aabd66e0b26b43137ea22039a9a3f066}} and observational studies of the jet circular polarization {{cite:0ba43391e488dc6de6936ea8f7fb3d6536f6000a}} suggest that the dynamics of powerful jets is dominated by protons on VLBI scales. The parsec scale properties of the FRI jet in the LEG M 87, on the other hand, are consistent with the dominance of a pair plasma {{cite:7df6bf3474cb08bca73e0a183208e0415168dd4f}}, {{cite:de8df493a280d97f38954e80e4ad16f00ca638ee}} and, in general, the assumption of light jets appears adequate for the modeling of jet deceleration in FRIs {{cite:1e9da1dc3c30c81cb3ceda39f1f7bfc4a2283311}}, {{cite:966cc4c7c7af1333fa8c50a5a29e9df9d598d7be}}. This possible difference in the jet composition may reflect a difference in the jet formation mechanism in LEGs (FRIs and FRIIs) and HEGs (mostly FRIIs and a few FRIs).
Based on the analysis of the jet collimation profiles, {{cite:fa827e377889b4242d5e3b072908e260dee0615d}} have suggested that a more extended and prominent disk wind surrounds the relativistic spine in HEGs, while the jet expansion profiles in most LEGs are consistent with a jet origin in the vicinity of the ergosphere, as directly observed in M 87. These results support the idea that jets in HEGs are more heavily loaded with protons on sub-parsec and parsec scales.
| d | d0968f1d1b725e6d15a6d1a9ce0fdee5 |
where (REF ) follows from {{cite:64e516502db0ad0524c8952cf1cc0eefb3c571f9}}, and () is {{cite:64e516502db0ad0524c8952cf1cc0eefb3c571f9}}.
| r | 1e26f0a94249e7c37d71339cac9bbae6 |
This effect may be related to both over-squashing {{cite:1d9c473cefa0863924212d6f309e32090bdad099}}, {{cite:a40f131ec44e97ff29caa5131e8eca484f09eb41}} of node embeddings and to the fact that our model uses a homogeneous GNN model in which functions {{formula:036ff5a8-2f6e-4207-aa9e-5a1807fda22d}} and {{formula:193667b1-1be4-4f4f-b174-030a76cd6fdf}} from expression (REF ) are the same for all types. Heterogeneous GNN models {{cite:35b74da72c62d67c04652ee0c2dafe3d61f67a16}} have been demonstrated to perform well in scenarios with multiple node and edge types and constitute a future research direction for our work.
{{table:2c643f3f-7cf2-42df-9e93-3d651ea0ccf9}}{{table:1e29d427-4b9d-4a40-acea-c23622400117}}{{table:5bef95f0-da19-4d13-b0d2-56d93e550bad}}{{table:343817cc-7065-448a-a8c6-577a80914ab1}}{{table:8afcd67a-e87e-4e4c-bc10-ca44c11d42e6}}{{figure:0725c398-9f68-48d1-835a-b1732401bcf9}}{{figure:096443d3-3186-4f69-8bba-18af50693353}} | d | 9c231a694b8532826c91ddb678d3232d |
There have been and will continue to be predictive successes for AI in science:
in biology, AI algorithms are used to predict biomolecular structures and to guide the bioengineering process {{cite:1ab3c0d047dce88c844c8e92d35f14b523d21801}}, {{cite:907c60d87953add7afe6dc2cabff747991d32470}}, {{cite:ed52b74effaaca596fbfa40a5257a4277bcf41fa}}, {{cite:a32b5e2787e39123a5c578ff1b1641d593a91569}}, {{cite:afb6943e29bd8e7ea404ec0ab61d4037556cc04e}};
in materials science, we are beginning to see benefit from AI-designed materials {{cite:79e9cd8c041a173255bc3547b1d4500213543149}};
in climate science, AI is used to reduce uncertainty in the prediction of future climate scenarios {{cite:697cd1597694e9ff7825f9326c7321b41264cedc}};
in cosmology, AI is used to conduct simulations of unprecedented scale of the early universe {{cite:4f129c2dfb273c9389245983c937477e542c0e3a}};
in neuroscience, AI is critical to processing massive volumes of anatomical connectomics data {{cite:a85f4c0c270dad9493829a6c4851da78a8fdb54c}}; and
AI is increasingly used for challenges such as drug design {{cite:aad82f6bbb8880e2e409a2331ca3a7fc66f2e95f}} and the management of cultivated ecosystems in precision agriculture {{cite:c162fac8317dce7ad429c6e5f3fb65e502a70ceb}}, {{cite:b9340daee82036e4edc89c026ced684011384482}}.
More interestingly, decision-support utilities and self-driving labs {{cite:90192c9987ad4fd15af25501d6bf618e43388e0e}}—wherein hypotheses are formulated, tested, and refined by AI agents—may herald a coming revolution.
The most important feature of the self-driving lab paradigm is not the acceleration of the tedious processes of discovery.
It is the translation of insights obtained from data by an AI model into scientifically interpretable hypotheses that can be tested—and ultimately understood.
These nascent systems aim to do more than predict what will happen, they attempt to offer insight into how or why.
| i | bde03c83b375959fa1a0e9a0383ae9a5 |
Our method is built upon the backbone of
the improved Multi-scale Vision Transformers (MViTv2) {{cite:8eb2f3027ab3ca7f0ffb0f355e20e56ff2ffbc74}}, {{cite:5391975b9dd3d40e84d616a11440e797e7f7d263}}.
Note that
our approach works with any action recognition backbones.
Given videos from multiple datasets during training, the model backbone takes the video frames and produces feature embeddings for each video.
The same number of Multi-layer Perceptron (MLP) as the datasets are constructed as model heads to predict action classes for each dataset.
To facilitate robust cross-dataset training, we propose two loss terms, namely, the informative loss and projection loss.
The informative loss aims to maximize the embeddings' representative power.
The projection loss, with the help of multiple cross-dataset projection layers, guides the model to learn intrinsic relations between classes of different dataset, hence the model heads can be trained jointly.
See Fig. REF for an overview of our framework.
In this section, we first briefly describe the MViTv2 backbone design, and then present our proposed robust cross-dataset training paradigm.
| m | b3f03d819573e851c201254b85dc87d4 |
Additionally, ref. {{cite:649e83a6eb48bb07090be381d5d74d04d276bd88}} provides for each UTM its complete set of permutations in all five momenta. In practice this means that we do not need to perform any crossing of momenta in the transcendental functions appearing in the basis {{formula:94e8e174-8b04-4c6d-b44d-4a6f6bbbc975}} , but simply need to identify the UTMs with the correct crossing that matches the crossing of the scalar integral. The variable {{formula:fc58a2e9-3725-4e6f-82cc-d4a113140212}} , however, needs to be treated separately under permutation of momenta since it is not part of the pre-crossed set of UTMs. In particular, its sign changes under odd permutations.
| r | ed5520b34a8cf97130f276076a5cacd4 |
In addition, we developed a numerical model to study mechanics of elastic strip networks. The numerical implementation combines several techniques from the literature {{cite:7e82dac6b086e8e3829157ce7e5df4032ca2c83b}}, {{cite:21431641c5737b6e2e2a9bb60e2590381588534f}} to formulate an elastic network as a TPBVP. Our formulation allows a convenient modeling of rigid and flexible nodes by either directly imposing the continuities of orientations in the case of a rigid node or including the constitutive laws of a flexible node. This is different from discrete elastic rods that require additional treatments for specifying rotations at coupled joints {{cite:a4fae6d9f54da9746bb2bf52c9b9d22dbd42cbfc}}.In Appendix , we apply our numerical framework to a planar bigon arm to address hinge joints. The method of formulating a MPBVP into a TPBVP is general {{cite:21431641c5737b6e2e2a9bb60e2590381588534f}}, and could be applied to elastic networks consisting of general one dimensional structures, such as Cosserat rods {{cite:bb080a53c186dd4281ef4f41823e0d35db2fccac}} and inextensible strips {{cite:0e029d2091bbb7e089c1d9a646d4bda7784c2da2}}.
| d | 394d7b90835f44f0325a6f91d8b8d85f |
The problem of the numerical computation of residues lies in the properties of the spectral
functions {{formula:d48d12b8-8023-4b0f-8895-424a48641a8a}} , {{formula:432e8fdb-932a-4d32-a826-c065d3420a33}} and {{formula:35ebfb82-7dc1-4bc3-a4b7-f4cf2e426d9b}} . When performing the analytical continuation of these functions into the complex {{formula:751664cb-4eb9-45d8-9bbf-3b3acefceb63}} -plane, we should carefully check when this operation is indeed legitimate. As stated above,
the analytic continuation of {{formula:5ec9500c-6a5a-4fa8-b2b4-4ce2f84da78b}} is limited, in general, by the rate of decay of the ZSS potential.
On the other hand, when dealing with the function {{formula:03aeafb7-fff0-4e2c-9006-cb718e77321f}} , its analytic continuation can be performed over the whole {{formula:b6e3fc87-fd5c-4b84-9f8d-fa9029eaa937}} {{cite:758619c393026b65fb3c29ce538efa9015c2ee75}},
where it is bounded and, moreover, {{formula:5a3f249d-30ff-4bb3-b603-56128a5d557d}} as {{formula:40888bc0-0d09-4ba3-abc0-04c93d97e2d5}} , and so this property adds on our motivation to use the function {{formula:374bffdc-4518-4e12-8483-e975388c71e5}} for the solitonic eigenvalues.
| d | 7e080c721ada3d80cdf4a443b68dce4d |
While the Nyström method has been applied very successful over the past decade or
so, in recent years, it also has been improved upon using better approximations
for symmetric matrices.
The results of {{cite:3454df092f614db71cfd8564a6fc0151c85d7f8b}} indicated that the Nyström method has advantages
over the random feature method {{cite:cf4372f31dfe1e6e7978f3a29cce15f19a04ca50}}, the closest competitor, both theoretically and empirically.
However, as it has recently been established, even the Nyström method
does not attain high accuracy in general.
A model which improves the accuracy of the Nyström method, is the
so-called prototype model {{cite:3cb0fafbdb13d3c279ac59b9d15ec8503845c41d}}, {{cite:08eaa5c8c991214451288fd3b4275ef11c8a21ca}}.
The prototype model performs first a random sketch on the input matrix {{formula:512aacb6-1933-42cb-9b22-84ce9c4dede3}} , i.e.,
{{formula:012a455f-b5ee-46b2-89a9-4bccffd58ffb}} , where {{formula:46e1ee2d-0cac-4131-9568-7b251d4e0ad2}} is a sketching or sampling matrix which samples {{formula:5963e926-d0d7-4116-b2b0-f081752a27f3}} rows of {{formula:f92829dd-89c4-4dd2-a966-a5773148e558}} , and then computes the
intersection matrix {{formula:6b930e3a-dc7d-4bbd-a013-ad63e217d216}} as
{{formula:75887435-60a1-48ba-8c1f-0cba58ae7946}}
| m | 99656cc502d0eee9641e2dfbd5c10e6a |
ESO 092 05: Using BVI photometry {{cite:25713029e18d5197977e06e5b84866750c7e5aa0}} estimated
an age of 6 Gyr and a distance of 11 kpc for this object. Our distance is
larger, 12.7 kpc, but the age is the same.
The CG20 analysis yielded a distance of 12.4 and an age of 6 Gyr, a good
agreement with both Ortolani et al. and ASteCA results.
The metallicity for this cluster is claimed to be markedly sub-solar in
Ortolani et al., estimating it at {{formula:d2901aff-1878-4dca-8f09-dfe6307ca996}} -0.7. We obtain a
larger metal content of [Fe/H]{{formula:045eeb90-6cf5-4ead-9d68-243e218e2a91}} -0.12. Although these values are rather far
apart, we note that in Fig. 5 of Ortolani et al. the metallicity can be
estimated to be very close to our value if the extinction is set to
{{formula:88186cc3-544a-4ce1-8b7d-5526b931caa3}} 0.11 instead of the 0.17 value given by Ortolani et al. The former is
the value estimated by ASteCA for {{formula:9d2d3763-0b1a-4389-906e-de1bf6f1d300}} , while the latter
is almost the maximum extinction value for the region given
by {{cite:24b3f1011aa7474c1712fc1af7af896ede94716d}}. The difference in metal abundance is probably a
result of an overestimation of the cluster's extinction in Ortolani et al.
| d | 283e5f22c092584d773eb4496813fb0f |
Riemannian optimization algorithms are in general local optimization methods, which start at some point of the considered manifold, and progressively “move”, using the gradient (and possibly the Hessian) of the cost function to decide the movement direction. They oftentimes derive from a classical optimization algorithm over {{formula:90231375-0170-4c18-82dd-4622be719595}} . For instance, two of the most prominent Riemannian algorithms are Riemannian gradient descent (which derives from gradient descent over {{formula:8c982298-3fc0-4267-a684-9781acef4fc7}} ) and Riemannian Trust-Region method (which derives from the Trust-Region method over {{formula:20541368-b9ef-418f-9b1a-77d1c4985007}} ). Many others exist {{cite:881f2b84eebcb106096ae3bf183306c194b86ce6}}. Each one of them can be more or less adapted to a given application. Here, we will try to keep our discussion general, and will not make any particular assumption on the Riemannian optimization algorithm.
| m | e27238313934b53b92ab9a4b6e71dc02 |
In proton-proton ({{formula:c527cede-ba36-456f-a0ad-c82138b6d0b2}} ) collisions a significant fraction of the total cross section is attributed to diffractive processes.
Diffractive events are characterised by at least one of the two incoming protons emerging from the interaction intact or excited into a low-mass state, with only a small energy loss.
These processes can be explained by the exchange of a virtual object, the so-called Pomeron,
with the vacuum quantum numbers {{cite:b37ddccc1cd56774a8552d5f677c14330fc020cb}};
no hadrons are therefore produced in a large rapidity range adjacent to the scattered proton, yielding a so-called large rapidity gap (LRG).
A subleading exchange of Reggeons, as opposed to a Pomeron, also contributes to diffractive scattering, especially for large values of the proton fractional momentum loss {{formula:f5f13190-81a7-4913-901d-5b1035dfee99}} , and is required to describe diffractive data {{cite:3c725063dee411ee751eaa9de18118a6a8ed9306}}, {{cite:cd5d17239c03c5605a3180f0816e992294f60efd}}, {{cite:14dceeb98fb35fce84d6a6f08368bd466502c697}}, {{cite:51c120a508edf2149d05a3b54d96b596acbffb6c}}.
While Pomerons mainly consist of gluons, Reggeons are mesons composed of a quark-antiquark pair.
| i | 8c9d3fd24cc4257af0ba9332845e3611 |
We compare energy scales associated with different sites using the potential landscape shown in Fig REF b. On graphene-terminated regions, the combined potential modulations from the graphene itself and from the substrate that permeate through graphene are expected to be weak. Density functional theory calculations suggest these modulations are of order {{formula:6d2cdc4c-dac6-4523-ac81-f9e72d140a5b}} meV {{cite:0020f49a14372dc403c5cd470b4b29ae6e0efdf2}}, which is smaller than the thermal energy {{formula:3af1b841-8e4f-4c8f-87c3-3dda00ba5393}} meV at growth temperature {{formula:df70facc-1c8d-458c-9d49-2fdb150cfded}} C. An estimate of the free carrier screening from graphene also suggests this potential should be weak. For a free electron gas with density {{formula:6ca3645b-02e5-4444-a678-6e98674f7c7b}} cm{{formula:9a294734-9d01-4c4f-9f60-cd75c7415ea3}} , the same density as graphene (graphite) {{cite:4847ad8f24b48197a416abfa85c5281c155780b4}}, the Thomas-Fermi screening length is approximately {{formula:f5987f41-a074-44e8-a5e3-7bf89b408aea}} Å. This is less than the film/substrate layer spacing of {{formula:a9baf15d-cb24-43f9-ada3-913bb4ccaadc}} Å{{cite:d3b628f0e07b517939334ab1929e4c249f9e052d}}, suggesting that the screened potential above the graphene is exponentially weak. In contrast, the potential modulation at a graphene pinhole is expected to be much larger, {{formula:20cd522a-e12e-4cda-ad54-6077ddac1f4f}} eV, since this involves adatoms making chemical bonds to the graphene or to the underlying substrate. The pinholes provide the low energy sites for nucleation and growth. Epitaxial registry is obtained because the pinholes are locations of direct epitaxy to the substrate. Given these factors that favor seeded lateral epitaxy when there is a finite density of graphene defects, it is possible that some previous demonstrations of remote epitaxy can alternatively be explained by a seeded lateral epitaxy mechanism.
{{figure:40407194-f8e1-4cb6-a889-30cdd5c53bfd}} | d | 8224dae3a814ddff56c8f02557595ca6 |
We focus now on the existence of a second critical point for {{formula:2b87a7e5-6118-4a80-9088-12b93772cc82}} . Denote {{formula:3a108b0c-0ac9-4c5b-86bd-7e3ad76ffcad}} . Motivated by {{cite:f87babaa9d9a7c4f5e7e4622a9950a2ba3c51e0e}}, we define the augmented functional {{formula:9b0c27e1-243e-4be8-9aba-c69ed0d16352}}
{{formula:f61467c6-0701-4c8c-82e1-99392b601fa2}}
| r | 22d838eb09c2e2aae9e2ea7965dc7a1a |
Comparison on VehicleID
In fact, the VehicleID {{cite:631a96ad3fec3d18a2d9d4e8a9b6f17c46d73449}} dataset has a larger data scale than the VeRi776 {{cite:298468bda01a05b780d77d6e2bc01ba1aa838257}} dataset.
However, the proposed HSGNet method still can obtain the 1st place and outperforms those state-of-the-art methods under comparison, as occurred on the VeRi776 dataset, as shown in Table REF . For example, the proposed HSGNet method is the only case that acquires a more than 82% rank-1 identification rate on the Test800 subset, which is respectively 3.22% higher than that of the 2nd places (i.e., Appearance+License {{cite:c9860c863d5d5de90b56b22943173e10a7ff622a}} ).
Compared with the RAM {{cite:2297d5c9212e646495f06052acb6d5b5ff8dc089}} of the best partition-free method, the proposed HSGNet respectively defeats it by 7.57%, 5.51%, and 8.95% in terms of Rank1 identification rate on Test800, Test 1600, and Test2400 subset.
Moreover, On the largest Test 2400, compared with partition-based methods (i.e., MRM {{cite:a45bb269266365df7641584336a420ef4a33c27d}} and Part Regularization {{cite:a7ab3e1fa23245003a2fd3f785926c4fc20bc6af}}), the proposed HSGNet still defeat 5.79% and 2.45% on Rank1 identification rate, respectively.
| m | 617121efd298f56f5d579619f9ff76c4 |
for {{formula:c1ebd633-8088-45e3-bbf1-fae4bcc991be}} . Classical results {{cite:9439a55fb53a07038689aa9d2eb76dd3a488fd0b}} state that for {{formula:e1e725e9-f06e-42f0-9380-ddf7ff84c251}} large enough, the solution to the optimization problem (REF ) coincides with the solution to the original constrained convex program (REF ). This justifies the exact penalty method as one way to solve constrained optimization problems.
| m | 142e0b866e178b89d9be6c0eb65dd926 |
Here, {{formula:651efe02-3f5c-4e7b-a6ed-60a2c2d3f12f}} again describes the measured number of events in each energy bin {{formula:230b3c9c-85dd-4511-ba72-ee904bac15e4}} and {{formula:3b11892e-3d46-4034-a2ca-67353cfd0c64}} bin {{formula:6da4b934-d2a5-4cbc-ad3d-719989955afb}} , and {{formula:8085249f-c178-4cbd-af6a-08909a757dec}} represents the Gumbel distributions for the respective bin as in {{cite:bf362e16338265d8192d7aec39ae8bd33d224f53}}.
| m | 58a7e40876f899978f374d0b857a8cb7 |
In linear approximation the effect of a magnetic structure on the transversal phase space vector
of a particle can be described by a {{formula:c2996057-2bc6-4748-ab95-653454d435ce}} matrix {{formula:ba6f2c10-17e4-45b6-80a5-0517594958d8}} , called the
transport matrix {{cite:5194d91361ffa7a5126de8bba1a69378c8d3ae64}}.
It becomes apparent that any transport matrix is bound to be symplectic, {{formula:01f9353e-06d4-415f-970f-35b88b43e6af}} .
Furthermore it can be shown that the group of symplectic {{formula:abc52c01-a211-41dc-a8bc-cfc1823b49e4}} matrices is a {{formula:af35e495-da06-44e7-ae82-549539439224}} -dimensional manifold {{cite:5fb18d2246ae47d57899e8bf8b01189d0e36b9f6}},
{{formula:a7aade31-e698-4cde-8fc5-b8626074ba4c}}
| m | 2a52f8d2a0776e5d43890917ef4db48c |
Our investigations were partly motivated by the celestial appearance of the algebra {{formula:85d74496-3ce3-4e25-beac-0ac749cfeaef}} following refs. {{cite:055ad6af8e92781072748a6de1f3361f94385b19}}, {{cite:fac6a4e3be5b59a4da805f720f1e6d8a8b73990a}}. We discussed how this algebra arises straightforwardly from the soft expansion of the self-dual kinematic algebra appearing in the `right copy' of the double copy decomposition of self-dual gravity. This point of view is a spacetime counterpart to old and recent work that describes self-dual gravity in twistor space {{cite:9c6b30b21432694cb42fbf32aa3ca50961cc9200}}. Following the same approach, we described the deformation of {{formula:b684b110-af43-4566-853c-57723bdcfc33}} into {{formula:94fcbfbf-bbc4-43e1-b6ab-82286f2f4627}} associated to a Moyal deformation of self-dual gravity. It would be good to connect this result to those of {{cite:13b23078e2cc2b8acd58cf2a8c7ea0c2098b2d23}}, {{cite:c9f0fc9a44a32630b70139edc3ce5dce5596512e}}, and more generally, to understand the space of consistent deformations and its relevance.
| d | 28631d1cd7a30adea3f18c4dcb9a4ae0 |
The first and second attacks we used are the model poisoning attacks from {{cite:80b2e532ca8500f72aec39f4ee4ef706080723b7}}.
The model poisoning attacks aim to increase the error rate of the converged model even facing Byzantine-robust protocols.
In these attacks, the malicious clients search for poisoning updates by solving an optimization problem.
We employ two attacks proposed in their work targeting at Krum and Trimmed Mean.
These two attacks are referred to as Krum Attack (KA) and Trimmed Mean Attack (TMA).
| m | e8048f31b9d56247b24bdba9ad077e6f |
Rapid progress has been made in both the Transformer-based architectures and the contrastive learning architectures. However, these two types of architectures are mostly developed independently, and almost all contrastive learning backbones are based on the CNN. A critical question arises as to whether the ViT can provide better expressiveness for contrastive learning than the CNN. Recently, some research works have attempted to exploit the advantages of both of them. For example, Chen et al. {{cite:828607a3d6da62f9a706bffdad90ca3916f3ef89}} studied the instability issue on training the self-supervised ViT. Caron et al. {{cite:b9f8dcd656ed5c41b471fed321ef5776396e379c}} explored some emerging properties of the self-supervised ViT.
In contrastive learning, a recent development is its combination with the image clustering task. Typically, Li et al. {{cite:50086ad40bb4ef1f17856ab55ff6e9f02e64e0e2}} utilized the contrastive learning at the instance-level and the cluster-level for simultaneous representation learning and image clustering, where the CNN is adopted as the backbone. As the CNN is more focused on local information, it is intuitive that the modeling of the global dependencies via Transformer may provide rich information to enhance the image clustering performance, especially for complex images. Yet surprisingly, it remains an unaddressed problem how to effectively leverage the Transformer (or ViT) in collaboration with the contrastive learning for joint representation learning and image clustering.
| i | c37009d1405f0de08b3b6c856c692a13 |
Let {{formula:12b27120-5a05-454d-8748-8bc101dddd4d}} and {{formula:c0969a3c-6a7a-4eb1-8b6f-1a9f6897156d}} denote the source and the target domains, respectively. We denote the existing annotated source domain dataset as {{formula:40eb43af-d486-4bde-ada1-560d63049e56}} , and use {{formula:0a8cca9d-947b-4401-a8d8-530c3cd9ce39}} to denote a small set of annotated images in the target domain. Additionally, another set of unannotated images in the target domain is denoted as {{formula:6c9673b8-88fa-4e38-92f9-47ad79504507}} , where {{formula:dbef1852-50e0-4349-b5b7-ee79b2519196}} , {{formula:34dcafcf-ef84-48ef-8f4c-0d5a425ebb27}} and {{formula:d4c3606d-c738-4fa8-98d7-a7c5681a09f3}} are the number of samples in the three datasets, respectively.
The proposed CS-CADA is depicted in Fig. REF , which consists of three parts: 1) a segmentation network with Domain-Specific
Batch Normalization (DSBN) that learns from {{formula:98ed9399-d479-436b-9707-31c9e8c9e402}} and {{formula:fddda89f-fe0a-4a03-bf6a-96183191fde1}} to provide a supervision guidance to bridge the cross-anatomy discrepancy; and 2) a Self-Ensembling Mean-Teacher (SE-MT) that imposes an unsupervised consistency on {{formula:1ed6ae32-6684-4f75-806a-5a7861e9968e}} to further enhance data efficiency, and 3) a contrastive learning that encourages the model to capture domain-invariant features. Without loss of generality, we adopt the classical U-Net {{cite:c8a71480c945d3f8f47e95907e3849ad5ff0b96a}} as the backbone for segmentation.
| m | 0630d01d824b71ef65ab501c82318b50 |
Facial renanimation methods use driving parameters from a source video to reanimate a source image or video. {{cite:e5f5cf5e463350b1e71ece39e0606bc941a46f5a}}, performs the reanmatiion using the parameters of a 3DMM extracted from the source video. In {{cite:c2176ca0366f3fed30a5e872e06ccd00a0be7bfe}}, Neural Rendering is used to reanimate videos using expression parameters extracted from a source video. However, this renderer is needs to be retrained for each target video. In {{cite:f68d26b47deaf687f20c88d4ba1f0a14043b141b}}, audio parameters drive the target video. In {{cite:5cb83f1d471024d81a2a15c3cab32ba0314b429e}}, facial landmarks from a source video are used to animate single images. While all these methods generate realistic re-animations, they do not model any facial geometric detail. More closely related to our work is {{cite:7dacebd0dacef260f021a6c2adfa213e1ae9b593}}, which regresses a set of FACS {{cite:04c93fc7e757e3db724f25cf861e2e94fc78a578}} based textures from single images and are able to generate realistic renderings in any desired expression. Nevertheless, they do not predict any facial geometric detail but instead generate the details in the texture space.
| m | 82b0591c9e3bad33b0825b0b3f4e06d3 |
Document intelligence is a broad research area that includes techniques for information extraction and understanding. Unlike plain-text documents in natural language processing (NLP) {{cite:fb0d07429e5ec2b1b95f5dcc891993dc8f50d766}}, {{cite:ae7e7bfc38d550b77c8c54a37fc8cbde5c4e1c18}}, a physical document can be composed of multiple elements: tables, figures, charts, etc. In addition, a document usually includes rich visual information, and can be one of various types of documents (scientific paper, form, resume, etc.), with various combinations of multiple elements and layouts.
Complex content and layout, noisy data, font and style variations make automatic document understanding very challenging. For example, to understand text-rich documents such as letters, a system needs to focus almost exclusively on text content, paying attention to a long sequential context, while processing semi-structured documents such as forms requires the system to analyze spatially distributed short words, paying particular attention to the spatial arrangement of the words. Following the success of BERT {{cite:b99962e42d92fe762c1555070d7d27b2b7512bdd}} on NLP tasks, there has been growing interest in developing pretraining methods for document understanding {{cite:d33ff90951e09da537df1f52187a42d23e7fb26d}}, {{cite:19f6ef4f8f1fbc326972a859700515d826a733a6}}, {{cite:151c2405ee715a2b979e10cc6ea0bbd3b1b622d1}}. Pretrained models have achieved state-of-the-art (SoTA) performance across diverse document understanding tasks {{cite:ea42ebba3ad8fc37b87503661bae4693a972c72a}}, {{cite:e84be289d9b6355a8911d0992c1c78c0555482fa}}.
| i | e1b00a60b6fa9b842c4679dd26017003 |
Beyond the lattice QCD and physics, it might be applicable to general machine learning.
Taco Cohen and Max Welling et al. invented gauge equivalent neural networks to for data with descrete gauge symmetry {{cite:a27a5f28f93c897be98d9cfe6fb5d9b35d0311d1}}.
Our framework guarantees gauge covariance for continuous group, thus, some data with local continuous symmetry.
If one implement recognition architecture with covariant neural network, performance would be improved.
| d | f39a2407e386ac132c61951200a3c526 |
Inference speed. Inference speed is measured for DPP and baselines on MobileNetV2 with a single-threaded core Intel(R) Xeon(R) CPU E5-2470 (2.4GHz) (Deformable DETR {{cite:8ef7e7ef681a9780caaa707a9521817950bfcbfa}} does not support CPU implementation). Results are obtained by averaging inference time of all images in the COCO validation split and shown in Figure REF . The latency is for the whole network, not just the head. It can be seen that DPP achieves a consistently better latency-precision curve and its savings in computation are clearly reflected in savings of inference time.
| r | 0d1b1141211f59655fbf1612de8de7e0 |
There is growing evidence that the transition from the L to the T spectral class happens over a small effective temperature range {{cite:b0da2af4f31f86a1535a9670d135fbc969892fa6}}. It is difficult for any model of a globally uniform, homogenous cloud to either sink out of sight or precipitate rapidly enough to account for the observed rapid change in spectral properties. Instead the transition may arise as holes appear in an otherwise uniform global cloud deck.
| d | e0f9e6ef69fc27ec829b641d524610fd |
We remark that there are a number of approaches to model
high-order moments of the wave field that are based on perturbative
approaches. Indeed, the derivation of such approximations is based
on the premise that the waves is only perturbed or affected by the scattering to
lower order {{cite:a8b0d7d18319c0b0f2ee9c3c559a29dc5a34d934}}, {{cite:18b5d968ccdba66f4de16a821ed04236390b288a}}, {{cite:aa398c490abe1d0dcccb5ac185db29a4f39a166d}}. In this paper we describe an analytic framework that gives a rigorous
scaling limit identification of the fourth-order moment in the saturated regime when the incoherent or scattered part of the wave field
is of order one and is as large as or larger than the coherent component.
In fact this description also captures the situation when the wave is fully
incoherent and the coherent part of the wave energy is essentially fully scattered.
| i | ef4a6d8fcb59e8794d04e46d3211cccc |
We also recall the following standard FTRL results (for proofs see, e.g., {{cite:23f6e5b4d410dd8b52e90eb900d1bcb5385ed05c}}).
| r | 793cfeeca44c6925bcfb530c3fdc9622 |
One drawback of the unstructured pruning methods is that they result in having sparse weight matrices, thus leading to inefficiency in speedup and compression on CPUs and GPUs, and also requiring having dedicated hardware {{cite:df82332160f0300874f52e83cdaf3f9b8c051931}}. Therefore, the pruned model did not obtain high efficiency during the run-time. The benefits of the pruning methods include reducing the total number of energy-intensive memory accesses and improving the inference time due to effectively higher memory bandwidth for fetching compressed model parameters. MobileNet-V2 and ResNet-20 student models offered the best trade-off between performance and speed when trained using KD with ResNet-50 as the teacher model.
{{table:c7fbd842-7062-46dc-85ad-9c2709478cf9}}{{table:52e09f9d-18e3-434f-84cc-16c48a185d04}}{{table:0c751fc3-b8d3-4362-82b6-825375aa3160}}{{table:138a0cca-cdf1-4af7-873a-3992624896e1}} | r | 1286ad7cdfa51c4c84f23117b2dd044f |
We slightly generalize McLean's spectral sequence in such a way that the component {{formula:3fcdd594-aabd-4d53-82d2-e74a662d4fa9}} , corresponding to the cobordism (REF ), is taken into account. The new terms appearing in the first page of the spectral sequence are summands of the Borel-Moore homology {{formula:d322211f-7baf-404c-aad3-267cf2f690be}} . They vanish because the cobordism (REF ) is topologically trivial: thus we get the same {{formula:0732d615-ed77-4452-ab46-f5d0699bbb78}} page as {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, hence the same characterization of multiplicity – now for the Milnor radius {{formula:7af1292b-253d-4826-9136-d373a0bcbbd5}} fixed within the family.
Our proof does not use Giroux construction {{cite:4b95f3d172d00f525d78c05d346d9416a5a2973b}}, {{cite:f95f65772dd0a8ed21ab2e3ab90ebeeb21ba2bcb}} or McLean result {{cite:97005650a8ac11991dc2e980883b47d01674846e}} asserting that Floer homology depends only on the associated contact link. We directly deform the natural symplectic monodromy arising from the Milnor fibration in the tube (REF ) to a radius zero monodromy. This way, for any fixed radius we get a model {{formula:7e00562b-8bb9-4f43-8b5e-fd0ebda4abda}} which is actually isotopic to the fibration {{formula:0506e3e4-be5c-4a1a-a449-b50128b552f4}} at that fixed radius, and has dynamical properties which allow to derive McLean spectral sequence. This is done without relying in most constructions in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
subsection2-.5plus-.7.5Outline
We will now indicate the sections of the article where each of the above steps is realized.
The technical core of the proof lies in Step REF , that is, in the construction of the symplectic structure on the A'Campo space {{formula:f6e15301-1fbb-4a91-9bba-a698b2aab5a3}} . It is carried out in Sections and . Definition REF of {{formula:4d32d691-531c-4b9a-8884-64dd45f57a60}} can be explained in two steps. As before, let {{formula:52570633-df72-4192-ad0d-571ab89c1f0f}} be a log resolution of {{formula:a270a32b-41bd-4e63-9d10-72dfd1a164ac}} , and let {{formula:5d106375-f8c5-4210-b984-9a3f81c80a50}} be the total transform of {{formula:272f5b5d-8a20-41ca-a420-deac9f393929}} . First, we perform a real oriented blowup {{formula:58fc1312-8bdd-4567-aaad-818b78bcba90}} which extends the fibration over {{formula:86ebbcfa-bc33-4151-8d9f-2214a4d50184}} to one over {{formula:3d205946-fe31-4e56-ac2a-6bb0fadc7aa9}} , see (REF ). This can be phrased conveniently in terms of Kato–Nakayama construction {{cite:11c332b55de10ed694e9d2e6daa26ac1890c04bc}} in log geometry, see {{cite:90bd4e1fb5380f5b8aca9f6702a6840f100c321a}}, {{cite:70adb23c36599a6e43b81099757a02eb33d0cc79}} for applications in this context. The resulting monodromy has the required behavior over each {{formula:df529905-8d8d-4d66-ab9c-b5563d29e4df}} , but is not continuous over the intersections, see Figure REF . To remedy this problem, we multiply the preimage of each intersection {{formula:1795a5bc-c68d-41b9-a343-45ab8d7b703b}} by a corresponding face {{formula:1c1f62dd-c2f7-4ddb-8c90-ee5c773eed4b}} of the dual complex of {{formula:a3c607a2-84f5-4556-8ff6-481f5bc50f31}} . This face is parametrized by tropical coordinates {{formula:fe85797c-4dc8-4002-b074-87b5f08dbe0d}} , see (REF ), which measure the relative speed of convergence toward each {{formula:8e89fc7d-f5af-4a6e-9736-d8566b1b17cd}} . The idea of such a hybrid construction, combining classical and topological coordinates, goes back to {{cite:6eff63b8f3f5dcce8783f1b47e6f60fd98efa244}}, and has been successfully applied to various degeneration problems, see e.g. {{cite:3e9cb059b0d6fb5a3dbd5a387bb3b13720ba2fff}}. For our application, we need to push this idea further.
Indeed, the space {{formula:670c4a80-122a-4759-b3ec-4f0f89be0e65}} constructed above is a manifold with corners. We need to choose its smoothing in a very careful way, which allows for introducing a fiberwise symplectic form satisfying restrictive properties REF –REF above. To do this, we introduce smooth charts (REF ) in such a way that both radial coordinates {{formula:cb67d4c3-0388-4ca6-80f8-f88150f93c83}} of {{formula:4b6c0398-1126-4c00-8fcf-212d49a32be4}} ; and tropical coordinates {{formula:404b190b-279a-4de4-a034-893d82079603}} of the simplices in {{formula:c68a8190-d797-4ef4-aea7-892b9dda5fdf}} , decay exponentially at their zero loci. This way, the local coordinate {{formula:8f7d3b77-a110-481f-b249-9da8acbd03d6}} corresponding to a component {{formula:eab568b9-44d5-4833-add3-891879a44364}} gets replaced by a global smooth function {{formula:6dad39a2-7411-48bb-a851-d9d82abe4910}} measuring, in a sense, a hybrid distance to {{formula:c81fb803-5d0c-42c2-b980-b237dd05f04c}} . Using these functions in (REF ) we provide an explicit formula for the symplectic form. The resulting monodromy {{formula:b70a944c-e0bc-4146-b7bc-a28678ec2922}} at radius zero is then a time one flow of the vector field given by the formula (REF ).
The guiding principle of the technical proofs in Sections – is that all choices made along the construction become irrelevant at radius zero. This way, the monodromy {{formula:0908b1f2-9fea-439a-87ce-55b67a2a094b}} agrees with the usual one at radius {{formula:f7438df9-1487-4624-9a0d-2bd39711c1ed}} , but at radius zero it has the same behavior regardless of the chart: namely, it is a rotation by {{formula:0e125f6e-4fa9-40bd-ba4e-b87fa9e3b3e6}} about {{formula:14d2b3a5-5e69-4769-9525-8e78bf791375}} , and an interpolating Dehn twist in between, see Example REF .
Step REF is addressed in Section , where we recall the necessary notions from the theory of Liouville domains and their fibrations {{cite:f95f65772dd0a8ed21ab2e3ab90ebeeb21ba2bcb}}, {{cite:d20ff9e1ba10f8d7a97dd150ef78cb99d18bc788}}. In particular, in Definition REF we recall McLean's notion of graded abstract contact open book {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, i.e. pair {{formula:e1a61ebe-8889-48d3-934d-523297d2b0ce}} as above, for which {{formula:5ec64114-3c41-4449-8708-cf908b1fd69b}} is defined. Then, we describe a procedure which makes a symplectic monodromy into a graded abstract contact open book, see Definition REF . Our setting is closely related to Liouville fibrations with singularities considered in {{cite:d20ff9e1ba10f8d7a97dd150ef78cb99d18bc788}}, see Remark REF .
In Section we recall the definition of Floer cohomology groups {{formula:906cf489-e86c-4770-8ca9-0df46bd40d69}} . To be precise, we will use Floer homology introduced by Uljarevic in {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, cf. {{cite:c580055fb2a31bbd41b397a216821656bfb54c49}}: in Section we explain how to compare them with the setting of {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. In Proposition REF , we prove a slight generalization of McLean spectral sequence {{cite:97005650a8ac11991dc2e980883b47d01674846e}} to the setting which includes the fixed-point component {{formula:b2bbec34-28b1-46a0-ab5d-a17134860778}} . In Proposition REF , we apply this spectral sequence to the monodromy {{formula:038f24f1-7822-4706-9c9f-57d2b43eb0e5}} constructed in Step REF , using its dynamical properties listed in Proposition REF . This way, we get a spectral sequence similar to {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, but with term {{formula:92e91b54-0f6d-4869-833d-5a122cde5145}} taken into account. Theorem REF is deduced in Section .
subsection2-.5plus-.7.5A brief history of the Zariski multiplicity conjecture
In his seminal paper {{cite:92143668e2055f8caed2343bf7d40699fc0ba78b}}, Zariski posed eight questions which had a major impact on the current shape of singularity theory. Out of these questions, the only one which remains open is the following.
Question 1.2
Let {{formula:32394845-cc85-48fc-9c8a-8466a6c0ad40}} be holomorphic germs of the same topological type. Is it true that {{formula:ef58775f-559b-4bd7-8c74-2517d2b05d14}} and {{formula:abc1e5eb-4daa-408f-b7ee-1270c7de09c8}} have the same multiplicity?
Here, {{formula:4770433d-d423-4d80-a71b-54d63d1dfebb}} are said to be of the same topological type if there exists a germ of a homeomorphism {{formula:53f9c1e5-33a3-41b5-8201-0e3b3fbc9977}} such that {{formula:0f65d651-6a29-4d1b-9e36-a759d1400633}} , where {{formula:e160e52a-9547-47a9-956e-2dc2c1dfc479}} denotes the zero set of {{formula:a3b952a4-45ea-47b2-8f13-ffae7405ec80}} .
If the above homeomorphism {{formula:0466887f-3319-486b-803d-8f89621a3edb}} is assumed to be {{formula:087900e4-66a7-4dd4-a5af-62f4a085e61f}} , then the affirmative answer to Question REF was given by Ephraim {{cite:7ec5670cbc4284e63553409184426e465c6c81cd}}, and, independently, by Trotman {{cite:a9ebe7d69ff27f3e35bbbcca5805901a7a959e1a}}. The answer is also known to be positive for {{formula:1c832deb-7629-4914-b377-e7bc8e96690a}} , i.e. for plane curve singularities {{cite:5fce510d44b67ba1ae071348590ec3aa6a91c905}}, and if one of the germs is smooth {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}}, {{cite:5fd425f7d8c892f7c0800aabd1b588478c51b3fc}}: in fact, the latter result follows from the A'Campo formula (REF ) for the Lefschetz number. It is also true if {{formula:58be2ca8-39ee-4003-b47b-2a54018c7b25}} and one of the germs has multiplicity two, see Navarro-Aznar {{cite:ee3554b048eca14c8ac6ae98e47f1210c5090c9e}}. If {{formula:3b9fd3f1-4184-41c3-ba92-d526e0e47712}} and the homeomorphism {{formula:081de49a-0350-4501-813f-87c8bdcb0557}} is additionally bilipschitz, then a positive answer to Question REF was given in {{cite:56a42e547672c00932e70d294a27848d2bbb3274}} by Fernandes, Sampaio and the first author.
Assume that {{formula:a5f6dba3-22b8-46d1-b733-e428fffa4be7}} , and both {{formula:6670aeae-04d2-4772-bb28-fe2d81932f10}} and {{formula:3074cc08-c9d7-4e0f-af2f-68f43492ed4b}} are germs of isolated surface singularities. Then, the answer to Question REF is known to be positive if both {{formula:4fa96e12-7689-4841-bf02-9f4e6fb1ef87}} are quasihomogeneous (Xu–Yau {{cite:cd131d2538504505aea679ee5f3802005ce998c1}}, {{cite:9da93a74b626afd56047807dc2ea814c8e3b1862}}); if one has arithmetic genus at most two (Yau {{cite:1f4fd33bf84142fd64d01e82295fb497f64122ce}}), or geometric genus at most 3, or Milnor number at most 26 (Yau–Zuo {{cite:ae63a4c489117684b4536db298129ad8fb89a42c}}). It is also true for suspensions of irreducible plane curve singularities (Mendris–Némethi {{cite:d4c4075cd86c0e67bcbb4eef5aac289e5f106bd0}}). For {{formula:04a2ebbe-23da-4c52-9b12-6037b9f8f174}} , a positive answer is known if one of the germs has Milnor number at most {{formula:f1ed9d07-7dca-4d1e-af47-8db3c3e08c6f}} {{cite:ae63a4c489117684b4536db298129ad8fb89a42c}}.
In general, if {{formula:14fe2b5c-3f30-47bf-8560-27da7b734591}} and {{formula:b74073cb-9f0c-4e51-86b0-ae438f7fbf8c}} are topologically right-left equivalent by bilipschitz homeomorphisms, Risler–Trotman {{cite:9e32ee364deb72fb1402e0f4a125654db1eefbeb}} proved that they have the same multiplicity.
An important special case of Question REF , which was also widely studied since {{cite:92143668e2055f8caed2343bf7d40699fc0ba78b}}, concerns families of topologically equivalent germs. Theorem REF gives a positive answer to this question in case of isolated singularities. More precisely, we have the following.
Corollary 1.3
Let {{formula:75ba1147-6199-4562-8018-87c565fd7672}} be a family of holomorphic germs of isolated singularities, continuously parametrized by {{formula:8bf14d2f-9a11-4642-b395-40ae39161b60}} . Assume that the topological type of {{formula:41bd7acc-0f16-4eba-99e0-cf0d83f74098}} is independent of {{formula:f9b2a350-bb1f-4ae4-aa61-51963a528925}} . Then the multiplicity {{formula:1fd388a7-1479-4aae-8c15-9e0ca189a068}} is independent of {{formula:11f7df38-1e24-4e3b-ab40-077b3a123e42}} , too.
Indeed, families of isolated singularities with constant topological type have constant Milnor number {{formula:2c5e7400-e79f-4733-8e91-6b8d99810317}} , because {{formula:864223aa-5847-4764-a8e7-1076be561b6c}} is a topological invariant {{cite:edcbc0062181eb9053bb57ecfae662692e9a247b}}, see {{cite:5fd425f7d8c892f7c0800aabd1b588478c51b3fc}}, {{cite:cae5328b201e297a2dc094a44f4363a6034c4fa5}}. Thus Corollary REF follows from Theorem REF . We remark that the question whether every {{formula:fc42209d-4189-4568-bd39-87ef470865a5}} -constant family has constant topological type has positive answer if {{formula:5ec184e4-d3ed-4a17-8080-b8e9f0aa0e97}} {{cite:2becdf6e265a68c3177b7f1043cb33dc4350d607}}, and is open if {{formula:49b2a84c-4509-456d-a0fb-888eb6dfc8d1}} , i.e. for families of surface singularities.
Up to now, Theorem REF and Corollary REF were established only in certain special cases. Gabrielov and Kushnirenko {{cite:30659a0eea8974191770d463881bb046991e175d}} proved Theorem REF for {{formula:56fb9e23-9423-4692-9eea-7af409216411}} homogeneous. Greuel {{cite:7cb11bb905d0951a6226429bb8f0be3685f9ab20}}. O'Shea {{cite:c416985a3ae30fb9e971078946563b2f684b676f}} proved it for {{formula:c6a22eea-91c1-4d7f-b671-23fbaf1c9468}} -constant deformations of functions {{formula:3f5c969e-d071-494c-966f-d3bd8e727b92}} , with the only hypothesis that {{formula:53af4209-a6d3-4ba4-bfd2-209bb4286a16}} is quasi-homogeneous. The case when {{formula:43c866e6-34ac-4842-b219-b8d85d574e43}} is Newton-nondegenerate was settled by Abderrahmane {{cite:050c5eb5ac940fa3cb7b158b962ea4c0aa2ed0c9}}, cf. {{cite:3eeb38c25de87531c97cdb7b3d9666b0ad02b221}}. Other particular cases were studied e.g. by Plenat–Trotman {{cite:30481d763160edf3540183cea73ee847a9f37e01}}.
As a final remark, we note that Theorem REF is easily reduced to the case when the family {{formula:e48f1fb2-8324-4cd0-b935-a5b534fa8fee}} is algebraic, i.e. each {{formula:fb771641-f1fb-4d76-89bf-248be40c3270}} is a polynomial. This way, Theorem REF becomes a purely algebraic statement, which can be formulated over any field. Nonetheless, even in the curve case ({{formula:c243a02d-614d-46a0-ad38-fc2b4dab726f}} ), no purely algebraic proof of this statement is known. In fact, the only proof over the complex numbers consists in proving that a {{formula:f20f2ff5-43ac-4707-8c9a-828bb95f1012}} -constant family is topologically trivial, and then applying the affirmative answer to Question REF , known for {{formula:c67fc1b8-e6ee-4c2c-b8d1-82ae135fb45f}} . For families of functions defined over an algebraically closed field of characteristic zero, an application of Lefschetz Principle gives the same conclusion, see Remark REF . Answering this question was one of the main motivations for development of the computer algebra program Singular {{cite:0ff729941fe91b0568c7530d9b3b4de055475fa2}}, see {{cite:6b0ed104e0393e259e4029f15a017c3151616d90}} for a historical account.
paragraph4.5plus.7-.5Acknowledgments
Part of this project was realized during the semester Singularity Theory from Modern Perspectives at Centre International de Rencontres Mathématiques, Marseille. We would like to thank CIRM for their hospitality and great working conditions. We would also like thank to Norbert A'Campo, Nero Budur, Ailsa Keating, Ignacio Luengo, András Némethi, Patrick Popescu-Pampu and Duco van Straten for helpful discussions.
Preliminaries
In this section, we review some standard notation which used throughout the article; and recall those basic definitions from symplectic geometry which will be needed to introduce the fiberwise symplectic structure on the A'Campo space in Section . Further notions, needed to define symplectic monodromy and McLean spectral sequences, will be introduced in Sections and .
subsection2-.5plus-.7.5Basic notation
For {{formula:a5a7966f-14a7-40c6-abcd-69cbf88cceaa}} we denote by {{formula:13e887fd-ed22-49e8-9526-cac7c0e0a71c}} the standard closed Euclidean ball in {{formula:0d59439e-f280-479a-9bc2-ae9200e6e008}} , and by {{formula:9196d747-a84c-46a6-81e5-5d3df0467b75}} r{{formula:dd5316a3-0bd8-47f2-9448-0dbd3a58d0ae}} . We put {{formula:b6a8f92f-8076-45d2-9e7b-365d12f42aef}} , {{formula:5f6d9934-192d-42d8-b66f-65ba755b2d49}} .
We denote by {{formula:c54a599a-c10c-459e-92bc-03885c649da7}} the identity map {{formula:af345595-0c3a-4d1a-ba76-91008865720c}} ; and by {{formula:80587241-e65f-4cbc-9a2a-dbd21d167fda}} the {{formula:873998ee-e76c-4cda-a384-a3fc4a3b9f86}} identity matrix. If {{formula:58399b0d-f48b-4a42-93ad-608276946ab7}} or {{formula:75cc6b73-3f13-4d4f-b27b-9d67a53c1db1}} is clear from the context, we will sometimes write {{formula:4dd508ab-c25c-4e0d-862f-df07ce720efc}} for {{formula:d7a8871d-f72b-445f-9659-b9604fff8e64}} or {{formula:e054be7b-3c15-4af0-9dba-888422b15cdc}} . For a map {{formula:74a4bf76-74f5-47a3-82df-0857713f44e5}} , we denote its fixed point set by {{formula:55ad20af-a7b7-4756-b72f-47ef6f6bdd40}} . We write {{formula:085dd42c-0a6f-40d0-86e0-2150eebcaecb}} , {{formula:356e2ef4-dc3c-4aec-aaf6-38dabd91aea2}} for the projections from a subset of {{formula:6a746508-9441-4275-9f0b-0240c94739fb}} to {{formula:69d80ab4-f6cd-45fc-ba2d-f78b7565a1a5}} and {{formula:01218b25-637b-47ef-94e1-f0251c0c6a4c}} , respectively.
Let {{formula:426b70f3-64d4-4f15-a1e6-e42df58a2476}} be a topological space. For a subset {{formula:097f6652-b101-4d31-9f7e-92d592ac068e}} , we denote by {{formula:9ba6e286-8b48-43e0-b911-2f2116b93156}} its topological closure, and by {{formula:7e51f150-0a4c-4eb7-9121-daac958c5812}} (or {{formula:2f5b37a8-3803-457f-8766-a6eb0e104920}} ) its interior. Given a sequence {{formula:d34ab220-812a-4305-a58d-0b3d048ab8d2}} of elements of {{formula:bcd14409-8fb4-4a25-bde3-308d946b1640}} (which we abbreviate by {{formula:6f822c4a-c239-423c-9822-ca09a11403aa}} ), we write {{formula:962656ed-65a4-445d-a24f-539b26717af1}} for the convergence {{formula:07bfe56d-0261-4325-a77b-896d57a5dc0e}} . Similarly, if {{formula:45625a42-00e5-47d7-be82-60e7f7bcad94}} and {{formula:54f5df89-b73b-4e18-872f-346235244b32}} and {{formula:6134054b-ee4d-49b7-8db9-33d690d1c2f2}} are continuous functions on {{formula:f92a0479-a576-4a2c-b503-4f47353333a0}} and {{formula:8fa67650-369d-4db1-8f99-16794df18098}} , respectively, we say that {{formula:51a9d144-49d5-413c-a0ec-c8630012ad5b}} on {{formula:70931846-e2c6-47a3-b41e-e083c45b1873}} if {{formula:819e963a-5fe0-4921-ba9b-dc2593d9b699}} extends to a continuous function which restricts to {{formula:eec71b98-a982-4008-bc1b-2e03cab3676f}} on {{formula:73f69d08-9d63-44f7-ac08-0e80637ce615}} . We denote such a continuous extension by {{formula:ef2a0c64-2731-4453-9aec-ab3b446af09e}} , too.
For a smooth manifold {{formula:127a4d2e-2e65-44d3-b72d-74567ae9dad2}} , we denote by {{formula:8061c089-6c15-425b-aef2-f37b32a5c850}} the sheaf of smooth {{formula:6526db1e-44a4-4743-be61-cd2febe23611}} -forms, and put {{formula:ea3b7ce5-5c72-4371-932b-a473d5eeade5}} . Thus for an open subset {{formula:ff857ebb-c30e-405f-b605-e5998140fe26}} , the {{formula:2cec96ed-6cc7-4871-bd3b-8f6aaae35c5b}} -algebra {{formula:9589cf6e-8473-4a36-a30b-78fbb74cabd6}} is the {{formula:b1dfe6bd-074c-454d-bd33-ce20b0f05580}} -algebra {{formula:a0523460-852b-40b8-aeb5-78856eb519e9}} of smooth functions on {{formula:00ef9b59-8d06-4b72-891e-55933cdd32b5}} . Whenever no confusion is likely to occur, we will identify a function with its restriction, e.g. by writing {{formula:976108a5-2124-4948-850c-17b9b3e2e403}} for an open subset {{formula:eb40c4df-0000-4726-9c29-d7c90a148bbe}} .
subsection2-.5plus-.7.5Symplectic manifolds
A 2-form {{formula:5919fdf7-3775-4117-8543-d5b0c759c42a}} is symplectic if it is closed and nondegenerate, i.e. {{formula:9edc2e70-afb4-4cce-9608-f7fe741f3fb4}} is an isomorphism for every {{formula:35b760b5-a39b-476a-9f78-92ad853ffa95}} . A 1-form {{formula:e9a55972-31b7-4fc8-9ff2-287032924e5e}} is Liouville if {{formula:05765aa0-5aee-4b56-8c86-c9a039ea4dbd}} is symplectic. Its Liouville vector field {{formula:1001ab7e-4c8d-4c21-bf5f-aca7f3da6b9e}} is defined by {{formula:a50193da-aec0-4040-8efb-83a27e72e86a}} . We say that {{formula:c94d9804-5fe8-4ac5-b9e3-099da8423b5e}} is A Liouville domain if {{formula:41ddaff9-a7e1-41bb-9ec7-ef0f9b304387}} is compact manifold with boundary and {{formula:1de4708d-169b-4f45-b88d-aeb32ce64c8f}} is a Liouville form such that {{formula:c8b7acac-4e1f-436d-afc1-9f546753c453}} points outwards {{formula:5640bdf6-48ed-42ec-b93d-b87d2ddff215}} .
Let {{formula:c7f81767-b1c8-434d-b452-71bbcedd0499}} be an almost complex structure, i.e. {{formula:27475430-331e-49e5-801f-f29689dcef0d}} . We say that {{formula:f4ca2b2c-6d70-4c32-b3ae-aae77d95afab}} tames {{formula:0dcccd6e-b095-4b9c-ad3f-09d62345c95a}} if {{formula:5efd85df-7966-480f-9e14-469a6b8eb1cf}} for all nonzero {{formula:0cba7171-2aed-4795-b8e6-7265f64ccbb3}} , {{formula:6dc79abc-9aa3-4de8-97d0-a2dcc55b6cd9}} : in this case, {{formula:96bf854d-33d2-45b2-8287-58ed480fc7d3}} is nondegenerate. We say that {{formula:b7d7990c-edba-4425-b01f-fafcb38502ab}} is {{formula:4721b72a-35ff-486d-b79e-024317f567e8}} -compatible if it tames {{formula:67df6316-f639-4eaf-999c-b0aa4f93d515}} and {{formula:00514399-f23a-4365-b2fb-99ce5240dbf3}} . Note that these definitions vary slightly among the literature; here we follow {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}.
Assume that {{formula:6997ab54-b304-4789-a242-0f148442e902}} is a complex manifold, and let {{formula:cca1cd8d-08b7-4201-a6ea-7578e814be16}} be the multiplication by {{formula:450d1960-b31d-47e0-9707-a811d9b58acd}} . A 2-form on {{formula:19ec2730-4dfb-4db0-afb5-b9c1ea3fcaf5}} is Kähler if it is {{formula:63646c28-f18a-4cad-a04f-a8a2a9220f05}} -compatible and closed, hence symplectic.
For a smooth function {{formula:00d6b685-f48f-4b0e-bf6e-b47bb65ecde5}} we write {{formula:343b7b91-dd2c-44e2-8802-35021d502b9c}} . We say that {{formula:3cca8c2c-036d-4469-bed3-84a162ad0f6a}} is strictly plurisubharmonic if {{formula:b74c614d-25da-41f8-ad9e-136aad20a264}} tames {{formula:ea44babc-4ee1-4aa8-b3a6-50f78f6478cb}} , see {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}. In this case, the form {{formula:0bf86d15-514c-4693-9c40-81b1a71d3da8}} is Kähler. A function {{formula:74f513f5-abc8-43e3-912b-598f538d18b3}} is exhaustive if {{formula:992dc017-1e32-4df2-a4e6-ef5872b6e2ef}} is compact for every {{formula:ee4ed198-caf5-4061-b491-50f11d8be96d}} . If {{formula:fa750365-1d8d-4177-9c60-4919feebbdf1}} is an exhaustive strictly plurisubharmonic function then {{formula:a1ed542d-9d82-4dc8-ba97-8abe0fea9e6d}} with 1-form {{formula:04b14702-efa2-488c-a8f4-03042b8787b4}} is a Liouville domain. Indeed, the Liouville vector field {{formula:cd1371a7-c5f6-48ae-b297-03cf92ab4d61}} is the gradient field of {{formula:de77fbdf-d4ef-4b7d-bb16-2ec6f80faf4d}} with respect to the metric {{formula:f0c76553-1285-460e-a02f-88c4a26e9b99}} , see e.g. {{cite:b1204f911b5b321f5d286fa4927a3bf3bb9cf2d1}}. Every Stein manifold admits an exhaustive strictly plurisubharmonic function, see {{cite:b1204f911b5b321f5d286fa4927a3bf3bb9cf2d1}}.
Example 2.1
Let {{formula:88382107-8892-4def-b660-ec9343ca2d81}} , and let {{formula:b2cd3f07-892b-460f-b5a0-e9c2a469b705}} , i.e. {{formula:bf9994ee-0335-44c2-b163-e038de86570e}} is a rotation by {{formula:255b2e66-4735-4cf5-b43b-8cdba6b65a64}} . Then in polar coordinates {{formula:94fc7a42-1580-4bcf-94e4-bb55bf0e298b}} we have {{formula:05e615d1-6211-4b88-94dd-9e49ad734f60}} and {{formula:b2c56fb2-7720-4c51-9d94-973fc7f699c2}} . Hence {{formula:226d2ab4-6448-461c-adde-5b2524585548}} and {{formula:439bb5ab-4785-4ef5-8c7d-a7dff53597e5}} .
Now if {{formula:e140d33b-2bd4-41e7-924d-093072e6aa9e}} , {{formula:4dbd5163-de41-456b-b149-8d9188b2a60d}} , then
{{formula:778b52d2-2061-484f-b084-188ffcc8a7f3}} ,
{{formula:4d68a504-cde3-4b0c-8327-1a6e2f90ac8a}} is the standard area form, and {{formula:3509a6c3-58bf-456d-bf18-0e862b0fe796}} is the radial vector field.
Let {{formula:c0e16d74-19dd-45ac-854f-0dffa766ac9d}} 0, and {{formula:3e3bd61f-c266-4ef4-8a2b-cc9bacbfab3c}} be such that {{formula:ce157aa1-dedf-465f-806f-64c147f8e22d}} . Then for {{formula:92d6334e-1a36-4e6e-9ba2-51525f65101b}} , the Milnor fiber {{formula:f0ee9bba-136b-4d0a-9459-3e8528c2c3a6}} is smooth, and {{formula:954d5f92-f910-4b9a-9ddc-5874fb36391a}} , so {{formula:a1e052c8-174d-4cc3-bebc-6f15e1829e90}} , is a Liouville domain.
For a more general example, replace {{formula:a123f6af-f9c6-45a1-9e1f-01f2c654b13c}} by a Stein space {{formula:f37e5563-470c-4bbc-a70c-b0ab68f43ddf}} , smooth off {{formula:4650c3ac-0ce0-4671-ace6-a6bad80b898c}} ; {{formula:4f344e37-b8c8-49ed-9237-49c7323385db}} by a sub-level set of an exhaustive strictly plurisubharmonic function {{formula:36734568-e34d-438c-a8a9-5252d6fdab0c}} ; and the standard Liouville form by {{formula:be49e4e4-5c3c-4abf-8773-412735970d75}} .
Let {{formula:d424a5bc-6ece-4ae8-8b03-b0b6f3313b05}} be a submersion. A form {{formula:a594e6cd-c61b-46a6-a7ab-eebd472ea1e8}} is fiberwise symplectic if it is closed, and its restriction to each fiber of {{formula:c4b21849-eeca-41f0-90bd-6f34de9cb7de}} is nondegenerate (hence symplectic). A fiberwise symplectic form defines an Ehressmann connection {{formula:1c5b6db9-3658-468c-9b2c-f8b4ae82ab41}} , called a symplectic connection {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}. Given a vector field on the base {{formula:012dab16-780c-43fe-ac52-67a3be6c60d8}} , its lifting by the symplectic connection is called the symplectic lift. If the flow the symplectic lift of a vector field in {{formula:685b5ed5-8134-4e25-873d-8ad4cc0c5e6d}} exists at a given time, then it induces symplectomorphisms between fibers, see {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}.
The A'Campo space: Milnor fibration at radius zero
Let {{formula:6489bd50-6553-483a-bd89-5de91781d85c}} be a complex manifold. Let {{formula:87e8bb55-293f-49f3-8fb4-768088a91f0a}} 0, such that the central fiber {{formula:0bfad4c9-ef77-4fd0-b1bb-b367ba981b12}} is snc. In this section, we recall the A'Campo {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}} model {{formula:46b3682c-8dcc-421f-9ba4-94639b3e5644}} of the radius zero monodromy of {{formula:39b38675-7185-4ade-abdb-4f65267fe830}} and construct a smooth atlas on it.
After some preparation in Section , we introduce the topological space {{formula:8fcc139e-0005-490e-8f9c-c684af267a64}} in Definition REF by a hybrid procedure using tropical coordinates and Kato-Nakayama spaces at radius 0, and classical geometry at positive radius. We list its basic properties in Section . In Section REF , we endow {{formula:4297d0cd-1624-44fa-a569-104b58584c64}} with a structure of a {{formula:1e183ce2-0e1c-48fe-ac38-a140032b25ce}} -manifold with boundary, which depends only on {{formula:38b53093-4836-4bfb-bfae-cb26ecaf871e}} . In Section REF we improve this structure to a {{formula:7528572d-da70-4960-a31c-4a99802ae290}} one, designed so that we can introduce a well-behaved fiberwise symplectic form (REF ) in Section . This {{formula:59b40a0f-69f2-4323-b023-943a603833c0}} structure will depend on some additional data chosen along the construction.
In a topological setting, {{formula:567fd652-e6fe-4ecf-a7e5-96e69f6144d7}} was introduced by A'Campo in {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}}, who used it to compute the zeta function of the monodromy. We note that loc. cit. predates the Kato–Nakayama construction {{cite:11c332b55de10ed694e9d2e6daa26ac1890c04bc}}, so instead of {{formula:7a982428-1abd-4011-bda6-cafba5d55df6}} it uses a real blowup. In turn, tropical (or hybrid) constructions have recently been used, for example in {{cite:3e9cb059b0d6fb5a3dbd5a387bb3b13720ba2fff}}, {{cite:d8143aa258464654ebf60c088bb018ee8a7b4bbc}} to define topological spaces related to our {{formula:27b47e35-b40c-4d12-bed8-f327c0cc48e4}} in Definition REF , together with some additional structure inherited from {{formula:b2f1bbc2-dc9a-4626-914e-e98e1fb5e04b}} as {{formula:069ecd1b-54fd-48e6-87cf-c4f1aab3bab8}} : e.g. a measure in {{cite:3e9cb059b0d6fb5a3dbd5a387bb3b13720ba2fff}}.
subsection2-.5plus-.7.5Construction of the A'Campo space
Let {{formula:40a0bb40-2b7c-446d-aabc-ff52b658e826}} be a complex manifold of dimension {{formula:bfc93a5a-2709-478a-80b3-382239b0bc08}} . Let {{formula:5233ecd2-4491-438b-a123-f696af66ca62}} f*(0)red{{formula:579a4bc7-f4e0-4df0-a0ba-bb3c8040c1e9}} f-1(0){{formula:8c401d71-0749-4b70-b40b-48fbec5764ee}} |f|<-1{{formula:ec3a1da4-75ff-462b-962a-4d04ba7966a8}} f|Xf-1(0)X*e-1{{formula:6c751eaa-f611-400c-940c-2bc37e99093b}} 0 is its only critical value. Let {{formula:23ff2c95-5621-4a72-96c3-46be7bd9eebd}} be the irreducible components of {{formula:3d8411f0-fbae-49ba-b41d-7507965ef4cd}} . Write
{{formula:9ddb80a2-692b-4943-ae1e-2d846d14dec3}}
where each {{formula:f67c4f58-960a-46c5-89dc-fe4fe5d5229b}} is a positive integer. Put {{formula:6c04dd3f-1fe5-4161-a333-2d4ce7d5d82d}} , and for a nonempty subset {{formula:b46d123a-ad1b-4fe8-ba73-7eba0460bc65}} put
{{formula:0e2b0e63-edf8-4e34-ae8c-ec8a0faa6dc4}}
Then {{formula:c7170b82-424b-445e-bc02-8c6c765939a9}} is a stratification of {{formula:49829c95-7d4b-4a91-8a1a-aa70b5201a75}} .
subsubsection3.5plus.7-.5Charts adapted to {{formula:74561f18-6877-449f-9364-e9265e13abe7}} In order to fix the notation, we recall some basic definitions. We say that {{formula:078f2f06-56af-4746-b973-15964beeeef9}} is a holomorphic chart for {{formula:313495df-7667-4f8e-93f9-b90c2c49aa1e}} if {{formula:f4d92e32-7b92-4a5c-98c3-066a8981d76c}} is an open set, and {{formula:9358252a-f8fd-4b27-9492-f58750ff68a2}} is a biholomorphism onto its image. The map {{formula:cf6b1012-c686-4aeb-8631-6f16cde0ce50}} will often be clear from the context, in which case we will abuse the language and use the term chart to refer to {{formula:5dff3317-efc7-4342-b111-dd30ab7ab417}} . We will usually keep the subscript {{formula:49416250-f6b0-4013-a456-846b848a8a64}} when referring to chart on {{formula:4ff747a3-7b10-43bf-958e-e99ac640c39a}} , and drop it when taking preimage in the A'Campo space.
A holomorphic atlas is a collection {{formula:b379a500-1766-4cc4-b6b8-4d736d25da49}} of holomorphic charts which cover {{formula:b3c657f1-3748-4db8-97cc-d8cd05b0e77d}} , i.e. {{formula:6ed5dde1-0dd7-48b4-9b94-b3c173c2739c}} .
A partition of unity inscribed in {{formula:d88c5fe2-2503-4d02-9261-f8d20915c2af}} is a collection {{formula:f2111a9b-18c3-44a3-bf50-eab6e9209115}} of smooth functions {{formula:fdebe01c-4cb1-4d12-9593-be1c8b19ad29}} such that {{formula:9bfd554c-dd09-4c8f-b8b0-c9616e6793ac}} , the set {{formula:2cdbdd2d-e063-4149-8d69-9dcf7556f48b}} is finite for every {{formula:dce7ebfd-6c1d-4ca5-b057-8a30fe9bd421}} , and {{formula:816d4eb1-e80e-40c6-9c5e-124dec5360dc}} .
We will use charts of the following form.
Definition 3.1
Let {{formula:536e5346-2d38-4450-8539-445c607f2215}} be a holomorphic chart for {{formula:51fcadfe-c443-4347-b367-755c61814c38}} . Its associated index set is
{{formula:db7ecce4-df53-4d61-a67b-5882c0f98a1f}}
Write {{formula:4db83202-b897-4c86-8fd6-ed55eb161c17}} for some positive integers {{formula:0e384d0e-4fdb-4b96-905f-72409791a560}} . We say that the chart {{formula:8378a60f-9943-428f-a7ea-8289e6fa9852}} is adapted to {{formula:67b7c8be-f968-447b-9f11-a8cd5c75fb27}} if the following conditions are satisfied.
For every {{formula:dd2d92cb-2630-4290-a1ee-ff5668fc1e11}} , we have {{formula:740e8dde-8c86-4424-90b0-ad1fb27b765a}} .
The function {{formula:24020739-451d-47dd-a17b-b2c8411409eb}} restricts to {{formula:3ecf61b5-8a2f-40e8-b334-660511b0064c}} .
For every {{formula:b7b68da6-84d5-410f-944c-00e803386f38}} , we have {{formula:7f0b46f0-f892-407a-b371-7ff6e5ba4a19}} .
The set {{formula:ef93c21a-4ab5-4779-9b87-9cc4f0095513}} has compact closure {{formula:cc806393-fe26-45e3-b643-ba18907c1639}} , and the map {{formula:71a057e6-0710-43fd-b702-31af04053deb}} extends continuously to {{formula:02718479-cfa0-43ce-b1d9-4a74817ad4c0}} .
A holomorphic atlas is adapted to {{formula:2b8be89f-af97-43ee-b464-7ade0d47ba27}} if all its charts are. We say that a pair {{formula:ef8a0dfe-9de9-40c1-8837-fcbbe720d441}} is adapted to {{formula:9512768c-91c9-457a-8912-673e46aad11c}} if {{formula:805345ca-9680-4130-8540-13bb4506a8a6}} is a holomorphic atlas adapted to {{formula:d094ec7e-7d9f-46f0-a6e1-ba234cd234f3}} , and {{formula:90bdc58a-86ad-4f1f-afe8-669f16738919}} is a partition of unity inscribed in {{formula:7286a3c8-a62b-4bd8-8f15-f9ded220c769}} .
Note that, by REF , the indices of {{formula:f933c950-f8ad-4066-9664-19f05a766f63}} 's are either in {{formula:9e08306e-7e12-4a43-b1cc-da389e9e217a}} – in which case they correspond to the fixed order of components of {{formula:5d5d9032-9da1-4292-a80e-f5dd2262305a}} – or are integers bigger than {{formula:c249685c-b065-4411-a7a8-fec4b49a9de4}} . Although a bit unusual, this convention will make further notation much more convenient.
Because {{formula:13d43bac-2fed-4a18-b50c-02f80161d83a}} is snc, {{formula:0c98f131-3402-4517-812f-de893d8ece71}} admits an atlas adapted to {{formula:47fd1794-beaf-466f-9883-ad5db42b5532}} . Indeed, the definition of snc implies that, after suitably indexing the variables, we can choose an atlas satisfying REF , REF . Then, conditions REF , REF can be achieved by shrinking the domains of the charts.
subsubsection3.5plus.7-.5The basic functions
We will use a continuous bijection {{formula:1e935ba8-902b-4b2b-9cac-2cd335894216}} , given by
{{formula:724dcd67-bec6-41d4-8573-03da138877c4}}
Note that {{formula:e9b0a85f-f243-4970-bd9e-21f8ac9e709d}} is smooth.
Since we have assumed {{formula:d7e55493-7dbc-4c3d-9ab7-69bc5d8a5887}} , we have smooth functions
{{formula:ec102868-2d49-4f3a-8a56-9cddc2198e48}}
We extend them to continuous functions on {{formula:30261b0c-130e-4984-b33f-2678ce701120}} by putting {{formula:993af1b9-b911-4959-9473-14d938da7893}} , {{formula:69b93e51-879e-45ef-be77-bf8b7818121b}} . We have a smooth map
{{formula:31dd49bf-36ae-48be-b6b2-ef50152295cd}}
Fix a chart {{formula:26300208-5c10-4b6b-afad-7fa5db895029}} adapted to {{formula:8a1891a8-04f0-4320-9d9c-000c4e86e462}} . Fix {{formula:4ca85cd1-f419-4eb1-afcf-010f45d7b215}} in the associated index set {{formula:021c439f-a14c-4693-a949-e7df9711aead}} . By Definition REFREF , {{formula:5df15b13-4e42-4082-902e-d9c648938388}} , so in particular {{formula:84026455-ad95-4f7a-ac21-c2549f8dc2e8}} . We have radial coordinates
{{formula:73a07aa3-4020-40b3-bf9f-eef15d84da6d}}
We extend {{formula:ab5ee681-796c-4ba4-af59-dd0c88120559}} to a continuous function {{formula:51dc5d40-2415-477d-b882-d0e1cbda8bad}} by setting {{formula:509fc547-a64d-4e6e-87b0-4ff219884655}} . We also have an angular coordinate:
{{formula:830e2592-089f-401a-94a9-97bebd0d6fc5}}
We will compare the speeds of convergence towards {{formula:d5d877a5-867a-468a-b5eb-33c4ee87ada8}} using tropical functions:
{{formula:74acdeae-cb52-403f-ac02-7fbd246c946d}}
Eventually, we introduce a hybrid coordinate:
{{formula:72763934-37f0-4284-a30a-65013122c276}}
To simplify the notation, we introduce a map
{{formula:f147796a-ac4f-46b8-964a-863be1dcdc24}}
It puts together those coordinates which have no tropical part: as we will see, they are easier to handle when defining a smooth structure on the A'Campo space {{formula:9887a208-a3df-4d8b-a424-7edb814b17e4}} .
Notation 3.2
Throughout the article, we will identify {{formula:e9afa4b4-2e0d-478d-8c84-e6e955a26cd5}} with the additive group {{formula:d3941717-2d25-4517-99db-62edcdee8e33}} . This way, {{formula:d93581a8-b7f6-485e-bb65-156649b5daed}} denotes the angle {{formula:94493b70-e0ce-4e15-9e69-392d634a74b5}} , and we have {{formula:315fefa7-3680-4ded-86d0-4a4d05d2ab71}} .
subsubsection3.5plus.7-.5The Kato–Nakayama space {{formula:62c68a48-6884-4de0-8869-9a103a0f624f}}
As a first step of our construction, we will extend polar coordinates on {{formula:1a9bfbfe-bc17-42c5-8651-3773ddd2da35}} to radius zero. This can be conveniently phrased in terms of the Kato–Nakayama construction {{cite:11c332b55de10ed694e9d2e6daa26ac1890c04bc}}. In general, it associates to a log structure (i.e. a scheme with a sheaf of monoids, subject to some additional conditions), a topological space. We will use it only for the monoid sheaf {{formula:027a9f93-695e-4b0d-b162-0fd3db9cc284}} , where {{formula:344a3f7a-7c4d-4e8e-98c2-37c1498c56bc}} is the inclusion. We follow {{cite:11c332b55de10ed694e9d2e6daa26ac1890c04bc}}.
For {{formula:eff59163-d718-4178-bef6-c0227cd16ae6}} , let {{formula:60c923df-e5a4-423b-90d6-9bd3ea75aefd}} be the stalk of {{formula:dc20abbb-c999-4155-86a0-36fb4cfc3ba8}} at {{formula:ff86ede7-0c61-47e5-a3ee-aea351e5c73a}} , i.e. {{formula:feaa2ee5-0e0b-4ae8-908b-15706b50e955}} if {{formula:f13681b1-d061-4b99-b60b-7f2203c5c9cd}} and {{formula:12c309aa-8473-4c8d-85e7-8eea81accc7d}} if {{formula:6b59918f-fee8-481c-9689-ddb44a34beb8}} . Let {{formula:83c0e414-8962-4130-92af-b838ff38903c}} be the group associated to the monoid {{formula:f63ce76e-9bde-4838-a2b6-39caa36fc0a6}} , i.e. {{formula:ea4c39ea-f243-4daa-a750-edb956828f7e}} with a natural identification {{formula:2ba57691-d0df-43dc-9b49-cbaf7c56cf1b}} . Let
{{formula:3dcc1144-7307-4859-9eb2-24e75bf62455}}
and let {{formula:717fcdfd-633d-4c6b-b9bc-eb305e7d5954}} be the first projection. To define a topology on {{formula:ee27cd6d-294f-4bcf-ba92-ce9e8437f193}} , fix a chart {{formula:320368a7-88a4-464d-8fc3-987bfdaf7f78}} adapted to {{formula:fb758e36-56f7-47e7-b1b8-9bbe7a2c94ac}} , see Definition REF . Reordering the components of {{formula:4f9c45ab-c01f-44ce-9025-ebd96efdec6e}} if needed, we can assume that its associated index set is {{formula:d36dd372-76b0-4cd7-8281-06311bda67c9}} . Put {{formula:503ea573-8ecb-405c-93e7-3e55dccba27d}} . A Kato–Nakayama chart over {{formula:87414802-40bb-43af-acb5-fbd63ad5863f}} is a map {{formula:5aaec52d-9ada-4626-a795-6304342013c8}} , given by
{{formula:25353683-89d7-4f1c-baaf-51ea8b49d01a}}
Note that the coordinates {{formula:7ffa6a6e-8f6d-41d3-8f27-35ad49835845}} and {{formula:1c3e674d-e416-4439-8797-f9a5f1b355aa}} extend to {{formula:a813adc3-5910-4410-a009-9447180ff4c6}} the pullbacks of the maps {{formula:cae50ec7-85cd-4f52-8d82-16b54d84087e}} , {{formula:0cef7ea4-9419-40f9-b98a-67a0d6350f18}} defined in (REF ), (REF ). The topology on {{formula:e5de901c-5407-4f5f-ae54-596f4c552e8a}} is the coarsest one which makes these charts continuous. Clearly, {{formula:c6640d3f-3e2e-47a0-8225-36d2123ded61}} is a topological manifold with boundary, whose interior is homeomorphic with {{formula:4c7fb702-c562-4757-a0ca-2ca6a43b85e7}} .
Applying this construction to {{formula:a7803369-3068-443d-9dd0-5277bbac5374}} , we get
{{formula:cc6644d7-7bc6-4291-a967-9c6dc7b7e5a8}}
so {{formula:1d2e685b-5717-4fca-8248-6e6c1855dc96}} extends the polar coordinates to a radius zero circle {{formula:0a179837-64d9-41e4-9c53-2f36cf2040f5}} , cf. {{cite:f624da0c0bf8108e59f2b1ae6888abbb5534af21}}. The holomorphic function {{formula:bccf2f09-7e30-4940-92c4-34e26d0ad0e3}} fX{{formula:85fa9b91-83e0-40cd-897e-268d957c5e75}} (r1m1...rkmk,m11+...+mkk){{formula:51a058b0-3c14-41fe-bc2b-c1054cd35c1a}} I{1,..., N}{{formula:a7b0c57e-f576-4c59-8787-38bbf401a8a6}} (XI)X{{formula:4f032eee-cd7b-400e-8d6d-636fa8ecc30c}} XIX{{formula:832284c4-f603-4092-9f16-3cb7bd4ff2f1}} (XI){{formula:c485c830-628e-4b13-987e-1e5a5ce9382d}} ri{{formula:b5e8d650-bcf9-4f6b-a083-70043b3be201}} iI{{formula:4de08160-8a0b-47f5-b253-e99d2cfdb1da}} ri0{{formula:b99b449d-27d9-4539-91c9-d6d2212d3f81}} iI{{formula:9bfa2e98-d3bc-4e1d-ab86-d452c7bf0e63}}
subsubsection3.5plus.7-.5Definition of the A'Campo space {{formula:091c2353-7e5a-4011-82e3-47a64228b5ca}} .
Fix a holomorphic atlas {{formula:457ab447-4c5d-4933-81b1-7eff45366de7}} adapted to {{formula:fb1f24eb-fe30-49d9-b5b7-ffb7ebcd3d0f}} , and a partition of unity {{formula:e28d448d-c126-4402-a1d6-ad0a476f2f35}} inscribed in {{formula:8e8c65bd-772b-48e2-8d3f-c4313b4cafaf}} . We write {{formula:61f8fe08-36c3-480e-b4dc-539b133d20f6}} , {{formula:6f9dcfc6-2d51-4160-a3c1-a74cabff8dd3}} etc. for the corresponding maps from Section . For {{formula:795753c6-1362-4c1e-a1d8-63c3b02243d1}} , we put
{{formula:ed348232-e331-4ed7-9e3a-0c07906e1028}}
and define smooth maps {{formula:873eb4b2-8913-4503-9ab3-e71e81a96505}} , {{formula:7732515b-ea12-4b1d-bfaa-082fa7dcb4e8}} by
{{formula:345ec67f-8d3b-4d8f-8ae4-ddfa08a9d73b}}
Definition 3.3
Let {{formula:d7d06c81-71f4-45ab-a6bb-600fabb25560}} be the Kato–Nakayama space associated to {{formula:a888af11-51c6-4e31-9d2d-c5c39a3ea3ed}} , see Section . Let {{formula:ac67873c-1844-476e-81a5-ffdee3662377}} be the closure of the graph of {{formula:1f033be8-a5f1-471a-9668-180610d24d56}} . The A'Campo space {{formula:730758de-05bc-4c39-9ec2-1a49d01e69ed}} is the topological space {{formula:d6045806-4a69-4ed2-be0a-b56194322649}} , i.e. such that the top-left square of (REF ) is cartesian
{{formula:22da567a-b746-40c0-bddb-9ef533bf2dfb}}
Put {{formula:7f79e084-7c21-4c50-8dfc-f66c45979cc2}} . It is a composition of the natural map {{formula:ff79a63f-a125-47e0-8580-140599e7c95b}} with the map {{formula:a7339cfd-1f43-40c2-8ae4-cd42087c086b}} induced by {{formula:327d1380-caf3-49c1-9d36-31ed12aa798e}} between the Kato–Nakayama spaces, see (). Fibers of {{formula:227e19a4-ae52-4842-9272-75f566e327b7}} agree with those of {{formula:99d458ed-0f9d-4dea-8edf-a6d4a8ceed14}} and {{formula:86760d52-c57e-416b-8215-8b5faf1367ff}} .
We pull back the stratification (REF ) of {{formula:5c8e0aed-74b3-4613-a5f3-6a710c273567}} by putting
{{formula:50a48bef-c5b3-4bac-a370-325be7a638ea}}
The partition {{formula:c6529a38-60cb-4704-8150-0df7363ac3cf}} obtained this way is not an stratification anymore.
We have {{formula:1e047394-2c29-4d83-93a9-106ed78bc80a}} and {{formula:0bd1c8ff-75bd-425d-9d9e-a9b1d18737a5}} . Thus, for each chart {{formula:5d7f0b81-7dfc-497d-b443-0bfc3a2584ea}} adapted to {{formula:7362144a-a7af-43ad-aa31-c4bec5360c19}} , the zero locus of the function {{formula:ae6e9866-db12-4a71-83ae-f52814041fa2}} introduced in (REF ) is {{formula:a8510c4f-6fcf-4f61-b295-89f42e164dd0}} , where {{formula:0d366a58-d4f7-4c84-a1cc-1b590efe8392}} . The deepest subset of this partition, namely the one corresponding to the associated index set (REF ), equals {{formula:9afbfedd-7683-4fbb-b560-bb9e1aa44990}} and is closed in {{formula:b5e1e4a1-2bd4-429b-b9ac-7ceebffcb560}} .
We put {{formula:c1808e12-8ef9-4f04-ae05-4261488a7480}} . In the manifold with boundary structure that we will define in Section REF , the set {{formula:9663766a-7201-47b9-9a76-6300484adcb6}} will indeed be the boundary of {{formula:17e6a4a3-5105-4a1d-bb37-472c0020b042}} . Note that {{formula:69b2c57f-db74-4b87-b0f4-cb33a78b4b0e}} is a homeomorphism. Whenever no confusion is likely to occur, we will identify {{formula:2f5434bb-9982-4baf-b5a3-2691701401ec}} with {{formula:2dde308b-6fff-40d0-b96b-c14b18c15d46}} . In particular, {{formula:1413e90f-1fa0-41a2-aef8-2145e95a603a}} has a natural smooth structure, pulled back from {{formula:18619017-b1c8-4def-9d80-30f3a4fa0762}} via {{formula:f02aabdb-208e-4e9c-96d2-5a7a491f872f}} .
Remark The topological space {{formula:f11e7fb2-b3f2-4410-9d8b-19280fbe28d4}} introduced in Definition REF agrees with the original construction of A'Campo, see {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}}.
subsection2-.5plus-.7.5Elementary properties of the A'Campo space
In this section, we list some simple properties of the A'Campo space {{formula:e3192905-5167-499f-be06-1771fc4de1e6}} introduced in Definition REF . They are summarized in Lemma REF , which studies the lifts of basic functions introduced in Section ; and in Proposition REF , which describes the structure of the topological space {{formula:dc941fc2-cefd-4cc6-a550-febb5bfb8d9c}} .
subsubsection3.5plus.7-.5Lifts of the basic functions
The following notation, which we will keep throughout Section , will help to make our formulas lighter.
Notation 3.4
For a subset {{formula:69eb0d77-8fc8-4221-866b-0568eb7b5429}} , we denote a map from {{formula:772f9d20-b7dd-471e-9079-02a6a1e1eabc}} and its pullback to {{formula:b61a91df-e253-431c-bc87-dda364ab87b9}} by the same letter. This way, for every {{formula:164aae6a-462a-451c-9c2e-5227d9f2a526}} we have continuous functions {{formula:e0af71a4-38c3-439f-88d3-3ab9c0b83716}} given by the formulas (REF ) and (REF ).
Let {{formula:1a43b73c-2795-4624-bca2-dd1a9d3d4e61}} be a chart adapted to {{formula:c170e2e7-aed6-46a7-b6c8-ee66bdfcb382}} . Pulling back the functions (REF )–(REF ), we get functions {{formula:6fc08bc5-935a-46db-81f5-95a2d5c82463}} , {{formula:b174ff0b-5d32-4c35-8cfc-5c1ccb1a2877}} and {{formula:89744c1d-6465-4823-a546-16b93e5f6387}} . In Lemma REFREF we will see that they extend continuously to {{formula:1cb6a39c-a7d4-48c8-b357-76cad897a267}} , and we will denote these extensions by the same letters, too.
Whenever a chart {{formula:a5181b59-0d9f-46a7-a4e8-7b29d350b913}} is fixed, we will put {{formula:5c6d40ed-10ae-4fb6-9064-c64cdf0929d9}} , denote its associated index set (REF ) by {{formula:4b1c04ed-aa3f-4082-ba7b-785be7506c56}} , write {{formula:3038b0dc-1639-4fc9-81e7-ceaaf35e037e}} and use the maps {{formula:bef81da4-842b-40ea-a3e2-6322763cb533}} introduced in Section without further emphasis on the dependence on {{formula:05ad90d8-ac2b-489c-b889-fb08f13f2773}} . If the chart is called {{formula:0cc66625-75bb-4c54-83b2-80f8ea1f5a6b}} or {{formula:a5302144-1865-4f7e-bda8-fb37a1e86d9b}} for some index {{formula:307e306c-424b-4250-add5-2f315d49a9b3}} , we will call the corresponding maps {{formula:c9bff496-f33d-44ad-978e-429083c112df}} or {{formula:d93138ca-603c-4cdd-825e-40aeef75fbb8}} , respectively.
In Section we will prove the following result.
Lemma 3.5
Fix a chart {{formula:de4a033c-f913-4b89-ba25-287a1cdcaf96}} adapted to {{formula:de033099-7e8d-4515-b34b-f8b83d331c8a}} , not necessarily one from the atlas {{formula:84e9aca1-a34f-406e-9d2b-9435f2fb70f0}} fixed in Definition REF . Let {{formula:15860bf2-06d7-4a9c-aa27-00ccb20ab997}} be its associated index set (REF ) and let {{formula:efd98e46-36a6-4a8d-a156-016a842cc5a5}} . Then the following holds.
For every {{formula:f7d9d8e2-edd1-4fcf-b8c9-942bd1d7f62d}} , the maps {{formula:2c1073c0-ed2c-48c9-bda2-a41bfc39a7d7}} from Section extend to continuous maps on {{formula:0050a552-e156-4207-893d-ece8343deb38}} .
Fix {{formula:ce28c473-7a5e-49ec-af2d-167f70aa5d92}} and a nonempty subset {{formula:8410593b-f895-4458-b87d-3763826925e7}} such that {{formula:ba88071e-08f1-44fb-987f-df176cbbd013}} . We have {{formula:73fb0896-5eda-4966-9f05-c911e3198c01}} .
For every {{formula:861947b5-9750-415a-b13c-5ad2f8fb45a0}} , we have {{formula:f791168d-084b-4ada-adf8-02c1e14f211e}} and {{formula:39ae19f2-5962-46a6-ba62-ff41722f8a16}} .
Let {{formula:ae7a0cc4-6372-41ec-b0d9-70c0d2a43801}} be another chart adapted to {{formula:7a806b26-2cd6-4297-9785-c34e715d697d}} , let {{formula:042658ff-4471-49ba-87e8-951b3b518080}} , and let {{formula:9255ba09-530c-41f0-9bb1-9bbba04239cb}} be the corresponding map from (REF ). Then on {{formula:e3d1beae-dec1-47c6-8fe3-a16f0f48b7fe}} we have {{formula:a7c49c52-c8e8-450d-a089-733ddf0bbca6}} for all {{formula:92574aaf-cd1d-42aa-9b4b-2bf022e1da14}} such that {{formula:88b0746f-06c8-4797-b271-610278546f71}} .
For every {{formula:c1cf150f-593f-408c-8b6e-3352151531a7}} we have {{formula:547db802-9cd0-447f-800e-81704f206b3f}} if {{formula:8b078cb0-b3a3-4aac-a20a-df05142acef4}} and {{formula:b517d87a-c6bd-4e00-a949-c243a898a5de}} if {{formula:02867733-1488-4dd9-89a3-5cbbeada635f}} .
Very importantly for us, part REF expresses the idea that at radius zero, the values of {{formula:5f078564-2ea7-4bb0-95bb-dade6122990e}} do not depend on the choice of the chart. Before entering in the technicalities of the proof, we proceed with the topological description of the A'Campo space.
subsubsection3.5plus.7-.5Rounded simplices
To state the next result, we introduce the following additional notation. By {{formula:b8c985d7-b8d1-445d-84fc-c59d6699c691}} , we denote the image of the standard ({{formula:5043e9cb-17a6-4400-8bc1-22f735f37c2a}} )-simplex through {{formula:47cf399b-8c6b-438e-a26d-3bb434ff89da}} , i.e.
{{formula:7df4bd56-eda9-422f-ba43-65c3e2f9c816}}
see Figure REF .
For {{formula:fd0ac5d0-b78f-408f-bca7-7ffc72e0eda4}} we denote by {{formula:b1b332a2-a460-4a9e-b1d8-f1450eaf077e}} (or simply {{formula:a62aa139-b7ce-4ea6-9a1a-1e4dbc38506b}} ) the face {{formula:649b482e-3bde-46c4-b7a8-09318ca7caec}} of {{formula:1c16cfb5-bc15-408a-8c0d-d2770fac11f9}} corresponding to {{formula:d4170d9e-149e-4219-8654-40839ef5e697}} , and put {{formula:2289bbc5-018d-48a6-a2a9-126d326d92c2}} . Hence e.g. {{formula:1ef0f739-80c2-4230-b327-f991faf14a35}} . Note that we identify a subscript {{formula:a1456fec-53cc-4021-9cba-708385d62870}} with {{formula:fe6a16e2-fd40-46c2-bb0e-5d9651d7d59d}} .
{{figure:792ab468-d967-4b4d-a2cf-e142690268f8}}Let {{formula:cf3254b3-73a7-4642-888d-31d8b2b02b72}} be the stratification (REF ) of {{formula:0dc6d8cf-92d3-49db-aef0-c310b000516e}} associated to {{formula:abadeaf6-04ba-4e75-be47-db132dbed27d}} . The dual complex of {{formula:7bf4533c-6e35-4178-bf13-eb9d436bcadf}} is
{{formula:0742966c-6beb-4612-a1dd-42985b1c0c25}}
subsubsection3.5plus.7-.5Topological structure.
The second main result of the current Section is the following.
Proposition 3.6
Let {{formula:6f23d6ba-ec1f-48a9-946a-7eb15841789f}} be a pair adapted to {{formula:40243b9d-bd5a-4cf7-b046-eec781609a8a}} and let {{formula:90e3f540-bf8b-4ff4-8c3c-bb3baeea4795}} be the A'Campo space corresponding to {{formula:056b5126-5299-4920-a073-cb0ddf87925c}} . Then the following holds.
The maps {{formula:3703339e-6a5e-47bd-941b-14230b765d0d}} , {{formula:5f3830b6-fb1e-40d7-ae44-d07ebd54f5f1}} and {{formula:829e718f-e7e6-4afc-a862-4fcfc6a0276d}} for {{formula:8c93ce92-5167-4e0a-96a5-2bc150b70bac}} are continuous.
The map {{formula:1742683c-7440-488e-a56b-65cff4ef25cc}} is proper. Its restriction {{formula:89561d37-753e-4845-8fd5-cd2cfddd3cd5}} is a homeomorphism.
For every nonempty subset {{formula:dcceef7a-d4af-47da-a3e1-bda087cc85d5}} , the restriction {{formula:58105736-a718-474c-b60b-9a02f9bc918e}} is a topological locally trivial fibration, with fiber {{formula:f66aacf4-233f-4f2e-94c5-1fbb82aa14f9}} . Hence {{formula:065308a7-f55d-4bb4-b4ad-183e15641c14}} is the dual complex of {{formula:50aaf01f-5872-469f-80e2-86159e49b15e}} .
The subset {{formula:ae7be48a-c6d6-4758-ba51-292d6b564225}}
does not depend on the choice of {{formula:f4fe1b73-fde9-46b4-a8e7-0a257d11f3e2}} .
Let {{formula:2d0ed842-9f7f-41b0-b38a-9e7b1dbbf83a}} be the A'Campo space corresponding to another pair {{formula:4a91219e-db03-4dd7-9a34-ea60a83f4b40}} adapted to {{formula:61ba3422-2bca-4580-8aa2-28591e933b52}} . Using REF and REF , we can define a map
{{formula:09dc3fad-a7f8-47fc-b4ad-4752c7a12768}} by
{{formula:a74731db-026e-4bdb-aee7-4e5fa7f32e37}}
Then {{formula:5d404428-fad9-49b4-a430-faf6ac3688d6}} is a homeomorphism.
Later, we will show that {{formula:c506a547-af13-47b0-a982-cb609d42687d}} is a topological manifold with boundary. We postpone this statement, since at the same time we will endow {{formula:e7cade23-2c36-4ee1-b1b0-9d1fc4542348}} with a {{formula:a13df10e-5040-43ff-a51b-841602c36435}} -structure; in such a way that the map (REF ) becomes a {{formula:2d9e9419-eaa4-4de6-a062-09764d2822ee}} -diffeomorphism. In Section REF , we will even make {{formula:0b108169-0e64-4641-983c-810c69a9ca91}} a {{formula:d4bc73ae-1b70-4064-9e9e-07db61230ff8}} -manifold with boundary. As we will see, the {{formula:3f2a17b1-bb12-40d4-9d31-05a5b794b42d}} structure is harder to produce, and the map (REF ) will not be a {{formula:c39de447-9778-4c7e-a19c-6a78207f0ed6}} -diffeomorphism in general.
Example 3.7
Let {{formula:78957bd4-bbe6-45fe-a17a-85435674c681}} for some {{formula:817d8648-abea-4abb-a64c-8adcc2dc3549}} , and let {{formula:b51e71c0-13b7-4ef8-b2ac-8d1c612704af}} . Consider {{formula:edca991f-e665-4345-ae7d-b5f31c5f03a5}} consisting of just one chart {{formula:bcd15a08-b1b9-44ef-889e-39d91156ce5d}} , and {{formula:26225ac5-6c5b-42fe-b479-e235f6831c1a}} . Then {{formula:e777539d-8220-4149-99f3-a8f88e7e2ad5}} is adapted to {{formula:8c719ab5-d20f-460b-8ef8-828fa5b9974e}} . The corresponding A'Campo space {{formula:ac72a408-11ee-4d19-b24f-3faf1381e027}} is the shaded part of Figure REF , multiplied by {{formula:d558784d-95bc-49f6-a891-1ab70da26153}} . Similarly, for {{formula:9bb0260c-40d0-46a6-9cff-f65f5077036c}} , {{formula:465daf75-fe15-4665-a772-4db52f37d73f}} and trivial {{formula:c959b3d3-5f4b-40e6-afa4-7ebbc9ed1920}} , the corresponding {{formula:f1e25c1b-c1b2-4b96-902e-e975592a0ce6}} is Figure REF times {{formula:dd77a761-5fe7-4bd3-b2b3-9d4632a45d43}} .
{{figure:220f6789-7770-4898-85e1-e17959f53745}}subsubsection3.5plus.7-.5Proofs of REF – REF
The remaining part of Section is devoted to the proof of Lemma REF and Proposition REF . We remind the reader to use Notation REF .
First, we give a precise formula relating {{formula:c273a0d9-bf9b-475e-ae37-6f5b4b984ba1}} , {{formula:45c63887-4dae-410a-b8f5-f5dee1b5f8c6}} with corresponding {{formula:d2344960-6ae7-47d3-94ed-64049962f225}} , {{formula:a0aa6769-80f9-4643-9cb2-c9c51ba56b4f}} defined by another chart. We will use it both to prove the equality {{formula:7c68aaa8-4663-451c-9341-7fa558168ced}} asserted in Lemma REFREF , and to make further computations in Lemma REF .
Lemma 3.8
Let {{formula:60c2c07b-a769-46b9-ad84-6467b55cccdc}} be two charts adapted to {{formula:faf22f8f-20b9-4518-9950-482adcde4c8c}} . Put {{formula:df3a2536-e2f4-4477-87f2-577656376b34}} , {{formula:d55aa18a-4549-4245-aece-298eb06a01dc}} , and use Notation REF . For every {{formula:ef1715a4-cf71-4c98-9863-b69272d589c2}} , the following holds.
The functions {{formula:4f581a39-0c43-443f-a0e3-582f3d450888}} , {{formula:25614096-f3d2-4615-a55f-affe4dc9c8f9}} are continuous on {{formula:940c4f93-1dce-4b2b-a556-98ae63142861}} .
There is a nonvanishing holomorphic function {{formula:4e47de00-9523-46e5-b92d-afb6d946827a}} such that {{formula:b550867c-a8aa-4f8f-bd5c-39fe6a0330bb}} .
Let {{formula:63a18adc-3249-41bd-aba5-f2e42bc46c42}} be as in REF , and let {{formula:6a007d70-810e-4528-9bba-920f881e63f8}} . Then {{formula:77aa1053-48ef-4bf7-9658-552e3fd228b4}} , and on {{formula:36a14920-eaf1-4b05-a2bc-469a1ef09c57}} we have:
{{formula:89de74cd-796b-463a-b3e8-4637da8f009b}}
The functions {{formula:603d990c-5ed9-4fa1-a321-f4b6059e50a5}} and {{formula:48680315-b10b-404f-b36f-f117389bc6ec}} extend to continuous functions on {{formula:9335849c-f3a7-442e-bd4f-fc9dde593cff}} , such that {{formula:841a684c-056c-4041-88f2-debecdffca4e}} .
{{formula:d17c0e5a-1b38-435b-b705-95a41ced709b}}
.
REF Recall that {{formula:6c29310b-e2bd-4b32-9423-8e03e355a384}} , {{formula:64474818-438d-4b93-ab9e-ff0d118c5cd7}} denote the pullbacks by {{formula:aa9e6cb9-7188-4ab8-99d3-38a5e35d279b}} of continuous functions {{formula:036d2991-a1ff-4e44-a99a-18960b749dc1}} , {{formula:2a6eab10-d2d1-4f1f-8a77-51328fd6ad46}} introduced in (REF ). By Definition REF of {{formula:07abf8d1-0910-42f6-ae1c-4c8482289f33}} , the map {{formula:5d1cdd04-96fe-48ec-9f08-4b6170e4eb68}} is continuous, so {{formula:2c7bf9b1-f290-4d67-be2e-7ab5c1914e3e}} are continuous, too.
REF By Definition REFREF , we have {{formula:3c624f3e-d4db-4d5d-a715-14045f21d7bf}} .
Hence {{formula:ed83c148-a927-4767-bc03-df432ae2bd75}} for some {{formula:185e7faf-1fe9-4959-bdfd-c48ccb41b25c}} , as claimed.
REF Since {{formula:993cb991-de39-441c-ad56-46d99a4ee7dc}} is smooth and nonvanishing, the function {{formula:3108059f-7b5a-4324-b26d-09b59042b2c9}} is smooth. By definition, both {{formula:ebf830f1-4368-4227-b56c-72fec88dd79a}} and {{formula:0ad66cd9-6423-4aa6-b824-5a0f3e0b24a9}} are zero on {{formula:cc002fa4-c6fd-4dd2-8f1b-ea52c2382a3c}} , so the first equality of REF holds trivially there. Away from {{formula:10f4bc30-7e8b-4e3a-8685-b831b63483c0}} , we have {{formula:6da11fad-1f39-4d80-b167-754c25c3fb9a}} by (REF ), so REF and the definition of {{formula:947d484c-7a58-48de-b086-9f41bcc5ff54}} give
{{formula:e0dc2622-3043-4b3b-9198-3cc783f5987b}}
which proves the first equality. We will now prove the second one on {{formula:905b7b7a-1824-4d30-92db-20f93ac6262a}} . There, substituting the first equality to the definition (REF ) of {{formula:3983dca8-981b-42f6-a8b9-70815f28b0f2}} gives
{{formula:60b51425-7742-4773-bcce-725335792379}}
Once we prove in REF that {{formula:a5fb1c91-2a49-47be-84fc-e4b232766f21}} and {{formula:f55491dd-49fc-4266-b74e-e902bdd8f08d}} are continuous, the above equality will extend to {{formula:4ff4bff4-a13b-4632-9316-67cf0e55a661}} .
REF As in the proof of REF , continuity of {{formula:5d4b5b98-5e99-4751-a71e-29764263df4a}} implies that {{formula:e4ab0178-8d30-446f-84b4-fe4bd5b3b273}} are continuous on {{formula:cb585b16-99cd-45ad-b8ec-703aa9a77b68}} . Fix {{formula:dc808b18-119e-4523-ade4-c837ccc3e784}} and a sequence {{formula:c03b3c83-5d52-41e8-ba00-97968f76ea27}} such that {{formula:815bc6a1-df45-4834-8409-69ce0c6bcf1b}} . Since {{formula:f21849ea-e4eb-4ee7-a4c2-4f7decba5e83}} , after passing to a subsequence we can assume that both {{formula:c99b73a7-2419-4423-943d-635477aaf555}} and {{formula:6a7b8163-43bb-4360-aca9-6bdbc9587278}} have limits in {{formula:54be0b32-c8ed-4f35-b97a-1fb2e7cbcfe9}} . Call these limits {{formula:93b79680-fec5-4741-aace-9097ad76bc50}} and {{formula:5eda845a-a50d-4f35-8ffd-13b018cfe782}} , respectively. Since {{formula:26cb4545-3fbe-4621-9831-2e8d0eb08d6e}} , the second formula in REF , applied away from {{formula:ea9148ba-0d88-419f-bf15-d9c858b463b9}} , implies that {{formula:b7c35206-82f9-4987-9942-0bff8439ec0d}} . Taking for {{formula:4c355c24-51f3-48ea-b088-f8e153abc209}} a chart {{formula:e631913c-ceb8-4bf9-b48b-36f27f9a0c79}} of the atlas {{formula:31676ea1-9d2a-4114-89f8-6d5196f4972a}} fixed in Definition REF , and writing {{formula:9d978e62-9d87-45f7-b8cf-1bea53e463d9}} for the corresponding map, we get {{formula:47e2e495-4830-417e-9e65-9ba1c4c66a9e}} , for every {{formula:62f799ca-f6fe-4a87-95a9-3912c7e120a6}} such that {{formula:e089f2bd-c4b9-4f65-b4d6-9a46a717517d}} . Since {{formula:5a438050-8204-4861-ac73-03a93163cc44}} is supported in {{formula:017c8df7-b56f-4c1c-b6a3-fa42662cecd6}} , we infer that {{formula:0fbf92fc-6ea3-4900-9dee-dc44c21b1680}} for all {{formula:a206012c-8153-4f54-afce-4fbb84e3ceda}} .
By Definition REF , the function {{formula:8387011b-6273-420a-82c7-cb26d11bb6ef}} is continuous. Since {{formula:14cfbc88-ceee-4ff9-9660-0b8eb1f7f63a}} and each {{formula:bfb7ea3b-041d-4bfc-a4e2-9ebff0452dcb}} are continuous, too, we get {{formula:eed1df79-985b-46ef-88d3-5877e420c59f}} . Bijectivity of {{formula:4eca995f-a44c-46c2-a82a-76e2c3992e64}} gives {{formula:0a4fb2c1-7970-4f37-86f3-e2c41d2f2b9a}} .
Thus from any sequence {{formula:a0423887-276a-4389-99bb-6830e0f869bf}} such that {{formula:48cb8053-783e-4ec3-862e-7e1d98d2de10}} , we can choose a subsequence {{formula:6dec4ba5-2f24-4c5d-9407-e1bb90be7fa7}} such that {{formula:51490963-bad8-4b2b-906f-fee8994088b5}} . It follows that {{formula:5db5b9d5-53ac-4d03-b269-15bbe22a3a94}} is continuous at {{formula:2570fe79-d8b0-44e8-9e9a-195f2a1f9fb0}} , and the value of {{formula:c337ba6b-c82e-4450-8385-9bd0f5429f88}} at {{formula:643116b8-ac28-494d-81bb-ae54dfd34ebc}} equals {{formula:4eb08a1e-7532-4f5c-bbeb-ad40bf339e8e}} , so it does not depend on the choice of the chart {{formula:9b126ba6-6d37-4930-a989-a52f3a7d81df}} defining {{formula:01718be7-802f-4249-b438-2a36f7bf1940}} , as claimed.
It follows that the second formula in REF holds on {{formula:ad095317-4388-4616-be0a-4be4cfbda5a3}} , too, which completes its proof.
{{formula:0e18fa8c-5ebc-4d1f-b3e2-aec914fc5cb6}}
Proof of Lemma REF .
REF Applying Lemma REFREF ,REF to {{formula:26063b81-f6ab-4ed4-a15d-cb1e2e274473}} , we infer that {{formula:c2ed0de6-a7b4-4059-98ce-743406010db5}} , {{formula:36821451-77c9-423e-8cdd-89457fece588}} extend to continuous functions on {{formula:f3b17383-fea5-4683-82a8-fc60321b9f06}} . Since {{formula:d5bd7d22-5b23-4397-84b7-c3093a49673f}} is continuous, the functions {{formula:db1e2842-cede-4cf6-9b7b-7fdda6204ea1}} and {{formula:9d430a36-91e2-43b7-ab5c-3f08f88d79d0}} are continuous, too. Eventually, {{formula:d2301cba-aeaa-4f67-86a4-f5aaa30be259}} pulls back to a continuous coordinate on the preimage of {{formula:2ed753a1-2c12-420c-98fd-7bd48eb5f074}} in the Kato–Nakayama space {{formula:4b2f6e1f-e6ac-4d05-9789-43c0049292f5}} , see (REF ), so its further pullback to {{formula:3732bf2c-d7a9-4b9d-b369-a58f94e74bf5}} is continuous, too.
REF Since {{formula:6939a1d0-b4b9-4021-918b-be5a642bf7b6}} , we have {{formula:7c01483b-6934-4939-9410-94ec674ccef9}} , so {{formula:6f930b24-a0cf-46b7-9a3e-b9f19ed0031a}} . For every {{formula:41e3708b-7483-4d08-b8fd-16e207c89034}} , definition of {{formula:7daa2a50-d8ab-45bb-a134-393c7d57636c}} gives {{formula:ab707d52-8e5b-44d5-ad2f-f8a4790d5806}} , so {{formula:f2d57678-0dd0-49dc-a6db-aba520f7eb5e}} . By definition (REF ) of {{formula:d9a486f4-eee0-43f2-9bef-9fcb1189b698}} , on {{formula:8148fa21-f044-4a68-bf7b-5cf21faea705}} we have {{formula:84a810d1-1329-4153-8b99-9e8ec854de3e}} .
REF Because {{formula:8dc29f33-d85c-4ed5-ac8d-e5789434c14f}} is adapted to {{formula:56ba068c-5c7b-4d0e-9892-a9aecbdf2627}} , by Definition REFREF we have {{formula:a9102c20-da5f-4a98-a49f-dbc8573bcd2a}} . Hence {{formula:8adf2dc2-f745-4dd7-8d10-d6c7cd474f78}} . On {{formula:8a7c987e-bd2f-4571-a95d-bb4dad2567f6}} , substituting the formulas (REF ), (REF ) for {{formula:2c176583-480b-4ea0-8388-de7423b48a8e}} and {{formula:18acae52-0ab1-4521-bc8c-ab82e43ed8fb}} , gives {{formula:2742856d-7d7a-43b1-b872-336e77408dcc}} , so {{formula:1f548a84-78c1-4506-80d1-1c9b34513dc1}} by definition (REF ) of {{formula:35ef71c8-cfc5-4c4b-a9be-70421a57d1e0}} . Since by REF the functions {{formula:756d1dad-9f26-41b7-9eb8-a598ec064f39}} are continuous on {{formula:43884517-a1ec-4bbe-a952-3fa4ffaefb51}} , we infer that the equality {{formula:b8280241-df21-4233-a6c4-5d87d5890d8a}} holds on {{formula:350ea8c2-2528-4cfa-a721-847a14dba4dc}} , too, as claimed.
REF Put {{formula:d778f73f-be2d-417d-9359-fc8c79085b01}} , and fix {{formula:fb8db851-51f9-4568-a636-ab8114235989}} such that {{formula:1bb69195-c988-4000-9f77-aa23e223d5c3}} . The set {{formula:52b63435-ccdc-45e7-8eac-9c760a5132a3}} is a disjoint union of {{formula:2effd32f-5e99-4ea9-8ca5-5898652c7f3d}} and {{formula:d4032abe-056f-4c69-b9a3-d2d183704497}} for all nonempty {{formula:0a280049-4984-4710-89a9-0dd03b16eca6}} not containing {{formula:b5c1331c-f584-4ebf-9d88-2fe950e782a1}} . By Lemma REFREF , {{formula:0bcf89e5-9396-4783-871e-9f5b526b45a5}} on {{formula:25e14b3d-2660-4199-818a-33245f90cb26}} . In turn, by REF , both {{formula:96d978cb-eb27-4918-bae4-22cec1b63922}} and {{formula:2ebd6f25-df12-46a4-812a-4463b4cd499b}} are zero on {{formula:b66f8ac1-b2f9-413c-9ba3-4a990ec7fd4d}} whenever {{formula:445b45f1-b9ac-4887-9f20-fa44e4e1a695}} . Thus {{formula:000ef131-dca0-4738-af25-c24b354fab53}} on the whole set {{formula:7c22061f-5400-4760-8c54-c002bd4cab61}} .
REF Fix {{formula:a3ec1bd5-388d-4db7-87ae-00502a758abc}} . By REF , for every chart {{formula:c2f5d0c2-57f6-4e25-9191-ef407866bcbb}} of the atlas {{formula:c32a884d-2e70-42be-8206-c462af1f47ce}} fixed in Definition REF , the corresponding function {{formula:4cda5ba7-4af5-4154-b0a5-321bed332d01}} satisfies {{formula:ba27509c-116d-44ca-a197-262ccd683bdc}} . Hence for every {{formula:16e66734-f21a-49e2-be1a-a9418a2a2774}} , on {{formula:68fe3d84-32bd-4a6c-909b-40c1484bdde0}} we have {{formula:0af0a0fc-d083-4911-9915-fba08d82528e}} . Thus {{formula:8c265428-ab61-40fd-9ca6-882d54a1e199}} on {{formula:b7a20def-742f-4ff6-a26e-d4280fe01fa5}} , as claimed.
Now, fix {{formula:d1ec8226-4373-4ef2-98f8-ec98c9461e18}} , and {{formula:976bb1b9-b5de-46bc-8350-cf9e5abbd881}} . Since {{formula:a196590b-5b3f-4b78-acdc-edcfc4658df8}} , we have {{formula:dac79cdb-94f7-4f31-896c-b2100fcfcb4c}} for some nonempty {{formula:f39d9faa-7ac3-4cbd-afd8-b39185156d72}} not containing {{formula:3cb70a74-fe12-4c2b-a244-61393fd55d42}} . Take {{formula:a794c3cf-164d-4e52-88a9-70221cde934a}} such that {{formula:114f3582-9ea4-4808-aeb5-67fded0c41d0}} , so {{formula:84c1ef19-9f84-467a-b919-22daf8bc5c36}} . Applying REF to {{formula:94918fc9-b75e-41bd-b6a0-98f8a32d9f38}} , we infer that {{formula:5b0305ad-7631-4901-b211-c489a42e5e9f}} , so {{formula:b22c61e3-160b-413c-9ba6-6730fc0fb9b7}} . Therefore, {{formula:bcb47667-cb28-4aca-bdf2-a885378c71e5}} , as claimed.
Lemma 3.9
For a nonempty subset {{formula:50fb6cd0-320a-407f-a7dc-11af61ef07fe}} , the following holds.
For every {{formula:fff81bba-adaa-47e8-9d34-d3f5c77e05a1}} , we have {{formula:ba3fce57-cb64-4cde-857e-b4bf2f5b3ab2}} .
The projection {{formula:f368f1e6-252d-480d-a8f1-2343cd5bb3c5}} is proper.
The subset {{formula:364fbd92-2eef-4b9d-87cf-01cf5cf55436}} equals {{formula:ef299e2a-ddc9-4445-952a-5c48b750f0ce}} .
{{formula:5fa8f2fa-c2a3-4e3a-bc22-6af94a71c986}}
.
REF Let {{formula:f500477a-ae71-4a62-9f6f-6b839bc04d84}} be a chart adapted to {{formula:34f30fd3-cbb7-4755-97ff-6bc5f6c270a3}} containing {{formula:1954cef1-19bd-4415-acca-1784be1eb819}} , let {{formula:a0de6c30-5302-4c64-9619-1a2f7022cc82}} be its associated index set, and let {{formula:6e90350c-18e8-40a1-8d16-7881859bd98f}} . Fix {{formula:c1d38bb7-414a-4e07-8fa1-9a99c9d26d8b}} . For any {{formula:6dec4ae9-3819-4cd0-b5c3-3d5f62852eab}} , we have {{formula:fc8a6f81-b20f-4d6e-ae5f-3ec0b5ab0ed0}} by Lemma REFREF . Fix {{formula:cbaa8c36-1541-42b3-82a5-0609e3c11e6b}} . By Lemma REFREF , we have {{formula:2e70d5f2-011a-4415-9ee4-0da87630c544}} . Thus Lemma REFREF implies that {{formula:ce850e1f-d1bf-4a25-90e9-966e031546c2}} . Moreover, if {{formula:f5237da4-fcd3-4e32-aeac-de847b59f9e4}} then by Lemma REFREF we have {{formula:e5fe4208-0f3c-4c4b-ab94-bbc07e231d1e}} , so {{formula:d7d09a17-dace-48c2-9779-778cae6f416f}} . We conclude that {{formula:a2076626-2529-4057-ad55-d6708a882a89}} , so {{formula:5922585a-76c8-42ab-880e-fd469e5ff705}} .
For the reverse inclusion, note that the subset {{formula:1eae5988-1f8e-4f55-aa3b-d8c6f61169f2}} is closed. Indeed, it consists of all possible limits of {{formula:4a0e931b-d9d9-4efe-9f11-231ff702e88e}} , where {{formula:b5c3ecaf-587f-4f8d-b551-66158b129864}} is a sequence converging to {{formula:65b6fcf4-3768-4418-bbc0-fb85ba87243a}} . Therefore, it is sufficient to prove that {{formula:67e27365-0e2d-4908-9c02-ef3f667892ed}} . Fix {{formula:2fa141bd-7973-405f-a15f-3d09434eb93d}} . Since the function {{formula:04d4fe99-bd7e-47c2-97e6-a661add56635}} is bijective, for each {{formula:c6bc56b7-ee69-4d1e-ae58-928eb45d135e}} we get a unique {{formula:e0059b49-fc95-4de1-9985-a443ce2e4484}} such that {{formula:771b3add-6d3d-40eb-bc0c-31aabb483583}} is the {{formula:364602f2-8fba-4377-ad91-b42c90298698}} -th coordinate of {{formula:70d132f8-7d3c-43cd-874b-ad7fd889edb6}} ; and {{formula:dcb7031c-5589-4cbb-9137-5ae7b7e6cf7c}} . Put {{formula:958de824-f1cd-4609-8676-a6656d3d8f88}} for {{formula:35dd10ad-a11e-4f22-b27b-83ee32811a08}} .
Recall that {{formula:ee8d2694-07a0-4c24-a5a5-05150898b1fa}} , for {{formula:244b976a-337f-4289-b10e-1f95eea695ae}} , are some of the coordinates of the chart {{formula:3e07c927-1dca-41e1-bb86-1fde418cc66e}} . Hence for sufficiently small {{formula:41e14e34-0e00-4fa6-8695-fb64973ae19f}} , there is a curve {{formula:78b33403-c455-4ea9-8bfc-e61546e2099c}} such that {{formula:b5a33e75-cc75-489d-bc01-9bbec075ce03}} for all {{formula:6bdfa4bc-c60f-4528-a587-f889210f10d9}} and all {{formula:167f7958-fe68-4702-a79b-e7d9bd38537e}} . By Definition REFREF , we have {{formula:d0488dbc-309b-4898-9c17-0f931303424b}} , so {{formula:f728075b-a3ea-4df3-abcf-961d9c690d31}} for {{formula:f3d4577f-af08-418b-9b21-afcd5097d230}} . By definition (REF ) of {{formula:ae7175ee-48b7-49c4-ac4f-6356c885966e}} , for {{formula:e4695ec3-c234-4f02-8013-a4508e0c0c34}} we have {{formula:49100f01-01d4-4643-8211-b4e814003a88}} , and by (REF ), {{formula:b4fa5890-8710-4324-a8cb-2da6118279df}} . Therefore, {{formula:b54e1251-839f-4b51-ac36-872c4146d229}} for all {{formula:6242ca5f-ed15-4996-975e-811aaae3c5a8}} . Passing with {{formula:7751a644-36c0-41ab-a752-8f9edfa2283b}} to 0 we get {{formula:5113ae06-739b-4f2b-9c1c-55d1d15a921f}} such that {{formula:c1bbf4ed-4b38-43cd-bf60-fbcc1162c261}} and {{formula:b53312c2-96f4-4374-aa4e-8f41c999fc1f}} , as needed.
REF , REF Follow immediately from REF .
{{formula:4839d6af-c13d-4cd1-94d3-d1759cf34b8b}}
Proof of Proposition REF .
REF Follows directly from Definition REF .
REF Properness follows by properness of the maps {{formula:b77c622e-383f-4734-9ad5-fc06955ad910}} (see Lemma REFREF ) and {{formula:3a9ed83d-fbfb-4429-8817-28666146409d}} , and the rest is immediate by the definitions.
REF Let {{formula:5e8ee61a-0f69-434f-9da7-975dc5e94aa7}} be a chart adapted to {{formula:35deb6cf-0639-46fd-9329-231b5dac0180}} . Let {{formula:7a585eea-747d-4130-a278-c16d5d943f48}} be the natural map from the Kato–Nakayama space. In coordinates (REF ) on {{formula:e8ea3aee-9ab8-4688-b944-7ce27f3cc5e5}} , the restriction of {{formula:1db99976-b981-4942-98c6-53f984fb2d5c}} to {{formula:ca5bdc39-2e83-46bc-aac0-07d077fe3f85}} maps {{formula:0378aaf9-ead8-4b83-9640-5d27b079f770}} to 0 for {{formula:86a1a20f-f78d-440c-b432-38c4924c0e56}} , and keeps the remaining coordinates intact. It follows that {{formula:6cf0a5ee-78d0-463e-8b63-2d3f3e66cc2f}} is a topological {{formula:5cc33b4c-4ce3-4a25-bd45-f38898d13935}} -bundle. Thus the first part of REF follows Lemma REFREF . The second follows from the definition (REF ) of the dual complex.
REF By Lemma REFREF , for every nonempty {{formula:4a21f045-03b3-42b8-b610-16d51500ff1d}} , the subset {{formula:cd046e2f-9e5a-4046-8765-cac7e1392a22}} equals {{formula:1e9531cf-cc06-4628-8001-35c7fb0cb371}} , where {{formula:ad30b49f-ad46-4735-8e61-3a805578abfa}} is the natural map from the Kato–Nakayama space. Thus {{formula:d1a51954-e71d-405a-8990-3ff623f4bcd5}} does not depend on the choice of {{formula:56097f31-2728-48d3-8e5f-377b5b05f01e}} . Therefore, neither does their disjoint union {{formula:24d5764e-d179-4c51-9a2d-8b7f62fe0464}} .
REF Let {{formula:b6bb2359-598b-4256-bb0e-a35088098429}} be the natural map induced by {{formula:a7a478fe-4ffa-4cf5-a663-45dc245baed5}} in Definition REF . Let {{formula:4d1f63ff-cdd4-4e01-a148-3d6cbff69747}} , {{formula:c54af273-6e11-4cd6-85b2-567a52ece232}} , {{formula:3931cd22-83e0-4a20-982a-c6f9e14e5c70}} etc. denote the maps from Definition REF associated to {{formula:106c122e-096a-45f7-b003-602c119ec43f}} . By REF , the subsets {{formula:4e888d78-385d-49c9-928d-9ad525d67dcb}} are equal, so the restriction {{formula:431da42b-3eab-4159-9f03-e3318a24cca3}} , defined as {{formula:ecc34e24-cec8-48cd-a121-e62cacbe68d8}} , indeed maps {{formula:cad0c0bc-8eb8-46d7-ba5b-912645002ada}} to {{formula:18071abb-5601-475c-85ea-73e9f2502408}} . Clearly, this restriction is continuous. In turn, {{formula:d924fc16-149c-49d8-a735-d5a46523d3f9}} is continuous by REF . Since {{formula:07c8979d-eae6-4b22-a7fc-68ed5707fb4f}} is a closed subset of {{formula:2211b5af-ad46-4273-9521-9c188b2023c3}} , to prove continuity of {{formula:56d1ca17-61f5-4936-94bd-129a8764d39a}} it is sufficient to prove that, for every {{formula:bf7b4e4b-2bd1-4203-b1ed-c1478b3fc537}} , and every sequence {{formula:e7bcfc35-e800-4bcb-be10-ef7dd9499c7d}} converging to {{formula:fa903420-e59a-451f-85a9-fee84eb668e4}} , the values {{formula:6244d8ab-d706-4515-8563-f00a246d9296}} converge to {{formula:8ca8dace-c238-4d65-b196-1c0860e52a34}} , too.
Since a point {{formula:f1de7181-2a34-49e6-91ab-6037c4ee18e7}} is by definition a pair {{formula:ac137849-d048-47d0-a369-84adfc227c5c}} , we need to show that {{formula:6be650b9-2659-468f-a957-68fb6426825e}} and {{formula:e7f13df8-a66c-4dd4-8d1e-45dfc0952ed4}} . By definition of {{formula:0e015c6d-35fb-426c-9bae-e290d67649b9}} , we have {{formula:4ddccc43-629f-46bc-94b9-e4a2ea3c6be6}} , so {{formula:bc99ceb7-129a-4d0d-8f05-c009cd2ddfb9}} , because the map {{formula:d2e74c37-048b-470b-a419-a5ca87955498}} is a homeomorphism away from {{formula:6af49635-0080-42ac-89f4-87f58768edba}} . This proves the first convergence.
For the second one, choose a chart {{formula:9ad3c521-5647-4884-8341-30b030b00514}} adapted to {{formula:246f8da9-20b3-440b-b456-0ad37b074038}} containing {{formula:091d97d3-78c5-4f76-bcda-7241507fdc32}} , and fix {{formula:1ae607ac-7eed-4b73-b912-9c7f72f2f9f7}} in the associated index set (REF ). By Lemma REFREF , the function {{formula:6599785d-ab69-4554-b871-dffbcd8e59db}} from (REF ) extends to a continuous function {{formula:18b2b15b-bbb9-46e5-9b13-0ab5b4535444}} , where {{formula:9a3edb8b-6607-442b-b1f9-de9467e28aef}} . If we denote by {{formula:b3106e11-5ece-4b2c-9158-ec8ce238e345}} and {{formula:b91866a3-fdad-45b7-aa82-4644c0ee2f85}} the images of {{formula:221359a4-e7d0-46e8-9365-78c167de04e5}} and {{formula:90efd74a-dc04-4f83-a94b-113c753156fd}} respectively, then we have {{formula:6595a35e-6461-4da2-8d3a-94ac242d7869}} . Therefore we have the equality
{{formula:ee717990-7a9b-42fd-ae57-8831524650a4}}
Consequently, using Lemma REFREF , we have {{formula:3a996aaa-1d86-496d-8b1c-1667224faec9}} as needed.
Therefore, {{formula:9e7b7bf6-97ff-46ec-b8ea-b0b67101178c}} is continuous. Continuity of the inverse follows by reversing the roles of {{formula:1dd8f5d7-296c-47ee-b80b-d4484dc9fa45}} and {{formula:135ffad1-bce1-41db-830f-2e45eb6e4763}} .
subsection2-.5plus-.7.5The {{formula:cdc0a44e-4096-496d-8e5e-5d8cb9d54b89}} -atlas on the A'Campo space
Fix a pair {{formula:42435fab-7f69-4b3c-a58c-c4b95374bc7c}} adapted to {{formula:e726c888-b986-4b90-b0ac-1a72e8c71115}} , see Definition REF . In this section we will introduce a {{formula:fd57c08e-3d53-43ba-a952-e03f55039c4a}} -atlas on the A'Campo space and we will show that if {{formula:bc902469-a0a5-416e-9605-9605149bcc60}} is another pair adapted to {{formula:39010279-5c4c-4a61-8efa-1fdd8c378d45}} , then the homeomorphism (REF ) connecting the corresponding A'Campo spaces is a {{formula:7c183355-03c6-4321-975e-b390c54bdb31}} -diffeomorphism. As the reader may expect, this {{formula:17355094-7b08-4261-8029-d561525165ef}} -structure agrees outside {{formula:f671239f-4827-4ee0-8ada-45608f4e4105}} with the natural smooth structure in {{formula:968ed2bd-2c55-4dba-97f4-a4449cd16839}} , i.e. the one pulled back from {{formula:2a1d8e8f-7db7-4b39-be61-d8fcc096e2ba}} by the homeomorphism {{formula:ac13e2e7-6380-42d1-97bd-f4b685ee5dd4}} .
subsubsection3.5plus.7-.5Definition of the {{formula:a3e284cf-a084-4656-910c-262139a1fbb3}} -atlas
Fix a chart {{formula:0a87c044-e5ef-47f0-9278-b7fe4774cd1d}} adapted to {{formula:68277ec2-1e51-4dcc-b8c0-5e3f6c2b4477}} , not necessarily one from the fixed atlas {{formula:c7f2e843-4abf-4084-8547-1347cf965b53}} . Put {{formula:351d0a2d-8fbf-4bda-a19b-9431bf33f60d}} , let {{formula:6d9b40cc-7219-4549-8db9-5373df07499d}} be the associated index set, and let {{formula:5a83bed9-b5d7-4369-b59e-0388dba43082}} be the maps introduced in Section . For every {{formula:2800e392-6fa0-49bb-9ad7-14072cb72445}} , we define
{{formula:9db1e511-0f55-4428-ae90-995811e3216d}}
By Lemma REFREF , the subset {{formula:5c6860f2-3cef-488a-937e-6c63df83ca9e}} is open. By Lemma REFREF , the open sets {{formula:df9598ce-edba-4bad-89f0-28f05faa3145}} for {{formula:28e26147-0b94-4fcd-9420-612cd94900d6}} cover {{formula:29bc3eaf-26a7-4975-90e2-7d8ee6156782}} . Put {{formula:c89277e8-7d5e-4de1-a4b4-3061b68721e9}} and {{formula:b9a5af72-c443-47ab-b3a0-e3649c37f6fe}} . By Lemma REFREF , for every {{formula:55531e64-c463-4e94-a55c-1605cdab34e3}} , we have a continuous map
{{formula:2ae90bf3-61d1-4be8-abe9-19b068d9bac1}}
see (REF ), (REF ) and (REF ) for definitions of {{formula:b72fb39d-1a16-408f-acf0-93103ae33e39}} , {{formula:f590d835-a4ba-45b9-b86f-1339ca94b44d}} and {{formula:5d03ad87-ff04-468e-af14-9368af12d1b2}} , respectively.
The pairs {{formula:342102aa-ff65-455e-9157-33b26cea64cd}} will be called the {{formula:c36ee6f9-e6ab-4a59-80b2-82d6df701567}} -charts corresponding to the chart {{formula:e20f0cc5-d3a6-484f-abaa-3190e8c2e3ea}} adapted to {{formula:0b513809-6ed3-4b97-a350-bfef681ab0be}} . The following lemmas, which we will prove later in this section, assert that these charts indeed make {{formula:e95592c0-4273-4cdf-b33e-5a3c20beca25}} a {{formula:4444fafa-d8d6-4c49-aa28-79600fadbe22}} -manifold with boundary.
Lemma 3.10
Each map (REF ) is a homeomorphism onto an open subset of {{formula:7ac6206a-89c4-4e71-893d-c853271c0030}} . Moreover the restriction {{formula:2223378d-58a2-49e8-8263-e0fd5f395e18}} to {{formula:3f5d5336-8e18-4776-9f18-8a3583e2e5ec}} is {{formula:ec8f92a8-6322-407f-af85-3a4b65b74a1f}} for the natural smooth structure in {{formula:d003893c-da85-4a65-9e3a-5505530004e5}} .
Lemma REF endows {{formula:033a8af0-3e87-42b9-ae42-da521401db69}} with a structure of topological manifold with boundary {{formula:13f2143c-d9a7-4b58-a824-fd1d422bb33d}} . The {{formula:99279b23-cdc9-45b0-a909-2dbac3131ae0}} -structure is provided by Lemma REF below, which studies the transition functions between charts (REF ), possibly associated with different charts adapted to {{formula:41b45e8e-385e-4392-94a5-985f93da79b8}} .
Lemma 3.11
Let {{formula:0a0c09ae-b874-4089-b6f5-c4b1c99a01a3}} , {{formula:24679994-0739-4553-b926-af9a6ce9a6b6}} be charts adapted to {{formula:a60f5c60-709c-4217-abbc-f79c5ac61dfe}} , let {{formula:bb619c5b-7266-41e3-b44f-2ac9f364fdb1}} , {{formula:ff96b27a-0b05-48bb-90bc-e3b94c800779}} be their associated index sets (REF ), and let {{formula:c2f1fcd2-82d9-4217-a94c-f102599b74f4}} , {{formula:b3aba91e-48b2-4f6f-a056-2bb5c56a55b1}} . For {{formula:6f120515-f4f1-4d04-987b-217ac103585b}} and {{formula:f8fb853b-60a7-4d41-bb2e-3d3baa7683d8}} let {{formula:65fbbdcb-2e0d-4057-9f83-0e1d36b08dcd}} , {{formula:f432552b-c8f0-4b14-85e2-1b24e916d0d2}} be the {{formula:3b9c9d36-82af-4742-ae27-9de813ce00a9}} -charts defined in (REF ) above. Then the transition map
{{formula:76de6bdd-b523-4877-a6fd-882f156b3137}}
is a {{formula:668db2f9-9033-420d-a170-12f66d499035}} -diffeomorphism between open subsets of {{formula:762e66bf-be64-4bea-8f19-0376a6e847c7}} and {{formula:c25f34d3-af2a-4cb6-9069-499eadbd50ac}} .
{{figure:1d2c58a6-2d7f-4a72-8d0d-636de31c1d18}}Since for every {{formula:ab7e55da-5dd5-4026-a44f-327ccc0c9349}} , the space {{formula:c3f84ea3-1ebd-4ab9-b904-fbfaba8b33cd}} is a {{formula:43c19696-37ed-453c-be9d-08138a053d66}} -manifold with boundary, refining the {{formula:02030f4f-7885-4e5e-8d82-d5b123a176b6}} -charts corresponding to each adapted chart {{formula:6dd3dc7c-1313-483d-ba63-3283bbe04274}} , we obtain a honest {{formula:2792d059-057a-4ebb-b695-f48fd49d7807}} -atlas for a neighborhood of {{formula:a07d3968-f225-4c29-b0f3-b6956486382d}} , whose charts have images in half-euclidean spaces. In order to have a {{formula:c8bfcb6b-05f5-4ddf-bf6d-a9a37bf01298}} -atlas of the whole {{formula:ca027a57-bec4-4a34-87ee-a9c1195d8696}} , we notice that by Lemma REF this {{formula:52e4513b-8709-49d4-b772-22f7f5d0b216}} -structure agrees with the natural smooth structure in {{formula:b1cf826a-70b0-43b1-bedc-47106c65e44a}} .
Proposition REF below allows to speak about the A'Campo space of {{formula:00ae1dc4-ba45-4611-9def-3c55636250e5}}, as a {{formula:b324eecd-4845-4b49-9471-0e156958f3e8}} -manifold with boundary. The key properties of this space are listed in Proposition REF .
Proposition 3.12
Fix two pairs {{formula:3691863c-94a8-481d-8941-55e0506be393}} and {{formula:a0aa469a-11fd-4b46-88c7-f77f07436465}} adapted to {{formula:7bacbd1d-8392-4583-a4e0-3ee03561c309}} . Let {{formula:f4289d14-76ba-4505-bcc4-39def4337e19}} and {{formula:e0fa89f6-6926-430a-a20e-868656750ed8}} be the corresponding A'Campo spaces, each endowed with the {{formula:4677d80e-9507-4dea-b9de-cab82866e071}} -structure defined above. Then the homeomorphism {{formula:b35a347c-d7ab-437d-80db-5c4931744473}} introduced in Proposition REFREF is a {{formula:b6ae3f41-e5eb-473c-b304-61e0eb414bfc}} diffeomorphism.
Proposition 3.13
Let {{formula:8565ab6b-f4a5-445f-964d-76be20dc99ad}} be the A'Campo space of {{formula:ccc97ea5-4268-4ee2-bd99-174771b2eecb}} , with the {{formula:402ae807-1075-4122-bb0c-758eec5e4317}} -structure defined above.
The map {{formula:2c8fa717-9cd2-493a-a4ee-df67d9666c8b}} is {{formula:ee718ee1-5622-438e-bae2-a46e859f9242}} . Its restriction {{formula:a9b2cd87-d234-4bab-a896-fa6f08e40e6b}} is a {{formula:f4f35929-fe43-47b5-b737-cb56c5403285}} -diffeomorphism.
Let {{formula:866da47c-2f44-4f68-b7d4-0a6313dfcbf7}} be as in (REF ), (REF ). Then the map {{formula:23119c7c-f53c-4185-bad6-eb500363053e}} is a {{formula:ab27e3b4-da30-4b68-a0f0-67e839bbaeed}} -submersion. In particular, the map {{formula:722eb99d-4e26-44b1-8e2c-0e6bd2f2bc42}} extending {{formula:bded61a5-a035-4ee7-8ed9-4ee05eb9d823}} in (REF ) is {{formula:344f3ea5-bc12-4ca5-b7c1-af571f7a2e65}} .
For every {{formula:64f2f16f-1a0d-4b57-89be-4e0fb0c3790e}} ,the function {{formula:6792604f-3a13-4442-94e1-7b08433e18ab}} associated in (REF ) to the pair {{formula:5f0010a7-adf8-4872-8e34-a6e2e24ddf16}} adapted to {{formula:702f9dd5-54bf-40b1-bb54-1a3dee609c4a}} , is {{formula:84156da2-c331-44c7-94b6-a1fbf7690440}} .
Example 3.14
The {{formula:ef21f0f6-5bf1-46c5-bdb3-8fa91b929365}} -structure on {{formula:80b72de3-6800-4650-9263-ebd87d880002}} constructed above may not be {{formula:10bc04f9-9c3b-4d9b-9fac-a88275ca901c}} . Indeed, let {{formula:b327fea4-30f5-413a-9478-d046bae8e844}} for some {{formula:1e8e6e2c-bac4-4f59-b858-f18d9781f51b}} , and let {{formula:5bae43ff-0983-42ca-972e-a95cc2b17362}} . Consider two charts {{formula:dbaceb8c-5153-4777-a789-894d12915397}} , {{formula:0074a501-601a-4920-a85d-2fa513bd53b1}} adapted to {{formula:e36552f4-d9bf-4ac5-8310-abcdecf0d534}} : one with the standard coordinates {{formula:4f60afdc-bf32-40c2-8a5a-b7fefd9036b3}} , and the other with coordinates {{formula:2a8f4de2-eb49-47af-9131-48ed5c280755}} given by
{{formula:d396f16f-31e3-4cbe-a7c1-2ebb974fd08c}}
Let {{formula:692c80b3-0c98-4a67-a851-282dd674b9a3}} and {{formula:0fa16b0a-8792-421e-b30c-abcf481b148f}} be the corresponding {{formula:45cf2bdc-b558-4f99-9339-60f9f9665320}} -charts. We will show that the transition map {{formula:c3237cba-7eb2-4e93-8f26-78e0fcef18bc}} is not {{formula:daead07c-aaf2-4569-b5da-d2f9bc3f7d2c}} along {{formula:ead8185c-c72a-496d-bfe4-c9687f19c0fc}} .
Using definitions (REF ), (REF ) of {{formula:2504c325-182c-4da6-a31d-c9aec4095a08}} and {{formula:c82489f4-b053-4dbe-a3a9-36adceb40f85}} , we compute
{{formula:74a509d4-071d-4abe-8982-8b3d117fef9a}}
so {{formula:16b8ef62-f694-4132-988c-e222d35ecc85}} and {{formula:84e6b42e-26ce-4fa4-9f51-eb5bd2b94380}} , cf. Lemmas REFREF and REFREF . Substituting these equalities to the definition (REF ) of {{formula:c31b7a87-b022-4b5a-918c-f5e933fdc353}} , we get
{{formula:4f1d4c1d-e173-48c0-a344-8390e998df0e}}
Now, we will approach {{formula:06f21b26-51f7-40a4-886d-000535fa565b}} along {{formula:d83e243d-1f09-4b55-a5fb-185e52c0e701}} and {{formula:b77c2647-47da-450f-b21a-ee6b0c2fd6a4}} . In Figure REF , these are, respectively, the circle-arc and the bold vertical line. If {{formula:b1aaa708-38bb-4faf-b464-695607189fad}} , {{formula:589e1057-761d-49a4-b096-71961fc1a858}} then (REF ) reads as {{formula:5a14cd08-b9fa-464c-b002-0f50fbd909f5}} , so {{formula:d4c93d7a-3c15-4cb4-83f5-a8d9811a662a}} . In turn, if {{formula:454df487-cab0-4263-971f-f7ccf684652a}} , {{formula:fd7eb9fe-cd1b-45bc-ba5d-93dcfc191b7f}} then {{formula:0be0154a-d88a-4a53-86cb-708457fce01c}} , so (REF ) reads as {{formula:3e4444c3-9151-49a7-886b-4e9a9f1f47eb}} , and therefore {{formula:e06d437d-0aad-4ddf-8b82-ac2bb72521d2}} . This shows that the transition map {{formula:1ac68416-1170-424c-8143-779c4ca37b65}} is not {{formula:4919fee8-cc48-4b37-b19c-0f4941cf7c86}} , as claimed.
In the remaining part of Section REF , we prove Lemmas REF , REF and Propositions REF , REF stated above. First, in Section REF we prove Lemma REF , which makes {{formula:58f42051-0d39-4057-8410-308d92476f0f}} a topological manifold with boundary. To prove that this manifold is {{formula:57461fbf-996f-43a0-a936-a036a080f237}} , we will need some preparatory results, which will be useful for the construction of a {{formula:85bda3b0-c328-42ee-a51c-333de2bf6ee3}} atlas, too. In Section REF , we gather some useful identities. Next, in Section REF we prove that, as we approach {{formula:e897a704-9d37-4aaa-b97f-710377c0ec8d}} , the {{formula:230b029d-0728-4f53-85e9-107a8a0a7dfe}} -charts corresponding to different charts adapted to {{formula:47f63e8a-0cc9-480b-84e3-4f4c40b21479}} become {{formula:0e4147e0-1ac9-45c9-82d8-374b4079f96f}} -close to each other. To state it in a precise way, we introduce a technical notion of differential forms bounded from {{formula:cd4bd668-27a8-46df-8a9c-5221a7631201}}. With these preparations at hand, we will prove Lemma REF and Propositions REF , REF in Section REF
subsubsection3.5plus.7-.5The A'Campo space is a topological manifold
In this section, we prove Lemma REF , which implies that the formulas (REF ) define topological charts.
Lemma 3.15
Let {{formula:7e6a90e2-35f8-4023-a517-a1978869ef44}} be a chart adapted to {{formula:16af8894-243b-4ded-9eda-b43c918557f4}} . Let {{formula:77f4907f-b860-44a7-acd2-68d8913d2ced}} be the associated index set (REF ), and for {{formula:3eb65c58-948a-4e3b-a99b-2880e72aa5e3}} let {{formula:2e94efd4-3e1e-4fa1-b72b-9d27fd7c28c2}} be the open subset of {{formula:254172de-ed35-488c-9762-c7c02fdfed28}} defined in (REF ). Then
The intersection {{formula:3c39b8d3-b2fc-4dc0-9ea7-ec4ddf877b95}} is an open subset of {{formula:7affaa1b-14df-40de-aaa9-48761a315d41}} .
Fix {{formula:4d8606ce-ff96-4e34-ab12-e505e55c4263}} . Then {{formula:4f521990-64dd-416c-9c45-edeaa27ca6b5}} for some {{formula:99129e01-b9f2-4e08-9ff9-41c92124f9df}} containing {{formula:7be57d05-4e52-4d34-83ea-a3be687b6390}} . Moreover, for every {{formula:e388ea0c-c192-404d-b51d-76a870fd5a63}} we have {{formula:667d7674-0fec-4773-a4d3-d63c69127de8}} if {{formula:40ebc130-6033-42be-80b4-4d3d61e0fc36}} and {{formula:db61a830-700f-4603-a326-d50e8f369004}} if {{formula:ffd54e93-d0ff-40af-820a-b6b5d548ea73}} .
{{formula:4152245d-6498-4e6d-a114-45ae7dbfadf4}}
.
REF Openness in {{formula:6ea2c7ac-4838-4e21-b2bd-fd86d38dce47}} is obvious by the continuity of {{formula:77c6898d-33ea-4503-82aa-18800f52fe1c}} (Lemma REFREF ). Fix {{formula:1d52742d-ccba-4554-93cf-fe694ed11e30}} . Since {{formula:6316da0c-ec1b-4b75-a1f9-1a9c5c48ca46}} , the formula (REF ) implies that {{formula:0a307535-5ea2-4034-ab5a-fc27d0bdd13d}} . If {{formula:700a091a-9a1a-4f30-81f9-5230951ee199}} then {{formula:187b06fe-4ab8-44e9-b40c-86693e5d5700}} , so we get {{formula:2a144b2a-1380-40f3-a2b1-09a7b3dde184}} , i.e. {{formula:a4e3ce52-9f74-45bb-bb2b-665ed35233b8}} , as claimed.
REF By REF , we have {{formula:1b353359-6bbc-4873-9602-5da1f46ddd9e}} . If {{formula:9f27d1d2-7b9b-4ba8-984b-6f8e5811bf36}} then by definition (REF ) of {{formula:06a6fb34-f113-4e4e-8d6d-e6165d9ac986}} we have {{formula:9fee5463-65e1-4763-bb08-a2998d1db1fe}} , so {{formula:39a2834a-bb29-4042-a234-b41cf7f56340}} by (REF ). If {{formula:19ce7b70-9f4d-4e5f-a78e-2b8942e96f60}} then {{formula:4d3c6447-c1f8-4961-8ef9-95df28bdcaf2}} by Lemma REFREF , so {{formula:b24f5fff-f7e6-420c-8c8b-d84e2ae29845}} .
The function defining the hybrid coordinate {{formula:e7cfc932-de64-436e-a256-aae993974f51}} in (REF ) has the following useful property.
Lemma 3.16
Fix a positive number {{formula:eb3756c7-d0b3-4e54-8ca3-228e4866879a}} and consider a function {{formula:ff843c88-b202-44f0-9d13-ff7dd1ca147a}} .
The function {{formula:0d3926db-d62b-4d44-8543-105bf7824742}} is strictly increasing. Its image equals {{formula:0f5b11c9-6c7b-418b-b7e9-76b2f56b849c}} .
Let {{formula:0d5ec7a2-5be7-4ca4-a5aa-50e980044ee2}} be a chart adapted to {{formula:9112bc70-924d-4bb5-b33e-4bf4e8fd11bd}} , and for {{formula:905f3d17-65f5-4f47-9bde-603001ed39d3}} in the associated index set (REF ) let {{formula:37c512a0-3517-4eeb-9286-b19d4481d5dc}} be the function defined in (REF ). Then {{formula:4aad75d4-4ce8-4219-8ba0-99b15f5aa3c4}} for every {{formula:deedce90-cdae-4dd4-9bd4-d4086d35e904}} .
{{formula:fe516a0a-3a6b-43f7-9c9d-123b3c5f21c0}}
.
For {{formula:c3e3533a-3c09-4ba2-95f2-6db4b6698d4d}} we have {{formula:a1c59c12-52cb-4a44-a00a-b8f1bf4b364b}} , so the function {{formula:17d70af3-65c3-49e5-a9e3-d4798e76c872}} is well defined. Its derivative is {{formula:52497ee6-aa48-4a06-bda5-eb6fd5993c14}} , since {{formula:6c2ea34d-996f-4dad-909e-fb433ddf4a3c}} is strictly increasing. Thus {{formula:9558c309-4ee1-4c24-a760-4ad8567e58e6}} is strictly increasing, too. Since {{formula:5e382995-2b56-4534-acd4-b3a804b478ad}} , this proves REF . Part REF follows directly from the definition (REF ) of {{formula:16f4a863-574c-4af7-abea-a7a1dfd3db66}} .
Recall that on {{formula:d7b1fee1-61bd-49eb-a1b4-4d835375f556}} we have the natural smooth structure pulled back from {{formula:7a6d5a8a-fce2-478f-8941-6d72abdcd7cf}} . The following lemma asserts that the candidate charts (REF ) are compatible with this structure.
Lemma 3.17
Let {{formula:4079d968-f06f-417f-a6e8-6d802a20c725}} be a chart adapted to {{formula:b70e05bb-1b91-43c0-ab2b-a3482ee6d842}} , and let {{formula:80088836-4543-4777-8295-48bd6ef7a133}} be an associated chart (REF ). Then the restriction {{formula:fb06abd5-26cf-4899-8efc-910b912a6b81}} is a diffeomorphism onto its image.
{{formula:e7cdbd11-aaa0-4684-9501-5a88639576c5}}
.
Reordering the components of {{formula:a5a01f63-69fa-4174-ae06-aa11bb234f8b}} if needed, we can assume that the index set associated to {{formula:6d6e4a57-af5f-421b-bb49-3374c037f5a7}} is {{formula:f81992e0-377e-4adf-bb17-b5a589725fe0}} , and our fixed chart is {{formula:b36b9646-5b81-4a9f-a1c5-2d93b8af0d85}} .
Recall from Definition REF that {{formula:162e9c8b-def2-423f-b198-a986d5ccd5c7}} is a holomorphic coordinate system on {{formula:17c69ca8-6209-4c22-a2fb-dc797ca50f28}} , and {{formula:046f93a8-cccf-49ef-8f53-a28f8afbdded}} is the zero locus of {{formula:e129b453-5bcb-4644-b893-5a3515527d89}} . Therefore, the map {{formula:b0748cd1-34e7-4e60-b236-e079ef257e1d}} given by polar coordinates {{formula:b036b579-7263-4a97-9de1-0d03d96ecc2a}} is a diffeomorphism onto its image.
Since by definition (REF ) we have {{formula:959f11b0-9fed-4faa-87a7-0caecfb012ae}} , the same is true for the map {{formula:7999bbd7-1e03-4483-84d3-16da0c326840}} . Thus {{formula:652b7d96-5300-4cec-9539-8fdd83189efc}} is a smooth coordinate system on {{formula:afcb4a30-8643-46b5-9893-ed8968542f78}} , for the smooth structure inherited from {{formula:63639869-659a-435f-8e95-21468ca36f54}} .
We claim that the map {{formula:6d0c5889-dd26-4068-b0cd-ace50e30dc0a}} is a diffeomorphism onto its image as well. Recall that {{formula:95487539-291b-4f02-8480-7cc4596d1f89}} , {{formula:62c50c66-b389-4169-ace6-350b8dfed203}} and {{formula:4646f671-cc0c-4aa1-a0a9-bb03ce61bd61}} , so
{{formula:742467f3-aca5-498f-b03c-4a3faf528431}}
thus in our smooth coordinates {{formula:e27e1fe9-d1c4-44b1-a9f3-36e1f1b11edf}} we have
{{formula:74db66b4-c35b-41c0-98c8-8a3064774b6f}}
Therefore, {{formula:04bbfd66-0fdf-44ca-a055-e6325437ed71}} is a local diffeomorphism. Suppose that for some {{formula:ae94fffe-cc4a-4456-8480-82e33000e2ca}} we have {{formula:a1e3a69f-2071-490e-ade2-40739416ef99}} . Then {{formula:520dbab1-849c-4fb3-b53b-e411ed8493bf}} and {{formula:f331fb36-4795-472b-b89a-31e277080b4a}} for all {{formula:ce7b93e5-5b94-4a0d-9b62-921b100a1bbe}} , so {{formula:81b9930e-16d1-45fa-b94d-9e3a453d87cb}} by (REF ). Since {{formula:20c21731-35e1-4e28-9318-8500fe8c1c8d}} , too, we get {{formula:3e0c044d-f129-48ec-8ec4-d941ae8fdafa}} . Thus {{formula:6e5cb0cb-80c3-468a-9f67-6905d95814d4}} is injective, hence a diffeomorphism onto its image, as claimed.
The above claim shows that {{formula:a3c48ffb-e787-4bae-9385-fe3afcfa40a1}} is another smooth coordinate system on {{formula:6a8a3898-474e-4d73-bfe9-e061806be598}} . With respect to these coordinates, we have {{formula:1b547cee-0306-47b9-9273-c6124d005fcc}} by (REF ) and {{formula:7f20e68c-7d67-4e61-81f1-120642273a96}} for {{formula:bd49140d-336d-4228-9d79-eac367ac6074}} by (REF ). For a fixed {{formula:35704198-0d4d-41a1-9cf2-bc52ab0cb11e}} , the function {{formula:ca60ff2d-a09e-4364-a1bd-3bc98f20f2fd}} is strictly increasing by Lemma REF , so {{formula:660d2b6c-875f-4573-96d6-278d61ecd910}} . Thus the Jacobian determinant of {{formula:79e51b0d-1be8-4f21-b3fa-06e71f9d2638}} in coordinates {{formula:e8b42220-c220-4298-9647-a98435656c97}} is {{formula:be25c9e8-abf5-4569-a8ff-8017127a95b4}} , so {{formula:0eb6a558-07d5-4c4a-a5c2-ee32a88d28f7}} is a local diffeomorphism.
Assume {{formula:057d7c21-7c55-434d-96f2-9ef145771436}} for some {{formula:a8299662-1739-4b07-9567-f85dc0358a43}} . Then {{formula:c9b73c8f-ebe8-4659-a953-61a5a9ac3286}} , so both {{formula:c06866fc-7bc0-4672-a649-ef02690d11b1}} and {{formula:62883bae-f53d-43ad-bc1c-518871e2b3c4}} are equal to some number {{formula:00c7f198-3e16-4962-90bb-4015a07c9c47}} . By Lemma REFREF , {{formula:f2b415d8-72f2-48e6-877d-785d4000c79c}} for every {{formula:a2c0bd9f-78e5-416b-a3ae-69d02d5766d8}} , and similarly {{formula:fe7d7281-d148-4059-a2de-199c0c42b017}} . Since {{formula:1cc8370e-8a50-4a42-bd26-1fecbe48b7a9}} , Lemma REFREF implies that {{formula:192543c3-3546-4159-b00a-7d636d12a84d}} for all {{formula:cbb5c441-73c8-4bed-95c3-8246e61a6c2f}} . The equality {{formula:c47cabb5-1592-4a7a-b30e-1aaf06cde9c8}} implies that {{formula:e15adeec-4c21-4d27-9d61-e5c2c246b0fd}} , too, so the values of all coordinates {{formula:82fb61f7-684b-4674-b57d-9ce182bf4059}} at {{formula:d812be15-1fc1-48a9-8a0a-28c687e8935f}} and {{formula:90a8ff8f-d97b-4f57-9626-06f048f6b245}} are equal. Thus {{formula:33b3f4e9-c753-448f-86d3-513ceab6bfca}} , so {{formula:17b263d9-5eb0-4421-bde5-f95d4ad422a3}} is injective, as claimed.
Lemma 3.18
The image of each chart (REF ) is an open subset of {{formula:24c82c25-7887-4a26-855d-cc0374b73557}} .
{{formula:480dc70d-3803-45e4-ac34-613da32e9c66}}
.
Fix a chart {{formula:09649edf-35af-4d3a-8690-a7a1940c34d5}} adapted to {{formula:de507e2f-d2d6-461d-9e3c-2c3d01d7849e}} . As before, we order the components of {{formula:05c114bc-77e5-4b6e-9f53-0d5a03fd61c0}} so that the index set (REF ) associated to {{formula:dcdb09ac-e4dc-48bb-b1f0-f45c82fa1aa9}} is {{formula:ee10cfab-a93e-4ce0-969d-91825d64fe30}} , and study the map {{formula:f034824f-d90c-49a3-8709-d24d5854fbdf}} introduced in (REF ).
Suppose the image {{formula:97050dae-903a-409b-88ea-2d3d835aed3a}} is not an open subset of {{formula:38ac800a-f9cc-43aa-96e5-95914a15e120}} . Then there is {{formula:3d9a26c0-f843-432a-8758-1264771b56b3}} and a sequence {{formula:5f5e3129-7153-4f79-9823-55bddae79490}} such that {{formula:29183b5f-509a-4465-a81d-9378f4f2c41a}} . By Lemma REF the restriction {{formula:6a0d07ba-6669-4c7d-a74c-0f09484305fa}} is a diffeomorphism onto its image, hence an open map. Therefore, {{formula:aa524e32-bed6-4f77-bb86-e65904c2f80b}} .
Write {{formula:0034ee10-49a6-4ca6-8dd6-e43ff6f4c22c}} for some {{formula:8bd1656c-5028-4a76-8a3f-383113520dc8}} , {{formula:b2ff0d4f-e05e-4311-8412-e4c99b6a0981}} and {{formula:7b7c6949-f970-4835-8132-57346314e058}} . Then {{formula:d06708cf-1869-4f7d-9eb2-30239e7d88f1}} , {{formula:27695f7f-4fa4-40c4-934e-8b327da091d6}} and {{formula:337a9181-1c38-4749-a740-8fef59319f8c}} . Passing to a subsequence, we can assume that either {{formula:edce898f-d24c-429f-b24b-4737f5dc70b1}} for all {{formula:00b4ba37-a708-43e0-abcb-43fb85ad5faa}} , or {{formula:0a92ff6c-48a5-4e10-9e30-f3b26e0d14f9}} for all {{formula:f0b6216d-6ca1-426d-88fd-038eabe9e95e}} : in other words, {{formula:14048b37-53df-43d0-9586-c45782bf57dc}} approaches {{formula:3cc84940-fcb5-4639-9954-2b493e62e00d}} either from {{formula:e95f5734-c687-42c5-9d33-f63b91b07a30}} , or from {{formula:b438d480-4101-4bf8-8a2d-d73665ee50a1}} .
Consider the case {{formula:252cde37-0c20-417f-8e78-7824dd784ad5}} . Put {{formula:09d57fec-fe12-4811-9162-b5275c566882}} . Since {{formula:82ee91aa-2c03-4de5-99c7-6fb59add8575}} is continuous, we have {{formula:f9e9c4de-b687-470b-8fd8-599ff847d14c}} . By definition (REF ) of {{formula:8871a5e4-72a8-4119-961b-0f25dbd43007}} , we have {{formula:6a4beb86-0d11-44fb-a0ff-0c64cc2eb3c7}} , so by Lemma REFREF , for {{formula:b31968ca-8abb-4f4a-8bb0-1d005aa40614}} we have {{formula:1d40faca-207b-4da2-9002-d3df9f9b9cbf}} . The formula (REF ) for {{formula:2c8aa123-2139-467f-b714-815566f2c28b}} implies that {{formula:0404e998-8c0c-44cc-94a2-332ffaf8875a}} . Hence there is a positive number {{formula:3c41382b-0972-44c2-b724-06a4d18e9a48}} such that, after taking {{formula:3a3aa4ab-4e5f-4949-b329-2f5403c7116d}} sufficiently large, we have {{formula:5d7fc7ad-dbb4-4f3f-bccd-7c3802fcd331}} and {{formula:943d08e1-5682-4623-a537-ad54c800814f}} for {{formula:0b1d93fd-e3ee-4975-ab21-1e132a398f33}} . Therefore, each number {{formula:fd7f1b7f-a767-4291-9b69-6a375e8cb032}} lies in the image of the function {{formula:2ba363d9-bd39-4e7d-a476-ced42b48ea41}} from Lemma REFREF . In other words, there is a number {{formula:e6193bf7-b85b-4dee-891f-f69c109d9715}} such that {{formula:3099607c-6e42-40c4-a043-3fa400fd2c54}} .
We claim that for each {{formula:56176ce6-20b7-4901-8e88-456437b6a84d}} , the sequence {{formula:700cbcd9-e271-4c56-8834-0b237f95e42a}} is bounded. Suppose the contrary. Passing to a subsequence, we get {{formula:25a47c9c-2d3e-4885-8208-128653752061}} . Since {{formula:cd468ddb-3135-4fd8-8f6b-b4f4491faff0}} , we get {{formula:55d1a06c-9ab4-40d7-ba99-c28cba3b4b51}} , so {{formula:eeb17a9b-00f4-4fb1-a642-53bb1c14d252}} . Now {{formula:c27b0961-dd7e-4b2f-842f-b9e879b22082}} , a contradiction since {{formula:f4038813-1446-4cbf-8039-3d0f81d99887}} .
Thus passing to a subsequence, we can assume that {{formula:de7685a4-53ff-44d3-b1ed-a97d06aab326}} for some {{formula:01797aea-b98c-4fe0-b42e-9bbacdffa483}} . We claim that {{formula:474be70c-501a-487a-85e2-b45251b86c2c}} . If {{formula:72eeb3c7-65df-4971-ad3a-aa4ca10d52d6}} , then the sequence {{formula:91578685-fdc6-4f5e-ab12-c958195ebd51}} is bounded from below by a positive number, so {{formula:43769070-f2a4-4f6f-91ef-db72b5bf9f3b}} , and therefore {{formula:ccf08c85-f854-4ea7-9952-bd8bf16c3f2e}} . Hence {{formula:0ae161b4-f86c-4bba-b5ed-0a2b11eef334}} . In particular, {{formula:fbaa7f46-8a47-45d1-85aa-7672dc8acd25}} , so by Lemma REFREF we have {{formula:0e67f9f2-1b11-4fbd-9337-a3957da00d55}} , and therefore {{formula:ab125b0b-dd74-46b0-ac49-abce78c172ec}} , as claimed. On the other hand, if {{formula:38bb6972-bcbe-4f6a-acdc-9993f540e287}} then {{formula:3108e17e-ecde-4e3e-99d3-51c4650af0f4}} , so Lemma REFREF gives {{formula:69ad990d-0486-4633-8f14-72da79595f14}} , as needed.
Thus for all {{formula:04d9f43c-b184-4456-8447-1d9fca6ff244}} we have {{formula:da724af6-285e-4cbe-9046-06a0e87209fd}} . Put {{formula:2619db51-dcfa-458e-bbae-5ca8b21c026a}} . By definition of {{formula:39d4634e-731a-42ac-b7ba-f53ae70c1f41}} we have {{formula:9c980b47-37a1-4198-b624-bd504f663053}} , so {{formula:5619b364-35af-4292-b606-d8972d65f360}} by definition of the function {{formula:15a10266-1f7a-43df-b2bd-5429a6940728}} . Hence {{formula:dafd955f-76a4-48e5-b5fe-2747e8acb999}} .
By definition (REF ) of {{formula:b40ad1d5-5f0d-440e-b958-63008b0196a7}} , we have {{formula:590a2368-4721-42a9-aabb-486f588c7f99}} . Lemma REFREF implies that {{formula:420b5ab4-8061-4040-96ea-33dec9228b67}} . Hence for {{formula:1af9b623-6058-48a6-bae5-3ffa117a49ef}} we have {{formula:4d1f92ec-98bb-447a-8b5a-32251a63a7a3}} , too. By definition of {{formula:2998321c-2991-4c37-9702-ba479c8198ee}} we get {{formula:dde10d79-7b1c-4cd5-9014-428757af5bfd}} , so the positive number {{formula:ffc7d4a3-3753-4def-afcc-e20720e62daf}} satisfies {{formula:d38f6c56-bed5-4981-8b7c-305398095f71}} , and therefore {{formula:af1d3010-4560-4e3b-8317-ae4a67d6a2cf}} . Since {{formula:650efc95-d9b4-4b6c-a187-7c4212b2b445}} , we infer that {{formula:d4e5e60a-ed2b-46ca-a905-491b3322938f}} , too. By Lemma REFREF we have {{formula:ab7169a9-c221-442b-acc9-31ca842e8f0c}} , so the convergence {{formula:35470dce-2837-44ee-bd5e-38f1e16e4bc1}} holds for all {{formula:45f94e3e-a79b-47a2-8718-542fd02b5116}} .
For {{formula:b58d2fd9-45a3-452d-9ae0-bf3a245a052a}} put {{formula:7615af07-3dfc-43f1-8a3a-8c54eb54dd93}} . By (REF ), we have {{formula:ee98d2d8-0cda-48c0-9355-19e1c10d0e6b}} if {{formula:145bfef9-62c0-4703-b70e-7eafb5d5b8d8}} and {{formula:211dd266-ecc6-4239-ab29-a96131342ca3}} otherwise, so {{formula:a103f46e-3e60-40fe-83fd-3fd76a83b6b1}} for all {{formula:7bd999f1-a554-43d0-b0e3-881d15ffe7c1}} .
Let {{formula:9c08373c-7cda-438a-9651-7b46015523ad}} be the image of {{formula:4ac76635-48dc-4a7a-96de-2de92ae48402}} in the Kato–Nakayama space {{formula:44e79fde-9d58-4d91-a4ef-952321d12f09}} . In the chart (REF ), the coordinates of {{formula:3ac4eadc-e900-4533-804a-90dea9680716}} are {{formula:6da46f89-64e1-4c43-aa78-fc88e9161988}} . Since {{formula:637593e5-ab9a-437d-9c6f-919c29f789e3}} and {{formula:8f1275a8-ec51-4863-abb8-173543927d37}} by definition of {{formula:8a94288c-08af-43f8-81f8-4806d6fd92dd}} , for {{formula:f76e0b05-3175-4bc7-b937-5e7bdf1ce647}} the point {{formula:92f5e9ca-b69e-4537-b200-ec3e5e6be68c}} lies in the image of that chart, too. Let {{formula:f0e4199d-bf58-49d3-9ce5-9080ba4d5b42}} be the corresponding point of {{formula:1a0b3044-3e9f-472c-83ef-e85fd69f9aa4}} , and let {{formula:bd518cb7-251f-48c5-a741-bf5d45a65db8}} be its image in {{formula:3923d429-ad8f-43d1-a2c8-d81f1d09d948}} . Since {{formula:d55b90fb-4b61-4ef3-b508-db472ec598de}} for all {{formula:82f928da-3f1d-4dbe-a00c-6ad3c85eba61}} , we have {{formula:e522252a-7316-4305-a91b-52420e2f1b94}} , so we can identify {{formula:a66c8e30-4ef9-43c0-a933-a58ffd9582c7}} with its preimage in {{formula:79581c26-6fc1-4403-89b6-65cbb42a7915}} . This way, {{formula:4913273a-094d-4db9-b12b-d49f502a74a9}} and {{formula:297cc232-9ed5-4655-b54d-9b6888a376ed}} , so {{formula:0ad042ab-d662-42ad-9458-b62bb3151b10}} for {{formula:e58362cc-03a5-464d-a86f-d4450a7e2049}} because {{formula:777bf12e-bc68-4eab-9aff-7a6e644de118}} is open. Therefore, {{formula:cfaacd9f-4ee2-4402-8b38-dcf096518410}} ; a contradiction.
Consider now the case when {{formula:bdf56274-c058-4bf8-a75f-0c1d951f8907}} for all {{formula:6c5cdb5c-07ab-48d0-9926-8e1b9352b9c9}} . Say that {{formula:bdf23633-94d4-42e7-a133-253eb8987278}} for some {{formula:a856fb12-d55c-4646-bf2b-8b184f3f3b84}} . By Lemma REFREF we have {{formula:11bb546b-2b95-4f4c-bcb5-967b81fa85eb}} , {{formula:ea6e61f5-4f5a-403a-8d9e-5b4f41b48232}} if {{formula:faf24017-aee4-4094-8ec9-3c3e61b95f20}} and {{formula:a4fdf0ff-2efa-4619-9c3d-f2f560ecb0fe}} if {{formula:752e02ab-4f25-4a99-a381-810c1a3ba47c}} . Since {{formula:22316c99-d15f-48fd-ab2e-f3f34b7336bd}} for all {{formula:a7a9baa0-c029-45e6-a678-fbf2f7e42ec9}} , after passing to a subsequence we get a subset {{formula:dfeb97cc-7a29-4487-ab47-0181ea899bed}} such that {{formula:ae00a5b2-916c-4374-86b1-d11f6f61a33f}} if {{formula:22a69de3-6d7e-479e-b8a2-6baf20fe9c8d}} and {{formula:74ea5713-610e-47e2-977f-b1a3ddf029d4}} if {{formula:35e7af4c-e5f8-4fc8-86d6-8c087fa8b185}} .
For {{formula:ac5a6ba7-6cb6-45e7-8de3-4c1253843057}} put {{formula:1cfa805d-688e-4349-8b09-6eaecc2b9966}} . Since {{formula:9435d5ff-35f5-42a3-b5d1-1dfd6c627536}} is continuous, we get {{formula:f5cb0e48-f294-48f1-8ed3-848efb09c14b}} . For {{formula:d7b66755-65a0-458b-a2d4-0a2f6c04c908}} we have {{formula:5f619802-5fc2-42dd-b383-a187585a163e}} , so {{formula:e6d420c2-7dd3-4703-951c-4d6a03ed68a6}} . For {{formula:a642aebd-dd4b-4ac2-adb8-7bf7aece35e8}} we have {{formula:5e3edac6-c767-4696-8835-62318dc1f38c}} by Lemma REFREF . Thus by Lemma REFREF we have {{formula:8de0f1ce-5275-4a34-a5b3-8b8acc60a8b4}} , where the inequality follows from definition (REF ) of {{formula:1188558f-3b11-48e6-aeca-059b3b38a087}} . Since {{formula:1620e5c8-e412-4626-9854-f4ec0777b82c}} for {{formula:48ca4959-afb6-487d-b9cb-6207f02ff3dc}} , we infer that for {{formula:cbed0da6-a91d-47d7-b249-a794e3017d04}} , the numbers {{formula:29e7c2e1-677e-4a05-b7cc-ac29717d9d34}} are positive. Put {{formula:6b21e596-fa0f-47a7-98a3-ab1ccb5d75c4}} . Then {{formula:1b714b8b-3d36-4a7b-b0b8-e8fec5133835}} by Lemma REFREF . Let {{formula:e757f952-10bb-4002-b8ad-22a641e1326d}} be the point of {{formula:649951e2-3f8f-439b-807a-38688df299b8}} whose {{formula:ee6b6347-312c-44d4-8fcb-7fcfb728524f}} -th coordinate, for {{formula:f3f9ea88-7a05-4dc4-aaa0-ab9ddf6d37d9}} , is {{formula:ea2ea095-214f-4342-9af3-92745f672d30}} , and the remaining coordinates are zero. Then {{formula:b8a01ca9-4db6-467c-ae9d-0251ca439119}} .
For {{formula:569cb31d-be90-4f64-8e11-85afed6389dc}} put {{formula:01952840-892f-4d01-946e-80e3b66923d8}} . Like before, we have {{formula:10fbc96e-454d-4a50-8762-87d22c66673a}} . For {{formula:4ebc8575-0bf2-42ec-9848-3d947201f2b3}} put {{formula:6611dec1-a4f3-48ff-81ec-3e8d8b797172}} , so {{formula:caec16c4-4108-4793-ae59-d67540cfb2ea}} , too. The image of {{formula:2fb74b4a-0143-4a03-ac59-4d2a29d055e5}} in the chart (REF ) of {{formula:0ef4e101-c043-4610-836f-794a04592e63}} has coordinates {{formula:6b7448ea-29c4-4817-8b84-9c4a8ba07f9e}} , so for {{formula:9cbae363-3465-4215-9aaa-4654587f44a8}} the point {{formula:1820a100-49ad-455d-b714-247688f2b8af}} lies in the image of this chart, too. The corresponding point {{formula:e3a26ddc-ab65-4d6b-8efd-e00a45bb8225}} lies in {{formula:ae01b5f6-eb34-4d31-b93e-e052aeda2a56}} . Since {{formula:780605c1-87ef-4116-abcb-80dc3e0013f3}} , the pair {{formula:c367c37a-279b-4f3f-b088-cede765fef0a}} lies in {{formula:6b6922cd-13c6-4741-9e97-609910f572e4}} by Lemma REFREF . By construction, {{formula:4a7c6a07-a71e-464a-86e7-0455b075b086}} and {{formula:9b967a04-6469-4097-8f31-2b6c5d6919f9}} . Since {{formula:dca420aa-f6c5-4ebb-81b8-588d1f43969e}} is open, we conclude that {{formula:aa52159e-28ee-47b2-b103-3d5b26d9d773}} for {{formula:be2d1113-de78-4ba1-a211-3e3bdf1175cb}} , so {{formula:8a7edf84-9e75-4015-9c7f-22f76b3a84b0}} ; a contradiction.
Lemma 3.19
Each chart (REF ) is a homeomorphism onto its image.
{{formula:788e3115-12f8-4bca-b9f2-fa962f0dee5c}}
.
Fix a chart (REF ) corresponding to a chart {{formula:200db990-da9b-4549-add0-3168e0f70c6d}} adapted to {{formula:59d4e1ce-5a66-4c6c-bafa-ef56042ab227}} , say {{formula:21c9c224-cc3b-45d2-8eed-40a15ad56006}} . Let {{formula:923d0f69-b2c5-4cd0-910b-7f99d20cf4a5}} be the index set (REF ) associated to {{formula:fec80b90-4bcc-472e-b79b-732a8a197ec2}} . By Definition REFREF , the coordinates of {{formula:660ef727-25ef-465f-b25d-630c98f1d803}} extend continuously to the compact closure {{formula:0b085197-f789-4608-a7f4-3e71fbd3ccc0}} . Together with Lemma REFREF , this implies that {{formula:89648a16-d619-474e-90a2-8460607352f5}} extends to a continuous map {{formula:de3a280e-ac09-4375-8a60-4d6601004936}} , which we will denote by the same letter. Since {{formula:de9d3625-3506-4c88-be75-05651092baaa}} is compact, it is sufficient to prove that this extension is injective.
For a subset {{formula:1ea62189-fe1a-4179-97f1-767eefc2f658}} put {{formula:8b710e60-db06-4076-bd3b-5a380ff8e7f8}} . We have {{formula:bf04b3d4-edbf-44d1-9a5c-db743c95a9dd}} and {{formula:8eefd2a5-7038-44aa-8da2-76ce9e567432}} for all {{formula:cbb19430-055a-49ea-9ee7-6ba724ad400b}} , so {{formula:7e5ff010-3cd2-46d2-97a1-15e52e5e9693}} . Lemma REFREF implies that {{formula:16aeead4-764a-421d-bcb7-b7b9a116d62a}} for all {{formula:8dee1a41-62d9-4de7-ba0a-a73785776598}} . Therefore, it is sufficient to prove the injectivity of each restriction {{formula:a37bcf80-f83a-4b8e-b0df-8c1ecfcf4a2f}} .
For {{formula:b75c98b6-2d3b-4f01-b0d2-ee59714fb65a}} , this follows Lemma REF . Assume {{formula:26098e67-9fd2-42ee-8e10-ac98c0ca5b1b}} and fix {{formula:2c9a4ff8-2f87-429e-bb2a-3a5447c9fb52}} . By Lemma REFREF , the point {{formula:527ae48f-7c4b-4067-87d0-cc09353f58b5}} is a pair {{formula:008586ef-07fb-4210-b6b2-077374facaeb}} . In coordinates (REF ) on {{formula:e50351bf-9242-4d4f-a0e2-8930915ac336}} , the point {{formula:ccc4eda2-41ef-4249-9127-175e83f44447}} is given by {{formula:71eda906-f6a8-4ca9-adab-2cb808729857}} , where {{formula:df4d19c3-3f05-4889-bf20-2289f75327d8}} if {{formula:d8875ffb-232c-4319-971f-d6072a9e8fac}} and, by (REF ), {{formula:5aed08d8-794b-4a29-a8bd-1dd8e5d6ff16}} if {{formula:49709309-c7ca-43e8-9eb1-43f22dde60bd}} . By Lemma REFREF , for {{formula:50d06271-c200-48bb-a002-26eb08dea53c}} we have {{formula:16006883-043e-48bf-935a-daa8cbd53d1a}} and {{formula:2a202b8d-a20d-4917-8ecf-a982af2994df}} , so the point {{formula:4d2a78a3-899d-4a02-9f1e-338449ce35b5}} is determined by the value of {{formula:448f0b1d-d167-46a8-8806-515ca245e389}} at {{formula:87827af5-b824-41af-91e0-5f217ec20012}} . In turn, the {{formula:77829419-203c-40d1-96aa-3b0f27c8bd99}} -th coordinate of {{formula:f4a6297d-e5cf-4de3-9573-2f174cbcbdce}} is {{formula:c7d2bae4-1526-42d7-b8b4-aa76d2c15c3b}} if {{formula:32e8fd0f-3a32-4d14-9f8a-38f53ca062c9}} and zero otherwise. For {{formula:8054fa2d-ff32-4de5-893e-fb9df5485ce4}} , we have {{formula:fde1a7cb-5ac5-4bc8-ae3b-c0980b05191b}} by Lemma REFREF and {{formula:c67739fa-1b50-4b6b-b68e-d11dd2ee2210}} by Lemma REFREF . Hence for {{formula:4c08e40a-b43c-4628-ab0c-f7169bf1f05f}} , the {{formula:d9730655-e2ae-4431-a27c-000fccc84f50}} -th coordinate of {{formula:14700a45-4222-4e59-9f26-cbbda16f0b89}} is determined by the value of {{formula:2f93cbe5-84a8-4f9f-bc52-b4503ac442ab}} at {{formula:5b0ac5dd-eb23-4460-ac92-85b6e3d5c01a}} . The remaining coordinate is determined by the fact that {{formula:1d161892-86eb-445a-b957-4ba87ba6ea85}} lies in the simplex {{formula:c7585211-54c2-4b35-9b53-9b593bfb17ab}} .
{{formula:911c1002-d6f8-4a85-bc84-a44d72dd0833}}
Proof of Lemma REF .
By Lemma REF , each chart (REF ) is a homeomorphism onto its image, which is an open subset of {{formula:13982ede-cfdc-42c0-af02-5a7ba7f5d71e}} by Lemma REF . By Lemma REF , the chart (REF ) restricts to a diffeomorphism away from {{formula:ed9fa6d4-866a-4057-accf-71e6c7e8c9ac}} , as needed.
subsubsection3.5plus.7-.5Useful computations
By Lemma REF , our atlas makes {{formula:16699e2c-4e0d-4e4e-a946-d4b9550932b5}} a topological manifold. Now, our goal is to show that this manifold is {{formula:2de5ac47-4bde-4501-a8f7-ae5a87e30b99}} . Lemmas REF –REF gather some technical computations which will be useful throughout the remaining part of Section . To state them in a convenient way, we introduce the following notation.
Recall that in (REF ) we defined {{formula:6ddf6a32-3f8b-4324-8e13-7c1bdf99da7f}} and {{formula:bb0de3b3-185c-4cd8-ab5d-cda0ecaebf0b}} . We put {{formula:e4ee94f6-74c2-402e-adfe-275cb74365b1}} , so {{formula:cf71b279-a52f-4d89-86a8-20d6ba759602}} , where {{formula:7132744b-2992-4606-b151-162780a612eb}} is the derivative of the inverse of {{formula:7e0e59e3-64db-42d9-99f9-9710b50ca1c7}} , see (REF ). More generally, for every {{formula:2c9ac310-d92e-407d-9d7a-51a45667ef93}} we put
{{formula:c3e8e99d-d1ce-4dc8-9b1b-ab8b666d6509}}
Let {{formula:e66bd96c-a81a-4b46-9a84-dfc99c6ec832}} be a chart adapted to {{formula:4a2029d1-499c-44d8-b970-4a502d257ab3}} . For every {{formula:010a09d8-277d-4c29-b42c-d089cb7306be}} in its associated index set (REF ), we define functions
{{formula:de3b98fb-cd1a-4b00-b34f-7621dd9bf220}}
where {{formula:21146444-5d3e-4b62-8f73-769144731abf}} are the functions introduced in (REF ) and (REF ). Note that {{formula:2c151318-f06f-4469-989a-3dd0a21f5f9c}} .
Lemma 3.20
Let {{formula:5b2eae5e-4040-4bc6-afe4-95ecd1c73d0e}} be a chart adapted to {{formula:77d732a9-459c-4a2e-a12f-7fef4e7efbfe}} . Fix {{formula:e67b6cb7-5656-4f85-bc31-bbbaaef9f947}} in its associated index set (REF ). The functions introduced in Section and in (REF ), (REF ) above satisfy the following identities on {{formula:6762ca0c-be7e-4f3d-a655-3d57c5db4bc2}} :
{{formula:8a0f8b63-3cb7-463a-bef9-9c28b73e886c}} ,
{{formula:5142d568-11f3-47fa-9bf0-bc68e92f6525}} ,
{{formula:64d20d02-b514-45bb-a109-ae905e21beb6}} ,
for every integer {{formula:659a976a-9dd8-475f-abe5-a8cb8cba09be}} , there is a polynomial {{formula:bb852a5e-e575-4c8d-bdae-b23f40b50735}} such that {{formula:d737b7e6-a67c-4106-ba69-74eaf25fbf39}} ,
{{formula:3341a10e-efa6-4300-9395-8c559f803f55}} ,
{{formula:33fa1baa-a497-43d7-acd1-a5a725b9954e}} ,
{{formula:362b3223-01ec-4733-9ffb-176384580791}} , where {{formula:4660c450-9466-4adf-a539-c01885b4fc92}} is the function defined in (REF ),
{{formula:0e543ef5-779a-4b71-bd79-037d127649bd}} ,
for every {{formula:d0941e14-e82d-4fc0-9ce6-33c26a5d7b05}} , there is a bounded function {{formula:59e7d30e-679c-40ff-b249-ba7cfa0fb2b8}} such that {{formula:0673757a-888a-4a24-bf18-7493f946be73}} .
{{formula:2d3ade4d-ab18-4fad-964a-65c7711709c9}}
.
REF In (REF ), the function {{formula:594f1f5b-0f33-420c-b401-4b45791fd672}} was defined as {{formula:f3b06c16-ef55-4237-8a4a-91520267d99f}} , so {{formula:9456a7e2-8967-4957-bea5-df53c64e1503}} , and therefore {{formula:46772c31-fcba-4f3d-924f-c217b31cfb29}} , as claimed.
REF In (REF ), the function {{formula:f9826981-8cea-4aa6-a645-1b4883fa79d4}} was defined as {{formula:f9c91933-69bd-4327-a2a5-d4a205e36abd}} , so {{formula:5d7f1d52-8fa4-44c8-8049-05532f2408dc}} .
REF Recall from (REF ) and (REF ) that {{formula:8b84c002-5d37-4b59-b0af-3a48cdc06fd6}} , so {{formula:1a1a81a9-e8d0-487c-ae4a-4abf576253e4}} . Now {{formula:cdff1682-14c5-4dff-a098-c36fbc346faf}} , as needed.
REF We argue by induction on {{formula:c1d83813-ce2e-403a-a466-b5acfd12f347}} . For {{formula:a2ba4306-7d73-4587-8096-078a465f6966}} we put {{formula:eeed25b4-9c6b-44a6-b065-507b7b4bb63a}} . Assume that for some {{formula:03bff725-fd0d-465f-adff-ae881ba12833}} , there is {{formula:1aa349d0-e500-4dec-a19d-33f8938d855f}} such that {{formula:c877b688-5c53-4c17-a91b-f6f59d315fbf}} . Since {{formula:c338ec2c-79c7-48e3-80a7-2bdbc7309d38}} and {{formula:bd496ce0-3b26-43b4-8878-0a3ce42f6bb3}} , we have
{{formula:51691f57-6238-43d6-9f84-29b0cd207b1a}}
The inductive claim follows by putting {{formula:a252fa8f-d242-4a82-b0b0-922d1587b2a6}} .
REF The function {{formula:762a1038-315b-4ac6-962f-83091ed50ebe}} is defined in (REF ) as {{formula:ffe239c0-9fc9-42fc-8fdb-188df72f9b88}} . By REF , we have {{formula:0957b501-b78e-48f3-a646-4494377a5c8b}} . Since by definition {{formula:0e93ce42-74e9-4199-ae71-3b8ee2dc0ce7}} , we have {{formula:bbaa4e29-701b-419d-9749-3d650eccbf45}} . Thus eventually {{formula:868165bf-7701-4572-923b-1e16265e1bea}} , as claimed.
REF The function {{formula:d0d0ac4a-f767-4788-882a-0d2ee7c98de4}} was defined in (REF ) by {{formula:9c02bf5b-bb81-40aa-9fd9-c66aa1a37db8}} . Substituting the formula for {{formula:1dcd9caa-a486-422c-b575-fbb4f7a60528}} from REF , we get {{formula:9c7e3f55-c61f-4133-9918-f7308b617cb5}} .
REF By (REF ), we have {{formula:9e8b0f6b-9c52-4d51-82b1-9ddce6c98a30}} , so {{formula:698f2425-dee3-4b0b-80ea-d7033e6814bf}} .
REF Substituting the formula for {{formula:36437cda-8f7c-433a-b6dd-0d235d176842}} from REF to REF , we get {{formula:92c4990d-d04b-421a-9b64-647f6021409b}} . By REF we have {{formula:01af5da9-ab6f-4cbc-baf9-222736fa40b7}} , so {{formula:84c69bd0-4f50-4818-8e54-2c7201d50f93}} . In turn, by REF we have {{formula:7b7ef41d-d9ce-49aa-b827-eab3a0c92f91}} , so {{formula:b39ba790-cb1a-4b41-bd40-ac0eceb5cf0d}} , as claimed.
REF Recall from (REF ) that {{formula:d1880ba9-425c-4b93-83f6-41ee34d54254}} . By REF , we have {{formula:65d9b99c-76c2-4953-ba05-0b23efe7d2f8}} , so {{formula:f9f2bdca-ee7b-46af-9cad-77b03281a79a}} . Substituting this formula to REF , we get
{{formula:faa9a6fe-681c-48ce-9fd5-1296e6907ebd}}
Since {{formula:c057adb3-2877-4097-843a-c8edcf87f7f7}} , we have {{formula:45f04728-1ae5-4972-87bc-847f3a08822e}} , so the function {{formula:d1c14417-14ee-41b3-9456-e5577e4ddd25}} is bounded. Fix {{formula:101328e8-26e8-464c-ae03-e5b3457eaac8}} and put {{formula:8a4fe5a5-4e6f-4f6c-bd88-902e19e917f1}} . Then
{{formula:fa82dc51-b30a-4f25-a617-cca456ddfcf4}}
Since {{formula:a035c198-163c-4c02-8bf5-6ad57d07075d}} , we have {{formula:fc90aa8a-941a-4f49-85c7-36f6eeecb06b}} as {{formula:c22c739b-d55f-4273-ad5c-0f9c274dea36}} , so {{formula:6181ba80-c38a-4bc9-a4d7-0a4bd3acb31f}} is bounded, as claimed.
We write {{formula:c5912a47-e250-42d7-8ff8-71e97cc50def}} for the exterior derivative. We will apply it to smooth functions on (subsets of) {{formula:9cd8a0af-9e39-4e6e-ba76-9f23f7b26bf3}} , and study the behavior of the resulting forms as we approach {{formula:eb8da572-4eba-41c8-973e-c09ff3b6a3b6}} .
Lemma 3.21
In the setting of Lemma REF , the following identities hold on {{formula:9d407ac4-edf1-47fa-bbbe-a74f945f2ae4}} :
{{formula:4a0e2d69-8935-4229-86af-47321ac33138}} ,
{{formula:56fed3a2-345c-49f6-9407-504032c82700}} ,
{{formula:bfaaa319-83df-4480-89df-09a295daa18d}} ,
{{formula:9998be0e-1ac7-4745-aade-6481dd59a166}} ,
{{formula:d6074900-8d51-40f0-983f-71445f86fbce}} ,
{{formula:d83f5f9b-2be3-4c6e-98cc-a7622f248c8f}} ,
{{formula:a2ae5999-b82d-45eb-96ed-795ab82d31d5}} .
{{formula:b3dbfda8-f7a8-4a4a-a405-537b671f678e}}
.
REF By Lemma REFREF , we have
{{formula:e02ad800-7768-4285-a27a-d1408567bdee}} , so {{formula:31d52fce-8d3f-44da-b93e-980b6a83c7c3}} , as claimed.
REF The functions {{formula:6b0e598d-40ae-425d-9722-d58fcb023c78}} , {{formula:f8491a15-8866-4af4-93e4-ca50d7fb79f8}} were defined in (REF ) and (REF ) as {{formula:730fe8c2-540c-4138-bbf1-37c3979431a6}} and {{formula:16637890-1649-41da-98c9-add7b5d7913c}} . Hence
{{formula:488590d0-e9b3-4de1-a622-8758b9b6683f}}
By Lemma REFREF , we have {{formula:b7f9eeaf-57f7-4748-9074-5c307b3a32c8}} . Since {{formula:1f0f455b-cac8-4d8b-b96d-68fd8c30bac3}} by Lemma REFREF , we get {{formula:512dbd75-7e07-4ece-a623-40a5b6af3183}} .
In turn, {{formula:b250ce90-e72c-4163-9b64-357a2e911ca4}} by definition (REF ) of {{formula:768b9e79-438c-4861-98a0-30ee65ca7d91}} . Thus {{formula:0501f673-bf7b-41a7-ba53-693817f87e1c}} , as claimed.
REF Solving REF for {{formula:02c044b2-665e-421c-b3ea-c0842724d4d1}} , we get
{{formula:e9c9af2b-0c3a-48bd-b17c-f0f2d4addb28}}
We have {{formula:8cdf93fe-27f1-4924-a1f9-00331be1e6ed}} by (REF ) and {{formula:c2858d9e-65d7-4634-b26b-afc7e6d752b8}} by Lemma REFREF , so {{formula:2f603f07-4a40-407c-a34c-d7f98bdd0dc3}} , as claimed.
REF Substituting the formula for {{formula:30e807cc-93ef-4998-aa80-cba10731f12b}} from Lemma REFREF to Lemma REFREF , we get the identity {{formula:39049742-8ea3-440e-9086-42dd750b21f9}} . Substituting it to REF gives
{{formula:d9c6c909-a177-4415-aed8-30de799556ea}}
The claim follows by substituting the formula for {{formula:a1314e1f-1cc4-4f2d-924d-0931b62b7bad}} from Lemma REFREF .
REF The function {{formula:10ca7911-66b6-4a31-a63c-3219a5505940}} was defined in (REF ) as {{formula:ec2c63fc-5fd2-421b-b47c-0d4a1f0f518c}} , so {{formula:9368b550-868d-4325-b3f1-fc14c636d2fc}} . We have {{formula:2209421f-766c-46ae-9b6f-88a30c683588}} by REF and {{formula:61655194-958d-4553-a80d-7e248215b19f}} by definition (REF ) of {{formula:104f541e-6e92-4cc2-b97c-6c7a92307e9e}} , so
{{formula:3cb32496-8a8e-4156-a7f2-e4e91fe2fb92}}
By definition (REF ) of {{formula:e5b728ba-aac6-46bd-8b58-00a63dc144ad}} , we have {{formula:7a8806d9-b914-4ce6-9af7-a3a9b28e0feb}} , which proves the claim.
REF By (REF ), we have {{formula:2b54f5bb-0484-4042-a736-a1b262317bdc}} , so using REF we get {{formula:aa13957e-4a3e-4f17-b091-c9a7fc000fed}} . By Lemma REFREF , we have {{formula:a429f163-9bfc-4e67-9da0-4581b151d671}} , so {{formula:c71f3db6-7320-412b-98f0-afd4f4825a53}} , as claimed.
REF Substituting REF to REF , we get
{{formula:8306316f-e333-4df6-aab0-c90867d16612}}
where the last equality follows from definition (REF ) of {{formula:5c78fc85-68b6-452e-8d0b-269583efa72f}} .
subsubsection3.5plus.7-.5Forms bounded from {{formula:b05f7962-2944-4727-9587-c764d754efab}}
Once we endow {{formula:4606a7df-0a04-4320-9b5c-5fb99d1281a8}} with a smooth structure, it will be important to distinguish those forms on {{formula:5222d7fb-819b-47d9-994a-cb4b22714594}} which come from bounded forms on {{formula:ab6e3960-ed5c-4665-865c-706007f33397}} . To this end, we introduce the following definition.
Definition 3.22
Let {{formula:8f286d02-dd0b-4145-afd2-c8750461d1e7}} be a chart adapted to {{formula:8fc2d144-eb44-4a9f-b785-e566f333d700}} , and let {{formula:5a0c30dd-2d7d-45a2-8560-dcb9dfb8f741}} be an open subset of {{formula:b7180d7f-b0c7-4866-9435-571b7b62e37f}} . On {{formula:e5c4430a-46e0-4315-a3f2-4aac8302de73}} , we consider the nautral smooth structure pulled back from {{formula:320a235c-5c61-4bf5-802d-3f21528a0696}} . We say that a form {{formula:e8f791db-6a5b-41bb-a93d-ea625dee80ac}} is bounded from {{formula:5555777a-c127-4487-b1c7-a881b045ec0a}} if {{formula:96eba0e5-acef-449a-87db-40083328f527}} for some {{formula:32bbc92c-d52f-474d-9965-b209b84166a7}} which is bounded in the usual sense, with respect to coordinates {{formula:ddfe1981-0a5e-4bcc-8b10-46e276f224a8}} of the adapted chart {{formula:c50579f1-6b85-4cf2-863f-cbb4b427bd95}} .
Remark 3.23 Let {{formula:6dc94105-dbd2-4085-a5cf-c7e911e2a0c3}} be a domain of a chart (REF ). A function on {{formula:4feb1770-7c04-45c6-aa59-a9d394739875}} is bounded from {{formula:34553a82-5083-4937-8a2b-7210d703444c}} if and only if it is bounded. Since the closure {{formula:3600b280-2521-4899-8051-377236a41aa5}} is compact by Definition REFREF , all continuous functions on {{formula:9c5ae4dc-fedc-469e-80ee-e6d2d9ce527d}} , in particular those from Lemma REFREF , are bounded from {{formula:0d54bb37-ba63-471c-94c2-4b49534ceed9}} . If two forms {{formula:de694135-b513-45b9-8a4c-edb80f4f3bfa}} and {{formula:69a3227c-4c33-4464-b0a9-bbcbab352a27}} are bounded from {{formula:a04707ba-18a1-4755-8019-47c618b16497}} , then so are {{formula:09b0ad95-8a56-426a-9efe-7ac9ee0f88a5}} and {{formula:da3c2581-3426-44e5-a6e8-3fa707a29751}} . Nonetheless, {{formula:377a447d-e2bf-47be-9f4a-bf702f22110e}} might not be bounded from {{formula:d6182836-8c9d-4ba7-b537-0473e4b7b5c6}} . For example, functions {{formula:a9bbe387-d682-4ed4-a15f-3a0717e0cc46}} , {{formula:6cbfbfd3-4c49-49e9-9c22-a293f58be4a4}} are bounded from {{formula:d5076f6b-ae30-4acd-8d5e-2ed3d2d258bc}} , but the 1-forms {{formula:a4196d26-6a9e-432c-8479-2766cf6c87eb}} , {{formula:c5b41258-803c-43dd-b3e6-ea540d80f543}} are not. Indeed, {{formula:18a38420-0c3e-4db3-aeff-c066c7e5fe16}} has a logarithmic pole along {{formula:9b1404ab-6057-426b-bf00-914fe06c8266}} . For {{formula:262336ad-37eb-4a9b-a5ec-4ad25dd42cc8}} , note that by Lemma REFREF we have {{formula:a8d45a21-423e-4ba2-b5ea-4803ac2315c7}} , so Lemma REFREF gives
{{formula:2d32528b-30f5-4614-80f5-88e7685983ae}}
In principle, one should think of 1-forms bounded from {{formula:4774cada-66d2-4574-a4e8-3deb1024b312}} as the ones decaying exponentially fast as we approach {{formula:ca40d326-b551-4a1b-bdcb-506655fd9986}} . One way of making this precise will be given in Lemma REFREF .
In the next computation, the following elementary observation will be useful.
Lemma 3.24
Let {{formula:7b38eb8d-a548-4ac8-abd8-f85699b1c481}} . Assume that {{formula:b64723db-5239-4816-b6e7-1b587c3611de}} . Then the following inequality holds:
{{formula:c1eb8a1a-0a0a-4aa0-b405-7c386aeaabca}}
{{formula:1027a5c7-6691-4145-941b-56075e7d0f1c}}
.
By symmetry, we can assume {{formula:67d41f64-8612-448c-b137-b42fe94cbbe7}} . Then {{formula:91918b3d-8a74-4131-9f3c-0231d6e8b1b3}} , so
{{formula:cdabd230-f42d-4e5c-b400-ee63e367c602}}
We will now use Lemma REFREF to compare the 1-forms {{formula:8a39b6e4-f754-4602-90b9-ee1ce495438e}} defined by two charts adapted to {{formula:e4674111-4d9e-4e80-a9f2-91f215535d2d}} . We warn the reader that functions {{formula:9222d2b3-a16e-42a4-9c57-0125e2a2cfc8}} and a 1-form {{formula:6070cfc1-b443-4287-ad6a-431794df42ac}} introduced below depend on the fixed charts {{formula:07e91a87-8743-4445-b1cc-58c4c96d4e40}} , {{formula:00eab713-8ab7-4a38-8345-067b9dcc52d4}} and index {{formula:33f9dd39-4b52-4443-ac12-0a28eb990203}} ; even though the notation does not reflect this explicitly. The same warning applies to functions {{formula:38bababa-044c-4e9d-8a39-10a22656ce83}} from Lemma REFREF ,REF .
Lemma 3.25
Let {{formula:cac7c94a-464e-413c-8871-da95400d2667}} be two charts adapted to {{formula:4674ffb0-ba59-40b1-b66d-852e46efb96c}} . As in Lemma REF , we put {{formula:c70c8286-17dc-4b1d-9230-9ab94e28442f}} , {{formula:46922608-4917-4a2c-b01e-d86be4c56055}} , and use Notation REF . For every {{formula:ac80238e-52e4-4517-9b25-64d6667f55b1}} , the following holds.
The map {{formula:a9116dc8-add8-42ed-9061-ba74cdef5c00}} is a pullback of a smooth map {{formula:233fb033-3d10-4366-b857-9435c6e42e63}} .
Let {{formula:73551fb1-cf11-4255-8dbe-d09d99dd83f8}} be as in Lemma REFREF . Then the functions {{formula:03e077cd-a378-4f99-9bba-f00836e9109c}} , {{formula:8b56565c-d9de-4a25-98a2-933896733cff}} , {{formula:0110c0e1-2627-4834-abf2-d3ca6e3debae}} and {{formula:a81c639d-f116-4482-a099-d75a00d56484}} are continuous and bounded on {{formula:125c45ac-e912-4c22-84d0-a2086ff4163e}} .
Let {{formula:fd8f3e02-5695-4930-ba00-65c12519a3b8}} be as in REF . The following identity holds on {{formula:86cb2046-0f5b-490b-895e-edd194a54497}} .
{{formula:8477e0a5-3b1c-40b2-9c46-413a2e6709c7}}
There is a bounded function {{formula:33fe0c8d-de61-4bdb-a593-454e3d8ab960}} such that {{formula:8a88d75d-2135-48bb-a72f-8558554eb733}} .
For every {{formula:b99ec470-f1e9-4399-a65d-abe43d17c721}} , there are bounded functions {{formula:39a5a3ff-f37b-4220-a953-494b47d33f11}} , and a
1-form {{formula:54da5d1a-322c-45b5-b9c9-fbff82afe2a9}} bounded from {{formula:aa06f1d7-1e34-4833-b84b-8083ba999a00}} , such that
{{formula:ae942f7d-4bb8-4dda-8ee9-d89d012b1919}}
{{formula:e54996cf-1812-4969-8d5a-a0c17c481045}}
.
REF By Lemma REFREF , there is a nonvanishing holomorphic function {{formula:4031db32-07a2-44e8-8dca-ca094ab78608}} such that {{formula:ee65d4d1-e2e1-4d5d-bbe8-177426dd86ae}} . In polar coordinates, we have {{formula:14e42ef4-89e2-4b35-ac2d-f89aebfd85cb}} for some smooth function {{formula:ab9245c1-343d-4637-a837-e4ac2fab4f86}} . Using the additive notation REF on {{formula:4e73fc0d-72fe-421a-b970-6aa7aa8bdb36}} , we get {{formula:0519a399-1922-441a-be38-51e133998b7c}} , as claimed.
REF Recall that by Definition REFREF , the coordinates of an adapted chart {{formula:8239dbe3-d9c7-434b-a0ce-acf53c6fa10a}} extend continuously to the closure {{formula:38f65cd2-571e-47a9-ac09-7870e4f840cd}} , which is compact. Therefore, the function {{formula:f45b8aff-58e1-4da7-86dd-2ceedabcea6f}} introduced in Lemma REFREF extends to a continuous function on a compact set {{formula:ef81b12b-0a79-4503-985c-3be3a5c63c0f}} . In particular, {{formula:66dea1a7-e796-4c34-8cb4-09c8da2adbd5}} is bounded.
Similarly, the function {{formula:803a3dbf-cd09-4789-9123-735e71afc2df}} extends continuously to {{formula:025c3ce0-fc15-4b5b-9c29-c8140c1bcaa9}} . We claim that it does not vanish there. On {{formula:4ccefc3a-6620-4832-87b3-7ea6add6acdd}} we have {{formula:91b2341c-0d70-47fe-9f2d-846e4313b05b}} by definition (REF ) of {{formula:faa33759-ab68-44ae-a362-c7dbb20d354b}} . In turn, on {{formula:2577e58a-9745-488a-a15c-acd8032ca7fd}} the functions {{formula:8c2c3346-1313-4214-9be8-51d4ea9c2a84}} , {{formula:3d514427-674f-497c-b1f4-a9d315618d68}} introduced in (REF ) do not vanish, so by Lemma REFREF the function {{formula:ef6555e0-9a6b-4d0e-b267-b3f181a84002}} does not vanish there, either, as claimed. It follows that the functions {{formula:ee6637f2-7ab9-4efa-964f-2c1e79811e5d}} and {{formula:aa7c7a43-58c3-4e43-95e2-3d40afbc186a}} are continuous on {{formula:4c37918d-0fe9-448c-b141-2143b82a6aea}} , hence bounded because {{formula:700fc536-2b87-411c-9632-61f94a287514}} is compact.
On {{formula:1aafd299-6e4e-47d8-aeb7-8839df325683}} , we have {{formula:890eea25-f2da-4318-9c7a-f61fa26a116c}} by the formulas (REF ) and (REF ). Using the second equation from Lemma REFREF , we get that on {{formula:f46bebf2-a011-433f-ad31-01eb85ebc786}} , the following identity holds:
{{formula:aa23f391-56b3-4089-8385-a476795b2c3b}}
Therefore,
{{formula:4d09f1a3-e653-4f9b-9d8a-68d2812176c2}}
On {{formula:bd8fb2a9-02a9-46fc-95bf-f41bcf5682da}} we have {{formula:6bc931eb-4374-4dbf-9f39-e53917c96240}} by definition (REF ) of {{formula:0f8923da-f1c5-4f7a-bc58-297b105bb657}} , so the formula (REF ) is well defined there. By Lemma REFREF we have {{formula:40cdea7d-993a-44bc-aa64-7ba190cfdee9}} on {{formula:47ba91ab-3fcb-4412-a945-7feccb161e9c}} , so both sides of (REF ) are zero. Thus (REF ) holds everywhere on {{formula:a4ed56a0-ac8d-4741-a46b-1d3c5bd31ce8}} . As before, it extends to continuously to {{formula:141016fc-667c-44d6-b19f-c7f965af375c}} . In particular, the function {{formula:3e85123d-c38e-484c-a1ae-778ba3163881}} is continuous on {{formula:a7ca3e4b-57e6-44ce-9b87-31ea8804cec9}} (divide both sides by {{formula:ddd032f7-e661-4bd9-975e-3b5601d8815c}} and use each side of the equality when {{formula:9e04687e-621c-4d24-8af0-a4e43faaa2c9}} is close or away from 0 respectively. Since the map {{formula:7d9512f6-a884-46e1-804b-7babc165feab}} is proper by Proposition REFREF , the set {{formula:27f53c8b-9699-46bb-ae51-d93351c051ef}} is compact. Thus {{formula:8f885a0d-2766-4efa-827e-116985154dd7}} is bounded, as claimed.
REF
By Lemma REFREF , we have
{{formula:e773965b-a3bc-45f4-90a1-1322ef743ebe}}
By (REF ), we have {{formula:e4348ce8-b171-427e-ac9c-7a8c5bf04d17}} , so REF follows from (REF ) and (REF ).
REF
By REF , the functions {{formula:cee69242-df58-44ef-91dd-a47a2d51679d}} and {{formula:21b1b175-2fc3-4a98-adc8-1352076323d6}} are bounded. Since {{formula:0b1ce3b6-9a34-491d-9095-05f909688d2f}} is bounded, too, we infer that {{formula:c5b43b66-217f-4191-8e3f-6185abfaad3a}} is bounded on {{formula:9e50e7d4-543a-467e-9d85-b4453b523b95}} .
By REF , we have {{formula:f50c16a2-b7f2-41b2-8b09-9209b96507d7}} . By definition (REF ) of {{formula:45a1d6d6-e5a2-4e46-a86f-c27005820142}} , we have {{formula:db649212-5520-4e15-9df9-07e8684c3900}} .
Since {{formula:66fd5b0f-6ea0-4f3b-8ce5-e72f832b44c2}} , Lemma REF implies that
{{formula:6352a96d-e551-44b9-b0ef-75ebf9c85dd5}}
Therefore, {{formula:1e63bba5-561a-494a-999a-4ebbeafd8940}} for some bounded function {{formula:7468d41b-252b-4d0f-8699-b37ee0a2c34f}} , as claimed.
REF Applying the exterior derivative {{formula:7a989506-e5ec-4f8d-8428-bb9d4e58db3e}} to the formulas (REF ) and (REF ), we infer that there are bounded functions {{formula:a25aeb02-8bf9-47ba-9f86-2d9d3e91cba7}} and 1-forms {{formula:5459f56e-5ec5-4d23-9f6e-b87b9bb0d5c7}} , {{formula:4cb0fdf5-d2bb-47f7-8e7d-50132e8da80f}} bounded from {{formula:54f680b8-14f3-45a5-ab5d-f23a0c86cecf}} , such that the following identities hold on {{formula:0ecc5f23-4043-456d-bd33-1e6f0a0b943d}} :
{{formula:8a35f324-5dd7-4876-bb12-a32d0ad87810}}
By Lemma REFREF , {{formula:61d6b894-9990-4f17-b9e2-51cf4b551e6c}} . Fix {{formula:92d75607-6eb6-4b3c-bead-7e44a5720bfb}} . By Lemma REFREF , {{formula:f922f3b6-2824-4397-ba9c-028d3c82ad02}} for some bounded function {{formula:a49d5e67-13bb-4361-86b2-09dbf790be49}} . Therefore, {{formula:3d2a111f-3319-40ff-bae1-ba68efcf9cb4}} . Substituting this to the above formula for {{formula:75bf331e-f973-4c72-87e5-44f5a45448d6}} , we get that for some bounded functions {{formula:e87796b0-5c74-4b47-a4e4-fa0fd49c95cf}} , {{formula:7c515358-e569-49fe-9f79-d05c45945d4b}} , the following holds:
{{formula:dcbfcee9-a5da-49f1-b501-08046157a16c}}
Adding this equality to the above formula for {{formula:40c0803a-699f-4b74-b5f9-d888c700175c}} gives
{{formula:efbb3a75-8ad3-40e7-a7fa-377c31706f9e}}
By Lemma REF , {{formula:f873e3ae-be50-4644-be6c-15c9ad958ba8}} , so {{formula:2f0ce397-af8a-44ee-be33-f853a5bb55c5}} for some bounded function {{formula:b5d81470-3b86-44c5-9c58-a1d206b7d21d}} . By definition (REF ) of {{formula:396fa4e0-a781-4b84-857c-8f29df7eeab6}} , we get {{formula:23262f47-373b-48ad-8fe1-b0d4ee54ed21}} .
Recall that, by Definition REF , the holomorphic coordinates on {{formula:99cb74b3-96fd-49c0-8160-fc111c5f8e15}} are {{formula:893f5a96-c71b-48ea-83f7-e17ab53f389e}} . Write {{formula:68eca210-9cb6-4493-96d4-ceecbe5a1a4c}} , {{formula:3ede4022-e4ad-465b-87de-4079bfd29f30}} for the real and imaginary part of {{formula:e3e95cd9-9a3b-4b60-a386-0a7fcad68dc1}} , respectively, and put {{formula:1888e6fb-cde5-4504-8c7d-195bfa3c6df7}} , {{formula:c4181e9e-bfa5-40b7-98d6-79ce232800b5}} . Then we can write {{formula:b715c03f-2ede-4f1c-8130-e26228f4d8bb}} and {{formula:66e26ead-4000-48dd-afdd-70b2d0554545}} for some bounded functions {{formula:e1f17641-5e2c-4c57-acf7-8273a6e7fcfa}} , {{formula:fe3e2d09-3dd6-4532-9392-a46001f8a1fc}} . As before, Lemma REF shows that there is a bounded function {{formula:dae9f75b-23a7-4b22-bcd3-1000b0b2642e}} such that {{formula:991ed6ad-cd58-4f9b-a22e-463fff024d4a}} , where the last equality follows from definition (REF ) of {{formula:41fb125b-8013-40b3-a162-e7b8e3bc0dc8}} . Therefore, {{formula:643285a0-7a36-40ca-a09f-f3410ac10da9}} for some 1-form {{formula:c009bb50-b26a-4f3f-8756-9b85359b8b2c}} bounded from {{formula:972c47a6-99e0-400d-beaf-eac8fe6e0dc9}} . Substituting the above equalities to the formula for {{formula:80584c4e-3c6c-46f4-88a7-fdcc25ea7759}} , we get
{{formula:c942af5a-5e4e-4ebd-8673-c2e987b1caa2}}
which proves the first equality in REF .
To infer the second one, we substitute the formula for {{formula:ae000b4d-6691-4566-8426-56f315714789}} from Lemma REFREF . Since by (REF ) we have {{formula:c8eb7c95-90dc-413a-83c7-e3a281eea4f7}} , this substitution yields
{{formula:2e5fe323-523f-4d40-a863-fc734142ec9b}}
By Lemma REFREF , {{formula:4345ab7d-735d-40f8-bd89-8e8207ca32aa}} , and by Lemma REFREF , {{formula:dba31de8-fb49-42d7-90b8-80c11daf6f0b}} for some bounded function {{formula:3ac78fc2-8939-4e97-b532-6605d7c67ac6}} . Hence the coefficient near {{formula:05a06011-9486-4180-a9a4-177c2c1c955c}} above equals {{formula:de904e90-aca7-4fd9-91d2-7fe2c97aee92}} for some bounded function {{formula:0ee1b322-f87c-459f-8802-d5ef38fea681}} , which proves the second equality of REF .
subsubsection3.5plus.7-.5The functions {{formula:9c70a3a2-2ef2-4ad6-980d-892b52851a41}} , {{formula:397e7357-3ae5-4407-8149-5bddd48b602d}} are {{formula:9288d9d9-334f-4723-9226-98ac18d765b3}} . We aim at proving Lemma REF , which asserts that the transition maps for our atlas are {{formula:28c38ee7-059f-497d-a5b3-ee8ce7045e91}} . The key step is to prove that in a chart {{formula:9149e460-1ae5-4444-af83-ddae3158d1ed}} introduced in (REF ), all functions {{formula:15f34d39-4d51-42a7-a30f-0656b9774795}} , {{formula:0b27bde0-8298-4a10-ada9-32e8f3234917}} for {{formula:48b5ef2c-6448-419f-b5ce-8b2798dabb0c}} are {{formula:4ab745a6-7be0-4f6f-8343-0eea6e0dd6ca}} . As a consequence, in Lemma REF we will infer that the function {{formula:08ae4972-0bf5-44c6-af45-bfcae244f7bc}} , which is not a coordinate of {{formula:077928e0-a683-456f-aefe-f2cd42087de5}} (see formula (REF )); as well as the pullbacks of all smooth functions from {{formula:73f79e18-6a88-49b5-9d1d-fec988277928}} , are {{formula:23c3870d-545c-4e28-87d6-28d8dd387fae}} . In Lemmas REFREF and REFREF we will prove a stronger result, replacing {{formula:bb60f32a-736e-4e37-83a2-f785249d886a}} by {{formula:085bf3bf-9551-45a6-9dc2-a60900904a27}} . Nonetheless, we provide an independent, direct argument here, too. This way, we give a complete construction of a {{formula:bf806356-1937-4985-b583-453d3ec57daa}} -atlas without introducing further technical tools, and illustrate computations behind the proof of Lemmas REF , REF , which is more complicated.
Notation 3.26
Fix a chart {{formula:bb32499c-6af4-4a19-af3b-ddb241ad7748}} adapted to {{formula:b20c07bc-dc6c-46c5-b511-1c6dc59e129f}} . By reordering the components of {{formula:50ef6c7e-d38c-4371-a067-a1f03da7d897}} , we can assume that the associated index set (REF ) is {{formula:0774bfcd-3d59-46bd-b385-5afad6543f5e}} . Since by Lemma REF the map {{formula:b2d96d43-5a10-4fb7-911e-4515f2cf7a9a}} is a homeomorphism onto an open subset of a smooth manifold with boundary, it endows {{formula:22b2b750-6c0f-43a9-ba48-db088bdd1271}} with a smooth structure.
In other words, we take {{formula:91f0bcb1-4ba8-4bec-9c57-adcb94ffd19a}} as smooth coordinates on {{formula:32e9cdfb-2bf2-4804-ab1b-03be216545b0}} .
For a subset {{formula:a4c7c67c-1db6-48d2-854c-7af75c3e50ad}} , we denote by {{formula:94427e1a-9791-4561-b844-ef7ff4cfe682}} (respectively, {{formula:437362c8-4b47-4eb3-8b9d-d2b715a02c44}} for {{formula:fc1508f2-ec26-49bc-bbcd-148602f42cb0}} ) the {{formula:2da5c99a-a028-4933-9351-31d992fc58de}} -algebra of functions on {{formula:e1c4e095-8dc9-4e7f-8360-31c4c21c89aa}} which are smooth (respectively, {{formula:b948a99d-cf2e-4b03-9eb2-05b1bcbecc61}} ), with respect to this smooth structure. We do not use the notation {{formula:95b34069-cbe6-463e-9e00-959c46aee243}} to avoid confusion with the eventual {{formula:6ee4d442-1611-4dc2-a0b0-bccc27d32263}} -structure on {{formula:80ff443a-db29-4256-8c1b-b4def1ee47f4}} , which we will introduce in Section REF , whose restriction to {{formula:eef24071-2a83-4dd9-8c42-082599f117a4}} which will differ from the {{formula:df871403-f0ee-4757-ade5-c6ad84fcdbab}} -structure that {{formula:6d53d813-46dc-462e-b5f5-fd4f3f59176f}} inherits via {{formula:20076bc2-a709-42b0-981a-c895bc296df9}} . Since by Lemma REF the map {{formula:ca80993c-092d-4413-8019-267e9d6af8ab}} is a diffeomorphism onto its image, we can – identifying functions on {{formula:c154b20e-bd32-4b43-bb8d-ea57fd640620}} with their pullbacks to {{formula:739a7a7f-5c5b-4672-adc8-3222738e7e2f}} – write an inclusion {{formula:d4fe1ff3-ac06-484d-88a7-299de3f414f7}} .
For {{formula:1a8d8be7-4c74-45dd-86a8-216fca918e8c}} , we introduce differential operators {{formula:7f06d113-73d0-43e8-99e2-a5ea487c294f}} by
{{formula:a55c2760-c476-403d-9c4d-efa4bf2c8b9e}}
and for {{formula:3265a54f-0f9d-48e3-9fc9-00ddb075d428}} we write {{formula:00cb4d54-d35e-4ac5-904b-1829d7adfa98}} , {{formula:aaed87a2-5906-4c26-a981-8d21a21279e8}} .
For {{formula:21de5d06-ae23-443b-87ff-ed2f55fc7722}} , we put
{{formula:67612492-46da-40b1-8260-f052870a1545}}
Using the above notation, we have the following immediate corollary of Lemma REF
Lemma 3.27
For every {{formula:510584e6-cda6-4d1c-8091-22dd40ab5c5a}} , such that {{formula:d81f4afb-6b7d-4d4e-a4f1-257bbe57f58c}} , we have
{{formula:18c7c03a-d498-4b53-8b04-d378265a1e29}}
{{formula:7788ba93-cc43-4611-a450-9c5fb1dafcf7}}
.
Since {{formula:e417af15-c81a-44ca-ad21-c42801ebfd3c}} are coordinates on {{formula:8f661596-07ec-427b-b1a2-987b80626937}} , we have {{formula:9fb422d3-95f0-471e-9a06-d16705e0072e}} for {{formula:be30c4a7-2e49-4adc-bead-6708d78d207f}} and {{formula:b014eef5-cd96-4cc0-9c25-69dc702d48ae}} for {{formula:acfdc2ed-23b4-4fda-a7a2-9f20871876c7}} . This proves the first equality. The last one follows by definition (REF ) of {{formula:1804dbe4-f982-48a7-b8f9-8f4552b6eac0}} .
For the remaining ones, note that the basis of {{formula:e54ed110-ed2d-44a0-aee0-c52e797ed1ba}} at any point {{formula:4287acf3-234a-49f2-88b2-b1f7c9c03c9e}} is given by {{formula:17b019c1-8876-4c61-8281-d77987d07245}} and 1-forms {{formula:38fdb260-3b78-4a0f-9252-0bb37adf84eb}} , where {{formula:f2c89dda-2255-4c5b-b0e5-c627fd3d872b}} ranges through the coordinates of {{formula:b0a4bbb0-8762-472d-b19e-304030f35124}} . For a function {{formula:e1d45ff5-5dc6-4bb4-a847-56bc664e2ef1}} , the coefficient of {{formula:39f2d25a-644a-4848-a7fa-de844416e392}} near {{formula:86985f62-975f-4096-be08-ab849b872c6d}} is {{formula:3b41753e-7cd6-4e52-a196-d1355f2aa629}} . By Lemma REFREF , REF , REF and REF , this coefficient is zero for {{formula:9ca4baf0-8799-44f1-aa04-4463de2fb23a}} whenever {{formula:c2ff03a3-1476-4ca5-aa56-3c3c6b36f659}} .
Lemma 3.28
In the setting of Notation REF , we have
The function {{formula:21e9d847-4f58-4aff-9e53-8aabec42462d}} is smooth on {{formula:63453602-ecbe-4d0e-ad41-8f1d21759fb0}} , i.e. {{formula:7a4533be-12c1-43b9-8f80-36eea2ae1360}} ,
For all {{formula:8ec1b27a-9777-4d5e-823a-a536ee6defab}} , the functions {{formula:9fe22051-e660-45e8-8a19-c1879ec82529}} are in {{formula:16846521-7bdf-410e-a24f-b0bab98f7bbb}}
The pullback of any {{formula:63dfd91c-33f3-47f0-a2b9-dff17d5cd7e7}} -function {{formula:2c165d90-b4b4-4380-88fc-d486b7d011ca}} on {{formula:283920ea-8d95-4293-a2b3-5be4507efea0}} is in {{formula:77058590-002e-4523-b11b-957e0e3fa91c}} , i.e. {{formula:f2ee1cdd-277f-46f5-b6a2-45833ec7ea8e}} .
{{formula:b27c1333-c336-4ecd-afa4-e071d8675961}}
.
REF Since {{formula:8549cebc-0c80-496f-8d05-d4754162ae3d}} by (REF ), and {{formula:60bd3915-d2a9-4c0c-912c-350cc58a9a9d}} is one of the coordinates on {{formula:1be7c446-b3cf-4795-a606-fd53296c81e6}} , we have {{formula:150e1f06-0de9-4c7e-84aa-ddfb79bc2742}} .
REF Fix {{formula:ae5c0f55-c689-499c-9c27-df1c679f58f6}} . By definition (REF ) of the coordinate chart {{formula:4c3b5a1c-223e-45c9-87d9-a4efe5d1f10e}} , we have {{formula:8b855ad1-37ed-498e-8025-5b41ff9094ab}} .
By Lemma REFREF , we have {{formula:5602fdce-bbb3-455a-978d-cb295c053856}} , {{formula:871c5f5b-335c-4d4a-82f5-1dab13342669}} and {{formula:d19d96e7-1679-4b11-9ed1-f94a148dd26d}} for {{formula:2b086f48-e81e-4c47-bd3d-662d7565c2ca}} . Recall that {{formula:0142dc68-5b09-4b37-a8cc-ef9aed233546}} was defined in (REF ) by {{formula:14803d56-4d5e-4dfe-854a-e02bc3b918f4}} , so since {{formula:582ee813-b6af-41f5-811e-6a68f456864f}} are continuous by Lemma REFREF , the function {{formula:1ef7b089-484b-4484-8165-da47e4115814}} extends to a continuous function away from the common zero locus of {{formula:6e2c9204-8bcf-4415-8d4d-a0fd0ea8329a}} and {{formula:cf65fdc6-025c-4ef2-90d0-af0f3f0336a8}} , i.e. on {{formula:ba9010ce-92a4-483d-9cb6-c66e9362f6bd}} . It follows that {{formula:975d5cc4-3ccd-4389-8cde-2014d88c4cf6}} . We claim that {{formula:17b97ad0-b5e9-45f7-97f7-8c2c9cf0ac89}} .
By definitions (REF ) and (REF ), we have {{formula:eee2e207-f541-497a-9ac3-b4cbb3c401aa}} and {{formula:f6f13e7c-63ee-4239-b0bf-b8008296a1eb}} , so
{{formula:268e769b-9b5e-4452-9ad1-a43694f8f7fc}}
By Lemma REFREF and REF , we have {{formula:b0ba2a96-e45c-4be6-bcf7-574eabc4617b}} and {{formula:b74220bb-c11a-4bfb-baa5-287d39385fa7}} , so, using Lemma REF REF
{{formula:82f926d2-5dba-4ea8-b82f-c7886f43b1ff}}
Therefore, {{formula:d4eb3484-ebe3-414c-8393-4d0d117a1471}} and {{formula:e0f57ba9-d3b3-4279-9e5f-83288bc3a8cf}} extend to continuous functions on {{formula:072cbbfc-1fda-46bf-8f90-5420877451d5}} , i.e. {{formula:424d8ce3-10a2-4f6d-9a7c-02eed89946bc}} , as claimed.
By Lemma REFREF , we have {{formula:d8020eee-d68a-4463-b7c1-b17ec7d756ac}} , so {{formula:ac6238c8-3182-419c-88cc-ec96faf9ba79}} , too. The remaining function {{formula:6b7660aa-e036-462d-add9-cf78c9e92154}} is defined in (REF ) as {{formula:6444a242-56e4-4eb2-9e97-c59581f9fb70}} . Recall that {{formula:a8b42a72-fa4c-4900-8307-43e79a1e1db5}} by REF , and the restriction {{formula:e0abbb9f-2b65-41f4-94bb-3855761ca12d}} is smooth. Therefore, since {{formula:7d09ded8-bd52-4f59-a285-56a3ebaf0436}} by definition (REF ) of {{formula:87711d68-11a0-4206-aaff-1b1911933fd4}} , we infer that {{formula:acdbe9e7-2019-4b85-9b27-d9f5e6a40a00}} , as claimed.
REF Using polar coordinates and applying the chain rule to {{formula:ce4efcbe-9691-4a99-b438-b3f80e35e693}} we reduce to show that {{formula:d8447ae4-5af8-4c28-81c9-fa08a94e3c05}} for all {{formula:05aa8476-5020-48ae-a4ad-8203b05044b1}} .
Consider first {{formula:984dea81-62cf-4b6a-a18f-c3bfebb85ef6}} . Lemma REFREF implies that {{formula:71237948-c950-4000-aca0-dd983e8f1383}} and {{formula:9a576741-91f1-492a-bcb6-208bef684081}} are continuous on {{formula:847ac4b3-2aa4-4be9-9187-fa8ad3c2a309}} , and {{formula:521fa286-5174-4d55-929f-c6a3285fcd38}} for {{formula:4eedddb3-7788-4b26-a3e0-64d429f4966e}} , so like before we conclude that {{formula:78df67e6-605c-4c5c-a29f-4f38f2cfedc4}} . By Lemma REFREF , we have {{formula:389e3763-f7fd-456b-905c-6631625317e3}} , so {{formula:8bcfbc99-a520-4e73-bce0-57876b7fb9cb}} , too. We need to show that {{formula:34dc98f7-8dd6-4cc0-880b-d1a5247c4a44}} is {{formula:91df02f3-8890-4b62-aa99-277014f3f08c}} on its zero locus {{formula:741f784c-c5e3-4566-a053-06aa5523f5fa}} .
First, we claim that for every {{formula:5cc18538-20c6-4be2-bd71-fa7ba090cae1}} , the function {{formula:8a7351ca-8c6a-4bb1-a3bf-c88723e55859}} is bounded. By Lemma REFREF , we have {{formula:79b88626-c060-4e77-b815-0309ca7b9792}} , {{formula:a9d89e32-e86e-486e-82d0-a9d18ff94f0f}} and {{formula:460d3613-ec8f-42ff-81de-4b87a263be40}} for {{formula:a4bec570-8bd9-4e5c-99dc-dfd73f595d84}} . By definition (REF ) of {{formula:c9bf5ad2-259b-4089-bddc-c3107214f618}} , we have {{formula:16a786fe-153f-45d8-9ee3-e340184b0517}} , so {{formula:e38a90fe-f0a0-4b0d-849e-34b3512922b9}} is bounded, hence {{formula:1970f11b-ec15-4699-8b11-b2f540962e96}} is bounded, too, as needed. To deal with {{formula:ff131eec-3dc1-4496-9989-9428b98791a8}} , recall that {{formula:3f764f1c-3706-4036-8a66-919cf4f8c839}} by Lemma REFREF , and by Lemma REFREF , we have {{formula:6d856ce6-3e58-468e-a7c5-e0616e9a7087}} for some bounded function {{formula:d4ed06e2-583c-47e7-80cf-877332617e9e}} and {{formula:7ac5991a-9a51-4d1e-a4be-a9342d4090fb}} . It follows that {{formula:f040511b-d1de-4e2e-9e35-8b2dad4c1800}} . We have seen that {{formula:54c02adb-5c0a-460f-ab55-10ea4d3e5682}} is bounded, so {{formula:7ce13533-a069-4d13-aca4-c2d2feea7e85}} is bounded, too, which proves the claim.
Now by Lemma REFREF , we have {{formula:52da238c-edfb-46fc-8010-42ba77111b55}} . By Lemma REFREF , we have {{formula:ad306328-d926-48f9-bbe3-6fc5f04fe0fa}} , so {{formula:defab105-e07d-45c6-8e29-d15592943b87}} as {{formula:b9060759-67d5-4c47-9a0f-5138b9c0bb84}} . Since we have shown that {{formula:1c4e0f83-f50b-4419-bea6-ffaff02abbe1}} is bounded, we infer that {{formula:fa33a58a-24d6-40d7-9382-093616c94e2e}} as {{formula:97b0dd11-928e-4308-a455-928c3cbf5caa}} , so {{formula:ad5b0b84-f2c1-4056-a65c-1f4ff8f8822e}} , as needed.
It remains to show that {{formula:bdae68b7-aacb-4079-aaa3-a8fa59d152ce}} . By Lemma REFREF we have {{formula:05a0274d-8307-4da7-a8ad-ec04b2811d62}} , and {{formula:550c4dff-1bc2-491d-ba38-e67066b71f6b}} on {{formula:d7000aab-44d0-44e8-8682-43e80f9f60c9}} by definition (REF ). By REF and REF , we have {{formula:766b9986-6a55-4841-8c0c-1fba549ce0e7}} , so {{formula:8e272050-1431-4b3c-82ba-502c59e82288}} . Now Lemma REFREF gives {{formula:38c90c26-8d4d-461a-89ca-660a7e1e855b}} , as claimed.
subsubsection3.5plus.7-.5The A'Campo space is a {{formula:364b9ac9-5c68-4b6d-99b7-5a7df3cbb5f3}} -manifold We are now in the position to prove the remaining results stated in Section REF , which endow {{formula:6866cc9c-17cc-4ae9-8d2b-c71db77ac844}} with a {{formula:6852fe2a-2504-4370-8bec-8bc215a846f5}} -structure.
{{formula:b8de9d5f-d829-496e-bf97-6c1fa30bee4b}}
Proof of Lemma REF .
Fix two {{formula:29ef3633-7bb1-4c94-803a-41fcdd5a7ed9}} -charts {{formula:4831469e-8927-40c3-bc21-1aaf992a675e}} and {{formula:41ba13b5-f967-4579-a12f-28b0cad83ea4}} associated to charts {{formula:70c738e9-e19c-4d83-a067-9cb12b173858}} and {{formula:687cbafd-1e13-4896-a61b-73711dc887fa}} adapted to {{formula:de44b244-2f54-4905-aaf6-8965ff9f5732}} . By Lemma REF , the transition map {{formula:e6fb84c9-4955-4e37-a67a-b9d675ff4f12}} is a homeomorphism between open subsets of {{formula:b5f5a5af-cb29-47c5-a850-8ddff658f721}} and {{formula:b10ef8b7-c16e-4097-a8f4-beac1b6ad6f7}} . To prove that it is a {{formula:8e834693-34c2-4d76-a01d-69a1a7c3c6e0}} -diffeomorphism, we need to prove that it is {{formula:17ec46a8-aa46-4843-af7a-7728559c9568}} at {{formula:9e51c4a2-28e3-4882-915a-d3cb6e433e2a}} for any {{formula:28ecb783-baef-498a-b3f5-b536f0852d4b}} , and that its Jacobian determinant does not vanish there.
If {{formula:3569b201-de1e-4d36-9fab-7fbb641ca380}} then near {{formula:a2175272-89bf-4b79-8b7a-a6469429486e}} we have the natural smooth structure pulled back from {{formula:056fd013-2dc1-4502-b938-12662fbce80e}} , and by Lemma REF both {{formula:b1871eb4-d4e1-48bf-834f-6fd3fb16b755}} and {{formula:2ab5053c-f969-44a7-b9c4-f49051efaf6d}} are diffeomorphisms with respect to this structure; so the assertion follows.
Assume {{formula:b7aeefec-6e3a-4783-828d-64c4dd4c0d65}} , so {{formula:f07eff84-89f7-47f1-8797-c6bcd0c5a744}} for some nonempty {{formula:320060e4-e8dc-42f8-8268-4172d89264bd}} , say {{formula:04366767-975a-4d04-bd24-33710c2769dc}} . Without loss of generality, we can assume that {{formula:6faed4ca-3f0d-4b97-b590-0b273a2a4a78}} , and that the associated index set (REF ) for {{formula:d4155325-e5ed-4f86-9446-941e42363cce}} is {{formula:7a8f8c07-d8f2-47ef-a3a8-a13a8fc09d99}} .
Assume first that {{formula:11d45d99-b230-4318-8b14-5069f7ebab96}} . Then the index set associated to {{formula:ad0624d0-5362-4474-83c4-92fc8e040537}} is {{formula:bde9e665-b9e2-495d-928d-3c5f30b8f0b6}} , too. Say that {{formula:a8271267-e18c-45ac-86a7-7cfef2121b8a}} , {{formula:334b9515-7b86-4417-aa9b-c61b2a1f301d}} , and put {{formula:ea38738d-8bdd-49d9-baa6-eeb6af7f4d77}} . The map {{formula:1b0d077e-e0ad-4739-9155-30731f64c5a1}} is the smooth chart used in Notation REF , and {{formula:37e704e4-3ad0-4253-8401-e38bf64d5d14}} differs from it by replacing {{formula:3dace7c4-256e-4e08-8e21-309f52c52202}} with {{formula:1d4816bf-f332-4c1f-8d26-eb04951c12f1}} . Thus it is sufficient to show that {{formula:36a61d36-1b20-4f41-bb1b-93fd71aafc31}} , and that the Jacobian determinant of {{formula:e95b72c6-f868-45a1-894b-aec0c5ade775}} , which equals {{formula:227f5b48-1605-4bea-9f0d-7cb87168c547}} , does not vanish on {{formula:fd89516f-b6b2-4c6e-98bd-83c7eae08be5}} .
The first assertion follows from Lemma REFREF . For the second one, recall from (REF ) that {{formula:859737c9-ee96-4de0-ad8f-fc5c74bfb697}} , and this formula is valid on {{formula:8a5b4340-6e16-4d6e-a4b2-eddf173fe6b4}} since {{formula:fda08bd8-1538-495b-9645-7b86928900f0}} by definition (REF ) of {{formula:406ec189-04c1-40d7-9ef5-40cdff6fbfd6}} . By Lemma REF we have {{formula:507775dc-6eac-4fa4-bfb1-17ef0a3bd6fc}} , so by Lemma REFREF and REF :
{{formula:0a61a6f0-b1a5-405d-a512-59e028b9aa3c}}
where the last equality follows from definition (REF ) of {{formula:f833555d-5df6-4215-9488-8c56d7c95e6e}} .
By Lemma REFREF , we have {{formula:5f3fa611-8478-4933-8e20-4967a8c69799}} , so Lemma REF implies that {{formula:db11f6a3-e68c-4538-9796-b93c7f557961}} . Now by Lemma REFREF , we have {{formula:faaaf249-7db9-4ea1-9aa5-ae6904a12659}} , so {{formula:493a743b-cec1-4605-8420-2da2c99d8d98}} , and therefore
{{formula:81572a60-5709-4114-8b30-0f0f8854d648}}
By definition (REF ) of {{formula:c63d2d6f-b9af-4205-82a9-46d2e75d71ee}} , we have {{formula:a3c50295-93a0-46b8-8767-9ca749f7a069}} on {{formula:eddac901-909d-4028-9060-09d918ff023a}} , hence {{formula:b548f96f-9919-4d8f-8192-09f19f4e08a3}} on {{formula:2d21ff99-292a-42f6-bfbf-892389f2050a}} by definition (REF ) of {{formula:b3ef76d6-a558-4124-a5ec-345fa35dc189}} . We conclude that {{formula:bcd8bac6-b44d-4e6a-ba2f-9e8ea826cce8}} , as needed.
Consider now the general case, where {{formula:f67fb15d-b9e2-405d-93df-80be4c5c5837}} may be different than {{formula:2d07af71-fb37-4ba8-8c1a-fe57371adc53}} . Since {{formula:1995787b-e494-4f82-ac6d-bbdc47a786e1}} , the index set {{formula:81c379ab-604d-43e4-b5d5-7202b7af7b5e}} associated to {{formula:bdd43ec9-472e-47fa-8204-9fc000f394e5}} contains {{formula:27e3d629-5408-43bd-b3db-5c48ee0061a2}} , so, say, {{formula:f69bd31c-1b40-4970-b67e-f14bec48103a}} for some {{formula:e4cc9745-ec9c-4899-afe3-9ecdacf554c6}} . The special case considered above allows to assume {{formula:4fa48995-0cac-419f-b8d4-175e10d02889}} . We can write the transition map {{formula:38a8bc72-521b-4105-a578-a7b13734de10}} as a composition {{formula:ac382086-9e2e-4028-a10f-5712d2451f3a}} , where
{{formula:bc716ba3-64c8-41d5-bc51-35e01351b6a7}}
We will first show that {{formula:c0c53572-6cdb-4a87-81d6-c90b7025a299}} is a {{formula:8b45cdde-dbbd-459b-b20d-689f877b227b}} -diffeomorphism at {{formula:840cb566-81dd-4c5c-b019-995fa8a9212e}} . By Lemma REFREF , the pullbacks of smooth functions on {{formula:d759d90d-2ffd-4d85-8d1c-2348f2a10992}} are in {{formula:1ca30c68-ec9b-4430-9a54-18b9095d0037}} , so {{formula:23a34853-e40c-4e1d-9934-ac01be8e6ceb}} , and, by Lemma REFREF , {{formula:fc817003-632f-4e3b-b566-142df72d1a41}} for all {{formula:4e2696ec-7b2c-4645-b214-18c5dce13f2e}} . Since {{formula:13325e29-9d82-4193-b302-d933b2df19df}} are coordinates on {{formula:71ca6d55-da07-4ef1-a00f-5831a02865c7}} , we infer that {{formula:45fe2460-8dbf-4c4c-a383-7549b8a31485}} .
Fix {{formula:03ed51e4-06de-4d78-9067-1e96da1705ad}} . By Lemma REFREF , {{formula:158bdd54-6710-4cfb-a3ef-5514513a215a}} , so since {{formula:98304cef-9eb9-41eb-971a-1fb720b86b81}} , we have {{formula:3d0c6486-758b-4ef8-9217-35c2c9b66f89}} . Thus Lemma REFREF implies that {{formula:78c7c899-004c-406e-b31f-f19b1dd76279}} . In turn, by Lemma REFREF we have {{formula:01c33590-89a5-4dc5-819e-748635e37189}} as {{formula:7aa11e51-f1af-4935-9825-24f1b4d938f9}} , so {{formula:638cfa76-f112-4da1-bf3b-1df97354247a}} , and therefore {{formula:c5e91650-aeef-4d3c-be44-59b54aea40bd}} , as claimed.
To prove that {{formula:8969ae80-4818-472a-87a2-a715c5e9289c}} is a {{formula:2b360010-d02c-40f3-9b59-e4868186db01}} -diffeomorphism at {{formula:2e3a01de-d5b2-4bf2-bd7a-453f94c01827}} , it remains to show that its Jacobian matrix is invertible at {{formula:2167e068-ca22-4b50-82d1-81d371a74e40}} . To this end, we will first write the Jacobian matrix of the transition map {{formula:97be77a1-7356-465a-a74d-827bf160b563}} .
By Lemma REFREF , for any {{formula:513c6fd9-d204-4941-a6c6-a6d7b211ac17}} there is a nonvanishing holomorphic function {{formula:f20efeb0-6c50-41bd-bd3d-4f9940f32cb7}} such that {{formula:babe024a-0d5c-4a37-8bd6-4488f8c06520}} . Hence for {{formula:4da0f5a5-b96d-4cc8-8399-efd9b49725f5}} we have {{formula:ab2435f3-5f65-4c55-a10b-f83819db4867}} , so {{formula:4517a74e-3d49-46cb-a014-ed1b9a348e5a}} . It follows that the Jacobian matrix of {{formula:89163c79-140f-494f-b00e-dbdeb66e4c02}} at {{formula:095a64bb-2780-44af-aa2f-3057352616e6}} has a block form {{formula:59d521eb-7b05-4e93-9bfd-1ce36d974cc9}} , for some {{formula:93000b98-0d6f-4b95-9c35-5a7c7fb0358a}} matrix {{formula:c8a1451c-468c-497d-88a0-57f0ee67b20c}} . Since this block matrix is invertible, so is {{formula:39f6f890-c2aa-4e71-867d-6fe1004ba20b}} .
Now, we claim that the Jacobian matrix of {{formula:28e5df23-68f6-4ae7-81a0-a12ca873644b}} at {{formula:e8c0d594-5f49-4505-82c6-71ff49befab1}} has the form
{{formula:22557c18-999c-433b-8c08-08a44969cb36}}
The first row consists of partial derivatives of the first coordinate {{formula:5bfc2cd0-2cd6-4b34-b61b-689b48b2322f}} , so it is the first unit vector. For {{formula:e3781a88-6d23-4de8-9f78-f7113897994f}} we have {{formula:c877be36-cb5b-4e4c-81fc-06abc17886c8}} , so {{formula:0f998ead-a3f8-4723-84b1-0f79d9142d3b}} by definition (REF ) of {{formula:572b8bfb-8384-4dc0-a639-510ea0152ac7}} . Thus by Lemma REFREF we have {{formula:7e1764a7-0c03-4911-8141-52823ea8d94f}} at {{formula:bd1c7af6-04c4-4884-9d45-34fed3c11eea}} , so the {{formula:e05b2c1c-65a3-48d5-b41a-c290fe3cf70b}} -th row is the {{formula:b0cdffef-7f09-4610-b3f7-424a1595253d}} -th unit vector, as claimed.
The middle {{formula:e5c86f49-3677-47b4-9643-3f7c0a378da4}} block is {{formula:576223e7-5dd4-4097-83c3-c808b45f5e07}} . By Lemma REFREF , we have {{formula:adac65b0-8513-43a7-87e8-88755240e555}} for some smooth {{formula:77371223-b177-415a-99c5-bd6236e7787e}} (recall that we use additive notation REF for {{formula:a4d66dcf-3296-4263-ad89-f4eac2e60d8d}} ). Write {{formula:847e00a0-ff42-440c-ba5f-1a4dcaaccf09}} . Then {{formula:e431a78c-5264-4148-808f-2b80a954020f}} , so {{formula:7e5cde07-e963-4f01-9098-530a1be21a2c}} at {{formula:30e4e5e2-b018-4b98-b4d5-8ba4bb3964eb}} . Thus the middle block is indeed {{formula:79e009f2-def0-4160-9f5b-6f9d8f451fee}} .
The last {{formula:dec3efad-e9df-48d9-9d00-38307de96dfe}} columns correspond to the (real and imaginary parts of) the functions {{formula:3cc67bff-1921-4faf-b909-4e40fbc956c8}} . The above computation shows that their partial derivatives with respect to {{formula:835a90c9-25bd-41d5-beb8-b15319bafa7c}} vanish. The ones with respect to (real and imaginary parts of) {{formula:4560773b-3007-4e24-83c7-0f714891dbe8}} give the matrix {{formula:f4fec83f-8c34-45e4-b092-0cc42f05e171}} . This proves that the Jacobian of {{formula:a863cf77-e497-4a29-bff4-053407ed61fb}} has the required form, so it is invertible because {{formula:fa66ebf1-55d8-489d-8473-a77a65d2df4a}} is.
Now, consider the map {{formula:e3689481-aeae-44c1-ad09-adb71119b22c}} . Its inverse {{formula:80b901e6-f811-422f-ae9b-739c392c0814}} is defined away from the zero locus of {{formula:e9458e2d-9d3a-4271-aa45-95f55b042762}} , for {{formula:4d348441-cfa1-4e75-8f0a-42dbd90a5620}} , and it replaces the coordinates {{formula:7c6ba947-d1ac-41dc-a65d-33cd65d54c90}} of {{formula:a29a77a9-5949-4598-9570-b9da1d0c7689}} by real and imaginary parts of {{formula:11f61ce6-971c-46c0-803c-d8c07737dca6}} . We will show {{formula:999b41b6-087a-4f11-a0da-496348266dc6}} is a {{formula:59ba8937-6708-4b74-8ec7-36e60a0ded20}} -diffeomorphism. It is sufficient to show it for the map {{formula:913c4b55-4550-48f9-a45e-7f10c1c9632a}} : the remaining part is passing from polar to standard coordinates. Since {{formula:1200df78-ca3a-43a0-8abe-5bcef3f40e98}} is a pullback of a smooth function on {{formula:c2d5fd13-605e-475e-a020-ffe9c606e309}} , by Lemma REFREF it is {{formula:53c93f05-021f-40d3-a8bb-aa82192df0d6}} in the coordinates of {{formula:71fe8467-c03e-479a-b553-1e6d2bcad376}} . Hence our map is {{formula:5059cf1e-a03e-46ac-9b87-77da44308b0a}} . By Lemma REF , we have {{formula:61c43060-e08b-48a5-b946-0d3aa0c0b312}} if {{formula:49771c39-059d-4d53-ac8a-011a749200d8}} , so its Jacobian determinant is {{formula:532cd90c-a835-446c-8174-fcb2aa17e26e}} . By Lemma REFREF ,REF , we have {{formula:16f13f01-ce73-40b0-9c01-dabab0d09c55}} , which is nonzero whenever {{formula:42efc06d-df5f-45c1-8811-0d00bba0ded5}} is, as needed.
{{formula:c37a3fd9-62ac-4c88-b6ae-0b617c42c069}}
Proof of Proposition REF .
Since the {{formula:894fca1a-0f61-43e8-b0ee-c8463dd6aeda}} -structure on {{formula:9ef50bf3-46a6-4211-8af5-0eebeeeacf08}} agrees with the one on {{formula:49932856-977d-4c02-ab93-90b602c967f8}} , the restriction {{formula:3cf5d26d-d0ac-49d4-b472-5e26342b5463}} is a {{formula:1e43c12b-d728-4046-9976-a0a8be4321c4}} -diffeomorphism. We will show that {{formula:27fa2f74-6629-4a68-84af-eafe48ea1e62}} is a {{formula:4b0676bb-813d-4a84-8d15-58a28fdfbaab}} -diffeomorphism at every point {{formula:a05d62cc-5383-40e5-8962-8d22f75fcee8}} . Fix a chart {{formula:b69d3f37-8335-4d70-926d-c0f95d85f5ef}} adapted to {{formula:c13f42da-dd6e-4dd4-9f0b-20470eba8f37}} containing {{formula:c1ddc26c-70bb-4918-91c6-e6531a0bd93b}} , and let {{formula:489d3db1-cb04-457c-a667-846b51eacb42}} and {{formula:810ef7ac-7acf-4dfd-8fd5-0c4315fbdccd}} be its preimages in {{formula:9a7ceb21-9cad-4479-b39a-32a2f8a99c9b}} and {{formula:17e33553-8ad8-40b2-94c5-2fa03d20095f}} , respectively. Fix {{formula:505cb838-77bd-4b53-af96-91b6c3dd86bb}} in the associated index set such that {{formula:598b3089-33f5-442a-9862-771ae49a5dd3}} . Then {{formula:7500acaa-a058-4149-9744-ffbfa6038c7d}} lies in the subset {{formula:ac344a2b-72d5-4b71-936c-3bd47cffe88f}} defined in (REF ), as well as in its counterpart {{formula:19f05cb0-00b4-4740-b759-482d68292196}} . To prove that {{formula:8ff926a7-c1c8-46cf-be17-bca0a54ee328}} is a {{formula:588f6af9-2b92-4c1c-a090-6f892c860122}} diffeomorphism at {{formula:6327ad0f-9eda-4ef6-bf28-d1b84e84c763}} , we need to prove that the composition {{formula:a3ec78ca-b8fa-4e3a-a37b-bf45e52d6632}} is a {{formula:5821454b-6652-463a-bfea-124c685cfc22}} -diffeomorphism between open subsets of {{formula:46a74d0b-975b-4862-a75c-6892db5dc28b}} . Since both {{formula:a4e81d17-abd4-464a-8401-10f0a15a6edb}} and {{formula:e1afdfe0-5374-4e53-b549-e28e8be8a99a}} correspond to the same adapted chart {{formula:11400f9f-77e9-44c1-b0b2-5a7b89a80324}} , this map is the identity.
{{formula:9bc9415a-96ff-4c7e-8741-b27df3f9e6df}}
Proof of Proposition REF .
REF By Lemma REFREF , the map {{formula:7b79bbe8-bdbd-4c40-bf4c-6f9544706804}} pulls back {{formula:7ffb8173-cf9c-4a97-8e84-5a4199b82e8a}} -functions from the charts {{formula:a29be03c-db15-4ac0-8706-f0fe081b841a}} to {{formula:413a42bb-5314-4e49-ada6-c12a93cb09d3}} -functions on their preimages in {{formula:d7731fc4-0411-4d8f-930a-2bb905421076}} , so it is {{formula:5ca47d87-5776-47a6-adbe-7d25d6f09cee}} . Its restriction {{formula:750704e9-0da2-4c19-93f1-a921ec3f51cd}} is a {{formula:67e94b98-8d20-4faf-9b4b-138b6979b56b}} -diffeomorphism by Lemma REF .
REF The function {{formula:d0f06df6-9982-4feb-97a2-4b0cb95f45ab}} is a coordinate in each chart (REF ) meeting {{formula:e07e1d61-19bd-4f10-aabd-35d61d527e69}} , hence it is a {{formula:81ca85c4-8e88-4717-afbd-87a9cde2dda6}} -submersion (onto its image) near {{formula:ff821602-64ab-402f-9168-5a7e52c76dcc}} . Away from {{formula:40e6ce5b-a0b2-4e00-a5bd-c73f0b68db4f}} , it is a composition of a {{formula:f335e911-9258-48aa-8be7-f16a236746dc}} -diffeomorphism {{formula:ce634219-009a-48f2-9d7d-9b39a1b1353a}} with a submersion {{formula:a87c381f-50ae-40b3-be83-e00db4c3e8af}} , so it is a {{formula:baed21aa-9958-433d-aaed-cbc1e9337922}} -submersion there, too. Similarly, over each chart adapted to {{formula:cdfe5397-8a57-4b5c-8bd1-4500466bd139}} we have {{formula:2fb2f899-ea50-4682-b7ed-6506ec613c9b}} by Definition REFREF , so {{formula:74076684-fb06-41c7-8be0-4d4b74f90da7}} is a nonzero linear combination of coordinates of each chart (REF ), hence a {{formula:7e5a1e42-ae0b-4b17-a097-7369a3df489f}} -submersion near {{formula:3697a533-9f73-4b99-930c-f5b5a679ee99}} . In turn, {{formula:4917d840-cf16-4271-90c1-513a449dbc2d}} is a pullback of a submersion {{formula:be91c428-df04-4e2c-88e4-bd49f9f7f87b}} . Thus {{formula:8622b793-4e8c-4ec1-a234-b9e601c3f0fd}} is a {{formula:a30e4b73-5b35-48c4-bb2f-628736f01490}} -submersion. It follows that the map {{formula:4fc30565-6eda-48fa-b241-91fdec6e9df9}} is {{formula:63e625f1-48da-4c3b-ad09-6dbde20be420}} .
REF Let {{formula:52376b6e-f4cb-4006-b60e-28e3d7ef133d}} be the chart adapted to {{formula:636fdab3-d600-4d4d-ab8a-96d04f59e6f5}} from the atlas {{formula:e9fe7e0e-2f44-4b6e-8fa1-475b952c05ac}} used in Definition REF , and for {{formula:7ff45be2-d554-4633-97af-c7630b6ed7c7}} in the associated index set let {{formula:89199d09-6c44-43cb-b191-964acfa2c531}} be the corresponding function (REF ). By Lemma REFREF , {{formula:5941aa7f-54f8-47e1-a403-230b6dc7c596}} is {{formula:b06209e8-b9be-499f-9654-330ede6157cf}} on {{formula:f0672cd8-7235-4f2e-9b3d-77029be3afd2}} . Let {{formula:60526941-874c-49e3-a08c-47e3a78c1982}} be the element of the smooth partition of unity {{formula:d58c508d-aa57-4616-82e3-d6e3a886b756}} inscribed in {{formula:ec422605-64f3-4512-b856-e209262682a3}} , supported in {{formula:fbb8814f-b6d3-4695-95d8-a6c5e3f81fe7}} . Since the map {{formula:eb101a41-7025-4666-97c2-2bb06e443014}} is {{formula:bfcffda5-26b0-4e33-959a-1901236f064a}} by REF , the pullback of {{formula:179c4581-455b-4b8a-9f6b-cb927901b33c}} is {{formula:677b4181-bc85-46d3-ad64-d0cc03a6b590}} ; hence {{formula:8c02785c-be76-48b1-9a6b-4ab26dcfd7d7}} is a {{formula:2fc8fbef-45c3-46f1-b104-9cff7ae7ffea}} -function on {{formula:92b909d1-3e52-4ad9-ba52-e5bcc05cfa42}} . Therefore, the function {{formula:551ddd64-485f-4bc7-bba3-390131008545}} defined in (REF ) as {{formula:f266b551-7ba8-4afb-ac79-705fa8ab8dea}} is {{formula:f7f64849-427b-455f-b4b2-be0c55b53a37}} , too, as needed.
subsection2-.5plus-.7.5{{formula:b42e089c-4b3e-4b3a-a268-cf5b82070cfb}} -atlases on the A'Campo space
Let {{formula:349d2bc5-9e5d-43dc-867f-3d1b5ae1fc63}} be a pair adapted to {{formula:8c2c0c02-a157-4797-bd90-d83ad30ee02c}} , and let {{formula:3dc03541-c5dd-4d63-9d0b-3e29137c8764}} be its associated A'Campo space. We will now use the function {{formula:e7ce3438-3768-4e68-ae98-5d901944fdf9}} , see (REF ), and the distinguished global functions {{formula:24579df8-bd08-4fa7-97a6-2610ce897dfc}} corresponding to {{formula:2ddcfbe5-93fe-4771-a61a-0c6fa16de58c}} , see (REF ), to produce a {{formula:61964b2e-1d90-41a0-9b58-c214f421ceb1}} -atlas on {{formula:3b74e658-280d-426d-9ca5-7ed39cc798b3}} . This atlas will be compatible with the {{formula:ec1cf58d-1d3c-429d-b6fa-660f0ecd686e}} -atlas defined in Section REF . However, if {{formula:9fc6480a-1844-454d-b2a2-9bacafddbd51}} and {{formula:573bfc61-07fc-4819-b6f1-1536b28288c9}} are pairs adapted to {{formula:8724b1cf-20bb-470f-ae1c-8061df306112}} , and {{formula:77fac5df-e858-437b-bba0-5bf5eb570964}} , {{formula:c4f97c25-ac66-4e49-9983-45ee170f5bbc}} are corresponding A'Campo spaces, then the {{formula:2956c765-b9a1-42c6-b1d1-f3e8878ed49e}} -diffeomorphism {{formula:fdde79d7-98dd-489c-b09a-6650368c1ec8}} from Proposition REF will not be {{formula:b4f02f45-7b78-4475-a7ed-c01c8121a800}} in general.
subsubsection3.5plus.7-.5Definition of the {{formula:d2aba126-51c3-48b9-ba6b-5cf91fc291b5}} -atlas associated with {{formula:ce428269-035f-4818-b5fb-c065134cce96}}
Fix a pair {{formula:06fb9ec8-3d75-4a08-bb68-17a2e1705409}} adapted to {{formula:e91cdb73-6867-4cf0-86e2-69ae93bb82dd}} . Let {{formula:a1de3934-4a0e-4a6c-bebe-2c414d17a53c}} be a chart adapted to {{formula:7dc1f0b8-c654-4528-96e6-e84ac250dc98}} , which perhaps does not belong to the fixed atlas {{formula:e20f9064-c6a7-46f9-804b-167ef67fc58a}} . Let {{formula:8937b296-21e5-4d6d-a73d-76b79dedf703}} be the index set (REF ) associated with {{formula:08a19678-736d-42e6-8500-6c130770f99c}} . Let {{formula:91fdbefa-7678-4999-986c-38f3d251963b}} and let {{formula:4abee91b-fe3a-4c1c-8822-17d0046fdc2a}} be the covering of {{formula:64dac2f6-8a95-4ec6-b5f3-1eb8804a102a}} introduced in (REF ). For every {{formula:62822de1-52de-498b-b2ec-76348cab10f7}} , we define
{{formula:373169ad-c3a5-4161-8797-3d452224967f}}
It differs from the {{formula:6fe3e99b-3a8e-4311-933d-602555bb9e35}} -chart {{formula:3358917d-c5a9-4416-b9dd-8318810a6e3c}} introduced in (REF ) by replacing local functions {{formula:35fa2216-3098-4663-97a0-76d7497cc468}} by global {{formula:695efed8-b2ae-440d-beaf-f5d0b2068626}} . In particular, by Proposition REFREF , {{formula:9aa61da2-2911-4913-99e8-12b439cf4917}} is {{formula:2774f1db-3908-4673-9565-90c00512c1fa}} with respect to the {{formula:576373a9-0b06-43e6-85e8-b5ec5bf8dcf1}} -structure on {{formula:d8dc2ce5-7552-4049-b491-b7160e7ce29f}} introduced in Section REF . Note that the formula (REF ) for the function {{formula:4c694d97-be1e-4e12-bed6-edab2d61ee4d}} used in (REF ) depends on the fixed pair {{formula:af496150-a3f9-430d-821b-c685f2c20c8e}} .
We will now use charts (REF ) to define a {{formula:15bc55b1-afd0-484e-9e79-114fbc17a388}} structure on {{formula:cab60eeb-00e0-4d0e-bb67-1c89532f83c0}} . As in Section REF , we state the main results first, and postpone their proofs to subsequent Sections REF –REF .
Lemma 3.29
Consider in {{formula:921599e9-01ff-4b00-9c41-a1c7ba152276}} the {{formula:cd102389-7876-4e08-823a-69c76f39be56}} -structure constructed in the previous section. For every {{formula:ea336857-3fee-43f2-a887-2262f545b303}} there exists a chart {{formula:fc23be10-c212-48b9-a39a-085c1657edc1}} adapted to {{formula:d39ec5c3-8f41-4bea-acb3-7b6831bdc4a5}} containing {{formula:2ebf03d9-ef44-4183-b63e-6fc17758e7df}} , such that each map (REF ) is a {{formula:4e90ef57-010e-4a88-8569-c778d9f58fbb}} -diffeomorphism onto an open subset of {{formula:cbe34f54-72bc-43b8-b371-2ce5547a462b}} . Moreover, the restriction {{formula:4c05c7b2-afd9-4c91-83be-368ca0db4f82}} is {{formula:4caa6503-6227-44c7-aa97-9091b32fe6e0}} for the natural smooth structure in {{formula:76d24580-926e-45bf-9b1e-1d7bbadf27af}} .
By Lemma REF , pairs {{formula:5a360def-96f5-4f47-8a8f-71c84fe57dbf}} defined in (REF ) are candidates for {{formula:05c425c4-3b7f-4cdd-98c1-216c4304ee63}} charts on {{formula:19411740-a7e9-4bed-8e86-9e3fae8c6e33}} . We will call them the smooth charts corresponding to the adapted chart {{formula:1e9111a2-c0fe-45d4-b3b1-ca49c1766681}} and to the pair {{formula:788e7af9-b4a0-40df-b5b3-f8b506d78f79}}. As before, we abuse the definition of a {{formula:097ff0a6-d6a0-41c5-ac58-46b97501347d}} atlas by allowing open subsets of {{formula:1574dd72-0d63-4820-ba98-80f8edf65d83}} as targets for the charts. Nonetheless, the obvious refinement produces charts whose targets are open subsets of half-euclidean spaces.
In order to complete the definition of the {{formula:d2b9d512-c9d7-459f-a089-89833efee4ba}} atlas we need to prove that the corresponding transition functions are {{formula:3a150678-c3f0-4f08-a7d3-c2c016afe08c}} . This is the content of Lemma REF below.
Lemma 3.30
Let {{formula:9ee7f847-0018-421d-8c12-62a00025faf6}} , {{formula:9113109c-e042-4bdb-8f34-05c82b901992}} be charts adapted to {{formula:646789e4-9b60-4363-8572-2ca06c4fc497}} , satisfying the statement of Lemma REF . Let {{formula:6626f6ad-e199-456c-a864-353638f3cecf}} , {{formula:91ada831-eed8-49cf-be2e-45faf9c360ad}} be their associated index sets (REF ) and let {{formula:9295dd33-94e7-4701-8dd9-74bab3408986}} , {{formula:5f0c2561-8edf-4885-b197-9aeb5d884c46}} . For {{formula:70310f8d-86e5-49a8-bcf8-d9afa57f5683}} and {{formula:1bce1403-9aa3-4375-9c74-3edb221a8ace}} let {{formula:79beb579-478c-4246-b862-3b75b9133689}} , {{formula:d1beda4c-c637-492d-95b9-1e1cd6a79e38}} be the charts defined in (REF ) above. Then the transition map
{{formula:dc03ced2-7c58-41b3-9f98-47e713bc5f3a}}
is a {{formula:853a688d-b528-4505-9073-16a7b6f2d7f1}} -diffeomorphism between open subsets of {{formula:9c46b805-7a47-46e3-bbac-e5b6bd85fdd4}} and {{formula:da6596ac-fafa-404b-b14c-7e64e5a29002}} .
Lemmas REF , REF endow {{formula:9c942bbf-fa1c-4ef3-9941-2f1bb9c9a114}} with a {{formula:ab7b82c3-911e-416e-863f-0e7669a858d1}} structure, compatible with the {{formula:712ef8f5-2862-415d-8ad5-6ac416b506fb}} -structure introduced in Section REF , and with the natural smooth structure on {{formula:4fd37ea5-b611-4a73-8417-33208ec9c140}} .
Remark 3.31
Example REF shows that the {{formula:659605c9-60b5-4f18-9212-01dc318a7943}} -diffeomorphism {{formula:81b70f59-e59a-4a47-9488-b4da8958d73d}} comparing the A'Campo spaces defined using different adapted pairs is not {{formula:104db982-66dd-4e0b-82ce-4cba814259f4}} in general. Indeed, let {{formula:6ccdf910-c18c-4702-a4f0-fdbcba5715f3}} , {{formula:aa9da2d0-8017-4a5a-b293-9bdb0988de55}} UX{{formula:ac6cb374-26cc-499f-9c65-4e9a9691f34e}} UX'{{formula:5ecf6f9b-12b5-4fdd-8d9b-148aa13388c2}} f{{formula:aba2030d-8883-4288-ba51-7b44265fb375}} UX{{formula:ac94cfcc-f5fc-45f1-a2ef-a37cd81ea4fb}} (z1,z2){{formula:3f9b3d59-9f30-4937-a798-b077a23f02d6}} UX'{{formula:af24e24c-5c81-4550-9564-cfa2c6180d88}} (z1',z2')=(e-1z1,ez2){{formula:4db41e31-f775-4835-94c2-8c4e658daffc}} A{{formula:e598aa7c-4141-4c9b-b1a6-e147347aba14}} A'{{formula:b015cbd1-1ee9-4ac1-b6e4-85c7c6199874}} (UX,{1}){{formula:470d649f-e5a4-4b4d-8a02-694bbf3e3549}} (UX',{1}){{formula:45b44ed1-a752-464b-a818-01c34f5f42be}} C1{{formula:4cd2df3b-302c-48b9-9a15-569e849dcbaa}} AA'{{formula:9bedcce3-0459-4613-b698-f0dabee01a85}} |A=idA{{formula:a106ef3c-51f5-4e06-89bd-4cede044ad0c}} C2{{formula:c828f6b8-af37-4cf5-9baa-b36b326ad49a}} A{{formula:edfe7a3e-e906-4b6b-8a34-d5bbacd8007e}}{{formula:9bc98f1d-cb8a-4e37-9a00-7f26e014bc1e}} C2{{formula:a1a33fc9-294c-4e71-9597-d1e56f06ceb4}}
The {{formula:a1df9284-7b82-4ab2-bf6d-0a94bd28f62d}} structure defined above has the following important properties, cf. Proposition REF .
Proposition 3.32
Let {{formula:0c5032a2-2cee-4da1-aa07-1ad9f200e682}} be a pair adapted to {{formula:41c1377f-2e47-4f7c-9767-8a2587ff0c7a}} , and let {{formula:55ace117-3735-48ef-accc-2566c48733c0}} be the corresponding A'Campo space, with the {{formula:ff7a63d9-2edd-4fa7-b426-19ffabd44167}} -structure defined above.
The map {{formula:dfcfeb78-d155-487d-9d8e-b829454db09b}} is smooth. Its restriction {{formula:4969cd04-0e10-4634-8578-99257aab189a}} is a diffeomorphism.
The map {{formula:eac0e879-de4d-4125-97ab-960fad385660}} is a smooth submersion. In particular, {{formula:67faf0b4-cea8-43b9-a329-9b5469d16719}} is smooth.
For every {{formula:00442a4f-fb87-4e80-8cfe-68543de6f777}} the function {{formula:d30b5951-2f44-4d0d-b748-aa68578d378b}} defined in (REF ) for {{formula:f1bd65c8-58e4-43a5-8704-061935ca5a5b}} is smooth.
For every nonempty subset {{formula:9c63fffc-f193-4fc3-bba0-b933dd589f42}} , the restriction {{formula:694331dc-eab1-4efb-a2f0-d644eb7ae7da}} is a smooth {{formula:22a405a9-7b5f-4a32-ab03-a7d1b72ae336}} -bundle.
Remark 3.33
Proposition REFREF and the Ehressmann lemma give a diffeomorphism {{formula:9dab678a-8c42-4647-b5c5-097605ed4096}} for any {{formula:9e7765f8-460d-476d-b3dd-b836eca21bad}} . By definition (REF ) of {{formula:bc19245b-3f04-45a0-9c31-a13969dc34cc}} , for every {{formula:4643ce73-9856-4b1f-9404-7837cc86b461}} we have {{formula:1ef0eee2-8830-4d3e-b4ff-a6d518b8134a}} , where {{formula:a440cdcd-8293-4a58-b460-546ae3640173}} . Therefore, the A'Campo space {{formula:c3597864-6f1c-4478-9474-d21c60abf5e3}} is diffeomorphic to {{formula:59860129-526c-4aa0-8e53-6bd7a02f39e9}} for any {{formula:3279130b-2304-47e9-b490-013f84cc0809}} . In particular, its diffeomorphism type does not depend on the choice of {{formula:1f89983b-f08b-449f-90c6-74a6f49b6fd0}} . Nonetheless, as we have seen in Remark REF , given two A'Campo spaces {{formula:748eb95a-8cac-4079-80aa-5d5c169d1201}} and {{formula:3dbdbb05-1eb1-480c-a0d7-847ebdf9ce95}} the composition of any diffeomorphisms from {{formula:6cd6d287-24b7-4183-bf02-5ee78b931de2}} to {{formula:8957a911-3503-4187-b02d-0c276739aa10}} with the inverse of any diffeomorphisms from {{formula:7d085b86-4a3f-4457-9e45-25a0ba8314ab}} to {{formula:ba504891-83f0-4523-9243-ff6b2f0cb5a0}} can not be equal to the canonical map {{formula:4817dc58-9999-4290-99b9-66b590aff6f5}} defined in (REF ).
Lemma REF will be proved in Section REF below. The proofs of Lemma REF and Proposition REF are somewhat intricate and need preparation which will be developed in the next sections.
subsubsection3.5plus.7-.5Compatibility with the {{formula:e2d03907-c531-4578-af66-357b29cf9a22}} -structure
In this section, we prove Lemma REF . It asserts that charts (REF ) yield a particular {{formula:f648cb10-9c83-4db5-911a-bd9fc644ab72}} -atlas on for the {{formula:47667782-38b8-4b03-9079-096ae1a47b7f}} -structure on {{formula:edd568de-8e15-469c-957e-cb87530eccc1}} defined in Section REF . To this end, we will use the following consequence of Lemma REFREF ,REF . We use the notion of forms bounded from {{formula:b0b20ee4-d8c0-4344-841b-1f5af5608955}} , introduced in Definition REF , and the function {{formula:dc486b38-9680-4195-80e0-f129303d9a22}} defined in (REF ).
Lemma 3.34
Let {{formula:c04ea77c-806a-43c9-b7b3-7ade0a12e4c4}} be a chart adapted to {{formula:673e4ac0-7381-4586-9f5f-6dde970d61b5}} , with associated index set {{formula:47603621-b198-4a64-a5d1-7547ef8a1bd7}} . For every {{formula:c2f9988b-98eb-408f-b8d5-7a52dce25ac7}} , the following holds.
There is a bounded function {{formula:abe5036d-91b1-46e5-8928-7d3c425150c4}} such that {{formula:e77c4cd8-b3c4-4dd9-b6cb-ec9d3017c7ef}} .
For every {{formula:3f880359-3e27-457c-9281-3b41b104674a}} , there are bounded functions {{formula:9d02b045-163f-4f6a-b6ac-8dac6be448b6}} , and a
1-form {{formula:c1c22e7e-5a91-4783-8729-700d36b49f55}} bounded from {{formula:76b20866-2b51-43da-90ea-438de62fc3a2}} such that
{{formula:405c53f2-2238-468a-9435-603fafdb3d63}}
{{formula:0f4aba2f-8f6e-4d66-92e3-165137683a3d}}
.
For {{formula:63aade90-3504-4a62-8120-79f0f143bbc5}} , let {{formula:7593aec0-acff-4590-8a5b-29b0094fbda2}} be the corresponding chart from {{formula:691fcd39-8e6e-492b-82e0-5b91ecffb595}} , and let {{formula:a51c6854-eae3-4385-be5a-ca9466804acf}} be its associated function (REF ). By (REF ), we have {{formula:a7c43707-b5da-4aeb-8c2d-265d7fa365d6}} . Put {{formula:0ca54345-4f75-487d-bb66-ad53cbba6d6e}} , {{formula:9a936fb3-e668-48d5-abd0-0fd51c49bb80}} , {{formula:14440f44-031d-4e15-99f5-2a08250e0987}} .
REF Since {{formula:3aa4604e-2943-45fb-ba2e-ef3200388ba9}} , we have {{formula:41d34248-ba51-460c-ba55-b38bba1bab80}} . By Lemma REFREF , for every {{formula:0463a7d1-6740-4237-847a-94948816af5f}} there is a bounded function {{formula:733bbdc0-a2f7-4eb1-9258-1009a4a9315b}} such that {{formula:5a97ad77-dc31-44c7-83ed-8ad118487ad3}} . Since {{formula:965954f8-6cf6-4807-83d6-a5df2ef3c70e}} is supported on {{formula:cf503ef7-0c58-4718-b083-f8bb9a28a915}} , the equality {{formula:d18d1146-706d-4a45-89d6-1a5cb419c64d}} holds on the whole {{formula:04d60982-8d00-4926-ba35-dc8b75ba684f}} . Thus {{formula:4e350ad9-2d63-459a-bcd0-ddace8f0eacc}} , where {{formula:2c1e68a6-f7ac-4dd6-b9f3-93f9a7970ed8}} is bounded, as needed.
REF Since {{formula:a2cfd633-0ae3-4e1a-88a7-177817bf3081}} , we have {{formula:560d28b4-8cf5-4875-bc59-a34dc2968624}} . By Lemma REFREF , on {{formula:2d6e42ea-35ff-4a03-96f7-9f967e21dd5d}} we have {{formula:965ed6f9-13f3-42f4-80e5-ad765c96e14a}} for some bounded {{formula:3e46e1da-a04d-4461-a293-8f241466b0f5}} , so on {{formula:d74bb16f-b58f-41c0-ad16-24b8ce052bd4}} we have {{formula:640a6098-c670-4b74-ba6b-8ddca1679041}} for some 1-form {{formula:c35db1d9-4c48-4037-8255-8744af16c795}} bounded from {{formula:dc131475-c7bc-4a6a-ad4d-0b285845f328}} . By Lemma REFREF , on {{formula:4129637e-e0a9-4ff2-ac0f-3b856304176e}} we have {{formula:bf921992-1389-462c-a321-1ad3e3a9f9c7}} , so as before, multiplying by {{formula:b9c3ce8a-87ad-4655-a85f-730b3bf9c3d0}} and taking the sum we get a formula {{formula:217c0df0-2eec-4241-8111-ca6de8f049b2}} valid on the whole {{formula:95369ccc-ed37-4eda-a8bf-b46c369bc9ba}} , where the functions {{formula:281c75b4-c7bc-4ba7-a4aa-1986837c4b3e}} , {{formula:ca2c517f-4381-4aa6-b1d0-453997b41812}} and 1-form {{formula:a9f7bb4a-2c23-4735-acd0-f924f3f3fc72}} are bounded from {{formula:444a2daa-d0f7-44b6-b6f1-df2978a29a80}} . Adding these formulas together and taking {{formula:0b4870af-af9e-49b3-aad3-a813dd408aa5}} gives the first equality of REF . The second one follows from the second formula for {{formula:1a7a1470-c02f-4c46-aca8-25328f51f63a}} in Lemma REFREF by exactly the same computation.
{{formula:5cad33b3-e584-4e50-8b9c-10bee945f9c0}}
Proof of Lemma REF .
Fix {{formula:839e8310-cea3-43f0-b57a-220e525edc3a}} , so {{formula:29090ec8-ec09-47c5-b61c-0572c04de205}} for some nonempty {{formula:0cbdbad2-a213-4959-a6b7-595f4f02e5ac}} . Let {{formula:9a70bd36-55df-43ca-9183-af7b6d6e9af4}} be a chart adapted to {{formula:99317a5d-8922-4f1b-87e8-e9daa612dc6d}} containing {{formula:0b6a774d-5af5-407c-8a86-d7b1350bb24f}} , whose associated index set is {{formula:618d80b9-200a-44f4-93e0-a2c9baea8c4a}} . For {{formula:05e2a9e1-f07a-42cf-bc68-6af8de5cbc73}} , let {{formula:7423c0e7-d4aa-495f-bcbf-fedfc45a0c69}} be the domain of the corresponding chart {{formula:8501fe41-fa70-4726-a3df-72fe0af4fb8b}} introduced in (REF ). We need to show that there is a neighborhood {{formula:69eac076-da7d-408d-9c91-64e5dad97143}} of {{formula:7f783032-62a1-465a-bfdd-4dd0ffa52721}} in {{formula:7b6bb456-4c00-4af2-9360-dbdf661fcf1b}} , such that for every {{formula:26560567-fa02-455a-8e92-0aa7c9084433}} , the map {{formula:79d8e0fe-07d9-4f72-80e8-644aecfea25c}} is a {{formula:42320019-b662-467b-afb6-5e9130fb08dc}} -diffeomorphism on {{formula:5117eeda-dc0f-43aa-b036-3a5f9fa7b919}} .
Fix {{formula:9fa33965-9643-46a2-b5f5-46c245e731d6}} . Then {{formula:884e4dc1-8117-469f-b0fe-cc9e3040ae4a}} for all {{formula:40fbef6e-ad2a-41d3-a122-66002cc71159}} , so {{formula:6f53faec-0472-41a0-a220-a74b4d94c1e7}} by definition (REF ) of {{formula:01d9645c-9d89-48f9-94dd-aa24c44bd0b5}} . Thus by Lemma REFREF , the forms {{formula:66f26e66-42c4-476b-a635-e0cf87219723}} and {{formula:ee2259d8-0d97-430f-a0e9-f5d4b01cbb04}} are equal at {{formula:e3374bf9-71d9-462f-a674-577180383421}} . Since the map {{formula:53d858b4-053f-441c-9392-572494107cc4}} introduced in (REF ) differs from the {{formula:7c4679e3-3566-4e29-a1e1-fcacc99de6d3}} -chart {{formula:31f31696-98ff-4607-b27c-f34e21dc2639}} introduced in (REF ) only by replacing {{formula:e30bac6b-3f2a-4883-8f49-6f651ce4b43e}} by {{formula:62f67664-58d3-422b-ba8a-73b1de67da8c}} , we infer that the Jacobian matrix of {{formula:d32a8bc8-002e-494c-be51-dbe9ec82e1df}} with respect to the {{formula:64dbeb23-d132-4728-ad69-764c2423fe37}} -coordinates {{formula:5579b0a0-1461-44f2-879f-4280372dc28a}} is the identity at {{formula:85309f9f-e82a-4adf-a6a3-8938ce14e4e0}} . By the inverse function theorem, there is a neighborhood {{formula:71cdd072-a913-42c4-aab0-424278387add}} of {{formula:d7b95cac-5079-4fad-a13b-96a854f11c9f}} in {{formula:8decc3e8-9587-4978-a507-6f55d73020eb}} such that {{formula:682adb1c-7800-44fd-ae6d-e42d9edc1e67}} is a {{formula:57276c01-89a0-4dc4-b5d9-c376e9ac09b4}} -diffeomorphism.
We claim that there is a neighborhood {{formula:1c6779a8-2340-4b08-8e8c-a034aeef261f}} of {{formula:6e3d2a82-94b8-4d4f-96c4-f6fe736374e9}} in {{formula:7297ca15-4c5c-43a1-997f-03f00cb635ec}} , such that {{formula:d50be234-2ee6-4299-b4e8-36486427a3e5}} is injective. Suppose the contrary. Then there are sequences {{formula:4b3f28dd-25ae-40ea-bed7-d03ac85ec8b4}} such that both {{formula:85c90b09-e359-4c56-b175-6ff546079a72}} and {{formula:2a4ca148-1db0-4c03-bd4f-aa02fffa285d}} converge to {{formula:4c507b37-5534-4b91-abb1-98d6fe94e16a}} , and we have {{formula:0bd5b71b-e441-46a5-af1f-ab81e3b6bc3a}} , {{formula:41d3d938-3932-49a2-99db-d7609ba0a7a7}} for all {{formula:357afe0d-46ee-4401-8450-51da6c849a34}} . By Proposition REFREF , the map {{formula:da7c8528-0c3d-4b39-9420-b2d49e866082}} is proper, so passing to subsequences we can assume that for {{formula:bca066d1-c0d1-4a00-9912-8ee0e8addf3f}} we have {{formula:0a5daaf7-855b-437e-bd23-c37e2ceaa765}} for some {{formula:bec43ce2-3da5-4ca9-b2df-b9fb72c5c230}} . Since {{formula:7f5ae95a-f6ac-4a50-996f-7a528485c3aa}} is continuous, we have {{formula:921ede73-0ee5-4df0-b6bd-f2e323d505d7}} .
Since {{formula:2c8ee816-7caf-43f0-b709-1093614040b9}} , for every {{formula:e213edcb-219f-4a55-8f8e-9066db2bae9e}} and {{formula:0c7a14f3-f909-4a59-8772-701ad59dbb21}} we have, as before, {{formula:064992dd-ccb3-4fd6-b689-39fda6fe16c4}} and {{formula:22f8be1e-bf10-43c2-ba0c-d12b7cb5e78c}} . Lemma REFREF implies that {{formula:865f8d30-1e5f-433c-baef-396343f0875c}} , so {{formula:0347750c-b131-4de1-8c7f-1032dfcf4881}} . Therefore, {{formula:5650f7ac-ac4a-46d1-b7a1-4fc5e938e488}} . Since {{formula:bf7ab3c0-13e6-42bf-807f-082e9259ed26}} is injective by Lemma REF , we get {{formula:65c1d52c-3cc3-4fdc-bc58-da4702199dd4}} , so the sequences {{formula:3f8a05bb-d5c9-46af-90f5-16a60c5512a7}} and {{formula:90a898b3-5126-4067-bd9a-e60f18ec9a35}} converge to the same limit {{formula:fd8285a0-3834-47aa-a412-6640a62a7f33}} . Thus for {{formula:9c4b94af-25d4-4b87-aa0b-3860a0a6efea}} the points {{formula:3f96c1d6-959a-4792-bb26-45a814084d23}} lie in the neighborhood {{formula:36faa4df-ae7f-422b-8a53-2e057e26fd0c}} of {{formula:61f0ffd8-9232-4fc3-a9f1-e9a6d13c8fd1}} where {{formula:b934b13c-6f36-46be-a6c7-9501f9367cbd}} is a {{formula:cea6dc2b-e2ed-4d82-ac04-1c97e61b3d90}} -diffeomorphism. Since by assumption {{formula:9fadfa95-2079-4fcf-aa9a-fbf97c9de378}} , we get {{formula:d771f217-e02b-49e3-9635-ae6b1661e805}} , a contradiction.
By Proposition REFREF the map {{formula:e306a1ee-1b34-4fb7-b41f-db9e03a3c49d}} is proper, so we can shrink the neighborhood {{formula:08cbe698-052b-493a-84f4-568f42b73d75}} of {{formula:2b978501-e478-49cd-966c-02aca61277a0}} so that {{formula:44160754-73e6-4c95-8676-4dba64c32133}} . This way, {{formula:0730da31-6f9b-4638-86d7-21237c836f6c}} is a {{formula:44e82955-880e-4131-9b2b-a57e3cbb514c}} -diffeomorphism, as needed.
The remaining part of Section REF is devoted to the proof of Lemma REF and Proposition REF . Throughout this proof, we will use the following notation.
Notation 3.35
We fix a chart {{formula:4bd38b27-09f4-4cdc-8385-48dc7270e17b}} adapted to {{formula:e1e7c2df-b6d6-446f-a9a0-4f24782c539a}} satisfying the statement of Lemma REF , and put {{formula:2f25158a-1f12-41ab-ab6e-8c8058231e78}} . We reorder the components of {{formula:b6476d8d-a7d9-40f2-b9aa-a0923ad77a6e}} so that the associated index set (REF ) of {{formula:cb411d6a-d81b-4e8c-afbb-f311462cc075}} is {{formula:2ec1a2f0-ff56-4d51-a5a6-3b2786986189}} . As in (REF ), we put {{formula:d2e826db-4a0d-48ce-94fe-243ab82c503a}} . Now on {{formula:0f024755-b1f9-4750-a44d-252445567d4b}} , we have a smooth structure given by the {{formula:37f2e770-91fd-4abc-bf3d-b8f0b92dec50}} -chart {{formula:4f9a2d9e-5173-4b14-ac96-dabd9fbb8411}} , see (REF ). We use Notation REF for this chart. That is, for an open subset {{formula:3acf15d2-3d58-4b74-bff2-5300c4c5b8c6}} we denote by {{formula:078acd05-8c6e-49a4-a3b3-8a9e0ff01072}} and {{formula:59cb5a47-8405-4471-87ab-5c2855fc73a5}} the algebras of smooth (resp. {{formula:35bfed07-b220-4b15-a9d7-8529bfbfc016}} ) functions on {{formula:97573317-d90c-450d-beb7-88563f6039be}} ; put {{formula:9310b027-ae5f-4a97-b974-700e662da663}} , {{formula:95b2cff8-a971-4014-939b-6ccfe4608067}} and {{formula:ae42c1bf-0fe9-421c-8533-935f16304a5e}} , {{formula:33bef686-7204-4765-a091-a3fa939fe2b4}} for {{formula:91786ef4-9a15-4850-9fd0-f1c340c94812}} .
Recall that by Lemma REF , the restriction {{formula:2bf2855f-2066-4cab-9f91-59609ca88dcc}} is a {{formula:d1874c2d-6d27-4d9d-93ac-572987e72fec}} -diffeomorphism. Thus, with our usual abuse of notation, we can write {{formula:7cbf6416-0056-48d4-9d73-0a8690c38d03}} .
We end this section with a simple consequence of Lemma REF .
Lemma 3.36
For all {{formula:cec320e1-b57c-4afc-836e-a9c2958a2e31}} , we have {{formula:a1bfbc42-7098-4167-8a57-713d4b014822}} and {{formula:edae325c-60c0-4dd7-a4d7-36f750b2651c}} .
{{formula:d446a17a-2f97-4fb2-8b97-f434a00b6a75}}
.
Recall that {{formula:5ea73303-7ed7-4f9c-abeb-80a4c14541ca}} are continuous on {{formula:3dcd8541-0bc7-41fd-b270-cf095f256e0f}} by Lemma REFREF . Assume that for some {{formula:672360cb-84df-4bf7-b4d6-b118a1a83c0b}} we have {{formula:266c3180-fa97-4c65-a4e5-5b7f5568f9ca}} , that is, the restrictions {{formula:0d4c1f82-5fff-4162-9617-2ed0ab50cd5c}} , {{formula:16114a3a-30ac-48cc-90e7-aea5c4601cc7}} are of class {{formula:0be189b6-5ab5-4e32-8887-119aaab327cd}} with respect to the smooth coordinates {{formula:3377b4a7-0d35-474f-9755-0734f1aaffee}} . Then by Lemma REFREF ,REF , all partial derivatives of {{formula:29ba814c-56d6-4286-a3c7-3b5cc3a5a713}} , {{formula:b88ffeee-c4b1-428a-9364-67dba55ef5ab}} are of class {{formula:56b4b75b-080e-45d4-a89d-0aa56495b94c}} , too. Hence {{formula:19734b23-eadc-461d-ae86-8685d1e1d3c3}} . By induction, {{formula:2c984569-999b-48e4-9fe1-93691199da04}} . Similarly, Lemma REFREF implies that on {{formula:71118565-1b05-46ef-8707-9832389f1a92}} , all partial derivatives of {{formula:0f2de9cd-ca71-4b96-953c-0838ef94dc17}} are of class {{formula:2bcff2f5-e2ed-4f07-9eec-f7b4c3d0d78a}} whenever {{formula:df679b9f-8f0b-47c5-b334-cc6f80cf44bf}} is, so {{formula:1b5b6e99-47c5-4ca2-9126-549dbaae5cf2}} .
subsubsection3.5plus.7-.5Flattening algebras
We will now introduce a practical tool which allows to extend the smoothness domain of a function along its zero locus. First, we settle the following notation.
Let {{formula:39bfc9db-42e9-4b47-9215-92a730d4aaaa}} be a subset of {{formula:336e8f6f-fa5d-43ca-8502-d6e00b629b67}} . Given a subset {{formula:4a56e40e-61a7-4929-8f45-2de7b19a3d15}} , we denote by {{formula:42036f9f-945e-4b43-858d-2d0968a3249c}} the {{formula:bfc797b0-7501-470b-8bc1-85759f991c5c}} -algebra (possibly without 1) generated by {{formula:ad8ff725-d7ae-44eb-94df-ef0b2ce877ac}} . For two {{formula:ca364844-936a-45c5-acba-4bb8d45359f3}} -subalgebras {{formula:28a728e4-4604-435b-b015-d6cdab63965f}} we put {{formula:56940489-9d88-466b-85fe-97fbf2de6c6e}} and {{formula:edc3c8b7-c8e3-447a-9921-d48cf177cc1c}} . In particular, if {{formula:542a6bf2-6318-43a5-91f1-75903636969f}} is a subset of {{formula:9dd52817-ff3b-4c96-b844-883d1794b9cd}} then {{formula:63b1c280-d80a-493f-8d1a-0426dd31111d}} is the ideal of {{formula:4f9e4ce5-72d8-49c2-a25b-7be0d380936d}} generated by {{formula:a492be8c-ec3f-4bbd-a846-6004db4b7bc2}} . For a function {{formula:ac123e10-c89a-497d-9caf-7a6586e9976f}} , we write {{formula:f58d12cd-f18d-4423-bb1c-94a41d0b525c}} for the algebra {{formula:568b92e6-4998-4af8-848d-14eee6dcc49a}} . Thus if {{formula:6427db9f-169a-4f62-80c7-359aee563a8c}} , then {{formula:9fbbd2c1-497f-477c-9ca5-6ce6069be932}} is, as usual, the principal ideal of {{formula:0a56f031-f5d8-4a76-8c8f-4657db29905d}} generated by {{formula:b73fe016-2ea7-413b-b092-f31e0ab52627}} .
Recall that we have introduced differential operators {{formula:1aa889a9-0dfa-45db-a622-606cbb7d0e2d}} , {{formula:d1135fb6-eb3a-4a36-a431-d57b60da0355}} for {{formula:8f45f344-4343-4082-bba1-85039c6ef8f5}} . For an {{formula:9389e839-405c-4c3b-92b9-0115ae33b680}} -subalgebra {{formula:82f55d41-6b91-4c97-9c7d-39bb7c347b8e}} we write {{formula:52151d3b-4c4f-442a-87e7-aab3b22592b6}} . We say that {{formula:3590d20e-2c1b-4564-920a-58c4c353da3f}} is closed under derivation by {{formula:687736d0-4b69-4f84-9237-70c85d5ec1d7}} (or simply closed under {{formula:2efc134c-cf0d-4c3b-9350-dac6fc9804b3}}) if {{formula:29db381c-6abe-4c72-a076-a7cf0b35abdc}} for all {{formula:feb2a69c-ebd1-44d8-93ac-77dc3701e76b}} . An easy application of Leibniz rule shows that an algebra {{formula:6c568b0b-52c8-41e3-ba83-829afa8707a6}} generated by {{formula:0871cb41-0258-42af-839a-8e0cfdae7164}} is closed under {{formula:876da2f5-109d-4b26-a2d6-2920bfd2bb2a}} if and only if {{formula:4f4e6246-259a-4e0a-a334-9736c896bef5}} for every generator {{formula:41b1e09d-bb59-4bdb-9216-54ba355e52e3}} and every {{formula:f083e9aa-3b9c-4d35-9537-1aee1ed0e0c4}} .
Definition 3.37
Fix a closed subset {{formula:e4d2f570-3887-4373-a661-ba5365b3ee8e}} and a function {{formula:9b943f1b-b721-4938-9b8e-51be5ee8949e}} . Let {{formula:232d9577-523e-42ca-8582-256fad4d80fc}} be an {{formula:a444ecb4-2459-487d-a9cf-1e82f776de28}} -subalgebra of {{formula:c3617221-9fb2-4139-b66d-3a3c4fd71f06}} which is closed under derivation by {{formula:79225f8b-4822-4813-b5cc-af8d092894b4}} .
We say {{formula:6c26cacc-a614-44d6-bdeb-922636f2051c}} flattens {{formula:6000a88e-a515-4608-a29f-a6746f4c7216}} on {{formula:d0ee3959-68ed-45d7-ae2e-dab99b4ac333}} if for every {{formula:ac63222d-9b91-4109-a18a-efceb072b568}} , and every {{formula:a605656a-6dd8-4de9-837c-6da7b988d115}} we have
{{formula:b83412a2-f61d-424a-9411-ab49bbcddd71}}
The following Lemma REF lists some elementary consequences of Definition REF . Part REF asserts that condition (REF ) – just like closedness under {{formula:fdb5bcbb-89cb-4609-a036-974e38852076}} – can be checked on generators. Part REF gives a practical condition under which not only {{formula:30cdbaa7-79b3-4d0d-89ff-4990c443da0a}} , but all its derivatives flatten {{formula:5ccbbd73-a6ab-45ba-b447-4904c640bfb8}} . Together with part REF , it will be used to infer smoothness of {{formula:034e0de6-bd5b-443f-a155-661e1e38014a}} .
Lemma 3.38
Fix a closed subset {{formula:72a43b4a-cfad-4870-8737-7eb07d9df495}} and a continuous function {{formula:adbfd209-80af-4bcc-92ac-e6852fd8ee14}} on {{formula:3efd34fa-841a-41f4-8c0a-bc7da08e4b74}} such that {{formula:f579d541-1c9c-4375-bfa0-80cc9430c800}} and {{formula:24b10878-2354-452d-b9e8-2f76f4fee1c4}} . Let {{formula:8ea0d350-751f-409b-87fa-c223b057bf32}} , and let {{formula:56fd84a1-c576-42f2-ba1c-b42cf5bf16f5}} be the {{formula:68062d82-f574-4bd3-92cf-cfe21166e2ea}} -algebra generated by {{formula:784dfd20-adf4-43a2-bd25-4e2c99e147cc}} . Assume that {{formula:877d2b33-6d58-4cfe-9df2-3a127a9f54b3}} is closed under {{formula:74366b08-4448-4d15-9859-cadc25345560}} . Then the following holds.
If for every {{formula:dc9a16f0-d1e0-4e29-97c0-9ce6c8332669}} and every {{formula:c583ba2a-8431-43c9-bdf7-0c8f09a3e0f6}} we have {{formula:d304ad6a-9972-411f-a8d8-813ee070fdfa}} on {{formula:47fb8c85-ddb7-4537-a7f2-759346463897}} , then {{formula:92abed24-8241-4531-b645-0a82120e72f3}} flattens {{formula:8df4b607-bc9a-452a-8249-87429da8b8ef}} on {{formula:ba3ee136-2402-4dcb-971b-ebec44bd74ff}} .
Assume that {{formula:c1946764-8894-4985-85ee-788679d50be6}} flattens {{formula:2fb37fb9-5d37-4fd1-b8ca-b942321231b6}} on {{formula:7abe8330-fc04-46cd-9b98-b434e4580913}} , and for every {{formula:93b60958-a27c-433d-aefe-344d58069b13}} we have {{formula:9f8a966f-a203-416b-9e6e-9380310816e9}} . Then for all {{formula:8c11c670-e651-42ee-871d-2f6be7330a35}} and all {{formula:d24c7942-baf4-4ac6-953b-fae45ec095db}} , the function {{formula:1c1e45a1-79ce-42c3-93eb-7272436a0e36}} flattens {{formula:5b21d439-487e-46ac-a654-ebf39da6d055}} on {{formula:fdb2df04-2c9d-456f-b843-417b4cf1c4cf}} .
Assume that all derivatives {{formula:80213fcb-7baf-4aba-b9b8-36de19ba17af}} flatten {{formula:34867966-6874-4753-bf95-b4bddb81487d}} on {{formula:baf544c5-a217-4e8a-9d49-78a7c9f9562d}} . Then for every {{formula:8991add1-d755-44e7-b9c4-a205725318e9}} , we have {{formula:2edbaf6d-cb94-4b23-b155-d9ef66df838b}} , and {{formula:cde688aa-5df5-47b3-aea0-a95b794c9b3f}} for all {{formula:861697f9-ea34-4651-a83c-e1a457f829e3}} . In particular, if {{formula:0ec0962d-b6cc-46c8-8f36-2e294f6a61f5}} then {{formula:6268d7cb-2b6a-443c-be95-df2812afe1e2}} and all derivatives of {{formula:5c08c589-4f06-440e-be9d-f2183ef866e8}} vanish on {{formula:16a5e662-08da-4e25-8f3c-2c2795d71e21}} .
{{formula:891c81f2-865c-41b3-a9b2-5e28e4ecfeb5}}
.
REF Since by assumption {{formula:8a958df6-c646-4a25-9c0e-1d8ac8194f4d}} and {{formula:72652017-b9c2-4bc9-b02a-6fcf8231c1a5}} is continuous, all constant functions {{formula:9c319296-7077-4293-ac59-79311214fa23}} satisfy (REF ). Now if {{formula:864b04ef-1117-49f7-ac32-5462dfc1bdaf}} satisfy (REF ) then {{formula:7f77d75d-5ae3-4f5e-a2a3-13e562c7b67d}} , and {{formula:b260771f-434b-4566-a61a-503e0b6a774e}} on {{formula:b9518aab-06c6-4944-943f-28f2387cb682}} , so {{formula:49b82da5-8176-40c6-80a1-959ff0db6234}} and {{formula:750c19ec-8838-40f7-91ec-453706a96eb2}} satisfy (REF ), too, as needed.
REF We need to prove that for every {{formula:3e4db7c8-ddff-44d9-9feb-41b7e01ba50a}} and every {{formula:87a0c149-fd83-445f-b0f6-66a894e4fe5f}} , we have {{formula:60a76023-81c8-4fb5-a635-2d6e0f6ce247}} on {{formula:1fed9705-49c8-4bdb-bc2e-90427c311827}} . It is sufficient to prove this claim for all {{formula:9586f19e-5a67-45bd-851c-581e34b31f6b}} and {{formula:5999807d-bfdc-413a-b6eb-aad7a78f29f4}} . Indeed, by continuity it is sufficient to consider the case {{formula:7de97a23-51d8-4fa9-8ce3-b20e56c6b53e}} , which follows from case {{formula:b5879790-17db-45f1-ab7f-956b7c60d01e}} applied to some power of {{formula:fe2881ac-3194-430c-bdf9-9121a7190ef8}} .
For {{formula:4b74cd2e-3133-487c-9b1b-ec3d19b99dcc}} put {{formula:92cc7ed2-362e-42d5-bf75-619590a772ea}} and {{formula:02e8b2af-10f1-4b39-8e04-c0d9d2d8455f}} .
We argue by induction on {{formula:7c54c148-c28b-4bbb-95e9-edc1223e2209}} . Case {{formula:1a635629-fec4-4be3-8f66-f77bd8f070d4}} holds by assumption. Fix {{formula:a353297e-8cb6-4eeb-9dd4-eabd9710647a}} and assume that the claim holds for all {{formula:cf5fad89-9855-4691-850d-ede1e474fcee}} . Fix {{formula:fc130f98-4336-43e8-9371-bfc6aec8d184}} and put {{formula:7c9ccc9d-c875-4eba-8926-9b173401685b}} . By assumption, the function {{formula:00b60621-f9dd-47ba-8204-b56195eb9897}} belongs to {{formula:66c90151-0648-4a6b-b3fd-69f83feebc42}} . Now {{formula:528335f1-4388-48e7-bc5f-0840b7b629d5}} for some {{formula:806d287c-ff49-460e-b621-f44c81aa5f6f}} . Since {{formula:9ee18255-9534-4271-87f4-86fd61f233b4}} is closed under {{formula:71b00e2f-56d7-41db-8a37-70282212322b}} , we have {{formula:2d790b2a-e223-4457-85b3-f7eda5d8c12b}} , so by inductive assumption, {{formula:ebb357e9-7df5-49fd-9536-5b162975cb23}} on {{formula:a95e81cd-d1ba-4556-87fc-dec02956d50a}} . Thus {{formula:3e95006f-15ed-494a-9ac6-fcb3be0eed98}} on {{formula:40fa96d6-d6b9-4349-90cf-8288d6e27979}} , as needed.
REF By assumption, {{formula:2fe18274-4f41-4aec-8cfa-66fe476e4c51}} . As before, for any {{formula:5ca0b551-ccc3-4d04-8d36-eff6769b3135}} we can write {{formula:8ef2fff3-59b2-4bae-8e75-2a2b95657a7e}} for some {{formula:0cd52a50-93ae-4b2e-b02e-4df13af4e520}} . Since {{formula:873a00dd-58e2-461b-9d8a-fb09083288d3}} is closed under {{formula:7f4a0d20-cebe-492a-bcc4-0cdf0d14469a}} , we have {{formula:7ec4ef3b-6887-44f2-8830-17fe65a2713d}} . Since {{formula:4cf4086b-b77f-4da1-b8cd-1f34fc163e74}} flattens {{formula:76398fb6-e042-453e-9817-ebd9df932add}} , we have {{formula:d10625d6-61b1-410a-8f63-f1129fa47d5a}} on {{formula:9643d637-18c5-4058-9d71-34f082a2464f}} . Hence {{formula:05916fd3-bbec-4d57-870e-f22a6c054806}} extends to a continuous function on {{formula:72457d64-a89d-472f-bbf5-7c7fb4d55c6b}} , which vanishes on {{formula:8abaeda3-cc54-4ac9-9840-a881e82bcaf6}} , as claimed. If furthermore {{formula:fd8b42a4-3614-4f78-945a-6723b4f3404d}} then substituting {{formula:52c5c235-8ee3-44b0-880b-20ce06e2707a}} to the first claim we get {{formula:2bdac72b-d513-418e-9e3c-e57011b50f71}} and {{formula:355ab338-e63f-4d2e-a834-b1951e795b98}} , as claimed.
subsubsection3.5plus.7-.5The functions {{formula:b1a524c9-6258-4c8e-96d3-e0c10b9ecda5}} , {{formula:7c45a44c-79b8-4713-9298-d1afcff98d7d}} are smooth.
In this section, we establish a smooth version of Lemma REF . Its most important consequence will be the fact that, for all {{formula:bc10d632-f9dc-4701-b44c-4454ab4cbda4}} , the functions {{formula:b372aab2-7f45-450b-9f24-c812283c7b21}} are in {{formula:0d8b5998-f367-4160-aa3f-5f135d94dc1c}} , and for all {{formula:099f4870-05f6-4350-958d-1872a037270f}} , their derivatives {{formula:aef1de57-b9cf-443f-a343-846b1521d642}} , {{formula:b81d0c85-1070-4fdb-85fb-5136237fca28}} , vanish on the zero locus of {{formula:7b4b0cbd-f009-4206-b74f-5bec69f09998}} and {{formula:8b725041-3a74-4b89-a510-008a50ab2768}} , respectively. To prove this result, for each {{formula:eb4a0808-bffe-4b5e-88b5-075762113782}} we introduce the following {{formula:80228f83-ac98-4201-8730-f4d16442bac4}} -algebras:
{{formula:df818891-a58c-4587-8769-cd8388bf8db3}}
We will study their properties in Lemmas REF and REF , respectively.
Lemma 3.39
Fix {{formula:201f3b28-a840-4f19-97f7-d0417263fbe5}} , and let {{formula:c4d2bb47-6ff4-4070-9d5e-b8d16747e247}} be as in (REF ). Then the following holds.
For every {{formula:c169a420-abb8-444f-8869-26a9b215a901}} we have {{formula:46f9726f-fec5-4542-b952-fe970475ab9a}} .
The algebra {{formula:a82c621a-c9ae-43a3-af95-50a206e4d57c}} is contained in {{formula:eb42ee93-2e5d-4822-8a84-583f68d63a49}} and is closed under derivation by {{formula:0200e08b-580e-425a-bd48-67e73cb747ff}} .
For every {{formula:92cc2a18-89fc-4ff2-9c24-9203c8a80e20}} , the function {{formula:1e7ec104-06e7-4630-8cf4-70661d53e8b3}} flattens {{formula:1bb1fc0a-0945-491e-a298-03915c8f7eca}} on {{formula:877ba8c6-f963-4efd-add5-0b13ec604d60}} .
We have {{formula:e152e56a-8772-4953-b0d6-cce316a2528a}} , for every {{formula:1b7ae580-aa91-44bd-baaf-445dfbe5ea54}} .
We have {{formula:d9dc753e-839f-471e-afdc-3e4ebfa33cc2}} .
{{formula:b75a1624-acd1-4bdd-9fe5-0e6c3b8d385e}}
.
REF Recall that by Lemma REF we have {{formula:bbacbe6f-a321-48aa-8493-474fbc5833e6}} for all {{formula:84c21179-6b38-437d-9b84-de68ac93fbae}} . By Lemma REFREF , we have
{{formula:407f031e-3f72-4d7f-a2f9-9a0fefcd36d9}}
REF By Lemma REFREF , we have {{formula:f77e5e46-10d5-479e-8f74-9ab0f3736f67}} for all {{formula:55004689-a2da-4238-80e7-f0b2a9ecabcd}} . By Lemma REF , we have {{formula:410abf73-1e9d-49f7-8df0-bec74be05c96}} : recall that, by definition, {{formula:981d5245-d03c-4b1c-8ea4-07784e0eb3e1}} is the complement of the zero locus of {{formula:904abe59-9cb2-430a-9a78-9e4a4178093b}} . Hence {{formula:fcb9ac1b-f511-4cdf-94a5-b7df34758e4f}} . The function {{formula:84f728a1-3ac7-4244-a2e0-ef2da4c78ba6}} was defined in (REF ) as {{formula:9cea38e5-00ce-4bb5-b797-407b7d690016}} , where {{formula:363e76aa-c722-4cd0-aa9e-bde42a7f6d26}} is smooth. Therefore, {{formula:ee759dab-d9ad-4f86-8ef0-0cf6259fdabf}} and {{formula:727b4f3d-ee3d-421a-b945-2dbfc359c719}} . It follows that {{formula:f935a53a-d998-458e-87c8-47b2d7393ba4}} , so {{formula:56efd2fd-3620-45af-94fb-2741e0ae4663}} , as claimed.
Now, we check that {{formula:1b5a0a74-9f4c-49f1-b8cb-28391f6ba291}} is closed under {{formula:0fc451de-b168-4fee-95f7-95d6f611eb4d}} . It is sufficient to check {{formula:8a7bc6be-1d2e-4c72-8d36-a5d4df81fb56}} for all generators {{formula:d82d6c34-a9f2-4a55-acc9-2a0b3ba36cee}} of {{formula:dedcd5b8-5110-4531-955b-5e20584a50da}} , and all {{formula:0c7a9116-9b47-40e1-8b6f-7f7b2bdffb2f}} . By Lemma REFREF , we have
{{formula:df9df82a-44c6-4c71-8391-684615cea7c4}}
Thus {{formula:99f75e6d-c7e2-4b7b-b743-33a14fe69593}} . Since {{formula:a34ab30b-5b09-49e3-a155-754aa80c6ed0}} , part REF implies that {{formula:fbfe0a9e-de9f-4ffe-ad14-d2d442609146}} , too. Recall that {{formula:3c8b51d7-e476-4d0f-b2ff-054c653e19ef}} . By Lemma REFREF , we have {{formula:36c97edc-9c20-4761-8c31-3d992d9f6938}} , so {{formula:39d74649-0a4b-4c21-b550-50ff4ff006c9}} by REF . In (REF ), we have defined {{formula:ed9b8737-22d0-4995-8ddd-d58847ceef86}} as {{formula:758395d3-d2ae-4bb3-a31e-6c87e677d566}} , so {{formula:3fba1379-ca88-430e-9c75-c44a16ab64be}} . Eventually,
{{formula:a2ed9ba2-d962-4494-939d-d05ea9927d69}}
REF , REF It is sufficient to show that {{formula:2e347f91-1b5a-4a5d-b743-181ac9041b8e}} flattens {{formula:4ec88695-6de0-4b3e-93a7-824b653fc817}} on {{formula:33ac68eb-46be-43dc-a8b1-90844da5114a}} . Indeed, since by REF {{formula:0dd22c12-63af-49e7-98e8-bd7087b5b817}} is a subalgebra of {{formula:911aedac-b0e1-4be5-ba36-da1bfcbcde6a}} closed under {{formula:42858471-5f57-4c9e-88ef-17c5d0245414}} , and by REF it contains {{formula:cfdf2da4-e2c9-4c96-a1e1-84d75c4098fb}} , Lemma REFREF shows that if {{formula:ad868769-a42b-48dc-bd79-1c6fd286cf14}} flattens {{formula:eb3abe99-5202-45df-b5b9-a38654e1b5a5}} on {{formula:9810e349-4541-4f5e-8c67-2eac16e15a15}} , then so do all its partial derivatives (taken in any order). Once this is shown, Lemma REFREF will imply that {{formula:b4e57ded-d0a1-41fb-9262-cbfed1193149}} .
By Lemma REFREF , it is sufficient to check that, for every generator {{formula:3b60a234-b31e-4586-908e-6a7f2f28581d}} of {{formula:e031c75e-f7f7-47ee-afca-889a26fb4745}} and every {{formula:8cbc6ea6-5257-4c79-b940-117f198afe0f}} , we have {{formula:bd0323ce-2216-45af-a517-42222873a56b}} as {{formula:d5128843-d52b-4b56-89a0-cb143e15f9bf}} . This is clear for
all bounded generators, i.e. for 1, {{formula:e70593ea-2557-4206-9417-b2d1572ff665}} , {{formula:e592a8e9-6c6b-4512-9df8-cfb31d5069a1}} and {{formula:d84e3c24-03f6-4ff1-a7e3-0b37a6b8e0b2}} . Recall from (REF ) and (REF ) that {{formula:99fb3f13-8bf0-45d3-aed3-c8b57cd5fad8}} and {{formula:988b97dd-2680-4644-9d04-0c49143a0576}} , so
{{formula:092c87c8-c096-420a-9822-e1f99f2b32ce}}
because {{formula:2b48a4f6-9992-4b50-9b94-f166eedec618}} . Eventually, by Lemma REFREF there is a polynomial {{formula:76ec47fc-0566-4f96-b626-a3a4af09a49a}} such that {{formula:f8251b64-97f6-4e3f-b620-07d1142c91fb}} . Since by Lemma REFREF we have {{formula:fabcbbaa-59da-4dab-b349-c849f2e1bd56}} , we get
{{formula:fd148228-bbd0-428b-80ee-058312f158ed}}
REF We argue as in the proof of Lemma REFREF . Part REF applied to {{formula:9bfca93d-5c6f-44f0-a440-c7af6ecdbf48}} gives {{formula:5e1ad48d-349c-4bc4-9089-24577e827e42}} . By Lemma REFREF , we have {{formula:01f883b3-c815-411f-9d9e-43b894e6a78d}} . Because {{formula:4238ca87-173f-4142-992a-a5d4750627d2}} by Lemma REFREF , and {{formula:26cc1838-f34a-4af3-95f7-fb2eda418c84}} by (REF ), we have {{formula:ced3f1d9-551d-48ff-9a88-b6c1bee2d34b}} , and {{formula:a288bf14-a60b-419e-b90d-552c4d265535}} , as claimed.
Lemma 3.40
Fix {{formula:2ffae094-483c-42ec-b39c-7062fe98c38d}} , and let {{formula:5d7bd74d-62a9-4fbe-8734-f48546fab0a2}} be as in (REF ). Then the following holds.
We have {{formula:07cc7a25-3435-4801-aeab-66f06c59414f}} , and {{formula:bae339ef-6531-454d-919b-d7e51e6f9937}} for every {{formula:0b5e33ca-022d-42b3-8bd9-e964c6cd2c70}} .
The algebra {{formula:c211b266-c4ca-4f70-b49b-ddd7551ea4db}} is contained in {{formula:3c78cc0a-9751-4c47-864b-ca03ba2b6987}} and is closed under derivation by {{formula:3c6afd3d-4342-4dd0-b7fd-c136ad10cfa6}} .
For every {{formula:a7cca6e5-fd60-4b6e-aa72-5e459279159f}} , the functions {{formula:ee1c7b9c-17b7-4224-8eec-cba53ef347bb}} and {{formula:00a715ec-fe19-4af6-a5ed-e8421f6cc6e0}} flatten {{formula:056fc8f5-2158-4de9-9fd5-58582fcf183b}} on {{formula:07b5bb0c-e338-4f5c-882c-a94a1324a072}} and {{formula:c6f41e68-96c0-45bf-85f7-2c2f3532d843}} , respectively.
We have {{formula:67bf2b02-f78d-490a-98b2-66b6395f96fa}} .
For every {{formula:2f637e77-b1ea-4c56-ba9a-abce1e143b11}} , the pullback of {{formula:f86abe4d-7ecb-4716-a0f5-29ea30eeb7a2}} to {{formula:659f3cda-cc60-40e3-9b6d-61c7a4b168c5}} is in {{formula:954fee46-7585-4d1f-8b47-21cbf1e2d97f}} (in short: {{formula:8aacd942-39eb-4115-a805-851de56b0c1b}} ).
{{formula:669509fd-60b6-4d85-bf64-cda73390d332}}
.
Denote the last two generators of {{formula:57cbcc56-df4a-4df6-9bbf-89604224ee89}} by {{formula:6ebd3285-78f7-487b-94b0-7f808fb474f6}} and {{formula:39aac4c8-f2fc-41e5-82a4-e7f31ba25382}} .
REF
By Lemma REFREF we have {{formula:97f58cdd-8398-4145-a01e-343c470551c4}} , and by Lemma REFREF we have {{formula:56becfe5-0f59-443d-8089-96c5f5aafdcf}} . By Lemma REFREF , we have
{{formula:263594da-7901-4122-86df-0d455ea9150c}}
Hence by Lemma REFREF , {{formula:a0689fe2-98ce-49cf-ba3b-787dcffc8573}} .
REF Recall that {{formula:70650268-7f15-48cb-a701-f0255851c503}} . Clearly, the coordinate {{formula:3da89df2-bb42-4fc3-a9d4-f863cf76cc64}} is in {{formula:4ada8b44-a0b4-433a-a03e-98a664f124c3}} . By Lemma REF we have {{formula:986c0dea-8392-4ac8-b93c-3ebdcc1e5c47}} , so {{formula:df110692-fcdb-44a7-ac0e-854a48bb6a86}} , and in consequence {{formula:0f5c4018-5ed0-436c-a589-baccb09de8e8}} , so {{formula:66363112-430e-4e26-a66d-666c7dcdf79b}} , as needed.
Now, we check that {{formula:c4b9e390-f031-41e5-a4a1-7e37fb307f95}} is closed under {{formula:3a4d71a9-128c-4554-82ad-df481cec8900}} . We have already seen that {{formula:276d3f9c-1fcd-4557-ab67-d1ed9706b08f}} . Hence {{formula:b80a74bc-3967-40be-ad79-8ce5ad8f3aa5}} and {{formula:a3038070-c5c9-4ded-a203-c55532318c9b}} . By definition of {{formula:9183347f-3759-4abe-a64e-b3f3d4dbe104}} , we have {{formula:6850a4ff-c4ef-4375-bf5e-d181b76f80f2}} . Eventually,
{{formula:5ce7fdae-157f-4c45-96f4-b94182a53ac7}}
REF , REF We claim that {{formula:3fc0fccd-abaa-4ba7-a7e1-2475911d4f0c}} flattens {{formula:8686eaf3-a1f3-4850-ae18-f643e612f9a3}} on {{formula:aa5f47f8-2125-4c2f-898c-7d8da0579e4e}} . Since {{formula:4a082444-7284-4f8f-b7b7-02e943e64523}} , by Lemma REFREF it remains to show that all generators of {{formula:6311591d-a5ed-47ac-b00e-edb7defc2423}} satisfy (REF ). Since {{formula:df3ede2f-6d7b-4f32-8876-9ca490e00b36}} , this condition is clear for all bounded functions, hence for 1, {{formula:eab5696a-6091-42b7-a9ce-af478e035f76}} , {{formula:f35be7b6-2b4a-4f40-8149-b45eb9f72572}} and {{formula:8ecaeb61-cc0c-42e2-9424-9f31ccfb5d99}} . By Lemma REFREF , we have {{formula:330cdaba-db16-4d50-a6c7-405e29ec25bd}} , so {{formula:7c984ffb-0aad-4d30-bf91-b81c2c243279}} and {{formula:f0f985d6-7c90-4b5b-a628-90b906134d9e}} . It remains to check that {{formula:730a6dec-b834-4590-8e8d-d1b8f7d79a99}} . By Lemma REFREF , we have {{formula:a21b291a-d61c-4fae-aa2f-494838df0a43}} , and by Lemma REFREF , {{formula:68306781-c7b2-4b8f-aa76-da372fe0dd6e}} for some {{formula:64c8f26c-5202-4a53-b572-0ea128945e34}} and a bounded function {{formula:7e86470f-9e63-4b5f-897b-27cfd87fc13b}} . Hence {{formula:51581127-5908-4930-b285-64ef22ec97d8}} , and therefore
{{formula:3ecda379-329c-4040-84c5-557ac525a98c}}
where {{formula:a7b35b19-b4d8-45d7-a3c8-2b7bc8f21aed}} is bounded. Hence {{formula:7564e930-8d57-4ee8-82c1-693441cea804}} as {{formula:ac48b44d-e537-41d6-938c-dd1eafe6a8d0}} , as claimed.
Thus {{formula:8013036f-9f01-4082-ace1-a61268a2b37b}} flattens {{formula:e9c1cdb6-6909-4ff8-94de-dfbfa13307af}} on {{formula:dbc031f3-20a2-4812-8f62-a7f50829dfa3}} . Since {{formula:3eb2db62-0bd2-4dd1-b1c9-d22cc6482953}} by REF , Lemma REFREF implies that all derivatives {{formula:bfce95e4-4dac-4b3e-b05a-04b499af2cbe}} flatten {{formula:04d07c17-0114-4810-9876-8c67c3989e50}} on {{formula:6a82fa8c-2ef0-4de6-b9eb-5060e5c8c8ac}} , too, as claimed. Part REF and Lemma REFREF imply that {{formula:8e2b70ea-a687-487f-b682-f96037e13b3f}} .
Recall that {{formula:81f5d177-6586-4ac3-8d14-8cf920488792}} , hence {{formula:014414c3-932a-45ea-a1a8-b8d5eff874d6}} , are in {{formula:9937ee28-1beb-4403-b2db-5bae84f66692}} by Lemma REFREF . We claim that {{formula:7dc0b7b1-2ca4-4b1e-8661-0ae38afde207}} flattens {{formula:fbb5fd63-c0ec-422b-846a-43983e635dfe}} on {{formula:36760d22-35b7-4f5a-920c-ded4eec7091d}} .
By definition (REF ) of {{formula:e1887f3b-fe44-44e5-93ea-61217eda8221}} , we have {{formula:337d18e4-a44c-4cd7-a119-a000300dc507}} , so by Lemma REFREF , on {{formula:6e42c56b-160d-4836-bed0-a41979609280}} we have {{formula:dd9251da-d8b8-48ed-a36f-4a53ea374941}} , and therefore {{formula:e275e1c7-91ac-48cd-b69e-6b22d29d36c2}} . In REF , we have shown that {{formula:4dba26dd-2c28-44c8-a481-94163c270200}} , so all elements of {{formula:c9cda4f1-ce75-4cbf-8cdb-117fa2f2a94c}} are bounded on {{formula:fe465023-9987-474c-9b6c-52d84bc2b9c5}} . Hence {{formula:d3296bea-f696-4834-9c5a-a307656fca63}} flattens {{formula:efe04759-d20f-4eec-bfc3-cdc9c948e462}} on {{formula:33b88499-6c02-45d7-a278-6238a17c81ce}} . We claim that {{formula:1e89a603-532d-4f27-b462-e7aa39c29772}} flattens {{formula:08c52199-7fdb-46cc-82b4-d123e0770a3b}} on {{formula:d93ae963-d68c-495d-b166-abb350585736}} , too.
Define {{formula:03024f8d-eb7a-4c51-aba4-e8221727a457}} . Because {{formula:d2e7b70b-3a22-4294-a07c-2b35ba3742d0}} is closed under {{formula:e55f43ea-7127-49c6-97eb-eba5acf6e288}} by REF , so is {{formula:46d05cd6-9c4a-441a-a369-f82ef39bc550}} . We claim that {{formula:7edc782f-6ccd-4af5-8f1f-1f39160c52f3}} flattens {{formula:e355b1e8-32d1-4de2-aa47-8870d87ae637}} on {{formula:e8ea8447-c2ce-4ff8-a054-4feead9c91c3}} . By Lemma REFREF , we need to prove that all elements of {{formula:e130b28b-97c1-438e-a217-44d257eba223}} , and all functions {{formula:539105cb-4aee-4707-8824-e96e953cf2b5}} , {{formula:87eed36b-5671-4c98-98b2-28f77e88abce}} satisfy (REF ). By Lemma REFREF ,REF , we have on {{formula:8b636c19-9f50-47b3-bfa7-220d8d9c21a7}}
{{formula:65b62bc6-d145-431e-a051-a8b5f821062f}}
For the last inequality, we have used the fact that {{formula:1c2b01a5-39b1-4101-9f62-dcd3066dc224}} by Definition REF of a chart adapted to {{formula:ac2ff271-0b12-4b2b-a0ff-389fe432860e}} ; {{formula:58c4f884-0203-4464-8adf-332221066271}} by Lemma REFREF ; and {{formula:aa872807-42a4-4dfd-a01b-fb138654b0ec}} by definition (REF ) of {{formula:ee513c5c-db5a-47ec-8dd4-26c9dbaf468a}} .
We have seen that {{formula:19621cb4-904d-4b16-9eb5-f53a03f21a0f}} flattens {{formula:140f2ac6-04f3-4e0c-ad2f-11c7de2cd484}} on {{formula:e0fdc0f7-0037-47e8-8054-82917e1a5543}} , so for every {{formula:e329f6ac-2435-443f-89af-80dbbc27e32d}} and every {{formula:955fa48e-97c8-4b54-9a6b-ff7fdb3bd963}} we have {{formula:76dcfd8d-781e-4087-9d66-03fd48f0662c}} as {{formula:dc3e0c63-33c1-4a39-80f3-2ef25163662e}} . Hence all elements of {{formula:018f654a-771d-47c0-a4a7-87c46ffef5df}} satisfy (REF ), as needed.
To prove (REF ) for {{formula:7099c703-2cce-49e9-b37f-e47d36e8e694}} , recall from Lemma REFREF that {{formula:300c002d-0111-42de-a4a4-6230dcc1924c}} , so {{formula:f924a955-7fbe-4364-9b3d-74a01c79c837}} . By (REF ), we have {{formula:4bf83eb0-fa72-45cf-9608-914e46f1e08b}} , so {{formula:28f35068-f708-458d-be24-ae0d1552fd6d}} on {{formula:d2fed539-c09a-42ee-8a09-3a968726b761}} , as claimed. Eventually, by Lemma REFREF , we have {{formula:df1527ed-5efe-426b-b083-f89d50ce89ed}} , so for all {{formula:9ba1d686-706a-498a-aa57-39d4ff7dde3c}} , the function {{formula:bbb9aade-4812-4c91-add3-b86f07fa96a6}} is bounded on {{formula:020087e1-1810-4c9f-a3a8-a22f70a06ba3}} , in particular {{formula:6dbcf84f-fa8b-48d1-812b-e7cdd1ba6843}} on {{formula:84ab2043-e176-405f-840a-df062cbf634a}} . Thus we have shown that {{formula:6cd3024d-b376-4b78-8460-b0aa3c31bedc}} flattens {{formula:0084e783-1aff-4a92-ae5f-a74a93a0e8a2}} .
By Lemma REF , we have {{formula:4a5d8528-6d06-43f6-b824-fefaf0a9e581}} for all {{formula:b49c9e63-0db3-4088-a3cb-85fc455b2266}} , so by Lemma REFREF all derivatives of {{formula:e3e2d92d-2850-424a-9a5a-17ef319c63b1}} flatten {{formula:16cb88a7-a41e-4ae5-99ca-cf33ace710ed}} , too. Since {{formula:a4051e87-6052-41af-bbfa-10409dcb3f60}} , it follows that they flatten {{formula:b72d1156-5171-4c1d-a83c-e32ae4a1f2b7}} , as claimed.
REF By REF , we have {{formula:ae011e5a-2b82-4d0c-ad13-f18579caadea}} . Assume that {{formula:53c83e79-9a4a-4cb2-a55c-a80be3d6aa02}} for some {{formula:8a355c77-c7b6-4386-9964-5742d38039b5}} ; for {{formula:afb530c4-168f-4011-8dd2-2f1357bb1e69}} this holds by Definition REF of {{formula:09af2885-9bce-4e26-86ab-8a874b264ed1}} . Now for any {{formula:05d8b88a-1cbd-4575-8f66-157bdf123493}} we have, by the chain rule, {{formula:93140e62-25a5-427d-9ddd-afea4eff3980}} , so {{formula:6f679694-92d8-49ed-8d18-213b88a1ef3c}} . Thus REF follows by induction.
subsubsection3.5plus.7-.5The Jacobian matrix of {{formula:7f8a1043-98c0-436d-a4ff-acaf90eec6fa}}
In Section REF , we have used the fact that the Jacobian matrix of {{formula:1062c3ac-297e-49e9-a88d-45456fa11ecb}} with respect to our auxiliary coordinates {{formula:38dc11a1-9b92-4d76-8540-f540bbd497ac}} is a small deformation of the identity. The difference can be computed using Lemma REFREF . We will now study this difference in more detail. In Lemma REF , proved at the end of this section, we will see that this difference belongs to a particular matrix ring {{formula:a44c24cd-8007-428b-b0ee-8ebe7c08f34c}} introduced in Definition REF . Properties of {{formula:1446b0cd-415b-42ae-8206-d8c0f9095352}} will allow to conveniently express the coordinate vector fields of the smooth chart {{formula:9f25cdc6-f78b-4e44-8536-7f548faee64a}} .
Recall that we work with a fixed {{formula:7e5a91d9-ba22-4954-abb7-cc81102f5929}} , where {{formula:b43a1c9c-5718-4776-9446-b4cc545f0137}} is an atlas adapted to {{formula:0d2891bf-5727-462c-813c-232264e1e5b6}} , and {{formula:30fd7bab-70dc-45fb-88ab-5ba9b6673bad}} is a partition of unity inscribed in {{formula:3294d317-59fc-46f1-8a56-73ef9867ecc4}} .
To simplify the notation, we will now restrict our attention to charts which are small in the following sense.
Lemma 3.41
Every point {{formula:26407493-d2c3-46a1-91aa-e1d15c42fc34}} lies in a chart {{formula:445cd598-2b0d-4bb1-986f-c6727788401e}} adapted to {{formula:26bf2323-83d5-4314-a20b-788d7e8f3f4d}} with the following property. For every {{formula:df19730d-a4b6-4b52-89c1-34d8f76f7453}} such that {{formula:f6598547-a771-4aa9-9acd-85e09e936a09}} is not identically zero on {{formula:d6c6ddac-ec8a-49f1-8217-54016695484e}} , we have {{formula:5b88f5b8-07ec-47eb-a8d1-1107985ddc0d}} .
{{formula:a9d70b39-6472-4143-9256-e54291a03b01}}
.
For {{formula:476615bf-fbd6-494c-ab18-78d0bacdc753}} let {{formula:56fd0022-c12a-4aaa-a42c-16fde33612a2}} be the closure of the support of {{formula:6bb444d1-44cd-4866-af5a-92d742600259}} , i.e. {{formula:8529e826-922c-4b37-8d0f-c300e512e230}} . Put {{formula:32747a80-9e96-48a6-bc2d-c1a2fda6080e}} .
Since {{formula:0111fa93-a3fc-4baa-951a-d43b6cb91b2b}} is locally finite, there is an open neighborhood {{formula:fda1c43c-ea43-4f3b-b308-7015fce3a03c}} of {{formula:3e0f6838-1825-4df2-a94b-387bf87e2515}} such that the set {{formula:a3847b23-78fb-4cd0-a88b-f519ef9b0774}} is finite. Clearly, {{formula:2492adfd-ed66-44ce-a3a9-22c2f69337ca}} . Put {{formula:fbb0f67f-abf1-47fd-ac39-ef799d3926e8}} . Clearly {{formula:d95ff949-d2af-4236-8caf-ada6b69e77b9}} , and since {{formula:c18cad1c-1ad2-4c95-ba87-c9f8a7bdb90a}} and {{formula:fd7dcc28-279d-4f5d-96a6-a9c36681d0c3}} are finite, the set {{formula:f231f37a-fad7-4afc-bfd1-223291925d3c}} is open.
Now, fix {{formula:ea2f44c5-0bb7-46b6-b45e-5bd320eb8307}} such that {{formula:166d7889-d94e-49a2-b005-4700ce55c4ba}} is not identically zero on {{formula:42be2e72-f83f-480e-99c7-4b812df01661}} . Then {{formula:b07e88d5-493e-44e7-9f8a-c7403bd36af0}} , so {{formula:479f078a-755f-4356-b7f6-4f3ba5d3f008}} by definition of {{formula:b85b985d-6473-452f-96d1-41fa9e074f0f}} . Definition of {{formula:3ea01709-5539-4cd4-bf2c-5ff3769afea9}} implies that {{formula:e6181958-9b3a-41e6-af94-8d67957ba270}} , and therefore, {{formula:cb69d659-8b5c-4051-9677-05d812a2de0f}} . We conclude that any chart adapted to {{formula:cc2cb811-fed3-4031-babc-1d0fed0efec8}} contained in {{formula:d5b08dbb-da7b-4916-8afe-e4c9aa6c168a}} has the required property.
From now on, we fix a chart {{formula:3598e275-babd-40c8-8002-e66ec4ec1bb2}} adapted to {{formula:b00da571-5cd2-4fd5-bcab-83955694875f}} , satisfying the statements of Lemmas REF and REF , and use Notation REF for this chart.
Put {{formula:0f763a62-133a-46aa-9cef-3a98bdb5042d}} . Then the statement of Lemma REF asserts that {{formula:96e873b4-5c3e-48fe-af84-92f4488c4b1d}} for every {{formula:115dd867-e186-4ead-a9a4-ad9d453b934d}} . Thus for every {{formula:bafd8434-d766-4c9b-ae3c-c90bdd33c211}} and {{formula:af8f0397-8f6e-4a85-938e-3c1f8e58978f}} we have a transition function {{formula:1259a929-1f1a-487c-b965-215924114f90}} defined in Lemma REFREF . We define {{formula:7a9b9f26-dcd8-44c6-9916-e815fe0b2410}} as the set of those functions {{formula:672a582c-a705-4421-a8f1-6a518f6d7540}} , as {{formula:89504663-eecb-42d2-b5b3-a24e8c007131}} ranges among the set {{formula:c2a83eb8-bbb1-4ae4-a66e-2b655d0a23a5}} . By definition, {{formula:0643ea1a-c552-4146-8b35-5381b3ed9d80}} , so by Lemma REFREF we have {{formula:5015db8c-b004-4b1b-9a75-5d796c4ce861}} .
Fix {{formula:455fcd78-2c78-4b38-9e53-d50faafa8e49}} , and let {{formula:deaaaff9-7fb2-41eb-a5d9-6eed089b81a0}} , {{formula:ffbea087-d35a-473b-be89-a089125d4410}} be the {{formula:e52e3624-b70f-4359-8001-371fec702aa0}} -algebras defined in (REF ). We define
{{formula:438e99cd-9f9d-47b9-8210-38dddfd36579}}
This way, {{formula:0d654116-480a-462f-a602-915d04c59fa7}} is an ideal of {{formula:352985a1-7b38-4079-8d00-9be62086943d}} generated by all the derivatives of {{formula:bba976d8-89d7-4ff1-ad6c-5b60211b98c0}} , {{formula:7a607c2d-6e9b-4c5e-8798-e7119821b418}} . Intuitively speaking, the ideal {{formula:89c7e204-a6e1-47c4-b41c-b75c9d257787}} consists of elements which decay exponentially fast with respect to {{formula:4b97f078-7b08-47e6-a83c-07c1aa8fcf9e}} . In fact, we have {{formula:7c9080e3-8d04-41c8-ba45-4a349ab4b029}} by Lemma REFREF ; and we have seen in (REF ) that {{formula:fe6669c1-3e9d-4246-adbe-5c84ad32c07e}} for some constant {{formula:eb6a84de-abdb-478b-9b6c-0a6d543aa38c}} .
We now list basic properties of the algebra {{formula:3dca949b-96b0-433f-b257-db0ecbafb978}} and its ideal {{formula:97a2b58f-5f34-43cd-8ebf-1ce818c76f91}} .
Lemma 3.42
For each {{formula:c050e83c-caf6-4c1e-83e0-ea16f01d6ad1}} , the following holds.
We have {{formula:fe35849a-d337-43cd-8e21-7fe3be2e1cd6}} .
Both the algebra {{formula:14405e2a-c664-4f1c-809f-b4629138a028}} and its ideal {{formula:00911cb8-949b-47f2-b5a3-9fc29b902991}} are closed under {{formula:0fcc773e-7432-4769-a2ca-3ce43313757b}} .
For all {{formula:0fd146be-9759-43cb-804a-297e3230daf0}} , the functions {{formula:fb7b8efd-9cc1-4aa4-9a68-0dd7f9f7df14}} , {{formula:1d153185-7563-41aa-92d3-5b11eec78608}} and {{formula:365deb65-a700-4e17-aed0-bb7d0d9c6561}} flatten {{formula:a876f2d0-bb93-473c-ba22-15836e5aa9eb}} on {{formula:744f065c-49d8-44d6-b45f-94158659718f}} , {{formula:4b1f84ce-60fa-496e-9506-6c64edbff538}} and on {{formula:65ed3331-d878-451f-a231-320093c4cb8e}} , respectively.
We have {{formula:7525c85d-21af-44a8-8619-f3d2e3a26114}} .
For every {{formula:39aa8063-7f32-4c56-8b7a-750ec951e9e2}} , we have {{formula:5c84918e-858f-4ee7-865a-3e4fbe799739}} .
For every {{formula:9fab2b9b-fe2d-4166-bfb6-d41b44a83368}} such that {{formula:7a51bb52-1f64-4509-afd7-489361c8002b}} we have {{formula:a36093fa-a3bc-4e93-8ef6-965c03feeade}} .
{{formula:06e6189e-cb66-459f-82b3-7f4a4b6182fb}}
.
Recall that {{formula:48f3c9d5-6f12-467d-822d-889f8e822330}} . Indeed, by definition (REF ) of {{formula:c50c4f1f-c292-4896-9e12-33c3bce2a872}} , {{formula:fbc89676-9dcf-4b15-9e87-b61a04054650}} we have {{formula:0fd0ab1c-3adf-4df9-a567-5e894d6e8f81}} and {{formula:a68e7a8c-058c-41d7-9fdb-338f98405155}} ; and by Lemma REFREF we have {{formula:c5e873ab-5f16-4ccd-8e38-81ea0af3541f}} .
REF We have {{formula:7efdf066-1240-4650-b154-b77b6469d404}} by Lemma REFREF and {{formula:062602d1-b082-4b34-b486-024a3d52b432}} by Lemma REFREF , so {{formula:3a1db324-5a8f-4862-98da-740ff1212036}} . In particular, the above remark shows that {{formula:7b4971c4-71c9-42f9-b3a9-175dec1f8dca}} .
By Lemma REFREF , for every {{formula:38039039-7db8-451c-8d6e-5029aaded018}} we have {{formula:d39ef1cc-1a78-470d-8f6a-875dbc6eae47}} . In particular, {{formula:20336071-bdae-4134-bfcf-27e06232d83b}} for every {{formula:494136fe-d311-4c85-a295-8700276f2ce2}} . By Lemma REFREF we have {{formula:0f089b2b-f0d1-47ec-b80d-1e09b536dab9}} and {{formula:468af5a9-ad6d-432b-b78f-a3b2dd7e9944}} on {{formula:11fb22c1-1d71-42a0-8e48-8165c20da5a5}} , so the fact that {{formula:536b5db2-6c52-4c41-85ca-6de27f061b01}} implies that the remaining generators {{formula:e0720de6-a132-4b32-b013-fd3dc5ae0fcc}} , {{formula:3af5ee98-a6aa-44f8-8e54-9dd288b0e1f6}} and {{formula:dcd167cf-bb79-40b4-a31e-1c40c5ef0ee4}} are in {{formula:0cfbdfe2-b73a-47b4-acd0-87c12d86d6b7}} , as claimed.
REF The algebras {{formula:bbeef938-fea3-461b-bb90-6732eff45838}} and {{formula:7f21eddd-5f10-41bd-89f7-becbb2251045}} are closed under {{formula:def96ba6-974d-4fe8-92c1-69868200bef2}} by Lemmas REFREF and REFREF , respectively. Hence their intersection {{formula:3b636702-85ed-4def-b3e1-51a81df30531}} is closed under {{formula:82585a9a-5903-46ef-8075-5aeb2d154cf7}} , too. Fix {{formula:af5cc91e-b09a-4c95-b694-467ea39e4152}} . For any {{formula:352df1d2-375c-4f20-ab7f-989d2fa33f3f}} is in {{formula:b0756b4f-7b89-4b95-ac1b-8c45d19c1ae9}} and hence closed under {{formula:bc390430-7468-49e3-aae6-ec52cce8014c}} .
To show that {{formula:7e92cfb1-7dea-4577-b2f1-8c958e368417}} is closed under {{formula:b5d4c757-0d29-4a80-9d2b-1f552c747478}} , it remains to check that for all {{formula:85029c49-60f7-48ad-83c1-effb8e6a9e7f}} , {{formula:084180c6-8fcb-4a82-8fa8-041469829aa4}} maps the remaining generators of {{formula:6fd433ee-7447-444e-8462-6cd4dafc0574}} to {{formula:db75b940-efec-4150-a390-b55644bfc34e}} . Fix {{formula:9d47ad4a-1175-4b5b-b3f2-fd57f265fc60}} , and put {{formula:1340d33b-f653-4b75-baa6-b27a02ff354d}} . Then {{formula:1f48a9f8-b3ec-4039-9bef-bd98ba3b06aa}} . Since by definition {{formula:417d4851-fd6a-4595-bcf5-fdc58d691151}} , we have {{formula:46b4d5ab-15f0-4b2d-8ea8-9e72cf182a89}} . Since {{formula:c0c62ae2-3b79-4ed6-b085-a0d58b7d14b7}} belongs to the algebra {{formula:446f844b-41ab-4c23-9c76-10b49b458fe9}} which is closed under {{formula:3488b1e8-b308-4ad2-8934-d9fa5d629b0a}} , we have {{formula:b8c729de-5ce4-4baa-8379-8e65e2c7e8c6}} . Hence {{formula:9359f839-8c82-49fb-9bed-5b0a951466a9}} . As a consequence, we have {{formula:c63fc8c8-163d-4045-8734-432a7d4506bb}} and {{formula:d98f3a0b-c26d-47b3-99ff-f971ff33a05a}} . It remains to prove that {{formula:7c6fe569-4365-4090-ad03-aebec232d15f}} . To see this, recall that {{formula:cbc39a93-8b89-41ee-b31d-8470726aa807}} belongs to {{formula:31e59b6a-40dc-4e50-ad44-55c894028ebd}} , which is closed under {{formula:4bb336c9-814b-469d-878d-1f1e52492448}} , so {{formula:2ceb2642-e41e-4fc0-b6b3-0947261fcd0d}} . Now {{formula:a34f2d06-b457-4209-9c76-72fa3562d4be}} , as needed.
Therefore {{formula:a9df12c5-354c-4de7-84de-e41838537e0b}} is closed under {{formula:db5e9378-84bf-4574-b7ea-6b13d85cb334}} . To see that its ideal {{formula:7a68afb5-297e-494c-9320-685f385dbb6b}} is closed under {{formula:b16a7eca-619d-4562-8031-2ce1a84a8911}} , too, recall from Lemma REFREF that {{formula:6e7d1a30-ae92-4f51-b03c-79a4d88c6e90}} , so closed under {{formula:d4892112-bb5b-4582-aedb-376bfe7bce7b}} , as claimed.
REF By Lemma REFREF , all derivatives {{formula:2596350b-5ce7-49e4-a8c9-c9077091153f}} flatten {{formula:4a0e8895-4da9-465b-bf6f-95980fb10e70}} on {{formula:350d85de-ffb9-41af-a91c-2f80271d2152}} (and vanish there), so in particular they flatten there a subalgebra {{formula:8b7d4676-2e47-46f7-a09c-d078051e1d17}} . Similarly, Lemma REFREF implies that {{formula:e3099e50-c602-4511-b5e8-a2083084aea0}} , {{formula:8df255e2-4b05-4f12-b70b-ef6abade6ccf}} flatten {{formula:7e9746a0-d11d-4713-a65a-8748150db0cb}} on {{formula:e59cc16e-19d4-44a8-a3ca-646cbf74484f}} and {{formula:51f2527f-4b77-4988-b53a-7cd3787f1399}} , respectively. By Lemma REFREF , it is sufficient to prove that all the remaining generators of {{formula:4d78ea02-1a9b-4b27-a243-a4f0223c078a}} satisfy (REF ). This is clear since they are all bounded. Indeed, for every {{formula:b9cadf2e-15ed-4cce-9786-38880d278887}} we have {{formula:6e5ffd03-a890-494d-a008-c9baa3d83810}} by Lemma REFREF , so {{formula:7035a687-b368-49a2-9c9c-1eedaba0dc61}} is bounded. The other generators of {{formula:72ed8d88-abe9-4a60-9029-598f6eb997ff}} are bounded by Lemma REFREF .
REF We need to show that, for every {{formula:96205802-7afd-4b7c-a357-a5354697c439}} and every {{formula:79db3387-65be-442c-81ec-efb0027ae96f}} , the functions {{formula:a38a47b3-eb59-4c70-a9ff-8baa79222b77}} and {{formula:0eb37ac5-5dc3-45ed-b373-92bc978604b3}} are in {{formula:730edf18-4d13-4d9b-afdf-986130931f99}} . Since by REF both {{formula:d174f505-7734-473d-8e30-f36e9fae0b5a}} and {{formula:5cb59b5f-d174-484a-8deb-61fe705eb765}} flatten {{formula:62908c37-f17c-4d5a-a672-54268403ba43}} , the claim follows from Lemma REFREF .
REF By the chain rule, we have {{formula:8cf0c565-0044-4383-bacf-df1621b46335}} . By Lemma REF , this sum equals {{formula:18584994-e03f-497e-8b17-adcf18ce58f4}} . Since {{formula:1aa35769-b040-4c4b-98ca-0b83d2c25ae0}} and {{formula:a84c788a-e5cf-4ac3-8a50-bddc44bdf7d6}} , we get {{formula:2911d52a-35b1-4e50-82ca-e6a5a2f510d1}} , as claimed.
REF Assume {{formula:c60cce82-a623-459a-ba20-161e3b33098e}} .
By Lemma REF , we have {{formula:60604fdb-977d-4d72-ad79-a12638732546}} , {{formula:a9c9637c-3ed7-43b1-9266-0d79d2ae7b7d}} , and more generally, {{formula:489dd35b-a3b0-4e94-88fd-17d5a6b20e52}} for all {{formula:bf6744e1-f690-4a4f-8281-907b0715ebdf}} . Thus {{formula:2a723348-480b-4278-98f7-9173e2840b35}} . By REF , for any {{formula:23c59cb0-2d00-4f2b-929b-d548c2d0b14d}} we have {{formula:b8250435-7f50-4a8e-b2b5-1b36b727adde}} , so for any {{formula:b0077b71-61f8-4d6e-9844-f0fbebecde7b}} we have {{formula:ba4b5aff-6bac-4a90-89e7-f3b4a3ec366d}} because {{formula:6c9206b1-ac8d-452a-a2de-a4720d985dc6}} is closed under {{formula:9333d4cc-efd3-48d9-a48b-39b665e3f629}} . Hence {{formula:d6ae4566-d250-4878-9562-f81274c7ba08}} .
Let {{formula:eaf0ef17-b055-48b4-9077-63293d09c82b}} be one of the remaining generators of {{formula:b1ab531b-dbdb-466d-8812-103e529603d7}} . Since {{formula:547f7ced-d066-4e43-9e3a-d5a3d21edf33}} and {{formula:d978d141-f708-46ef-a681-8f7acf259f5c}} , when computing {{formula:6b19e65e-4a11-41c2-85a5-8a0f94ee27a3}} by the chain rule, we get an element of {{formula:8310e22c-ea89-4be1-92f8-fbb6ea58ac3f}} multiplied by {{formula:35e48b77-9199-4a45-998b-8a2164078dbe}} , where {{formula:fb20e367-7e6b-4825-a06e-88509917de4a}} . By REF we have {{formula:4ddc8c9c-9cdd-4228-9e19-d335c49ffbb9}} , so {{formula:271f85d3-14f8-4c48-9e11-c93d19432bc1}} , as claimed.
We will now introduce additional notation. For {{formula:5148fa57-31c9-4be6-a438-8db19d44bd38}} , we put
{{formula:d1087ca4-b881-4f13-b17e-8d7a88ba8ec0}}
Moreover, we put
{{formula:0b4b4c88-a08b-481b-8dcd-d3e99f6fa445}}
Then {{formula:dc47ca76-0151-4c1e-96dd-a34bd8f7263e}} is an ideal of {{formula:248bc88a-6c50-471e-ab5c-e9e2ec6e5937}} , and {{formula:ba6c4d30-b674-45e3-8f1a-fdaf2eeb6632}} is a subalgebra of {{formula:5d30d245-283c-4112-a260-2a48c68279a3}} , containing 1. One should think of {{formula:7e77d2a1-6ea4-4206-8b7f-59fba6f1a885}} as an ideal whose elements decay exponentially with respect to all {{formula:4236ad9e-cd4e-41b0-a514-fefed1852610}} , see (REF ), so they can be used to flatten elements of {{formula:ab89a490-0816-448b-93d2-a6b5a0e760b5}} . In turn, viewing the elements of {{formula:00a192d3-e14c-40fa-a07d-d1a094fcb44f}} as those which decay exponentially with respect to {{formula:2a959c70-71de-4612-9ae3-c1079ecdf6dd}} , we can view the elements of {{formula:33fcf2c6-b07f-4a33-99e5-c54a4fc73d01}} as those which do not require further flattening. In particular, we will see in Lemma REFREF below that {{formula:94229305-ffbc-46e1-a8f8-9482d09b6d8b}} .
Lemma 3.43
The algebras defined above have the following properties.
The algebras {{formula:d467a06a-8fc3-44d5-9f94-f66370d81736}} , {{formula:4db31993-c1e8-4bf0-9a08-2532a69f67f9}} and {{formula:33160242-0efa-4f32-bfca-7c1688720922}} are closed under {{formula:4c052efa-d5ed-4219-9d7a-fdcae628d5f2}} .
For every {{formula:a3fd4519-97ba-40b8-a4d1-78b1f80c9e6c}} we have {{formula:7f55beb4-76f4-42b4-b6a7-3121bcc3f109}} .
We have {{formula:c931596b-1c6b-4322-ba9c-8c3c4a1e90da}} .
For every {{formula:07230838-d682-4f23-b25e-8f29ffa8cc30}} we have {{formula:62a7c0bf-f706-415b-8898-326d5be3fc48}} .
{{formula:207aadf0-bed2-4800-b0a7-dae0e5ed0da3}}
.
REF Follows from Lemma REF REF
REF By REF , {{formula:6487feb1-86c7-462d-a1ff-3dc24f8190a5}} is an ideal of {{formula:642c431c-e66d-4302-9e20-f3413c156f00}} which is closed under {{formula:1ef29dd0-9778-47da-a3a2-1f3c0619c792}} , so {{formula:b5c53ca8-6ca4-4180-a146-041c50c2a751}} . It remains to show that for every {{formula:120c7012-3659-4fcb-aa69-2c99ffe1c1c8}} we have {{formula:39c00e56-e538-4256-a302-7d0f1df692ba}} .
Assume {{formula:cc5abc9e-fe4b-40e6-b4e9-e78972c5082b}} . By Lemma REFREF the ideal {{formula:6ac4bc5c-7380-408e-a1f8-49f5a5ccd3a3}} of {{formula:835479ec-2e49-4d55-b70a-681cf8feac8a}} is closed under {{formula:01b1c9be-5aae-431b-bb36-6b5e2790da81}} , so we have {{formula:43676d87-8215-4c6b-91ad-9eff7e0c2283}} .
Assume {{formula:1d840336-c16e-4fb1-89f4-823c8443166e}} . We claim that {{formula:804d470d-f2d2-4475-8451-a8fbef290cc7}} . Let {{formula:27ca120c-1e14-41c4-8d26-d3a89f2b965a}} and {{formula:e2ffc276-ccd8-4747-b422-472ee3d38c3d}} . We have {{formula:1bb200e4-2805-4d5e-a62f-b2580c32df62}} , so {{formula:3bf4bb4e-32de-47bf-af67-2d7d051d6c5d}} because {{formula:c4beacdc-cd46-41f1-a4cb-ba4f1f3ade22}} is closed under {{formula:92de5ca6-a83b-43e1-92f1-2f156e9d416b}} by REF . For the second generator, we compute {{formula:7f27b459-428b-41ed-a4fa-dbad45d35505}} , because {{formula:8fb3ad49-f606-4f99-9393-157282a7c98c}} by Lemma REF . By Lemma REFREF , we have {{formula:c7f70ef5-db8b-4406-9c6a-3030ed7aca08}} , so {{formula:8637be76-8325-4ba3-be8e-c14268444534}} . Hence {{formula:060cf3b2-896f-42da-91a0-1cb2abcdf830}} , as claimed.
Now {{formula:75a6da6a-f849-481d-a224-f341b819fb4a}} , as needed.
REF We need to show that {{formula:0e7db7b1-1d2f-4003-bf3d-7f60461a709a}} and {{formula:79c225cf-76f0-4803-915b-7f0873c1e815}} for all {{formula:57e90e6d-489d-4caa-8331-d087d8c98f94}} . The second assertion follows from Lemma REFREF . For the first one, recall from Lemma REFREF that each generator {{formula:adfd7f50-66f8-44a7-987f-f26153d595d2}} of {{formula:f7237e23-b0ca-4ab8-9820-d5b29c195496}} flattens {{formula:c6ffe3dd-5418-412a-aa48-90d55f79a3d6}} on {{formula:7fff9f56-8884-4209-9830-25b2b7cb5d5a}} , for all {{formula:76ac1030-16c5-4c87-8285-08afdb4e1c5a}} , so it flattens the whole {{formula:21a9b52f-0a07-484d-89a7-0fc6388972da}} there. Thus by Lemma REFREF we have {{formula:12427787-2606-4bb8-8eaf-faf5032b9dba}} , and therefore {{formula:b94aad70-812a-4ab6-ac77-031fb075e3b7}} , as claimed.
REF By Lemma REFREF , for any {{formula:6fb0daa7-d9c4-40c8-b279-070feea79bbc}} the function {{formula:e0a5176a-7915-4c70-8a30-191093593d56}} flattens {{formula:ea6a3d04-f5cd-4dc2-8e9c-9cc9ed9baad1}} on {{formula:ae8d4008-3400-445d-b08a-71400b78c498}} . Since {{formula:09a0a6bc-f517-4e2a-ae09-10826d50e4db}} , Lemma REF implies that {{formula:08b2cfb4-439b-4139-a04b-04725de931bf}} , as needed.
Definition 3.44
Let {{formula:3a6dda30-1445-451f-8ea9-3ed1f6ab1414}} be the set of {{formula:767caedf-6c63-4479-bd4e-f7897c3dff72}} matrices {{formula:44c2935e-a371-4c06-9725-7f28e3a2a6d2}} satisfying the following properties.
For every {{formula:7d8a4a47-68cc-4971-ab6b-3ba7f7d21eb5}} we have {{formula:b4106172-b2d8-46b0-863e-b9d897462986}} (i.e. the first row is zero),
For every {{formula:944759cc-c56f-4cc1-b418-4e51be8ec19d}} , {{formula:b0f5b16f-3cd2-48b0-be54-fab397739bef}} we have {{formula:5460e410-0520-4870-a3c3-51f7d621180f}} ,
For every {{formula:b2c6a962-54a1-468b-a024-fa32293e7d2a}} such that {{formula:9bda9f39-2dbb-4d31-b22a-5d004de431c7}} we have {{formula:10ac93ec-6ca2-46a2-b11b-e8982ec828a1}} ,
Lemma 3.45
The set {{formula:9f2a9ede-4466-4c85-ba7b-23a96bd415d5}} is a ring.
{{formula:a9d7afac-d297-4d7e-8c96-4be981ee22ae}}
.
Let {{formula:a9c81660-6d2a-45bf-ad58-e8b86be0fa30}} . Clearly, {{formula:444a280a-2da7-4152-bb7e-ed46e38dbe06}} . To show that {{formula:5a58b2c5-ded7-4e3c-ad6e-71806f73bd22}} , write {{formula:3372af9d-3052-4475-9b31-902875703d97}} , {{formula:28e7e60b-3d8b-4dcf-babf-0537f3576d54}} , {{formula:c48e9686-7620-45f1-b9c4-58d6f1673716}} . Since {{formula:58246e37-5dfa-4713-a395-b8ab9054d700}} satisfies Definition REFREF , we have {{formula:553a1e25-c0ea-4069-bdb5-e44ab7fb1d11}} for every {{formula:395ba1bf-7aa7-4e03-8fb1-d4860497686a}} , so
{{formula:fa5e729a-13c0-4781-81f3-eab51351fce0}}
Condition REFREF for {{formula:475de5b8-7bf0-493a-a311-a5c4f0b11e20}} gives {{formula:9fe8bc21-0ae8-45fa-b980-0bddeb30fc70}} , so {{formula:c626b08e-1ba5-45d9-8969-d84048471cf5}} , too, i.e. {{formula:d304d45a-0538-4dfc-afe9-a2fe63a946aa}} satisfies condition REFREF .
To check condition REFREF , it is enough to show that {{formula:3af0daf5-bddd-4a50-9135-2b7d6bfe850d}} for every {{formula:16612c2d-9b76-427f-a162-d0c3aae7376b}} and {{formula:40b22fb9-4cd2-4c70-a60d-61488bb89454}} . Consider the case {{formula:35628a80-1e8c-44d4-a21b-04a272210c44}} . We have {{formula:4d5fd899-cd7e-4344-9f1e-78c3a4302948}} and {{formula:235eee48-3478-45d0-b0f4-f0531b8d173f}} , so {{formula:b9d1babb-7968-43f4-a008-7bed2f04c9ad}} , too, as needed. Consider now the case {{formula:440e873f-d5e6-4f48-9014-7f176b570d4c}} . Then {{formula:bc1842fd-8066-4ae5-bfcd-b8e14eb2de81}} and {{formula:e0baf574-f647-4f86-be35-76f5cf14d729}} . Since {{formula:20af37e7-e41f-481d-a4cd-65a72c72aaea}} is an ideal of {{formula:f161d809-31b2-4813-8bac-4213474f9cd2}} , we have {{formula:0e341345-e121-41d8-a728-112d08b9b632}} . Thus {{formula:43079a2c-9abb-454c-859a-b37a105f92fe}} because {{formula:b661f002-9999-4414-be71-8e799bb599b1}} , as needed.
Condition REFREF is checked similarly. We claim that for every {{formula:64294cc5-46ab-4149-99b5-24ba879db438}} such that {{formula:e8d81f7d-58af-47d7-a05e-a9bd38705067}} , we have {{formula:180eb663-e702-4ff6-a225-5fadea6521d6}} . Since {{formula:76493c0f-ebea-483c-ad94-d16045b13d9e}} , we have {{formula:00822f9d-8ae6-4458-895e-80143d3c80f1}} , which settles the case {{formula:f41e97d3-6474-4ba2-9ad7-2d68c96c430a}} . Assume {{formula:98f6e945-2494-40e2-9216-661776d5784e}} , so {{formula:411166a9-4499-4be0-bb36-1fde8d3dff68}} . If {{formula:6aac15a6-cd4b-4661-bea6-c4b591115696}} then the equality {{formula:5dc0024f-32a2-4be4-b7af-20723627a38a}} implies that {{formula:7fb6345a-4f83-4033-aa05-b57f36c36ec0}} , as needed. Eventually, if {{formula:73137153-00f2-47de-8287-aed62bb7b613}} then using the equality {{formula:46b11990-7a90-49da-bad0-d7f16734619b}} we get {{formula:23bbba37-cd3b-4e24-af06-8af63430b169}} , because {{formula:e2e0ae67-4e45-404a-a17f-c3d1c9494de1}} , which ends the proof.
Recall that our goal is to study the map {{formula:8b6ad772-d52c-4556-acaa-0786177b3ab7}} introduced in (REF ), in the coordinate system {{formula:bb6e96d6-16ed-44c0-8c78-0a2398a02754}} fixed in Notation REF . The Jacobian matrix of {{formula:07f9556d-1f31-4740-9973-39dbc8659ae0}} with respect to these coordinates has a block form
{{formula:38eb9e7a-b3f5-4be4-a871-115408276a7b}}
for some {{formula:521f09c5-0a8b-425b-920e-aaeb80e814cc}} matrix {{formula:61dfa83d-d568-424e-9df8-ab92d72ccf5f}} . By Lemma REFREF , the matrix {{formula:57ae31b0-55a7-463f-851c-18f0ed4287cf}} is zero on {{formula:d64408bc-d8b6-4795-94b9-6236d825fd22}} . Therefore, we can, and will, shrink {{formula:83df53bd-63f0-4457-a1a8-f0d2e3794a05}} so that on {{formula:11497e8c-2cc1-466e-ab92-470d41a7f25e}} we have
{{formula:ac0d3752-88e8-45ef-840a-6a91246836e8}}
Recall that we still require {{formula:f058d34a-3a28-4911-b4c7-a23175c79698}} to satisfy the statements of Lemmas REF and REF .
The following result is what motivates Definition REF of the matrix ring {{formula:7eaa01c6-d06d-411a-863a-bf2f8fe53546}} .
Lemma 3.46
The matrix {{formula:e129bf64-005d-4b90-9624-fc1bca287017}} defined above belongs to the ring {{formula:77a34937-3911-4d90-a02b-731f5b20befd}} .
{{formula:8b5c39b7-8646-4302-950f-2179c5a9c5e4}}
.
Recall that for {{formula:33793463-b37a-4aec-8b81-4941b66ae3bc}} , the function {{formula:eb45c7b3-d484-42a9-9b09-e5158a9c5a40}} is defined as {{formula:fedb4b40-bbf3-44df-b60b-b5478ea524cb}} , where {{formula:97f602c6-b19c-42c9-8c87-573f8405ebc4}} is the fixed partition of unity, and each function {{formula:1ebe7402-1178-4c48-ae6e-50473d6f15e0}} corresponds to the chart {{formula:072a6ed3-5202-4a4e-9de0-16e4d4145a36}} of the fixed atlas {{formula:10e97158-71d3-4119-b807-84d97f331205}} . Since by assumption our chart {{formula:c0507cc5-494b-416d-9711-f121923fb0bb}} satisfies the statement of Lemma REF , the above sum runs over the finite set {{formula:8ca2c88b-84e1-4d72-a674-bbab59a7bf16}} , and for each {{formula:aabd6873-0fc6-4d97-9084-36ed5df6240f}} , the function {{formula:1199e4d9-a47a-4465-98f1-337516447958}} is defined on the entire {{formula:bc4209af-4261-451b-9b3d-4678c21bf9ec}} . We have
{{formula:f1fdbefb-ffc7-4cab-99a2-f14c6b164ca5}}
so we can write
{{formula:023ee512-c67c-4c64-b24e-30c1599def28}}
where the {{formula:cbc5836c-ed0d-484b-9e59-e04d30eec2e7}} matrices {{formula:65083447-68b0-4cd9-80b8-325e849c03f3}} and {{formula:ee69d8f3-81d6-4812-8607-586439575742}} are defined by
{{formula:ef87cd11-236c-410c-8a83-6d8ece80fe1b}}
for all {{formula:82969014-5d0d-48e9-b07a-1892cf0fca6e}} and {{formula:1b088475-6439-4792-a5bf-efbb1f009a02}} . We will show that {{formula:e51e46a7-0297-46ca-8238-6d386b7daa77}} and {{formula:48b79733-5e66-4058-808c-b0eba35d6566}} . Clearly, the first rows of these matrices are zero, so they satisfy Definition REFREF .
We have {{formula:5b5b5414-0356-4133-8b0b-51f1578e9ab5}} , so for every {{formula:88edd854-cafe-469d-9415-e8c980fd85f9}} we have {{formula:efe22ef7-b30a-45a2-90ab-9eae015774a7}} by definition of {{formula:36be19ba-ee48-4bde-95e8-99c9ee9cb55d}} . Definition of {{formula:f99071cf-0607-4b34-b2ef-ee8c86322401}} and Lemma REFREF imply that {{formula:680667fc-d7ab-4379-a20c-c7d9280248a5}} , too. By Lemma REFREF , the algebra {{formula:43d9afa8-2108-43d2-aa09-1d181609fa9e}} is closed under {{formula:6a7df3bc-bf5e-4d09-9ed4-963fd7578047}} , so {{formula:d8852452-9bec-4d1a-bcd5-44ffcdea889d}} and {{formula:ca7f6d03-387d-4121-87ac-b6683ca42a79}} . Since {{formula:9429d784-c75b-4079-b8b8-2990d4e4ebbf}} , we have {{formula:d12497e5-68e9-40d3-9048-2e6f9845f59e}} , and therefore the matrices {{formula:154fc1e3-4f1d-4506-91b9-9409b2c68f8c}} and {{formula:9ff7cdcd-1b40-4760-b73a-cd4428d14375}} satisfy Definition REFREF .
Assume {{formula:eabdb92a-7594-4c5d-b62d-10a9c543b843}} . Then {{formula:a54833bb-48d6-41f6-bcc0-7c1ee37a9f4e}} by Lemma REFREF . Hence {{formula:f1fff160-1f37-4466-a148-4e6bb3e092f5}} , and similarly {{formula:6cd28151-baea-4c23-a95c-36a811af6a6e}} . Again, using the fact that {{formula:e7105ba9-1930-4633-9121-689617f0ec49}} we infer that {{formula:d4924fd7-ba98-467f-a7d7-ebc125c1c8ba}} and {{formula:9f46e8f7-0ff9-4861-811c-26e9c7a469b9}} satisfy Definition REFREF . too.
subsubsection3.5plus.7-.5Differential operators induced by {{formula:d3d0b977-50b6-4366-b812-8967436a14d7}}
Our goal is to compare differential operators {{formula:d06f6f52-674a-47a7-8f24-5dbc8d2d64d5}} with the ones given by the coordinate vector fields of our eventual chart {{formula:a525d198-3dbf-4618-b405-cc0950416488}} . To make this comparison in an efficient way, we introduce the following notation.
For a matrix {{formula:8cc41e0d-101f-4372-a784-d8ff0ea4cf78}} , define differential operators {{formula:519332e4-b725-4ac1-a56d-60ff8e188b36}} by
{{formula:7024f6db-2480-4e0b-9c87-e2dc8f5b32d7}}
We say that an algebra {{formula:c5f8c4b4-eaa8-4ee7-a787-78f734b81d08}} is closed under {{formula:05a4bd79-3210-4b44-b492-dc2ee3f150df}} if {{formula:fde6e75f-e965-45ca-b078-5d89b164c90e}} for every {{formula:c2a0fae7-e2b1-4a61-98df-d516a727fda2}} and every {{formula:ed878980-bc16-4b24-8981-aacf2775174b}} . For a multi-index {{formula:1480af45-b5a1-4346-8a16-e7698eb45fdf}} , with {{formula:d5e13ca6-b8bf-467d-981b-63c9ba0ab6a4}} , we write {{formula:f9c0089f-7bfd-4c71-a18f-7019ab77b0d3}} and {{formula:2c09b83b-c715-43a9-a0d3-442ba8553910}} .
Lemma 3.47 For every matrix {{formula:a669e7d1-06e2-44de-b7a7-c7ebac22646e}} , the following holds.
The algebras {{formula:ecc2b2d7-f5a9-4aab-b83b-bcff9c62f378}} and {{formula:8f795b4a-5bf3-4ea1-8949-35a48688d4e4}} , for every {{formula:402a831a-c8c9-46db-b6d7-6d8b929e235c}} , are closed under {{formula:f1eac5b7-0274-4b27-a08d-9279fb6722a5}} .
For every multi-index {{formula:a01fa72c-2ac6-4eef-aa30-869bfbc459f4}} we have {{formula:d101642a-2f24-440c-a01c-3103ec40a965}} and {{formula:b6d8fae5-2a25-47bb-bc43-5e8410391882}} for every {{formula:50b292d9-4e15-4857-ba33-88a8a0fee321}} .
{{formula:8724e724-d441-48ab-a5a6-cdfae2775c9c}}
.
REF Write {{formula:03ab186e-bc76-4672-9391-dbbf8d5ffb36}} and fix {{formula:15da9d6f-08e8-4410-8b65-90f4ec0cb008}} . Since {{formula:ae715a0f-e03e-41ff-b52d-963a9d84bb3a}} by Definition REFREF , we have
{{formula:4fc61636-3749-422e-a248-4090b4d41405}}
For every {{formula:51cdec43-b82f-4bd7-81ba-e62c9a789fdb}} we have {{formula:3cc5df9e-7825-4ca7-961b-6fcc92955d37}} by Definition REFREF , and {{formula:427e689b-b8a6-494a-b5e1-3adf382cc98d}} by Lemma REFREF , so {{formula:c04270f9-c159-4f03-84e7-b7cea8880b1a}} . By (REF ), it follows that {{formula:85cc4928-9bf6-4f4e-b330-a12ba56d8d26}} . Hence {{formula:a4cac9d9-224c-45cc-b8cd-beedff1d869b}} is closed under {{formula:a425cd79-6b16-4d35-be18-b73dd3072de1}} .
Now, fix {{formula:5d8d57d7-0403-4f67-8b5a-2e3047586478}} and an integer {{formula:8a722e7d-735e-4cb8-a2ce-9dc3d6dc13e1}} . By Lemma REF , we have {{formula:88620c29-80a5-4bcc-84e9-974dd5bca1f1}} for {{formula:da21b493-1944-42eb-8d75-8f02cde11762}} , so by (REF )
{{formula:3858103c-b62c-4312-b4fa-e9210a39f5eb}}
To prove that {{formula:f2698021-6050-48ab-b43e-68c2289ecf26}} is closed under {{formula:c433485e-5c7b-4ef6-bc76-9cd07a131e97}} , we need to show that all the above summands are in {{formula:f21a19aa-70a9-45be-9bb3-c5ec21d1133d}} . By Definition REFREF , we have {{formula:1ad37f71-ae93-4d8b-987f-9e97dd127471}} , so {{formula:836ac2f7-ec52-4f2a-9b90-caafc394f929}} , as needed. It remains to show that {{formula:fd2e2c80-6804-4f1b-993d-fe691f0fc321}} for all {{formula:13b70d7b-8494-495a-bc52-0a06d2727bfa}} .
Fix {{formula:1f895f96-1ea7-4b5f-ba40-c585e0056038}} . By Definition REFREF , we have {{formula:cd51e46e-e941-483e-af91-3ce9d7b340c9}} . If {{formula:50f3a2f9-d1bc-4452-b0ab-f6997b6e0463}} then {{formula:e193d781-b2fe-4f28-888b-b39a195900c8}} by Lemma REFREF , so {{formula:edc1bb20-0625-44e3-964a-09ffacd883f0}} , as needed. For the remaining case {{formula:3d61a6c9-199c-4b34-9854-f984048d691a}} , recall that {{formula:57f9aeac-2f9e-4cff-863f-4cb8421d1ed9}} is closed under {{formula:53360edd-cc66-4e58-9f49-74169a554aa4}} by Lemma REFREF , so {{formula:450389e3-689c-47b6-a402-02cb8587c863}} , and therefore {{formula:757cdc5a-1b53-4715-947a-1edf325988c4}} , as needed.
REF
Fix {{formula:e8bd0986-24d9-46a3-86cd-75d47453fb55}} . Since {{formula:c9c687bf-8e30-4eed-93c0-a932c10aecce}} and {{formula:506b26ef-1bfb-4d68-b6c8-944a49a1ed7d}} , we have {{formula:5d95d817-96f2-47d3-adee-52c5f48fbfdc}} . By REF , we have {{formula:c25fd369-9457-4552-9473-a7e4d6ce3328}} . By Lemma REFREF , REF the latter algebra is contained in {{formula:c978cf32-8b4b-4b8d-9a36-c899e660eb54}} , so {{formula:5ede2bf8-a326-4e5a-bbfa-a95d7a2617fb}} .
As in the proof of Lemma REFREF , we conclude that {{formula:6a607898-0096-47d6-a5d0-c7212839fa72}} . Let us recall the argument here. Lemma REFREF implies that {{formula:be91c843-ee0e-4c0a-9d33-8e55630da628}} . By Lemma REF we have {{formula:7d9e5104-bb76-4fe5-a759-4f7a881b1da8}} for {{formula:50b18e85-0d52-4ef1-b317-de082d8a4665}} , so by (REF ) we have {{formula:64bd9f70-1044-4eb4-bb9c-319d2840777d}} , for all {{formula:4720e162-e69c-470c-ad9b-a162b167819a}} . Since {{formula:ebf6e108-5966-43c3-a6ff-6c2521438e7e}} by (REF ), the formula {{formula:935b3cab-6d36-44fb-876a-6de5232252d3}} shows that {{formula:305e1f63-e847-46e7-bb1b-e750c663a36b}} , as claimed.
For the last statement, fix {{formula:d4464054-2f18-43f9-b869-9406e2d680f1}} . By Lemma REFREF we have {{formula:985492c7-9aa8-4b7f-9752-f90fca95812f}} , so the claim holds for {{formula:25c5af5f-33cb-4906-897d-673e5233048c}} . Assume {{formula:6bc051ca-b810-4865-899a-9ab233eaaa4c}} . By Lemma REFREF , for any {{formula:6a108f0b-23f3-453d-8feb-edfdfc1da082}} we have {{formula:756f7f8a-27e3-408c-aaf1-f75d169ba279}} . By Definition REFREF , it follows that for any {{formula:91345116-6da1-4ded-8d10-7b0bb36131be}} we have {{formula:14403a64-5876-4489-8a71-c848a20a8c04}} , so {{formula:27353790-276d-4adc-a1d6-eb308c79819b}} by (REF ). Since by REF the algebra {{formula:c2e2682e-cb69-4d5b-8214-14606838233a}} is closed under {{formula:c0a89a46-da46-4b47-89ae-cd73a0eafb35}} , we infer that {{formula:a9df68e0-b658-463d-b8c5-32b95a01ed3e}} . But {{formula:fccaa2f7-6825-48fb-af28-7055467fbab1}} by Lemma REFREF , so eventually {{formula:1d677356-dfe8-4274-94a2-40328920da89}} , as claimed.
subsubsection3.5plus.7-.5The A'Campo space admits a {{formula:a16a5d17-03f4-470f-b046-db0b5b8fbf51}} structure.
We are now ready to prove Lemma REF , which makes {{formula:c65f1231-63ad-4aa4-bd73-d773ff11e435}} a {{formula:d6c4ab3f-a05d-4a1e-bcfb-2ab49cde2043}} manifold with boundary, and establish its properties listed in Proposition REF .
Recall that in Notation REF , we have fixed a chart {{formula:35a40d5f-5d6e-498a-b85b-a14eb4590d0e}} , considered the subset {{formula:3c38e52a-dca3-465d-9b59-1b28e545471b}} with smooth coordinates {{formula:89ba1acf-9973-4d1c-a778-7f51bc9d10e9}} , and denoted the sheaf of functions which are smooth with respect to these coordinates by {{formula:6ade1828-1d93-4023-a4cd-8b4a1f505f4f}} . By Lemma REF , the map {{formula:afaedb9b-b11e-4a0d-9886-308dbbad09fa}} is a {{formula:28196036-8ad1-4e68-8784-9de13b35d4af}} -diffeomorphism from {{formula:1b0ada5e-28d2-47fd-9d0b-a0a305ea018a}} onto an open subset of {{formula:1f8cfc4a-ef1b-4543-b6be-9ed9305fe411}} , so it gives another smooth structure on {{formula:6dd718e3-045f-4fab-8cfd-84b959b3eb0c}} , which is (only) {{formula:5baf087b-82d1-4aa1-99ac-8f1ab19a7957}} -diffeomorphic to the previous one. We denote the sheaf of smooth functions with respect to that smooth structure by {{formula:06c5b1f5-0c6c-4c1a-8028-0865f29a8e9b}} . As in Notation REF , this structure gives differential operators
{{formula:3802006b-9f46-4c56-8594-1c7b9a1df0d8}}
Note that {{formula:447468da-3102-41ba-a747-e0ee4aa5ca03}} has different meaning here than in Notation REF , i.e. {{formula:ad37e794-0b9f-4c12-97d4-0e7be615138e}} may not be equal to {{formula:cec7a463-fc52-4446-95d4-d4b73321bb71}} : this is because the vector field {{formula:4eb82975-1409-45b6-be56-d884039846fd}} is tangent to the fibers of {{formula:a6fb5231-d99e-4ed4-aae5-01a97c0b1c6c}} , which differ to the fibers of {{formula:c69cf846-0be9-4fb6-93c8-4dce0168e98b}} , to which the vector field {{formula:5d4a89ca-9062-4b9d-8e53-083a3fbbe09f}} is tangent.
Combining Lemmas REF and REF , we get the following.
Lemma 3.48
We have {{formula:31f8a368-7af4-48af-95a3-7746d87822ed}} for all {{formula:b2012eb7-f07b-4d73-89f9-c1ab3ea9bb3e}} and {{formula:f2e0c706-2635-4670-872c-70f785c1c029}} for all {{formula:ab11922d-2f90-4ee0-b491-ea12ba944a08}} .
{{formula:d9a5a983-8b93-4471-b093-6e3ecbddcaaa}}
.
Let {{formula:eca58f45-af42-4183-b8f5-28cb1c76d69b}} be the matrix introduced in (REF ). Then the definition of {{formula:a7c64a8b-da6d-430b-a3f8-3f626801f264}} reads as
{{formula:908cc722-17f0-47e8-9e8e-c3f2d4dccc25}}
By assumption (REF ), the above sequence is uniformly convergent.
By Lemma REF we have {{formula:3cd86ca9-ea74-4b90-9a1a-c0c2396a59c9}} , so by Lemma REF , for any {{formula:51c6b684-2d14-453a-831f-e5ca7a673fc4}} we have {{formula:08889b75-f787-4cf6-9e01-ace18ad8218c}} , too. Therefore, by Lemma REF for all {{formula:ffa841c7-0396-4c54-996a-57d6fe06f92a}} and all multi-indices {{formula:82673812-6889-4ebf-b7ec-83604c33b0c8}} we have {{formula:fa6cc3e6-da02-42f8-8173-abc60edb9c7c}} , for any {{formula:951c615c-e8df-47c8-961b-c1a610866b69}} . Recall that {{formula:5ca58234-cf63-4d54-801b-bf0a10035726}} by Lemma REFREF and {{formula:ab9764b5-7f1c-4cea-aede-309f8014def9}} by Lemma REFREF , so the above result holds for {{formula:0ca4154b-9d74-4a6a-9ca4-363866e0cdaf}} , too. It follows that {{formula:2d828882-108e-44d1-aec8-be0bad4dbbfa}} and {{formula:e2d9e9d6-ae8d-4e35-b525-9b912e8e34f6}} , for any {{formula:41f8726f-d795-4e54-9728-6360e8fc4819}} , are continuous on {{formula:0f8545a6-7063-4ac5-9dc7-c45720fe14c2}} . In other words, {{formula:bdd12421-5998-4d7e-8e16-2cb7cb3775e1}} and {{formula:13589acd-f186-408e-bc2e-f1d537409cf8}} are smooth with respect to the coordinates {{formula:5132601f-430f-46d3-8c7f-40d28ce64392}} , i.e. {{formula:3c4429eb-6c76-4321-8dc7-d47f80f1d946}} .
Taking for {{formula:2b9536b5-b8a1-4707-95ae-f81a4cc21984}} the restriction of the {{formula:3313f27b-4678-4d33-9825-04bfdb0849e6}} -th chart of the fixed atlas {{formula:155857c9-ce2e-4c2d-a7cf-c71c3622f5e4}} , we infer that {{formula:8228dd95-99e1-4c12-bbae-d8cfcb2f062e}} , for all {{formula:0becc795-e7c5-4eea-8f29-9be69c7b2d08}} . Since {{formula:ec717bdb-c95e-4b01-aec8-f0c8fb8aff07}} , we have shown that {{formula:f6c7c296-7846-4813-9d73-0cb0f4862f97}} , too. Hence {{formula:13771ab5-e81b-4878-aa7c-970b788d8bb6}} . The functions {{formula:ccaf8c17-3561-4b59-86b7-b7034043df7d}} are coordinates of the smooth chart {{formula:f67573f8-e23c-4a4f-8ecb-8779a2b80013}} defining {{formula:4a5adc27-3910-42d7-9e21-d8c721bf1fc7}} , so they are in {{formula:dd193926-b332-4aa2-adaf-d88bc9cfeb67}} , too. Eventually, since {{formula:181ead02-175b-49de-8399-053809ccbe6d}} is disjoint from {{formula:d88431ac-a666-4096-993f-e6e805c83847}} for {{formula:ddb3d688-1e5d-4909-a9e4-04524d1c9f02}} , the functions {{formula:0a85d9f2-897e-4516-bf91-401ddf6e781b}} are in {{formula:2646ec7b-c276-47a3-88a2-df394833bdeb}} , which ends the proof.
{{formula:72db4bf1-861e-4b3a-a551-30f50f6c2d49}}
Proof of Lemma REF .
Since by Lemma REF each {{formula:a88df7b2-0047-4f47-b2a9-510578259dae}} is a {{formula:b855723a-cf61-4c1d-98dd-86870a6b847a}} -diffeomorphism, the transition map {{formula:43b819ff-0e66-428c-b191-6db745546dd4}} is a {{formula:905d65cd-ae2c-4d1f-9b7d-43bff4c27d68}} -diffeomorphism, too. It remains to prove that it is smooth.
Without loss of generality, we can assume that {{formula:04bd1a76-cae7-401f-82bf-892decc942d0}} , and choose {{formula:0bc64e1a-b836-4df8-b91a-a346e9dd003b}} so small that it satisfies the statement of Lemma REF and the assumption (REF ) above. Moreover, we can assume that {{formula:00dc1ef8-3dd4-4222-ab1b-9e4b90ea5995}} , so the index for {{formula:af4e50f3-6b9f-4704-841f-9864aac50f25}} is contained with the one for {{formula:0203f486-28a5-4960-8320-c18a4f48d2be}} . Call the latter {{formula:e8967837-6ff2-46db-a67c-3b8f1acb416d}} . The chart {{formula:a42aa772-bf0c-4e69-a840-8eeb66c8e89a}} is defined in (REF ) as {{formula:114ab89e-c223-4484-9d19-48c0264bbdb3}} . Using above notation, we need to prove that all these functions are in {{formula:6267ca26-fe5c-44e2-b51f-8fe0fed8d26e}} , i.e. that they are smooth with respect to the coordinates {{formula:bd6a47f4-1a70-4977-aafa-ea869ff4b49f}} .
Clearly, the coordinate {{formula:a3ce0fea-e1e3-4f51-9d4f-ce9bf4e206f6}} is in {{formula:b28f85a2-bd91-4a01-ae23-a5358dd109de}} . By Lemma REF , we have {{formula:3ecff31d-18bb-4942-b753-20b2e34eb127}} for all {{formula:6906a751-f603-43fc-95fc-c35792745d83}} . The first {{formula:acaeea11-c355-4f41-bced-a11acf1bbd84}} functions in {{formula:bf667ae1-ff0a-4f3b-980a-5c517f343493}} are {{formula:4c98e11a-372a-453a-abd6-dfe163d8d7b5}} . By Lemma REFREF , they differ from coordinates {{formula:e6e97a28-7e8d-48a5-af09-96bf812ab110}} of {{formula:4de32db2-39bc-4cce-a87b-bed0bdf6fa77}} by a smooth function on {{formula:2936b45e-227f-486f-b6fa-3ec306955e0b}} , so by Lemma REF they are in {{formula:da21c1d2-99d3-4e3b-91b1-8cb9004337e5}} , too. The remaining functions in {{formula:640a9357-cc90-4e04-842c-8b3c3df3ce66}} are smooth on {{formula:ba019c50-2e6d-47f8-8f94-6726095a8297}} , so again by Lemma REF they are in {{formula:9eb86f8c-475b-48ec-88a6-b7fbfb634a7a}} , as claimed.
{{formula:e973a2e4-42fc-4a03-ba77-a3ba7c381001}}
Proof of Proposition REF .
REF By Lemma REF , {{formula:1d3c18a6-6dd7-40b8-8a0b-e16c104797df}} pulls back smooth functions to smooth functions, so it is smooth. Its restriction to {{formula:1b831759-0816-4eab-b204-1b9f97c5ec38}} is a diffeomorphism by Lemma REF .
REF We argue exactly as in the proof of Proposition REFREF . The function {{formula:24c4d8f5-0850-4f62-8743-046b97c47749}} is a coordinate in each smooth chart (REF ) meeting {{formula:42aad078-a74e-41e8-911e-3e4d6291d4da}} ; and {{formula:1cd0c59e-08ed-43f9-8839-99378f0d0ba9}} is a nonzero linear combination of such coordinates. Thus {{formula:02fedbfd-0f7a-4fc0-801e-3de2b099f9e1}} is a submersion near {{formula:8e823e67-7b08-4ae4-b9c3-35f8d528ef71}} . It follows that {{formula:1ed40bc0-5d70-4af0-89f0-c325e58f0d88}} is smooth near {{formula:e9180f57-f01c-4dac-8054-5653af33b555}} . Away from {{formula:6f33dd84-4102-4838-8e61-dcc93e664c3b}} , the result follows from the fact that {{formula:564cd71e-bfb3-410a-9939-711590af470a}} is a submersion.
REF This part was proved in Lemma REF .
REF
Recall that for every {{formula:2f75c8f6-25b4-4afe-b121-9ab5cc70835e}} , the restriction {{formula:e8a8a41b-744a-4f18-a70e-1f4ba7d9f7f1}} is a topological {{formula:0bae2ee3-fe61-4c77-a062-f990436f18d5}} -bundle by Proposition REFREF . It remains to prove that it is smooth. The map {{formula:38f2b779-4d38-462b-ac77-ad6b936a0c67}} is smooth by REF . The {{formula:cabeba4e-e0bb-4c07-b772-f4cf4d1faec6}} -th coordinate of {{formula:5406938d-c1f9-4e44-90ec-92185d335ce2}} is {{formula:b7a2b0ba-3815-46e3-8919-2b3eac4207bf}} , so by Lemma REFREF , it is zero if {{formula:5b2a0071-2a0a-48dd-9525-e8095220d7d3}} and equals {{formula:dd05646e-dc31-461d-b89f-3d85354d3483}} if {{formula:c2a28556-a413-4f82-afa2-436c466ee341}} . Thus smoothness of {{formula:04002dad-7baa-4d13-b518-9803fa4a3560}} follows from REF .
Fiberwise symplectic form on the A'Campo space
Let us recall the notation and assumptions from Section . Let {{formula:e34ac690-322c-4354-bf5e-26b879c78737}} be a complex manifold of dimension {{formula:72f037ae-6e0c-438d-9c17-93b037994491}} , and let {{formula:5a90af4c-b38e-40b6-b208-0664f9edaa0a}} 0, whose central fiber {{formula:be8cfe77-dc74-4413-bc4b-c9231c703cad}} is snc. We denote the irreducible components of {{formula:4bd3832d-937c-4c1e-88ae-6ae043bd4b7d}} by {{formula:1834f80a-ed6c-434c-a92f-31549d2c089c}} , and write {{formula:10b1ff28-229d-47b1-8c25-f6408800144c}} for some positive integers {{formula:0150c6dd-772a-4119-b33b-737e33441fec}} .
Assume that {{formula:bd770c3a-8a49-4e86-96d6-344f3cff3091}} , and {{formula:3b499de3-994c-44be-af0b-a599572f402d}} is a submersion. Using certain additional data {{formula:a173b6a8-9722-45d6-8da1-1fea74c9ec72}} ,
we have constructed in Definition REF the A'Campo space {{formula:026d8912-d090-4cd0-9508-24dfde4134e6}} . In Section REF , we have endowed {{formula:c29d2a06-c4ed-4307-8f2b-93d5d9d31b85}} with a structure of a smooth manifold with boundary. By Proposition REF , this manifold comes with a smooth map {{formula:d593a6d5-ded0-4575-9f3e-77a91a50115e}} which restricts to a diffeomorphism {{formula:46b8fee3-7418-4110-9814-d4987772187b}} ; and a submersion {{formula:659b5c1b-9586-4d13-a01a-267db9750440}} whose fibers over {{formula:1fd9d44e-e4f3-4e11-a2e2-4785a228d048}} agree with those of {{formula:ba598ad9-be52-4bfd-9fa0-93f4c37fe099}} .
Fix a Kähler form {{formula:69d74e5c-f3e4-434b-8c90-62900c28533b}} . In this section, we will introduce a 2-form on {{formula:b3eec89f-abb7-4131-a1dc-48c88cf7dabf}} whose restriction to any open subset {{formula:a564c6f0-8b07-4024-a6ce-0dbf6ade3b47}} with compact closure, is fiberwise symplectic with respect to the submersion {{formula:8c031512-0894-4298-bad7-9af178a28e1e}} , and agrees with {{formula:f76a159e-d16a-4fdf-b510-6ef2d44f94ba}} on {{formula:8ec318a7-cbb9-4470-b7e3-74e30ad0fb11}} . Moreover, if {{formula:593eed1f-ad9e-43e4-b1d1-916dbeb8596f}} for some {{formula:b10d39b3-ec08-43cd-9eec-e03501f3ab2f}} , our form will be exact.
The restriction to {{formula:81e7ac03-5246-4678-897b-37e11cdd078b}} is a technical condition, imposed to ensure that {{formula:9d138cec-d74a-4d24-ad96-6746d11d5711}} does not degenerate at infinity. If {{formula:97988147-75f2-4c82-839f-79903c846ecf}} is a family of projective varieties, then no such restriction is needed at all. In our applications, the set {{formula:b21c78b3-457d-4058-a468-662feb09e1cb}} should be thought of as a domain containing a ball, which sometimes will be a Milnor ball. We do not require {{formula:ccb01bb8-b2db-4705-8cc7-1c94cb1e9289}} to be small in any sense (apart from being compact).
subsection2-.5plus-.7.5Construction of the fiberwise symplectic form
Definition 4.1
We will say that a holomorphic chart {{formula:70146699-f121-4c9e-ac39-c94f761f0f36}} on {{formula:9b63b2d8-2b7f-4e72-9f84-986b324eb378}} is fine if it satisfies the statement of Lemma REF , or is disjoint from {{formula:1cf1b51d-0691-497a-892a-6a0c55f9499c}} .
Lemma REF shows that we can choose such a chart at any point of {{formula:f71c5e23-707e-47f2-906c-71bdfb0eca0e}} ; and any such chart gives rise to a collection of smooth charts on its preimage in {{formula:0ce84277-812e-4735-84a3-faaf01da73e5}} . Fine charts are, by definition, adapted to {{formula:69017dd2-2ded-440a-9306-edde439c81b5}} , so they come equipped with functions {{formula:094db467-28fa-4888-8568-69eb4a3ed97f}} introduced in Section . We will now use the angular coordinates {{formula:270fcd08-4eef-4a6c-a65a-909f645a02a0}} to define global angular 1-forms {{formula:cfcdce27-6b25-40db-9e46-d6300add1309}} , as follows.
Fix an atlas {{formula:d3aa331b-f935-4f92-99f2-1034c26192ba}} whose all charts are fine; and fix a locally finite partition of unity {{formula:c1d3f90c-39aa-4808-9fe2-261bfa1b534e}} inscribed in {{formula:c413e615-125f-4cd6-be87-1aa7e4a71103}} (note that {{formula:98f202ad-eb6f-457b-b6de-70e6e9c9ac88}} may differ from the adapted pair {{formula:770e5dde-faa8-4438-acb9-4665dbcdcca2}} used in Section REF to define the smooth structure on {{formula:9fc5d949-80ba-4707-92cb-7c4d6d6bc229}} ). Since the map {{formula:360e3b32-bc2d-42a2-b527-99d78de0bfcd}} is smooth, the pullback through {{formula:a0cdd770-3503-4585-ab0d-571249a036f6}} of each {{formula:aeb50b2e-8431-4522-8e2f-848893842cda}} to {{formula:4545d336-8337-4921-b4de-63a515b22eeb}} is smooth, too. As usual, we denote these pullbacks by the same letters.
Fix {{formula:a1a01873-d54d-48d2-b14c-34361a583470}} and {{formula:16a96a00-24b3-45b9-b3da-f063d60ba163}} such that {{formula:9dedf9a2-bbc0-431e-979c-95dd755f5f61}} , so {{formula:dd5cf3e6-77d7-47fc-92d5-03f60a50b6b6}} lies in the index set (REF ) of the chart {{formula:a84dc449-b6d0-4eb6-b8bc-4cca40b514ca}} . Let {{formula:adc2e2f3-1039-4317-a66a-72362dcfe322}} be the corresponding angular coordinate on {{formula:2661f297-baa0-44a1-a62d-e34cd162a2bd}} , see (REF ): it is smooth by definition (REF ) of the smooth charts on {{formula:0d5cac41-ee93-4e8a-aea8-9966a020f4d5}} . Then {{formula:ee8e24e9-fc86-4b43-8a31-f01506c5b5ef}} is a smooth 1-form on {{formula:91e75aab-7f46-4e57-bfd1-29bb8a4e296b}} . Since {{formula:a8b60d86-6e7b-4749-a5d8-9ef349ea77d6}} vanishes identically away from {{formula:6ced8d3b-c887-4733-895c-4f83974719cb}} , the form {{formula:b8db50e3-b78f-4138-a384-f91521f2c4a0}} extends to global smooth form on {{formula:82135213-b9d8-4786-b72b-dd41e3a959bc}} .
This way, for each {{formula:63f20a62-d8c3-442b-a02d-b8b1e50d02cd}} we have a global angular 1-form
{{formula:d9d9903a-dd55-4e4f-a278-0ec80bfae5cf}}
By Proposition REFREF , for each {{formula:1840770f-1285-434b-9af0-a4818ccfccd1}} we have a smooth function {{formula:8fd4ebf6-a57d-49bd-99d2-5423b8f9fdf5}} . Put
{{formula:1e1fa5d0-e448-4a04-895f-8d4fe5c5cf59}}
Given any {{formula:a548afb1-820f-4edd-814f-d6068146c959}} we define
{{formula:2056e5bf-dce6-43bd-a6c4-feea28628eba}}
For every {{formula:1832878f-8639-4013-af64-8e92747ad6eb}} we fix a smooth function {{formula:1cf77daf-fb0a-43a4-a199-2655a9890f43}} satisfying {{formula:9800655f-bfa9-449a-bba9-3373eaa8739d}} and {{formula:a717e257-30a1-4904-b45a-abe4c4232f4a}} . We define
{{formula:0d7297d6-869d-4c06-a18a-803ee0101f2a}}
and
{{formula:e7313b23-b7e7-4270-96f0-a1dce01ebb06}}
Note that the restrictions of {{formula:516c2354-ed7d-481d-a152-5fb48b612e8c}} and {{formula:1ed68c20-2f6f-4051-b627-bcfbea265d0b}} to {{formula:b152efce-2e0a-4b45-b18e-c8ca28b7310b}} are equal for any {{formula:d96fa7f6-9ec5-4e52-a4bd-1320a9c8c3db}} .
Remark 4.2 If {{formula:f48bba9b-94ed-4161-bce9-04514cf4716a}} for some {{formula:0aa7ad8b-4069-4651-95d9-8062125e4a3f}} , then the forms {{formula:3edad1e0-5c73-458b-89d4-74840b59799b}} and {{formula:5a5912a2-cc1f-4913-9a63-d76024fa3236}} are exact.
Proposition REF below is the main result of this section. Part REF shows that the symplectic monodromy at radius zero is particularly easy to handle. In Section we will see that – assuming {{formula:f062690d-00de-4fb2-bba3-44278ae98840}} is {{formula:c7d98bec-da0c-4829-aeda-2497a0f62d87}} -separating – its dynamics are exactly as for the topological monodromy constructed in {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}}, and nearly as for the symplectic monodromy associated with a model resolution of a germ of isolated singularity constructed in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, see Remark REF . The main difference is the fact that all iterates of our monodromy will have one more component of fixed points, whose analysis is crucial for the proof of Theorem REF .
Proposition 4.3
Let {{formula:da733ced-ab54-433e-9a7f-77f2652b9a3b}} be an open set with compact closure. Put {{formula:c8b82b85-b54c-48ee-9df3-f7e559165c01}} .
There is {{formula:0a169dfe-0732-4966-804f-e0e02f9d3a7e}} and a neighborhood {{formula:d0d3ca18-52c4-4dd0-9c8b-e15f57b1e3f4}} of {{formula:aa25cc86-1fe0-4148-9615-e28bd7a1d038}} in {{formula:18adb7d2-86a1-4dfc-9968-9fa075b8c8ec}} , such that for every {{formula:35d413b7-387d-44ee-a61f-e6d6cef4a91b}} and every {{formula:43b1f62a-59bc-4340-89c6-d2e3c3c2ade0}} , the restriction of each of the forms {{formula:56390cbe-2e2c-44de-99a7-c7c7b71a6966}} and {{formula:9d5fb681-905f-44c4-9a73-721a977e38ea}} defined in (REF ) and (REF ) is symplectic on each fiber of {{formula:f7318a2f-a527-4a00-b9ee-52a46760c769}}
Let {{formula:52db6fd8-8aa7-4475-9835-67ada6a3725c}} be a fine chart such that {{formula:930d5b27-6990-461c-8af7-2d55cf851a24}} . Let {{formula:bae6b776-cdcd-4184-86d9-121f5ada9f02}} be its index set (REF ). The symplectic lift, with respect to {{formula:c9135d2a-13c7-48ce-8179-aefcf3031b33}} or {{formula:31c5f058-9c6c-4478-ab4f-e545c6bc8474}} , of the unit angular vector field on {{formula:38d079c3-aa51-498d-9b20-39fdf04fae94}} to {{formula:6941b8f3-e9b2-43ee-a13a-58d64ba1a056}} in coordinates (REF ) on {{formula:cc575933-d102-4472-b4b0-87a2de217b89}} is equal to
{{formula:7d50ed3d-510a-41d8-909c-172dd154760d}}
where for each {{formula:8029b63e-f639-4a2f-9550-4e94007c5332}} , the function {{formula:d8b714c6-cdc1-4df1-a818-9b4133522510}} is defined in (REF ), see Lemma REFREF , and {{formula:1e1691ea-9185-4a4d-ab8a-b31e56a89ef4}} is the coordinate vector field of the chart (REF ), see (REF ). The function {{formula:45966df0-9fb7-41ba-81f6-55315d67bddd}} is the derivative of the inverse to (REF ). In particular, {{formula:c81ad06b-1e59-4b93-851e-ccd227540f1a}} is smooth and all its derivatives vanish at 0.
The proof of Proposition REF will be carried out in two steps, as follows. Since {{formula:16df1c88-7fe5-462d-88cb-49c8d1f23c1a}} is compact, we can work locally, in a preimage {{formula:b38c43e5-3acb-4e49-9d34-ada550f1aa4e}} of a fine chart {{formula:a0d2d167-724d-4244-a3bf-211d8f0d8def}} . In Proposition REF , we prove that the forms {{formula:d6698b63-05bf-43e3-8bc2-772d322dcdbd}} and {{formula:f90f2ee2-27b8-44ad-8c6f-ecc08f9ae12e}} are fiberwise positive definite on {{formula:43987459-faaf-4768-92d9-d6cdab0a1a6f}} , where {{formula:9956b492-4574-455e-bd35-a9e40f06aa04}} is the standard almost complex structure on {{formula:3d91cbea-e68d-4185-b4dd-fb6b21c344fe}} , pulled back by a diffeomorphism {{formula:f3bddd70-9ca0-45b6-a538-8eea824c1c7e}} . Unfortunately, {{formula:da5cf126-12fd-4d9c-9a56-d8cff194836b}} does not extend to {{formula:74edd1ed-c6d7-4506-a8ed-3381869f115f}} , so this argument does not show non-degeneracy at radius zero. We prove the latter in Proposition REF by directly computing that the symplectic orthogonal in {{formula:cb1d14c8-2df4-4774-8011-c7385b7e3d5b}} to the fibers of {{formula:12871d50-7017-43b3-b69d-9bb06be0f2ee}} is spanned by the vector field (REF ).
subsection2-.5plus-.7.5Positivity away from {{formula:56636e4f-16ba-43b1-a65e-91485a284f57}}
Let {{formula:3c9e2c5b-994d-4179-85ad-11c4d5f87f8b}} be the standard almost complex structure on {{formula:cc5b3636-6937-46d4-a4b1-c6ead9dda634}} . We use the same letter {{formula:4d408d74-cf82-475b-bd69-c5d44a4aa057}} to denote its pullback to {{formula:598aef40-8b92-40f7-ba49-18e1a6754b62}} via the diffeomorphism {{formula:52542198-b2af-4fec-b199-b650f7b95578}} . We say that a 2-form {{formula:a46c19e7-ab3c-4642-9920-a60ca7ce9855}} fiberwise tames {{formula:f4357d29-3b84-4227-906c-2f4e33542233}} if {{formula:23387ebc-b0cd-4e7a-b8ae-b7be90c6a002}} for every nonzero vector {{formula:d9c04e70-3e83-44ad-b308-8fe1732e00c2}} tangent to the fiber of {{formula:870cbf19-c20d-4bc5-8937-13232f588b25}} . In particular, if {{formula:32b6ad76-e3cc-40a5-bae6-69a1602cbc10}} (fiberwise) tames {{formula:3cf9e5df-dfb3-4852-8e31-af58260316a6}} , then it is (fiberwise) nondegenerate. Note that in this definition we do not assume that {{formula:f83d01c8-1069-4344-9ba5-30a376830dc3}} , that is, {{formula:b0abb5b7-1c23-480f-938b-7f62e8e36e9b}} may not be (fiberwise) {{formula:b5cd1648-4342-4d8c-a091-5b3cc95f52c1}} -compatible.
In this section, we prove the following result.
Proposition 4.4
For every point {{formula:481154d3-36a4-4035-9002-ab40ca5e9c9b}} there is an {{formula:d39052aa-82ce-40fe-a8ba-b7ac6c13cf2b}} and a neighborhood {{formula:7a91eab0-88d8-4822-aa3e-f558b0103cc4}} containing {{formula:57cb958e-565a-41aa-8130-8a2cc0b54cc7}} , such that for any {{formula:846220bc-38a2-45a3-bf0f-d70541f4c333}} , {{formula:2ea24135-1540-4e06-8999-193dff77b2d6}} , each of the forms {{formula:adbe3dd8-da19-4925-9fcd-391568143325}} and {{formula:ad46a711-722d-4b39-b212-a97c27af24bc}}
fiberwise tames {{formula:d6d7ba67-9ff2-4d3d-a26a-e1c26bca91bb}} on {{formula:156ca82a-c54a-4ec7-bbea-bd114b60f917}} .
Remark 4.5
For the proof of Proposition REF it is enough to consider the case {{formula:3d9efc71-6ad0-432c-8cee-6dd530d52599}} , for {{formula:135c9342-0bdd-4925-bc01-61da52b0a015}} .
{{formula:1cda9a2b-09b6-4f1b-92f3-8b6f119a97b3}}
.
For {{formula:08a6df24-ead2-4e7d-aa43-18939f17ba2e}} we have {{formula:bb6cbd7f-9cbd-4871-b958-70c71cfe5fb3}} , which tames {{formula:2695919f-e8fd-4d0c-82ba-86610ed78459}} . Assume that {{formula:e1c7c5d9-2c97-4396-a727-7a415fa860b1}} fiberwise tames {{formula:b33233e1-1331-424d-8a72-3815e78463af}} for all {{formula:d22cb602-184e-4060-8472-89304e34b2e9}} . Fix {{formula:46b9e03e-2c79-46e6-9414-5f09d87337f7}} . The restriction of {{formula:0e15b5c3-c7b7-4c26-bf76-b5933e6c9c95}} to the level set {{formula:7aefbf95-7c85-489b-84ab-83992b4097ec}} equals {{formula:0c467517-169d-4161-ad5e-3ba9d201159e}} . Since {{formula:a36bf69d-6bdf-416e-ba2c-50d875cf4a6b}} we have {{formula:e5cae5cd-5531-46db-8d51-e624601214f8}} , so by assumption this restriction fiberwise tames {{formula:07b72a02-cecd-47af-b704-22068714c183}} , as needed.
We will use the following notation. Fix {{formula:14c20a16-47b3-46c1-a8e5-7ba221134624}} , so {{formula:5b314e60-ca6d-43a8-a9af-595f947dbaa0}} for some nonempty {{formula:e530b9d8-ab55-496c-a2c7-cf6d8d8d8bdf}} . We order the components of {{formula:e0ddf354-f746-4be8-a726-643e05ae84e6}} so that {{formula:ec4a7534-943e-46ee-9b20-1c8b74cd8508}} . By Lemma REF , there is a fine chart around {{formula:194b17c1-abc6-4a0c-ab57-5c3bfed97bed}} whose associated index set (REF ) is {{formula:1c7977b0-4ad1-4651-8a5f-d0ffdcd83916}} . We fix such a chart {{formula:44436338-ecfa-47dd-9261-cd9053b3c1e1}} , put {{formula:a1f8089e-3ec4-4bcf-a88d-5b017c6ca523}} , and use functions {{formula:ae2edd8f-6c6f-4fd9-a8a1-fc0a03633c8b}} etc. introduced for {{formula:f664ab2d-e24d-4a4a-a454-00f31f00c744}} in Section , cf. Lemma REFREF .
Recall from Definition REF that a form {{formula:95b883fb-3890-49d6-913e-3a134bfc9d03}} is bounded from {{formula:98130a8b-0303-41ac-ac86-7127cd0cbf34}} if {{formula:1e9957c4-cba1-4f72-a744-d4572bc9914d}} for some bounded {{formula:108828fb-ee54-405f-8c58-70c1974e433e}} . For example, if {{formula:88aa3af1-4083-493d-8d16-ac65074e541b}} then {{formula:0f0aec92-13bc-4d77-88a4-65c9bfdb38a8}} is bounded from {{formula:f5a92b2a-4966-43c5-b9cf-ee5e008e95f8}} , but {{formula:8ce3b7eb-ef9a-412d-a8a3-25607e53c119}} is not.
Lemma 4.6 The following holds:
For every {{formula:dbe7f9de-c108-4d26-aaff-b736efe4f6de}} there is a {{formula:26da343a-4d74-48fe-ba3d-f43547e43db0}} such that we have {{formula:886aaa67-4261-4753-845f-381184094180}} .
For every {{formula:53fd94d6-ef3e-4361-831c-17ecbb3b93c6}} , the 1-forms {{formula:575f941e-0b99-4e99-9af9-f48fe41789fd}} and {{formula:04257e8a-6e8b-4223-9f8e-b96fbd71c606}} are pullbacks of smooth 1-forms on {{formula:c1f4e264-954c-42af-a020-028f5cf49361}} .
For every {{formula:7039b311-08cc-4ee0-bd73-2d542c1061ec}} , the 2-form {{formula:3aae3cac-772d-47e6-9396-202d1d9527f2}} is a pullback of a smooth 2-form on {{formula:7091a0d8-db90-4264-a08e-f009b3d70d2b}} .
In particular, all the above forms are bounded from {{formula:b2de9c2a-e64b-46f5-8d1b-6d66e7acf059}} .
{{formula:08fe4c0e-4a96-460b-bc00-dd67197d04e2}}
.
REF
Fix {{formula:0744f0a2-a3ed-4722-a393-eb19427c5b5e}} such that the {{formula:79e679c4-cb88-400c-a55c-8da5d5666adf}} -th chart {{formula:b3fda235-7461-4480-b4d0-e5a63d230208}} of {{formula:92d0bd97-a615-4093-be9f-ba04a26c9dab}} meets {{formula:6734fb27-c9b9-4e20-9e61-a850ba4874d3}} and {{formula:8f71e0dd-25d7-4d10-ac41-74ee47f70aec}} , and put {{formula:369b5902-8dae-4227-a76f-2e9cfbba4065}} , {{formula:5fce9f99-132e-49f4-8763-85a20a4c7136}} . By Lemma REFREF , we have {{formula:8ea8ab43-3cd7-497b-98fc-f294fb674b6d}} for some smooth {{formula:3d7f99ad-71eb-4667-897e-af5f6af46f4f}} . Hence {{formula:1899e657-10d0-48c2-9783-d88aab7146e9}} , where {{formula:d575046b-e3d2-439b-ba36-46279d67440e}} is a pullback of a smooth 1-form on {{formula:9ab8e6a9-bc61-4058-bab0-4d4d45dc9f8b}} . Since {{formula:ffd61ed0-4da5-4626-9ff5-a1b557d16cea}} , we have {{formula:53f10e66-bfbd-4092-bb09-06009516de40}} . The 1-form {{formula:65779b6d-e107-4575-8be8-6af56d1c2ee2}} is a pullback of a smooth form on {{formula:7e85caae-ab85-4389-bdb0-41b77ce7b9af}} , as needed.
REF Since {{formula:687f1d8c-4f86-4335-ae56-c5d4d77b6b44}} , the functions {{formula:abf45dea-0d7b-45ff-a14e-b646a4f9b046}} and {{formula:a4de3b42-5b90-4d9b-946d-1f3d4d930e09}} are pullbacks of smooth ones from {{formula:f6ff8b4d-b784-4ec8-917c-021dd8ad9e9a}} and {{formula:ed0718db-9fba-4b4b-8820-403a62d004bb}} for {{formula:c7e4bad2-0dc4-4436-aba0-ade08d0681d2}} , {{formula:9c3aebb1-4ff7-4aa4-961e-7e65c600bdd2}} . It follows that {{formula:efa2f350-aef6-4826-8c89-1e23630000f4}} and {{formula:17ddb836-d5c0-4ae4-a898-1d4ea627560c}} are pullbacks of smooth forms on {{formula:00a25548-97ba-4156-88bc-3a886503be22}} , as needed.
REF Follows from REF if {{formula:275a134e-f4c7-4ed2-9e88-d371574fd778}} and from REF if {{formula:6a149322-902c-4725-a3d1-93f421f3571c}} .
We will now introduce smooth coordinates on {{formula:6ffa1e1b-0c0b-4401-93b6-d7b5d0514864}} which are well suited to the standard complex structure {{formula:33e69315-84ab-486d-a2cc-0a915d2d7bcd}} . Recall that our holomorphic coordinates on {{formula:0f4bc793-a69b-4afc-891b-7d33d7b42c25}} are {{formula:93b0dd58-68b5-4d4c-aaa2-a2b4a9b86636}} , for some {{formula:1bd4db85-e976-4068-806d-a26aabc72620}} . For {{formula:e647a193-826f-4770-8740-e2bff2840238}} , we have {{formula:55eb034c-503d-46d5-93db-b6f08289dec8}} , {{formula:063b59e8-ab6b-4da1-a94d-faec3914781e}} , see (REF ), (REF ). Put
{{formula:ff69206f-f80a-4eb8-a796-8a16685e920f}}
For {{formula:30fb357e-bb32-41c8-9468-8873837d43ec}} let {{formula:7fe6d343-5594-4ae6-a77f-14c76d4fbe87}} be the real and imaginary part of {{formula:bdb389f8-128d-48f7-95fc-c6b50ff61f46}} . Now
{{formula:6aacfde6-1adf-418f-af1f-f5670186855d}}
is a smooth chart on {{formula:d788aec2-e63a-4a61-adb3-a87e996d585a}} . With respect to this chart, we can define coordinate vector fields {{formula:465612b4-dcfa-4dcd-a3f1-4211e2575e12}} by the conditions
{{formula:3308fd43-e69e-49b9-a136-a6e3cf5e0b93}}
for all {{formula:92894897-68b2-4778-8cdc-0ae32b63bdde}} and all {{formula:6de62265-1ff4-46f0-b9a9-8a28c853e466}} .
For an integer {{formula:1c6d2273-0c17-47bb-9461-ecd23d1adf62}} we put {{formula:2027c51f-11e5-43f5-b29e-0ac484e322cb}} if {{formula:0cb1970e-1597-44e9-999e-691dce50c755}} and {{formula:8170945c-4701-436b-8182-78b181d1eb1f}} if {{formula:bb5211af-df67-49e0-8340-c21447571f68}} .
Lemma 4.7
The following holds.
For every {{formula:1ec08100-f0e4-4f28-b046-58036f335c67}} , we have {{formula:6592d806-9385-48f1-b563-057cc76bd0a7}} ,
For every {{formula:597a26ac-55d9-4ec7-bc87-d000345ff3e2}} , we have {{formula:a76f80ee-b6d4-4c0d-9055-c5291d949fa8}} and {{formula:36225fc9-f449-449b-bd13-38716171c601}} .
For every {{formula:1015defc-b7d7-4688-9e48-90110e9228bb}} the vector field {{formula:eb397fa5-7836-44f2-ad46-b9bb0a84bf19}} converges to 0 as {{formula:075c0e79-9a73-4349-9b92-b4d0a64d48b9}} converges to 0.
{{formula:37a6e7ff-ffc9-4306-bbc1-338a3e2ac7ae}}
.
REF The function {{formula:b28015cf-ae14-4e88-99d4-85ec56ec0792}} was defined in (REF ) as {{formula:0985f895-f717-4534-a6fb-ddb07a1cc301}} , so {{formula:524ed9d0-a9e3-43de-9c1d-bba8cc0d11bf}} , and therefore {{formula:dd72bed3-dd8b-43fc-9bf0-f520fa399e2c}} , as claimed.
REF We have {{formula:f5c8bd59-f172-4a79-8780-0d719e2de1b1}} . Recall from Example REF that {{formula:09027733-5484-4f1d-a3aa-40de20b2a4f4}} , so {{formula:0186e297-d467-4f03-88cc-091c49defcfb}} by definition. Now {{formula:b7a4da07-a3bf-44f0-8088-e8e339660312}} , as claimed.
REF If {{formula:03212f41-3a6b-46d4-8f84-f8e980093b44}} the vector field {{formula:459a0f97-f74c-4f46-bfd5-d92f343a7479}} coincides with {{formula:0b923a60-f0ff-4582-afee-6d581595199b}} , and if {{formula:34ee23bf-d1f2-4d39-817e-668c37a6c4fc}} we have {{formula:2144ddeb-e97d-45e1-b602-9fee47e7dc66}} .
For a 2-form {{formula:6592dea6-3f04-4bc4-8d2a-71cc6e003a54}} , denote by {{formula:f9e61bee-79b5-468f-b311-b4f58a2b162c}} the symmetric part of {{formula:973e24f1-2a35-43d5-8343-994bf72b84a1}} . In the next lemma, we compute {{formula:089848f9-0411-4b7d-a884-1821921fb970}} . We will use the functions {{formula:f3d1d2f4-9439-48a9-9f51-e9220bc4691d}} defined in (REF ) by
{{formula:5a6db9a8-37c7-4313-9b45-8c4bdc582e6f}}
Note that {{formula:03dfedbc-2674-47b7-93a7-20fc00ac44c2}} as {{formula:79e75878-71e4-46d5-b3c5-f2ba733393dd}} . In particular, {{formula:81dadeb1-169f-4de9-abae-3d4c5b59f393}} is bounded.
Lemma 4.8
For {{formula:2a6ae82c-d89f-4e6e-93fc-625883b75dd4}} there are bounded functions {{formula:92adcb6d-07dd-456b-ac83-f75fde0f2332}} , and forms {{formula:f34339af-47da-4df4-ac08-3a47301cc4e2}} , {{formula:3ca72e6a-a4c1-43fb-9707-dfe09af4555e}} bounded from {{formula:b034441a-4fe6-4e9f-a74b-7722881e4be5}} , such that on each fiber of {{formula:31bcba85-06e3-4321-a66f-173459c23f24}} , we have
{{formula:5a981bce-fb74-4523-982b-553d3c30e182}}
Moreover, we can shrink the neighborhood {{formula:666a4e4a-c184-4dc9-9fcc-efea83f53ef1}} of {{formula:0967799c-a20c-40b0-a51e-0a377e7902be}} so that {{formula:0514ac3a-001b-42fe-af38-cb554cf73eb8}} for all {{formula:f3fc4ca4-cd57-4f2d-899b-94df7f3a4122}} .
{{formula:3f3207cc-982d-42de-a6e6-5030317b00ee}}
.
Fix {{formula:be6cb141-fb91-4a04-a371-462cc0a0f927}} . Since we work with forms restricted to fibers of {{formula:629821c2-a115-4083-9486-181832a41d2d}} , we will tacitly use the equality {{formula:e2460542-2e19-4ab4-8d02-5c35dd6f884d}} . This way, Lemma REFREF reads as {{formula:c5d4e5fd-f726-43d7-abf9-4e45bb360809}} . Since {{formula:b48851a6-f21a-49a7-a948-b00b7a92a1c7}} by definition (REF ) of {{formula:c1f48f9b-6828-44c8-9743-2feafe3e4829}} , we get {{formula:544b5f47-030e-4fd8-8eca-7260500744b4}} . Hence by Lemma REF REF
{{formula:da878c84-33b8-4aff-ba0d-4080424478e6}}
Substituting {{formula:8a46db90-6809-46bb-95ce-11db0ebbd6e7}} to Lemma REFREF , we infer that there is a bounded function {{formula:31d28249-3dbb-4557-891f-8276226c76fd}} , and a 1-form {{formula:3763070d-01d7-4489-b1ce-1e556bde0c77}} bounded from {{formula:95bb0992-81f7-43e6-9da9-a322c167af3e}} , such that {{formula:e5169461-13c3-45b0-82a7-e270d7e8c1c4}} . By Lemma REFREF , we get {{formula:3232d90b-38a2-4290-a75d-17b73bb485c4}} . Substituting the above formula for {{formula:43244829-247a-461d-972e-5902d57774be}} , we conclude that
{{formula:a4a074bf-a43f-4b37-bfe8-233226aabcf6}}
By definition (REF ) of the form {{formula:81807cd4-5e23-47ce-8001-8ba3c3fc5e6c}} , we have
{{formula:44ac507d-961c-45a7-9a41-3624533dc263}}
for some 1-forms {{formula:aa899a7a-c63a-4276-bd5a-47b84f9a1d07}} and a 2-form {{formula:9ab3dd5e-5e34-4767-8796-e03b41dc4beb}} , all bounded from {{formula:a0be3441-a2d2-4df5-b2db-27938456a344}} by Lemma REF . Using the formula (REF ) for {{formula:c7af540a-a3ba-45ce-803a-4edaff36549d}} and the notation {{formula:e9f60a95-524c-42f4-98ae-46f98ed4d33f}} , we get
{{formula:8560b82a-319c-4b9f-adc4-ccac66baadcc}}
where the 2-form {{formula:7ed8012f-a30f-4f65-9606-f452ab39bc5d}} is bounded from {{formula:ac1be716-6270-46a4-a4f9-5e47e1769d57}} . By Lemma REFREF , we have
{{formula:bdd8b92d-dfb3-46cd-886a-b0c205a1f5f1}}
Since the function {{formula:8d2fae3a-205e-4987-8c92-e7ca037fb87c}} is bounded, and the forms {{formula:75830825-5451-4538-8f51-2b128ba96719}} , {{formula:5d4e4e30-c248-4fb4-b45e-32a1b38e500d}} are bounded from {{formula:1f73fc16-6699-4b82-9c93-21976ef1029a}} , the forms {{formula:d50bcbfd-9ad4-471b-b9bc-c124f3ff011d}} and {{formula:f033fdbe-0507-4913-ae57-4848ab5124bd}} are bounded from {{formula:7e278a5b-a480-45c0-9f6d-dd39366543f6}} , too.
Substituting these definitions to (REF ), we get
{{formula:d574f5f2-e713-4e0c-9b68-60560360c769}}
Put {{formula:73222f17-d68f-4a2d-9a55-ea289b7d0d1d}} . As we approach {{formula:9e07eeac-f6e6-45fc-9f91-c7487e810486}} , we have {{formula:59e59976-4bbe-4876-8795-74d55c2d40af}} . Since {{formula:7282a624-57c4-4296-b839-35b5323c78ae}} and {{formula:d97bd11f-ea53-4ab3-8502-16b6278c9046}} is bounded, for {{formula:da26d15c-e224-4a4c-9034-bfde5a7677eb}} sufficiently small we have {{formula:68e367e6-9ec6-4914-a341-0f13d7e63e19}} , as needed.
From now on, we assume that our fine chart {{formula:00519b18-093d-48b2-ba31-554a7550c735}} is so small that the functions {{formula:b4edc17c-7bf6-4c11-91c8-6efc7e0cd5ea}} from Lemma REF satisfy {{formula:831c3bb6-24a5-4428-82de-648711cbde6e}} . For the next result, we introduce the following notation.
Let {{formula:87c510a0-1c81-4c7a-9604-e802608e6b5f}} be the algebra of bounded functions. For {{formula:b5a27282-e32f-4254-bc7d-e688eb35c72d}} , we put
{{formula:01b3391a-467c-4b47-80a5-7bda2327031a}}
Since by Lemma REFREF we have {{formula:ed39d2db-013a-4464-82d8-e2ce5dd71517}} , the algebra {{formula:467c594c-b567-45af-84e4-cba2e61c81e3}} is an ideal of {{formula:e8a5670f-8418-4873-a92a-ab1274ce7ffc}} . One should think of {{formula:50a60236-eab7-4c4c-9002-9bcfda2af5a3}} as the ideal of functions which decay exponentially with respect to {{formula:cd3bf344-021e-4c9f-9d9f-17294f366882}} . In Section REF , a similar role was played by the ideal {{formula:00b74db0-e489-4b04-9ad9-5a1574b72abf}} .
In the following lemma, we use the coordinate vector fields {{formula:78ee0580-9db9-45af-8813-d64b363c23c3}} , {{formula:d16cd2b8-7511-4a1c-b11b-de9dfeef3972}} introduced in (REF ).
Lemma 4.9 The ideals {{formula:94ab7c2e-297a-4c8b-b12c-4b028d2bb525}} satisfy the following properties.
For every {{formula:2109783e-324c-41fb-b1af-8b30d4e669c5}} we have {{formula:02951a9a-f188-4e64-8dca-0ec412df3aee}} and {{formula:34b9c0a4-6df5-459d-b7fe-3a8a41194b3d}} .
Let {{formula:7ab27435-8467-4073-aa8b-5478fd438b8f}} be a 1-form bounded from {{formula:74a9a03b-fb2e-42b4-a20d-b6bdc7148113}} . Then for every {{formula:2dd19beb-80cd-4862-93b6-c8d8bd4d546b}} and every {{formula:8114da40-99b6-495c-98b9-bcade5296653}} we have {{formula:60430f41-6aca-4e4b-b4d0-e9f687aa4feb}} and {{formula:e32c3d60-e656-4c05-91ce-e1d642d26a3a}} .
Let {{formula:df28803c-e2fe-4c04-9938-820937b384ce}} be a 2-form bounded from {{formula:afb68f91-3956-4335-9d02-1b7c9f37116e}} . Then for every {{formula:9e2b9765-d86a-407e-af5e-317004922d57}} and every {{formula:2d39b373-dc37-45e4-9c04-25e705fd51ee}} we have {{formula:27c7e257-2c3f-4dbf-9b12-ab0a82bdb70c}} and {{formula:f2fe37f1-cfc9-47f9-9e86-39b73b04740c}} .
{{formula:74e4ad73-fada-40d9-8402-ac2fb6f61ead}}
.
REF The first equality follows directly from the definition of {{formula:d2818e07-6492-44ed-a4a7-8b3516abb764}} . For the second one, recall that {{formula:d95e3da7-654f-4c40-8f61-d0e9edcbf83c}} was defined in (REF ) by {{formula:aea1549f-4aed-48a3-b4ed-deb4bccb11e7}} , so {{formula:eb97d986-0590-49f8-89aa-4c1d5143c443}} . Since {{formula:bb2c4952-1604-4579-aa2f-55132cdce579}} is an ideal of {{formula:a8ad127b-7042-4257-997c-213771773269}} , we have {{formula:11b4e0c3-d209-4208-99e6-a9f1acffec30}} .
To see the other inclusion, note that {{formula:bba9d7d7-9c6d-4b56-8a47-74bc5b28b4eb}} . Since {{formula:184b3130-c6b6-4a92-8c64-d439337ccb83}} is an ideal of {{formula:11930309-f4ea-44d6-bbb9-837d1614d814}} , we have {{formula:d061e947-193a-49e7-809f-d71f289c636f}} . The first equality of REF shows that {{formula:7f71fe43-6d16-4351-b99e-37fcab0943fe}} , so {{formula:bdffc525-9d3f-4f81-802a-01c5219011a0}} , as needed.
REF By Definition REF of a form bounded from {{formula:a2ee2670-be5d-4df3-804a-c24673b8da4f}} , we have {{formula:178b5a2c-2cc0-4cd8-961a-eec79cdaed0a}} for some bounded {{formula:a5208a91-6a1b-4bcc-a391-87d1fb984e32}} . Recall that our holomorphic chart on {{formula:ff8d58d3-a474-44cb-baed-121306edc381}} is {{formula:cd09889d-83d1-4ecc-b696-bb61b6f730c6}} . Write {{formula:322fe2f6-2234-45b8-917e-af972179c646}} , {{formula:0f2808ec-414f-44ac-bf32-49e1de230b7a}} for the real and imaginary parts of {{formula:7d3282ad-cb4a-44bc-aae9-976ad1e6b310}} , {{formula:d06f7f43-9c6a-430d-8e10-cbf131842083}} . Then the 1-forms {{formula:95e4d237-ab3c-4f7f-b2ac-5aa1349fe12e}} , {{formula:2b6d94da-b814-44cb-aac7-383df782d370}} for {{formula:2c2bd2e1-afc9-4545-8ab7-325fe1c732a7}} and {{formula:916c1c77-af39-4903-98b8-67ba61867dec}} for {{formula:2c033645-2a77-4a33-8a94-9c5931fe35ad}} give a basis of {{formula:d8278dc5-ce6b-4fd2-82a7-64f1107a179a}} . Away from the zero locus of {{formula:b84823c3-664b-4e23-b6a2-4a56f0c29b63}} , we can change this basis by replacing {{formula:3deecb02-c9a9-4da5-b1df-28a4364a7243}} with polar coordinates {{formula:cccd499a-c1a5-4454-8bfd-2682e06739e4}} , {{formula:9882fefb-3a85-4695-8b7f-5acba850c0c5}} . Hence there are bounded functions {{formula:b706cee0-4fa0-412c-bc65-ed9ec115d271}} such that
{{formula:e9a7068b-1195-4b02-b75c-a26b2125b289}}
We have {{formula:48c16519-a100-4527-b8ad-da0cc78b55a7}} , and {{formula:43861ac9-b534-485d-bf25-0acda83aadde}} by definition. Since we denote the functions on {{formula:dc60d648-9176-4652-8622-1eecae65a1c9}} and their pullbacks to {{formula:185bd5d2-44c8-4383-84a3-6ff00963f408}} by the same letters, we get
{{formula:601414df-42c8-4de1-97cf-3b813e12d361}}
Now by definition (REF ) of the coordinate vector fields {{formula:53526743-5f46-4751-898f-701330361a3d}} we get {{formula:6ae2850c-2676-409f-96eb-75fdecc2f2f7}} and {{formula:7feba00c-35ba-4873-9632-b9f26dce5a13}} , as claimed.
REF As before, by definition of a 2-form bounded from {{formula:dfe11b77-cf11-4317-881a-efbb742d2440}} we have {{formula:f76e9599-da68-4e26-a2aa-5267d66fcc8f}} for some bounded {{formula:72f736c9-8b64-4d68-ba6e-b6e60b849027}} . Using some basis of {{formula:6bbbddc5-28a1-4ff7-a675-ed3c2c938f7d}} , we can write {{formula:418ef6e5-f508-4f1e-9461-24d8e64e47b7}} for some {{formula:6022286c-0b13-482c-9275-f33b2881c211}} . Thus {{formula:13d84682-bb16-4c42-89c8-37922c6da0d1}} for some {{formula:252983d0-97f9-4c45-8564-3b9d4368d0e9}} bounded from {{formula:7a6afe39-e4fd-427a-979f-6cda977bc4a9}} . Now REF follows from REF .
In a fixed basis of {{formula:44fc4911-d3c1-49cb-9b95-a46b9f56c037}} , we can write the matrix of the symmetric 2-form {{formula:53d04672-7056-46e6-a372-97e2c68b1d4a}} as
{{formula:9b7a8959-ac36-48ad-afc5-a4584da5f7d0}}
for some {{formula:d309de8e-10dd-4d2b-8270-e832578fff6f}} matrix {{formula:bce7640c-862c-4419-98b5-8aab602da0a6}} ; some {{formula:2e35a4a6-da5a-4be6-91e1-4435f162bbd2}} matrix {{formula:4063c373-f4bb-4c71-95df-d1ee38642f47}} , and some {{formula:3d93fef4-50b8-4d5b-94a8-09327c1dda24}} matrix {{formula:e019c435-6b7b-400b-8930-7a052773a7b1}} . The entries of {{formula:2a63652e-295b-4f51-95cc-eebf420042d0}} are smooth functions on {{formula:99ccaac4-55a2-4b02-bedd-feef9d65f26d}} .
The next lemma summarizes some properties of these entries for our basis {{formula:8676a2b3-1c61-4966-9231-68e25ee7e4ff}} given by coordinate vector fields (REF ). In Lemma REF we will show that these properties are preserved by Gaussian diagonalization of (REF ).
Lemma 4.10 The matrix (REF ) of {{formula:59ad658b-81e1-442b-be56-78ba33744900}} in basis {{formula:d04d8397-221e-4594-aad3-71810ac077c1}} has the following properties:
For every {{formula:7e0765c3-e72c-415a-9e1f-445e2648ba46}} , there is a neighborhood {{formula:8dbec5c3-d000-47bb-8979-f041ff9f0566}} of {{formula:8d328014-9a30-410f-88bb-951a2d5eeafe}} and a function {{formula:1c2be47e-0146-4d0d-8dea-8efbfd814878}} such that {{formula:c6cef5d7-aca6-4fe8-a042-936609ed78d2}} and {{formula:45382f19-08f2-4095-a137-52838e915d9d}} on {{formula:58c76e3a-0fe1-48f6-9bf4-881045141174}} .
For every {{formula:ac90ccf7-0ed5-4ebb-ba71-9f80d2e6c24d}} such that {{formula:5c8985aa-5ebe-4ede-ac09-c1c9cc12d95f}} we have {{formula:fdd3df8a-ab9e-41b2-a0db-4cb6f228315e}} .
For every {{formula:7340aa06-a9a3-4d41-a1e5-5d6df7196eb7}} and every {{formula:9b68ad9f-5769-4885-ab65-879cdadea51f}} we have {{formula:7d588a0b-1db8-42b7-b58e-796b8678d122}} .
For every {{formula:ebe26986-0477-4ac0-8951-c6c47b53c771}} the vector field {{formula:ee3e808d-685c-45cf-b1ab-28d4c4b9d06a}} converges to 0 as {{formula:db9c962c-4fa3-41ce-8876-ed72086f4e78}} converges to 0.
{{formula:78ebe9ff-ce21-4a2c-a1f2-ed3c96044ca6}}
.
Lemma REF gives
{{formula:c31eb15f-806d-408c-9ca3-f0056695d8b8}}
where the functions {{formula:70d47f11-8909-4fa4-9c25-4379a5468f10}} are bounded, {{formula:1d42253f-1976-4391-ac55-2f81a73c26a1}} , and the forms {{formula:3a6e2d56-20aa-49bb-a3b7-41647dd834ad}} are bounded from {{formula:d1e1ad93-0624-463e-9a36-bc9f7c3754b3}} .
REF By definition of the matrix (REF ), we have {{formula:6ff241b2-979f-415a-8124-33955d6ef549}} . By definition (REF ) of {{formula:9b0d69ef-2615-4411-9b1f-52fc90258681}} we have {{formula:b28f9166-c148-4a6c-9168-bbb6d4f50a70}} , for all {{formula:834d263b-87e4-49bd-bbb1-76e63e562e97}} . Substituting this to the formula (REF ) for {{formula:7b4182ac-4f42-4d7e-9621-5a4d8948ae07}} , we get
{{formula:d53cebb5-978f-44e9-9b6f-9044d3c217a7}}
Since the 1-form {{formula:5fb1d424-cec5-4e12-8202-159bc112f3da}} is bounded from {{formula:4ee419f8-3e02-4b40-9c01-ba92007b422f}} , by Lemma REFREF we have {{formula:b8630798-f8fb-4d01-b6bb-4958003237fc}} , so {{formula:13251bb2-0fd0-4717-ab21-e28145369598}} , where the last equality holds by Lemma REFREF . Similarly, since {{formula:74d3f167-c6aa-4e0a-a960-78f95f5d3221}} is a 2-form bounded from {{formula:b02ff2b0-edc4-4f7e-acfb-a53e8dbc9f58}} , by Lemma REFREF , we have {{formula:198882e7-99d4-436d-b20d-92324495dc08}} . Hence
{{formula:928ae3b4-5d8e-45b8-a4a3-61c1c955e6af}}
for some {{formula:f2746616-3c3c-4df5-b956-062f93de419a}} . By Lemma REFREF we have {{formula:6b5acd34-9b1c-454c-b5d0-6668119b9952}} , so we can write {{formula:154bf2c6-4450-4574-8f7a-03782f9952a7}} for some {{formula:cc5142a7-3224-4bf9-9375-e24c56af816e}} . This way, {{formula:96125bcf-f87d-43a1-a6b8-70b8031f57cd}} , with bounded {{formula:ba5ca12c-ac2e-4341-b38e-d7be38dc4359}} . Since we have chosen {{formula:ad9743a3-7f4c-4bb0-9d9c-4b18614e60e4}} so that the function {{formula:63e38bcd-9c17-4d22-8b11-a24e010dd267}} from Lemma REF satisfies {{formula:efd9e15a-d3ec-4c5e-8576-a175209b32d7}} , we have {{formula:7b63248b-8c68-4e58-823e-a6fd0a90babd}} whenever {{formula:0adbe569-af00-4297-942d-29c2aa56ee04}} is small enough. In particular, {{formula:a0027533-6fa0-4a21-a3df-5659efe32cc6}} sufficiently close to the point {{formula:6156034f-ece2-4d55-a2b2-c42a30ec5c61}} , as needed.
REF By definition of the matrix (REF ), we have {{formula:1b1ada5f-b401-4b29-a380-d9aff59b56c8}} . As before, using definition (REF ) of {{formula:af80730c-06de-420d-a07b-2503d0db9912}} and the formula (REF ) for {{formula:19a93050-0122-4dbb-858b-84158b718fe8}} , we get that for {{formula:67fdbcb9-f98d-47fd-9d95-b91c1d586394}} :
{{formula:dcfceab1-b049-4ae9-81b2-ccb74106a411}}
We need to show that each of the above summands is in {{formula:72b9a749-8725-4866-9fc7-24bf2b649ba9}} . Since {{formula:b16a586d-0a1e-4d8b-973f-866c9c700305}} is bounded from {{formula:a2131421-1a25-400f-a9a3-906d62efb6ea}} , by Lemma REFREF we have {{formula:3f7f6922-5207-4a30-934a-f3dc0398fa80}} . By Lemma REFREF , we have {{formula:c0bf9bac-34d3-4c5f-868b-e79137bc5431}} , so {{formula:1d60960d-0021-42ab-99af-099752922d70}} , as needed. For the second summand, we apply the same argument interchanging the roles of {{formula:36b318d7-01d5-47b2-9001-1a6a0bf479b9}} and {{formula:bdf3a963-aa1e-49a0-a7ef-e2c0ab83aca7}} . Eventually, since the 2-form {{formula:5afcac80-72ab-48a4-b299-a63a80c03cff}} is bounded from {{formula:4b59bda2-41af-4828-8b5a-7ef5cbb41505}} , by Lemma REFREF we have {{formula:bdc01555-83ab-42fd-ae87-b9ea3e8cf0b7}} , where the equality follows from Lemma REFREF .
REF The entry {{formula:709d9d57-6f5b-4959-b7eb-c66f2a13baad}} is defined as {{formula:97460408-70a7-4cc5-becb-fdea8590159d}} . Now, substituting (REF ) to (REF ) gives
{{formula:7f64e517-de65-4b13-b0a1-5c6c1a47f9fb}}
By Lemma REFREF , we have {{formula:9b4f147a-c65f-4883-9dcc-c3de8ff9ac38}} , so {{formula:d04b4c71-599c-454e-976c-5a154fa3935b}} , as needed. Lemma REFREF gives {{formula:0d6bd28d-23f1-4db8-8b11-78336fc55bcf}} . By Lemma REFREF , we have {{formula:8490faa0-ffe7-4110-9147-10fc144d8929}} , so {{formula:bce9febb-86e9-4959-9a2a-e7d31d364665}} , as claimed.
REF It is precisely Lemma REFREF .
We now apply the Gaussian diagonalization to the first {{formula:2f703d2d-588c-491c-a631-8002c0b0543d}} rows and columns of (REF ). For {{formula:ea401990-7f31-4f07-9580-1f97b5b813aa}} , denote by {{formula:8f99c6eb-b2cf-4701-bfab-a882b296a3b0}} the basis of {{formula:86d16206-2f9b-481d-a118-2dec3c7d981a}} obtained from {{formula:eb391687-8765-4a5d-9de3-77e2058c88d3}} after the {{formula:2f6263d0-8342-45ff-8ce1-83b3d212f85c}} -th diagonalization step.
Lemma 4.11
For every {{formula:24c23a58-f26d-49ba-b661-1b5ce7ed51be}} we can shrink the fine chart {{formula:e2249217-498a-4583-9f15-74a23418062d}} around {{formula:bcd0202c-7ae1-40bc-8ff7-b82063178122}} so that the matrix (REF ) of {{formula:cc4b11d2-d332-4c43-a98b-6a547da5a6aa}} in basis {{formula:4a1134be-a746-41c8-9607-8a71363ae22e}} keeps the properties listed in Lemma REF .
{{formula:a97a7fe7-18bd-46a0-8048-2e4ed020697e}}
.
We argue by induction on {{formula:5a994801-4f4b-4ce9-a4a7-ba42d10929ec}} . Assume that we have eliminated the off-diagonal entries from rows and columns up to {{formula:347dbc49-6cc1-46b5-a2c6-b2e2e3ef229a}} ; and that the resulting matrix still satisfies Lemma REF . While eliminating the off-diagonal entries from the {{formula:1f82277c-092b-4e1e-9b8c-9cfd651580c1}} -th column, we perform the following operations on the columns {{formula:1b04f0b1-6912-44dd-8cb8-f43a7e7e1c28}} :
{{formula:6e353ead-a6ca-44e8-b785-96527be1242e}}
Put {{formula:3b3e226a-d7cb-4e4a-870f-4fbd56bccf4b}} . We claim that
{{formula:8a9aa688-c398-4eb1-8ebf-a9040a472f43}}
Once shown, (REF ) will imply that the properties of {{formula:0faf13ec-4a84-4a88-9532-8e82b3f08576}} and {{formula:ae2b087a-aedd-4d43-82c5-b59868986955}} listed in Lemma REFREF ,REF are preserved.
By REFREF , we can shrink {{formula:6f77accc-bd03-4e67-bd23-adc767c0678b}} so that {{formula:d76b4d05-014d-4dcb-9ea7-71d7c8a1afa7}} for a function {{formula:1d0dd2b3-605a-424c-beef-d49994ef12c1}} bounded from below by a positive constant. This way, {{formula:9002b9b2-b7c8-4464-b60e-c79e421eca33}} , so {{formula:1dd8698f-53b5-4bcd-8bc4-24d75da34e52}} . Since we assume {{formula:bbdee59d-83cf-4577-838f-1d3cdaa46306}} , by REFREF we have {{formula:9fdfbfd2-c62b-49b3-837a-c3ca7ebb6222}} , so {{formula:5c765168-5738-4334-a7c1-7b570ade7362}} .
Recall {{formula:a736d2a3-1489-4731-96ae-6bb5089297c4}} was defined in (REF ) by {{formula:a8b204fe-e52f-4599-a961-ebf370250430}} , so {{formula:c4517992-a9a4-4581-9eb0-b5cf9918b396}} . Now since {{formula:adb4d3e1-2267-42d8-ae22-e1662184de6e}} , by REFREF we have {{formula:303a93d2-0c51-470d-acd6-4beedbc4e3e9}} , so {{formula:91848938-0fb0-42c0-8e83-281d8e2c08f7}} , as needed. Similarly, by REFREF we have {{formula:85a9ce0e-e4aa-4a45-af38-f2be596a41ab}} , so {{formula:173abc09-8bd3-4a6a-9035-f97fe84f4c41}} , which proves (REF ).
Thus we have shown that the off-diagonal entries in the first {{formula:0b803a8a-427d-4dfa-9fb3-3f608e7bfc5e}} columns keep their properties. For the diagonal ones, we need to show that REFREF is preserved. To do this, assume {{formula:ae33e217-fe6a-4f51-a24e-4bb32aeb346b}} . By (REF ), we have {{formula:f3b2128e-f276-4201-bfa8-a081e34070b3}} . Recall that by definition (REF ) we have {{formula:3f4a3c0b-4389-4909-9a75-046f39e7ce7f}} , so {{formula:e8a0fa1e-1da0-4677-aff9-bfb8415e8704}} , i.e. {{formula:33a38cbe-fee2-403b-ae1a-486988b62ace}} for some {{formula:8ec4674a-619e-4d3e-8ba0-fa0779b8cee1}} . Thus the new {{formula:a3c7ddda-fda7-48b3-865e-9deada74f621}} -th entry of {{formula:c0252d82-3a9b-4b1d-8919-22225f9735e2}} is
{{formula:5dbfb24c-5217-47b9-9ab9-844b8b9ceb82}}
As we approach {{formula:56b337e7-c4c5-4e9e-bf41-bdbd813528ea}} , we have {{formula:fc789432-b634-4265-ae0d-d6b84237a8ed}} , so {{formula:da3d4cc5-2a8b-4cb6-a1b4-eed4d41d46dd}} because {{formula:09ccd7d5-7171-4412-9627-210793d4e8da}} . Thus if {{formula:bcd17b99-6b93-4e07-9260-7e3c2f6cf1ec}} then {{formula:dae2b6a5-8cfc-4b59-8e76-874b9bf46914}} , too, for {{formula:4e094eeb-dd7c-4ef4-84ea-89c1ae58e04f}} small enough. Therefore, shrinking {{formula:c671610e-6b66-466d-a85f-8a8c999310d5}} we can ensure that the diagonal entries of {{formula:4596032e-5ee8-48bd-8450-869bb64b53d0}} keep their property REFREF .
Property REF on the size of {{formula:175a232b-6c20-4967-be77-a93b8063a7aa}} follows from the Gaussian elimination formula {{formula:0b98fdb4-7cd8-4ce4-a173-ebea7cdd1586}} for {{formula:eb85e0eb-7172-40c0-a466-c1499db4cf5e}} , and the facts that {{formula:cbecc7b0-b7c2-42de-8650-d73103895352}} by REFREF and {{formula:98dd0bd2-104f-419a-8a76-0595fa062dc6}} by REFREF .
{{formula:0812569a-e4e2-42b2-a879-ae1add4d5dcf}}
Proof of Proposition REF .
Consider the block form (REF ) of the matrix of {{formula:75c39d68-5c03-41e0-bc25-430ff70edb29}} after the first {{formula:a2735808-f501-4a68-ba3f-aadb36376a94}} steps of the Gaussian diagonalization, that is, in the basis {{formula:9c18320c-d61b-45cb-9dd3-9b2e4cd9ff82}} . Then its off-diagonal block {{formula:fa7e0433-69b7-4d34-b11e-f0d5ecc82aff}} is zero, and the top-left block {{formula:ef902054-eba5-469a-bb12-b0bd993f1904}} is diagonal. By Lemma REF , we can shrink {{formula:7179867c-0bd8-4b0d-a0f0-067d2eb1b873}} so that the properties REFREF –REF are preserved. In particular, by REFREF the diagonal terms of {{formula:fe89ec04-47df-41aa-9175-fc3e01b954ab}} are bounded from below by {{formula:58afa29e-10ca-4822-a700-87d49cb70e7b}} , and by REFREF the corresponding basis vector fields {{formula:c2314d03-f0ed-4529-994d-8975fccfb514}} approach zero as we approach {{formula:b325f436-f50e-4737-9388-e0f559b01fe8}} .
Fix {{formula:33ca50be-5890-41a5-979c-c3d21722b16f}} . Any tangent vector {{formula:66bceb44-139f-460f-88a6-5796f6d8ab29}} splits as {{formula:e5bfad82-10e3-4978-9d15-f0f80d45e321}} , where {{formula:7b45b471-ed7a-42a5-a09d-a4a4b78fd9bf}} and {{formula:64dc3f87-e088-4ad2-8abd-3a7bb7667648}} are linear combinations of {{formula:93bd6fbd-8129-4495-860c-5c1d4dd23224}} and {{formula:0ae4e4c4-99e3-400e-ac3d-ee0fd7554e19}} , respectively. Since the block form (REF ) of {{formula:586a3b97-7c49-4b40-8a54-af7609b3498a}} has the off-diagonal block {{formula:bcff4e4f-a385-4257-a9ba-f7db8c7eac9c}} equal to zero, we have the equality
{{formula:186c3b7c-df1c-4a14-9489-1767c07f4477}}
In the tangent space {{formula:a5677012-3ac8-4563-9f58-ba51ae818001}} , we consider the maximum norm with respect to the coefficients of the basis {{formula:1593960f-39d2-4694-bda1-40705b52ea97}} . Property REFREF , which is kept by the diagonalization, implies that
{{formula:f60d7b46-2614-4047-9e59-b4f07d717964}}
Fix a constant {{formula:7a5eee5e-d165-4ee8-a5ea-7fa2fa997bae}} such that
{{formula:2c96de30-805f-4e7b-b1aa-865261c5bf62}}
Now, we fix {{formula:c5ef086b-8057-41e3-a52d-3573603f854c}} . Recall that the form {{formula:c2b682c9-5e6e-42e8-a80a-a6671b4b8530}} was defined in (REF ) as
{{formula:ce535c5f-bbec-494a-ba42-e61b2095fe2b}}
We need to show that there is {{formula:77d2459a-3220-4668-adcc-23c16a53181c}} and a neighborhood {{formula:0a8d10aa-778d-4046-a533-29c293971b3c}} of {{formula:7ceedbc7-c5ea-44de-9d6d-268cff066dbf}} such that whenever {{formula:1bcb489c-5bf9-4904-9231-9209f637f41c}} and {{formula:38cd4ffd-1cc5-4bd1-ae82-6283ab76d94d}} , we have the inequality {{formula:84537162-3be6-4949-bbc9-274b1c1bde3d}} for every nonzero {{formula:6639029a-f100-4574-82ae-f181a8747843}} .
We split {{formula:fe9f88a4-6d3c-4111-a8c6-df70370cd072}} as above. Consider the case {{formula:127a0778-253d-4830-b1e4-3932c80fe217}} . Since the form {{formula:9bd7052b-40f5-437a-8117-ad73d2b8c55a}} is positive definite, we have {{formula:912b3597-d1df-4fea-92f2-34b72f6f4a4c}} , so {{formula:63539b4f-4833-4766-87f6-b18ea7ee0bd1}} . Substituting the inequalities (REF ) and (REF ) to (REF ), we get {{formula:ab38b5fd-96f4-43ef-aa8d-8a29013cb257}} by assumption. Thus {{formula:f3b7976a-b13b-4aa6-b0e5-ee2bdb2590eb}} , as needed.
Hence we can assume
{{formula:ba7ffcd1-b6b4-44d2-8e3c-653fc61a1a0e}}
Using (REF ), we expand the equality {{formula:a7d88016-cddd-47b5-9638-e1d681a951b8}} as
{{formula:e4e546a4-f80b-4803-9aba-0690bd77bfa2}}
The inequality (REF ) implies that {{formula:1fbab685-d3c7-4292-a8d0-59507e48a23e}} , so since {{formula:3941b350-219e-4ea5-8034-61a9d1bfa229}} is positive definite, for {{formula:0bf8287f-b38b-4fd0-bc95-66ff0f726a6e}} we have
{{formula:c6ddf292-6d76-4a1f-a71d-4be603ddc9f4}}
We group the remaining summands of the expansion (REF ) as follows:
{{formula:b3a121f8-b8f3-4b26-a5a4-9d92eb73f458}}
Let {{formula:1f59a561-149e-4873-bdb3-7753899b8d1b}} be any norm in the tangent bundle {{formula:69494d1a-7c25-4c3c-b04d-149212db6520}} . Since the restriction of {{formula:9a1e7445-38ec-4a4c-b9e0-977576060cd0}} to the sub-bundle spanned by {{formula:3b639e45-484e-4393-b252-32d68e8fb52d}} is an isomorphism, there is a constant {{formula:85b196de-0910-49a6-a9ab-65d452b5f40f}} such that {{formula:4789967c-0dc9-4919-a570-2275f000d58d}} . Hence by positive definiteness of {{formula:99807ca0-1f09-411d-9497-1bf863358aa1}} there exists {{formula:d48a7937-d44a-406a-9663-5e1fd3fc5fdf}} such that {{formula:657d96c8-0e1f-4252-8313-b2389625b0ab}} . There is also a {{formula:f06ba51f-346f-4dc5-b9a5-484f5a7daa79}} such that {{formula:3fc916b2-8c09-4c62-b25b-fd65c86705ab}} .
On the other hand, property REFREF , which is kept for the {{formula:170a6fa8-c54c-4ef4-b009-c15d8615a973}} 's, implies that there is a continuous function {{formula:949114b5-3006-44ef-8835-b58a2a80a5f9}} such that {{formula:7a46e206-984e-462a-be76-d23588706eb6}} and {{formula:971c26cb-9c32-423f-93bb-b63f8bcff1b7}} . We have
{{formula:9121741f-bd59-4a5e-89d6-d643691a3049}}
Since {{formula:2ecc2319-5558-48e3-8a4f-6d7a77bd85f5}} vanish at {{formula:6b2f3ccc-7af4-4c20-b2df-8dda89128b62}} , we can shrink the neighborhood {{formula:3252b768-dc7d-4a6b-b54b-69c717f5439b}} of {{formula:e51e1741-4daa-41f0-a3d7-42c04a301036}} so that {{formula:a98a6b22-e5d4-417a-ba74-29660c964404}} . After this shrinking {{formula:54c34606-7d2f-44da-b7d3-a09cf412d208}} , we get {{formula:680618d7-9b5a-4590-b68d-54497b59d2af}} . Notice that this shrinking is independent of {{formula:6084911d-7d73-466e-960f-5746e66444e1}} .
Now, choose {{formula:a6e3fdc4-1bb9-4a1b-b780-3cb06b240d0c}} . Then by (REF ), for every {{formula:0f56513a-6f00-49be-9208-f123ef198a4e}} we have {{formula:2edf9108-4b78-415f-a9c3-3860e6379369}} , and hence {{formula:13e49f15-2a0d-43c5-988f-ee6262addb03}} , as needed.
subsection2-.5plus-.7.5Non-degeneracy on {{formula:110cb6d4-b7c8-4e2b-b159-a47407384f41}}
In Proposition REF we have shown that the forms {{formula:89fa4041-be84-4c1a-b67f-e38d37603743}} and {{formula:36f6a5ce-924c-465f-b059-8411fd7195a7}} are fiberwise nondegenerate near {{formula:01dc6f8b-5942-4166-b705-bd51681a06d9}} . Proposition REF below, which we will prove in this section, shows that the same holds on {{formula:457905af-5d02-4b65-898e-5b707f913ee9}} . Recall from Proposition REFREF that {{formula:2a8551bc-847d-4aee-b686-7f4b7016271b}} is a submersion extending {{formula:59735ecc-ba31-4209-b400-b3396d814acf}} to {{formula:2b533000-1b0b-48a1-a024-fc13af412850}} .
Proposition 4.12
Fix {{formula:4ef00514-4a2b-42cd-9736-71bcb0dcd6f3}} . There is an {{formula:f88c9d12-5ed0-4fff-adf8-df96bd543375}} , and a chart {{formula:a883d2ca-531d-4387-81e8-f6c1a05ce861}} adapted to {{formula:87ece803-d8f8-4aff-96ce-2a1fa8c598be}} containing {{formula:f061da9f-9ea6-4a5b-8489-f2a46e339402}} , such that for every {{formula:4f4495ba-4adc-435a-92d7-a8d8a506af1c}} and any {{formula:be879d58-f72e-46dd-b8bc-5ba42085ca5d}} , the restriction of each of the forms {{formula:44a2bcca-014f-4d9c-9a4b-ae65e49e615b}} and {{formula:44967fb5-8c19-462d-8f06-e2ad04bedce4}} to {{formula:e3584093-2b36-4e90-85c6-a9c739a3337a}} is nondegenerate on every fiber of {{formula:962744c9-883e-44d7-9e4d-04fe40aacdd0}} , where {{formula:d74c47d1-ba3c-4d03-8e87-44bb0b902639}} . Moreover, the symplectic lift to {{formula:e2f86da8-1ed9-475e-80e6-baae65a97170}} of the unit angular vector field on {{formula:4076cbc2-3d04-490d-b6b5-0e6431b69b66}} is independent of the choice of the form {{formula:ad56e088-4a82-4855-b0e1-7178a2b88603}} and {{formula:4b0cd4e6-ec51-4c59-9b79-c8265e2968d2}} defining the symplectic connection and is given by the formula (REF ).
We notice that at {{formula:0b486951-d23b-4fb5-9270-f126d7e73e3d}} the forms {{formula:0cca52de-ca44-40da-9e5e-4102dc149efb}} and {{formula:aa4a1d0d-6759-4647-87ad-6421887f4e1b}} coincide, so it is enough to deal with the former one.
As in Section , we fix {{formula:eaa0258d-cbd5-4e62-a618-674536762703}} for some {{formula:c0ee2c61-9125-45ca-9cac-39e6bb430482}} and order the components of {{formula:34eec4ec-b028-4fd0-8324-5c33538d5ce6}} so that {{formula:3ff7a16b-239e-40dc-8c10-5dc501658e9d}} . We fix a fine chart {{formula:62cb60ea-bc58-494f-b2b4-977e3e3457b2}} around {{formula:233614a0-8265-435b-95e2-855887427842}} with index set {{formula:6a603b29-e4e3-4cc1-ad6b-19eb6f6c9cbd}} ; and put {{formula:da201e38-d54a-4592-a577-1e78e240c237}} .
Let us recall the definition of the smooth structure on {{formula:dbfa00be-b597-4062-b129-1cf643741f1c}} . In (REF ) we have defined an open covering {{formula:801902e2-7364-46bc-b16c-3a1080b385b7}} by {{formula:0228bb51-2dd7-4885-9366-37dbe13d6620}} . The smooth coordinate chart (REF ) on {{formula:3e5e160a-260f-4a4c-89d2-3aad51d856fb}} is
{{formula:7343f391-0fb8-4675-94ae-f4001d5ac216}}
where {{formula:1d4fdbe0-28f6-4112-9dd8-3b76e851e1d5}} is some subset of {{formula:6c131112-e4eb-4a8e-bcd9-ccc84ec4433a}} . The charts on {{formula:64de57f9-2841-4721-8b96-b3bba39212df}} are defined analogously. Thus at each {{formula:76886a9b-a0d2-418e-aff0-c3d8f4082d10}} , the tangent space to {{formula:55b78272-d5ef-4aba-aa73-daaab5b3ff6b}} splits as
{{formula:528c7c63-18b9-46f7-8dbf-d06b13f71ca4}}
where {{formula:76c70de0-13e0-488f-9680-59c6ff87909e}} is an isomorphism onto its image, and
{{formula:1b2adc72-2bc7-4c7f-be5d-99a85b9b0fac}}
where the coordinate vector fields {{formula:cbdc4904-15f5-4b6d-82ab-525b91939a3d}} , {{formula:977c85c9-8097-47c6-89c9-0a2a8f2031e6}} are defined by
{{formula:956534be-3164-476f-8a56-a2dcec3f627c}}
Note that there is no coordinate vector field {{formula:87d00ff2-f114-4bbb-945c-2a1a7c201ff0}} on {{formula:e910a35b-5d15-4fa7-aba6-4087ed22d902}} .
Lemma 4.13
Fix a point {{formula:ec1663ef-281c-4289-b561-00cddacc242c}} . Then on {{formula:4d7b1417-8b6b-4f76-a3e2-d531ac40cf62}} the following holds.
For every {{formula:69ca4a2b-b54f-4fbb-941d-553ee1ecea23}} we have {{formula:2e880d73-6067-42a3-9e2b-7a824cf57c3d}} .
For every {{formula:ac7d3846-2f33-4737-bfb4-ada826e73081}} we have {{formula:24edd597-d6dd-4e9e-b655-583a0e356eab}} and {{formula:fcd762d9-4f0a-4224-a417-5f4cd31e55cc}} .
We have {{formula:9c0948ac-cd86-4a23-a70a-d638c0afa458}} , where {{formula:3a99166c-27fe-4978-929f-00f10065408c}} is as in (REF ).
For every {{formula:8b71822c-5be4-4a19-873d-eae8c18c74d7}} we have {{formula:c693dcb8-5daa-493d-a44a-c323395496c9}} .
We have
{{formula:ddbc04dd-2ade-4956-ba5c-808e0395e8d7}}
for all {{formula:72580eaa-a795-4923-9b9a-d0c433c25d9b}} .
{{formula:9ad42b43-e5a9-4049-8fc4-835a387aefaf}}
.
REF The map {{formula:c0f8cc4a-1052-4ac7-80c0-4ce0e645c035}} collapses {{formula:b43701af-b7e5-4675-ab51-bd39480a9f20}} to {{formula:042bcee3-b43c-4ec8-b2c2-a1e2c3afa9ff}} , so {{formula:5411ebf9-a8f7-4ae4-812a-d67f92ed0ef8}} .
REF By Lemma REF , we have {{formula:9f829604-9173-4919-bd33-2bf47ae98b40}} for some {{formula:c78e7307-f812-4573-ab2b-d5ed4aa36265}} . Hence by REF , {{formula:49e28991-65d7-4713-b2fa-a046c8b1bc0f}} . Now REF follows from the definition (REF ) of the coordinate vector fields.
REF Note first that the restriction {{formula:bf691df2-f541-47e6-8cfb-4b6b7b355f1f}} is nonzero by definition (REF ) of {{formula:d9a434e9-8c08-4865-b81c-0ac3e61049b4}} , so each summand in REF is well defined. By continuity, we can assume that {{formula:89c6f2c2-c1c1-4026-9ac1-5ffc47b9878d}} .
Fix {{formula:2e0eba87-bedd-484f-930c-d6d5340ac11a}} . Since {{formula:47385d39-3468-466d-b40e-a886b69f1130}} , we have {{formula:8dcc9cdf-67bc-430c-a4ca-c881d361f01f}} on some neighborhood of {{formula:fc7b6455-4ca7-4d1d-9635-5a6f175cc287}} in {{formula:87fc8176-c002-4197-add1-41f91de3fd41}} . There, {{formula:e2719957-a13c-4452-9bca-47434749c06b}} by definition (REF ) of {{formula:7733c10e-ac01-486c-8d2c-72255a54acc0}} ; so {{formula:208510e0-cc48-4df6-a75a-c597f050e7e2}} by Lemma REFREF . The formula (REF ) gives {{formula:f9099150-7808-4f06-807a-b73e691d1a64}} , so {{formula:a8dea858-e8d0-4179-9730-a2422e1aab39}} , and therefore {{formula:3dceb34a-fa05-45ef-a57d-68d2bb05a4ff}} by definition of {{formula:72348425-b9db-465a-a6bf-d51144988f30}} . It follows that on {{formula:6d7a4717-b3eb-4a51-b35d-a6ff74fc68db}} we have {{formula:7c0011c8-b75b-44a5-a07d-0ed32381a929}} .
In particular, {{formula:5e29e137-2c52-410c-b555-8c59d7d35243}} . By Lemma REFREF we have {{formula:8eee4d4d-2041-4e32-b288-fc92f02927b6}} , so
{{formula:1ceea726-34e6-4b22-b44f-a69183a1f6e4}}
as needed.
REF For {{formula:e92073a8-120a-491f-a7bc-5f6e87bee866}} we have {{formula:30a43ecd-b361-4dd6-97c4-1a4349c794d2}} by the formula (REF ) defining {{formula:c9fa7bee-b5b0-4c4e-8767-8e926f82893e}} . Now, the remaining equality {{formula:144b2494-aa9d-42d1-90b3-ec9dd8d310d5}} follows from REF .
REF By REF , we have {{formula:bc31dcf6-6740-4b27-a92e-ef9a533b1417}} , so {{formula:199af9bf-1a27-4a6c-997d-2962325e7d4a}} . By definition (REF ) of {{formula:80005b8b-af8d-4714-a7ac-31261956bcc6}} , we have {{formula:0906540c-f1ff-492a-9ec0-1f95b34e9cdf}} . By Lemma REF , the 1-forms {{formula:43df939e-61ee-4ea3-805c-2c9d239e170d}} , {{formula:830bf1e9-add7-4e76-b4ee-8d1aa19d6d8c}} for {{formula:af25da7f-bcbb-4a77-9966-bece0b295eab}} , and the 2-forms {{formula:394375e9-ab72-4102-9935-a6b2f5707040}} for all {{formula:8f5ff5c3-3151-4ddd-9b2a-528122539ef2}} are pullbacks of smooth forms from {{formula:9e139bbf-7ff1-426a-ba40-070280b655a5}} , so again by REF , we have {{formula:0909a6ed-e326-4e78-a0d1-4d5f4cf0e1a3}} for {{formula:af5b162a-7772-496d-b5d4-5d5e13c46539}} and {{formula:fc8420dd-c832-45a8-8864-2e5dfde7a9b0}} for all {{formula:66af8d4b-6cf1-47ae-991d-b285b1457f18}} . We conclude that
{{formula:9b578517-5c8f-40a8-847e-ee443af29d0f}}
where in the second equality we have used identities {{formula:194bd976-1e2f-4afd-a16e-befab328ad9b}} and {{formula:008a4863-162a-4e41-8c53-73127755c116}} from REF , REF .
{{formula:b3baca88-735b-4193-9747-7af3ac32feaa}}
Proof of Proposition REF .
Recall that we assume {{formula:71d7a8cb-93ab-43d8-9e67-78983655854b}} , {{formula:cc04dd51-ab4d-43c2-a76d-33f3392118c8}} . We have chosen a fine chart {{formula:5f29823d-dc5b-44ee-a5c9-93a341a688cd}} around {{formula:e9f72e3e-1157-429d-8957-b92c95f03fde}} with index set {{formula:726ec378-5f35-4bea-812a-61b840fe00ff}} , and put {{formula:2df9c8ad-68bf-4776-98b2-409647112b5e}} , {{formula:da5c1778-881c-48c3-a9b4-20be91f262ae}} .
Fix {{formula:dbda90ba-4c4d-4b1c-8f84-80d9badd9d72}} . Let {{formula:17d860b8-3dc3-443a-9e63-bf59c3a26a20}} be the fiber of {{formula:387b5a27-2306-4016-8d17-0c7918041cea}} passing through {{formula:6b607897-2f5c-4d10-a7cb-7b5cfc20d259}} . Since the fine chart {{formula:ad17eb6a-cbb7-4722-8b6c-941e2f9440d8}} is by definition adapted to {{formula:5bb0299c-c34c-4387-a7b5-30c5577cea02}} , Definition REFREF together with the formula (REF ) for {{formula:159d0b15-aa5e-48ad-9ab4-2e8cab1efbb3}} show that
{{formula:63052aef-15d1-4130-97ac-d8caafcf56b3}}
where we use additive notation REF on {{formula:0e22b54d-c118-438c-9d52-da9841d39b15}} . Hence
{{formula:2b8ebb7e-f270-477d-8826-2d73ea6959fb}}
The form {{formula:718cc2f5-c1c0-40e7-a83a-ac80eba96755}} defined in (REF ) restricts to {{formula:fd7ea2ed-6e81-4a0f-980f-7c341c9de5c5}} as
{{formula:e0eb1bfd-fcc4-47f6-ab49-203e2980043f}}
where {{formula:470e75f2-e7d4-4ec3-96b0-aa489a5a6154}} . By Lemma REF , {{formula:8bdfde9c-cadc-4036-b8d2-7fa785ed29bc}} for some {{formula:b5980615-2e2e-4416-a81b-740dfc9c67f8}} .
By definition of the splitting (REF ), the restriction {{formula:4ed90ec4-0d0c-448e-8fcb-30555968e8a6}} is nondegenerate. Hence there is {{formula:1c2bf4b4-bac1-4465-95f1-3fb0a1cb1f51}} such that for all {{formula:8c4c4a7c-7735-4519-b328-1ad9bd1f9ecc}} the form {{formula:9390f112-0acf-45bc-a010-0fa2b9ea0766}} is nondegenerate, too. Fix one such {{formula:549c1652-11c1-452b-a435-640502c9ca4e}} . By Lemma REFREF , we have {{formula:3a2523c6-4525-4f08-886e-1d01a642b6fa}} for all {{formula:4c1d0cc6-d553-4535-bf7d-066811b95921}} , so {{formula:e5c409db-5773-4370-9279-beb12e044e0a}} . It follows that {{formula:43c2be56-4224-496d-b81c-20844526b9ba}} , so {{formula:34fd631f-b50e-478a-9fcf-f29f8055ed0b}} is non-degenerate.
Assume that a nonzero vector {{formula:656e3a32-7e43-4f09-8709-f31a36efcb95}} satisfies {{formula:1bcfc886-38c1-4e94-9f05-02242b969977}} for all {{formula:7bcb5b88-879d-45c7-a1aa-25d9ac340a79}} . To prove non-degeneracy of {{formula:26ac7e4f-b457-49ea-9b8b-ed008b2c14fa}} , we need to show that {{formula:7eb4b8d3-e920-4f4a-bdd5-e71f59babdbb}} . We will first show that {{formula:f0bc5b9c-3dba-4dab-9b95-53deb1772d95}} is proportional to the vector (REF ). Using the splitting (REF ) of {{formula:f934def9-2763-49f2-8799-a021a551a04e}} , we can write
{{formula:07674de4-469a-4abd-91ae-551fc3fd089a}}
We claim that all coefficients {{formula:d24ca7f7-c21c-4639-a287-85962d7e2e1b}} in (REF ) are zero.
For {{formula:d1600c4a-1778-43a8-9be6-a807f161248c}} put {{formula:433b0822-898e-4433-96ab-76e98454297d}} . Formula (REF ) shows that
{{formula:dc6e09bc-f839-4353-b0a9-82d14c24d038}} , so {{formula:9ae31f4f-1a90-4802-8466-234498567a2a}} by definition of {{formula:70ed8d87-2139-4d2b-ae73-5616a053f89f}} . Using Lemma REFREF , we compute
{{formula:cafd5504-cf46-4707-a68c-30c8781724ef}}
where the last equality follows from definition (REF ) of the coordinate vector fields {{formula:047f3204-37a5-4729-82b3-36b834699b5e}} . Thus for all {{formula:3266c653-9221-4735-a752-060c24f7de01}} , the numbers {{formula:1d9dac03-6fe1-4e6e-a786-0e5c7a08ca21}} are equal to {{formula:425c8be7-9e5f-429b-91af-b28c445bbdbf}} .
Suppose {{formula:5e5d1a17-a8b9-4998-8fe9-69603ab00846}} . Then {{formula:838ebd6c-780e-4ca4-9626-4934a86fed8d}} . Since {{formula:0b2aa6d9-8179-48af-a8f7-560e2dc2d788}} by Lemma REFREF , the formula (REF ) for {{formula:c378ebf3-419a-4165-bae8-5ff825779134}} gives
{{formula:9c854eca-5228-460e-8801-653c7bd8059d}}
where for the last equality we used the fact that {{formula:78e88fcb-8a23-4ca5-9a8e-d0afcb3e7d5a}} , and the formula for {{formula:c07a8262-8206-443e-99ab-f4d8257bd9d2}} from Lemma REFREF . This is a contradiction. Therefore, {{formula:f0d08c24-113a-4e8c-ab34-c19351fbe1c6}} , so {{formula:496c23dc-6068-4626-b8f9-2db1a5068d0c}} for all {{formula:4403a778-2448-4eae-a070-0735349b573e}} , as claimed.
Now, we claim that {{formula:939fda8e-8b4c-43cc-a9f7-efd1eeffe967}} . Fix {{formula:38a3f2f8-e277-4bac-bbf6-8c3e230cf672}} . Then {{formula:de57bd04-346a-4ebc-9a71-69367d2449e3}} by (REF ), so {{formula:72de4a14-dba7-4648-9ebc-32915b16ad0c}} by definition of {{formula:94d33eb8-684a-42a6-8a8c-3f05bee354ed}} . By Lemma REFREF , we have {{formula:964fda36-960f-4fe0-8c15-9b8b2fbf1545}} , where the last equality follows from Lemma REFREF . Since we have shown that {{formula:4c0bc6dd-7aff-4faa-9555-9f65567f5e3e}} , we infer from (REF ) that {{formula:d6033b56-7ed8-4652-888d-307c383c752b}} , for all {{formula:f64295d3-9876-4112-85ec-5c7b6d15e38a}} . Since {{formula:48858525-1bf3-4a97-9a78-57aeebeb0635}} is nondegenerate, we get {{formula:6de91021-b4c4-4fb2-b29b-f24cfa0a17f3}} , as claimed.
Therefore, {{formula:260fa43f-8353-4bae-b1e0-2a5ef7389c5d}} for some {{formula:2d5dc162-c9f3-4756-83ae-814542f10e7b}} . Since for all {{formula:1abf2756-2b1b-435f-9553-51dfb447681e}} we have {{formula:65a96ccf-679a-47ee-aa02-b15bf932fd7f}} , we compute using Lemma REFREF that
{{formula:92b12cfd-853e-49a1-8e90-2e345e535062}}
where the last equality follows from Lemma REFREF . We conclude that for all {{formula:9ec95223-73f9-4e66-bee4-aa8a9c541cd4}} we have {{formula:177cd621-c686-4ced-8ff0-54fb8c379309}} for some {{formula:0f62b3d9-a82d-41f9-9b45-9a17a8307eca}} . This shows that {{formula:b01b2501-031d-4a04-93e0-28e22e69016c}} is proportional to (REF ), as claimed.
Replacing {{formula:6093d4d3-d2fc-47cb-beb3-d14877b08bc1}} by {{formula:c4f09e0e-8f34-4a01-8796-d9039da7dba2}} if needed, we can assume {{formula:d2349b1f-f65f-411d-beee-95feeef790cd}} . Then {{formula:0cea202c-ca0c-47f4-85f3-96f973dd67c6}} for all {{formula:ad37338d-ccb8-4b96-8984-60f2b667f2b6}} , and at least one inequality is strict since {{formula:0fd458a3-410f-4790-8dab-d56cb664543e}} . Hence {{formula:3af30f07-566b-478d-873a-8b5d9c8b8c20}} , which by (REF ) means that {{formula:33f817e9-6b05-41ef-879f-c894a6ce733c}} . It follows that {{formula:e424006a-e135-44b0-989a-e84e51591c50}} is nondegenerate. Applying this result to all {{formula:01cd4bdf-d8e6-4bb3-bf60-fcd95ad6319d}} , we infer that {{formula:8045bf11-aec0-4073-8671-41728cac8a76}} is nondegenerate on {{formula:d9966b7b-3ed4-4848-b909-b945d3bb30af}} for some neighborhood {{formula:f7bcea91-0760-4da5-a747-3334c96576cc}} of {{formula:f3dee510-0a16-4a72-b891-8288a25a68ee}} , as claimed in Proposition REF .
It remains to show that if {{formula:ad40bd38-f2e5-4660-a535-55e38a9ac0d1}} is normalized by {{formula:9287b8fc-5a1f-4640-9880-9ff2115f623d}} then {{formula:fd2aea33-8558-43b2-aa9b-36bf0a76e14b}} is actually equal to (REF ). In this case, we have
{{formula:a543613c-661d-4165-af48-f7260ec73498}}
hence {{formula:20f77067-99a0-4985-b196-495a610a7cf8}} , as needed.
{{formula:dd24f898-52dd-432d-befe-6985dc016525}}
Proof of Proposition REF .
Fix {{formula:ff159b53-3ee0-47a5-9d2c-27dda0e324f4}} . By Propositions REF and REF , there is an {{formula:aac4f2ca-b65b-490e-bff9-8506423ca58e}} and an open neighborhood {{formula:8e5ea1fa-bbfc-4298-8509-f707ee825742}} of {{formula:c1e243c6-6f05-4c1f-96b7-a528e92b0e90}} in {{formula:e5a770de-ce07-4946-a1ed-8679d1a25532}} , such that each of the forms {{formula:bbcf7d4c-9bea-489c-b21e-411a3e8561e3}} and {{formula:a2f99833-cb63-4fc9-b7a4-c23f913b3422}} is nondegenerate on {{formula:cc9c0ac2-3a70-4e03-8063-37ca28516013}} for every {{formula:116af9aa-be25-4f81-8c01-e00560f7405e}} and any {{formula:2208e7cd-a593-4f64-b83b-f590afa705a6}} , and on {{formula:217d573d-fe85-4c1c-acfe-e57b2e09e98d}} for every {{formula:d9618102-4b48-427f-b31d-52e39f9d9f5f}} . Since {{formula:eb9b11fe-f3a2-4bed-b3ac-f1d249cb3883}} is compact, choosing a finite sub-cover from {{formula:cb1bec4f-3cec-472b-9eb2-42f038849af9}} we get an {{formula:00836b39-6957-4668-96ca-34f2c376b41b}} and a neighborhood {{formula:6edc89c5-e654-43dc-bea8-04fd22d4d4a8}} of {{formula:2111d491-0a34-46d1-8f60-13047e7a196b}} , such that, again, each of the forms {{formula:68e176c3-3fff-49c7-a66f-e641c2ebd292}} and {{formula:1e44caa2-80cd-444a-b6c2-8e42fe9731ec}} is non-degenerate on {{formula:b6a79e75-984a-4844-bc4f-14a2e30de381}} for all {{formula:5250f1da-a96e-4095-995a-b601cab55fc4}} , and on {{formula:5a0fe3ed-3eb2-4fc9-830a-bfa2826c7d23}} for all {{formula:0bcfdec2-c1e9-400f-85bf-0e75d6cb0fee}} .
Now, the form {{formula:6ce2bd6d-ef23-4420-9aa3-71f8bff17e4a}} defined in (REF ) is fiberwise nondegenerate on {{formula:bd86373b-6d11-44c6-a272-0ef89d7fcaac}} . On {{formula:52512129-1e38-4644-be3e-4c738abbeac9}} we have {{formula:3aa99fbc-d641-464f-8df1-834c2383815d}} , and {{formula:15b22594-6827-4c3f-b51b-a34ec4f3b286}} is a diffeomorphism, so {{formula:1de3a3a5-faef-476d-a4a9-66255dffe24e}} is fiberwise nondegenerate there, too.
We end this section with a simple example illustrating the dynamics of the monodromy vector field (REF ) at radius zero. In Section we will describe it more precisely in terms of the associated abstract contact open book.
Example 4.14
{{figure:9ff14dc6-1e13-43c7-9ed8-151136f3a43a}}Take {{formula:9fc53f39-2d7e-49f8-bfc5-01a05aaf3689}} f(z1,z2)=z12z2{{formula:2bf3c456-2ab0-4e28-91c6-6b04ff2d6ff0}} f*(0)=2D1+D2{{formula:1a0791e7-035d-4ec4-9115-6f19442562b6}} Di={zi=0}{{formula:68cffad4-8691-4071-8ca2-cef5b6264d7a}} X{{formula:2e10c07f-0b8a-4ccf-b7b8-5daae17e6e15}} f{{formula:43272b86-941d-497c-ad40-b8bb64cf4bbc}} D2{{formula:beb8b852-1995-45b0-9e01-e1001fc01d76}} D1{{formula:812304b7-8e9f-441a-bcb3-d02691d1de42}} D1,D2{{formula:b391f163-709f-49fa-8f4e-1a8c0136e5b2}} D1{{formula:0fe4cab9-2919-4613-803a-97f7904689c1}} D2{{formula:f4b108b0-16f4-441a-8d46-d3afe8ea7348}} Now, the remaining pictures are radius-zero fibers in {{formula:1572b4ed-d4fc-4bff-a10f-6e97c4fccbd6}} (top-right), {{formula:98c4ba5d-729b-4346-8058-465442d0b694}} (bottom-left) and {{formula:fb37ecd8-c320-4cef-9d14-662231496e48}} (top-left). In {{formula:2e918b30-9281-407a-92ee-77b002e4d780}} , we replace the origin by a 1-simplex {{formula:a0298396-74b1-49c0-958d-e2e9a05babfa}} , with coordinates {{formula:f9a4036a-d1e3-412a-8765-23d5642508c2}} , {{formula:05b7b7e2-a067-423a-946d-8d3c826cfc82}} . In {{formula:e488e040-f5ad-4105-883b-147be69dcdfe}} , we introduce angular coordinates {{formula:e47ff414-204d-4dd7-87a2-1ebc7960117e}} , {{formula:a9c029d8-1982-40ac-aeb2-55d935b4ca2a}} satisfying {{formula:1ca00729-8668-4171-baee-35fcc330a340}} , i.e. we replace the origin by a circle. There, the monodromy is not continuous. Eventually, in {{formula:ee2f3bcb-7144-49e0-af16-f67b54740359}} we multiply this circle by {{formula:9c10dc8a-e2f3-4528-b8ba-aa718903c0f5}} and make the monodromy interpolate between the two pieces, using a parameter {{formula:89324fa0-9645-4d4b-bca4-5b73234fc99f}} on {{formula:4bf10db9-c753-4084-8eed-567b31f0518c}} .
Symplectic monodromy and abstract contact open books
We will now recall further notions concerning Liouville domains and their fibrations. For a general introduction, see {{cite:b1204f911b5b321f5d286fa4927a3bf3bb9cf2d1}}, {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}, {{cite:64222017754251cca06cd4d00a71ce64eea9ea16}}, {{cite:f95f65772dd0a8ed21ab2e3ab90ebeeb21ba2bcb}}, {{cite:d20ff9e1ba10f8d7a97dd150ef78cb99d18bc788}} and references therein. Our aim is to establish a natural setting REF where we can produce or extend symplectic monodromies.
subsection2-.5plus-.7.5Hamiltonian vector fields
Let {{formula:9ceab106-ea37-41d3-a6db-d1280f9cc66b}} be a symplectic manifold. A Hamiltonian is a smooth function {{formula:23483f3a-e4f0-447b-b50c-e40f79a425bf}} . A time-independent Hamiltonian is a smooth function {{formula:25702997-a030-40b5-b750-220e55fbe0ea}} , which we view as a Hamiltonian {{formula:339d0aeb-02eb-46cd-8eb4-dd4934d8d3c5}} . In other words, our Hamiltonians are always time-dependent, unless explicitly stated otherwise.
Fix a Hamiltonian {{formula:cf6c1d17-ca7a-4a4f-95b3-dd7c37aa4b2f}} . Since {{formula:e1730237-cc28-4429-8f2b-d9b8c4cbc118}} is non-degenerate, there exists a unique vector field {{formula:73071ee5-8af0-4227-af51-72aae20cab32}} , called a Hamiltonian vector field of {{formula:56e7c9bd-7035-417f-8b50-70739daa1eb9}} , satisfying the equality
{{formula:1122bd90-1c5d-4977-82a0-a2aa86bb0317}}
The Hamiltonian flow of {{formula:9ce03305-48f2-43e0-817f-de99e4d1b6a5}} is a family of symplectomorphisms {{formula:c46ff8b2-86cc-4b80-b0c2-ede944b3645c}} , parametrized by {{formula:52bfd28e-e152-493e-96e3-d3676cdcfbf8}} , defined by the formulas
{{formula:a546bd2e-3494-4bbc-a88b-6a1047e4c290}}
Our definition of {{formula:db1eedb0-8a6d-489b-8eb8-6036158812c4}} coincides with the one used in {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}, {{cite:d5bc3ba72058ac6304d8c4621b90572df69b7cd1}}. However, {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, {{cite:97005650a8ac11991dc2e980883b47d01674846e}} use the opposite one, namely {{formula:70f3bab4-631d-4490-943e-d62b0e992fe5}} : thus for given {{formula:9ca587e9-ae9b-418c-9ba4-74b4fdc76a4c}} , our {{formula:e797a3b1-a1d4-4624-ba64-85387096a65e}} and {{formula:f79eee5b-994a-4e0b-b1d1-6590a07957db}} equal {{formula:61f55252-0cc4-4d24-be78-3dda451f92e7}} and {{formula:6f415ae1-0385-4a34-ab41-6aad95884a4e}} from {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
So, when we quote {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} we need no adaptations. However, we will quote results from {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, {{cite:97005650a8ac11991dc2e980883b47d01674846e}} too. To avoid sign mistakes, whenever we do this, we replace {{formula:4f1b5076-be1a-4cc2-a19a-b44040df7e95}} in loc. cit. by {{formula:1eb83be4-79a2-4663-a39d-46fe388585aa}} : this way, we get the same vector field {{formula:b2621c58-a7d8-4908-971e-753391fe0f81}} and flow {{formula:8fa496a0-9c38-4f79-9c0b-69e6f390478a}} as in loc. cit; so no further modifications are needed. In particular, to get the same Floer equation as used in {{cite:97005650a8ac11991dc2e980883b47d01674846e}} to define {{formula:eb5b703b-5e22-4952-b6d0-d797eae7864c}} , we will need an opposite Hamiltonian, so we will chose it negative near {{formula:4a205157-badb-41fd-a4bd-d2d22894681c}} , despite the symbol {{formula:3dae5035-f740-4387-9959-8aed24595623}} ; see Section .
subsection2-.5plus-.7.5Liouville and symplectic fibrations
Notation 5.1
Fix a map {{formula:4afdbd73-9c53-4d64-ad00-4ecf81227548}} . If {{formula:1ce7da2b-36ef-4be3-8c52-2f47d7fb0d1c}} is clear from the context, we will use the following, simplified notation for its fibers. For {{formula:b58ccfc8-350f-4223-a436-539592cdc226}} , we put {{formula:17128fb5-3a7c-4838-beb2-5b3e696f0f5a}} . Given a form {{formula:4999f31b-fa6f-4655-ad80-2f822a6deefb}} , we write {{formula:ac5eb307-b1fb-46d7-848a-c5a9c0d782e3}} for the restriction of {{formula:3c20ece7-20e6-4751-9636-7272fad5773a}} to a fiber. For subsets {{formula:2bd79518-2ce2-45ca-a579-930cadc3c14a}} , {{formula:4e2543e3-5311-433b-8ee1-92a98e3825b8}} we put {{formula:44b14d16-04bb-4133-bb77-888c17488eff}} .
A subset {{formula:9e9d28e8-1d9c-4186-ba66-cf4b93f46992}} of a manifold {{formula:23c3f606-4278-484a-973c-549934b717df}} is a codimension zero submanifold with corners if for every point {{formula:e5bf2a75-588d-4dc9-a011-791096b352e7}} there is a coordinate chart {{formula:36ee9762-dec0-49c5-af13-b0bb991d4eae}} of {{formula:2392d1dc-1e1c-4193-9a72-0decfcd0fa62}} around {{formula:6be6bbdb-10c0-4d96-8f0d-65ef69ca3e36}} such that {{formula:ff3d6d50-0fb2-4504-90e2-e7b2bd4e19d2}} is an open subset of {{formula:63f14aa4-a9c1-4a3a-876c-cb26da8e04bb}} for some {{formula:eb6c493f-8dcf-4849-9ed0-7f8efe76e109}} . A manifold with corners is a topological space which embeds into a smooth manifold as a codimension zero manifold with corners. This definition is compatible with the one given in {{cite:f24d18a69e9c818fe069ba78d0f67630c8ad8ab2}} by Proposition 3.1 loc. cit. We say that a map (or form) is smooth on {{formula:e30a67fc-be2d-4caa-b99c-56e8e570d725}} if it extends to a smooth map (or form) on some neighborhood of {{formula:5eac64af-0a2c-4fe8-b9bd-13becf133c93}} in the ambient manifold {{formula:ecd3c19c-ea9e-4a39-93fc-5527bf5ac680}} .
For example, the closure of each {{formula:73b9b2dd-5391-474e-b905-fc7679818b92}} in Figure REF is a codimension zero submanifold with corners of {{formula:ffb93207-1db3-4d2f-8083-f1b9e72ddab0}} , cf. Proposition REF . Another example is the total space of the Milnor fibration
{{formula:bd363174-f9db-4e90-8bee-af11ac63b20b}}
where {{formula:1c9c1f83-7b1b-4340-807c-8283bd7821a1}} 0n{{formula:f0d00371-b03f-487c-b02d-04db9b25f55a}}
Consider a smooth map {{formula:c9f7eb43-60d4-4b25-9ba3-be7d98ec82df}} from a manifold with corners {{formula:d7adfc56-2eeb-4760-a124-50a144292254}} to a manifold with boundary {{formula:b2ab95a3-dce8-4af2-97a8-e2bcad3ab4b1}} . We split the boundary {{formula:72982826-3079-4c46-8448-f54550e86b8b}} as follows:
{{formula:58ff4c8e-1d65-4a91-bb09-b4946fab4fed}}
We say that {{formula:a2c58de5-bdb6-46ff-9f27-db49b009d945}} is a fibration if it is smoothly locally trivial, i.e. every {{formula:23844740-0169-46f3-895d-470002a6b2e8}} has a neighborhood {{formula:5917a319-a16f-45bd-8cbf-5fa0e9ff8f3a}} and a diffeomorphism {{formula:192f91b1-bcfd-46b2-bac0-067d5fe41d1b}} such that {{formula:a8880df3-7f78-4f54-8690-2d15d2826af5}} . By the Ehressmann lemma, see eg. {{cite:36c0783a3ba6676e9efc8699625a977ac5912f2c}}, a proper map {{formula:eeee4b0e-1ef9-4744-9e0d-0f4f9bd35105}} is a fibration if both {{formula:82a82602-8db0-43a9-a2d9-61d9cb1ce4f2}} and {{formula:d69a32f2-0dc0-4109-bc5d-e6f85d6ce385}} are surjective and submersive.
A pair {{formula:65ce6f33-f2c6-4936-8175-22c8ed294cce}} , formed by a smooth map {{formula:22066705-2c83-47e4-a5f8-15684bcf8631}} from a manifold {{formula:23df91b6-7a31-4330-abb2-7db431e79c58}} (possibly with boundary or corners) to a manifold {{formula:ca167da5-765b-4a82-be3b-3c9cdf2ab2fb}} (possibly with boundary) and a 1-form {{formula:e5a07701-c342-4433-874c-0520ce923f68}} , is called a Liouville fibration if {{formula:6c03c4f5-4936-47bd-9fca-9c457e66f0fe}} is a fibration and {{formula:8354f964-f598-4085-a7b5-5de774b65f00}} is a Liouville domain for each {{formula:aadb759e-0d5a-44c6-b2ad-5acc619c55c2}} , see Section . The similar notion of a symplectic fibration is defined replacing {{formula:93aedfc2-e26f-4bd7-b89c-20aa8fb8cfb1}} by a closed, fiberwise symplectic form {{formula:f07f61c6-2b9e-41a9-8237-2eb5b3874c11}} . A Liouville fibration induces a symplectic fibration taking {{formula:f9d4df1e-9f86-410d-b136-d39299186a2e}} .
subsection2-.5plus-.7.5Definition of a symplectic monodromy
In this section we will modify the Milnor fibration (REF ) it so that its monodromy becomes a compactly supported symplectomorphism. Notice that such a modification is needed: symplectomorphisms preserve volume, and there is no reason why two fibers of (REF ) should have the same volume. The following notion, introduced (in a slightly different, but equivalent way) by McLean in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, abstracts the properties of symplectic monodromy that we need.
Definition 5.2
Let {{formula:6dd0b0c3-faa4-4e96-85a1-d88b1e989d77}} be a Liouville domain. We say that {{formula:11c16841-50a8-4a04-b844-f4a983f570e6}} is an abstract contact open book if {{formula:574a013e-22fd-436f-9fa7-77bae96ebf14}} is a diffeomorphism that is compactly supported, i.e. such that {{formula:54e5bced-989b-4015-83e6-46d7325a2f1b}} for some neighborhood {{formula:c9744af1-7698-454e-8c1c-8df9c29bdb73}} of {{formula:947a164c-f298-431e-9d8f-5d1c2955c11e}} , and exact, that is:
{{formula:1a281667-6312-4a74-8cf8-07912df0314b}}
for some function {{formula:f976b1b2-2c12-4cbc-8225-ea73f590582b}} , called action of {{formula:ee2b3a19-df01-4bd9-b3f1-2af306bdf6e3}} . Note that the formula (REF ) defines {{formula:bd772b7f-7911-454a-b939-22c7f6e8efd2}} uniquely, up to an additive constant. Moreover, (REF ) implies that {{formula:ffd3543a-2ecc-474d-bb53-8e5d833d841a}} is a symplectomorphism.
Let {{formula:1de637d2-be60-4392-86e4-07b71b5ba14a}} be a smooth manifold, let {{formula:cbf5cd2e-4d57-4205-8da7-159b08f822b3}} be a smooth map, and let {{formula:337908cd-3a8a-4773-8190-a00ffe4f1923}} be a 1-form such that {{formula:d4e3a4a9-a3c6-47f8-88ca-55b74574651d}} is a Liouville fibration. Let {{formula:a5a287a3-b110-415f-b7b3-7b317e173d02}} be a diffeomorphism which preserves the fibers of {{formula:01f9dd4a-5fd0-42b1-a28d-7c415e6bd9c5}} and is equal to the identity close to {{formula:55a83170-0c40-4273-937d-40478ab11b20}} . Put {{formula:2f9d3398-2c88-4cf4-ab37-6184da914fa0}} , see Notation REF . If for every {{formula:db451576-1cb4-4d6b-bd9e-1bbb8f61bd37}} , the triple {{formula:994ecfab-3a22-432c-8c5b-adb7d4b601c4}} is an abstract contact open book, we say that the family {{formula:628a2b0e-8d04-4a39-a7da-56d468bddedd}} is an isotopy of abstract contact open books. In this case, we say that the abstract contact open books {{formula:31d0f03b-46ef-42e2-99ba-75df41dfc35d}} are isotopic, and write {{formula:a755cdde-13fe-434b-9152-5b0881bac7aa}} for {{formula:4f899164-8cd3-4dd4-8d9e-4965604a9454}} .
Clearly, if {{formula:f962ab99-24e5-4792-8c65-41fa2b4c18c2}} is an abstract contact open book, then so is {{formula:5849e890-6389-434b-97f8-638975f242f7}} for any iterate {{formula:6fe12a8a-8b71-4738-86e0-2ef00a1f1a6d}} of {{formula:52be2d16-b9ce-4dfb-9d8c-49db659dffb3}} ; and if {{formula:b6f83f84-9e92-45cf-b668-78d0319fef6f}} then {{formula:8462aaec-5880-4b31-a9c7-541105461fce}} for every {{formula:90b44b03-43ed-4d29-be55-5f14d0f58eef}} .
In order to turn monodromies into abstract open books and compare their isotopy classes, we will need the following notion.
Definition 5.3
Let {{formula:c2b4e830-ba54-47a0-a87d-910ad9826d86}} be a fibration from a manifold with corners to a product, where {{formula:ac6f6269-0b0b-49b8-a2f7-4631ac68206b}} is a manifold and {{formula:fdad00d2-cfbc-4bac-89c1-bc799e8b1e58}} is a manifold with boundary. Fix {{formula:6f053a6f-f101-408f-b750-423e68e75a66}} and let {{formula:b9a3f341-03a2-424e-acaf-41406d79950d}} , see Notation REF .
A {{formula:c5ca355c-8251-4551-8ef2-e0372df2946f}} -fiberwise trivialization of {{formula:7f168e00-70a3-4c07-8ba2-cc51f491ddf2}} is a diffeomorphism
{{formula:13697ca2-1e92-4cd2-a172-d0b487412973}}
such that for all points {{formula:ebec8d18-41dc-4a6b-8116-14402602fafc}} , and all {{formula:815c025f-e36c-40c4-9202-ebe17f5aac70}} , we have
{{formula:5e5bd894-257d-40ec-aa16-2536a6835154}}
A {{formula:c83831cb-f125-4046-b4dc-eda4dc339891}} -fiberwise collar trivialization of {{formula:ba029f5b-5691-4b04-b1e7-4ad0bbbb025e}} is an open neighborhood {{formula:7a1111b7-fa5e-4294-a3c0-86b0c01382b9}} of {{formula:8fc43d63-3793-4fb9-ab21-e3cd2763f238}} in {{formula:f56a19e5-5ad4-43cf-81a0-c192c8c40d8b}} , together with a {{formula:478699f8-142a-4c9a-a24c-7e5002d52a0e}} -fiberwise trivialization of {{formula:fce1e665-951d-4d34-8d2e-90008c185de1}} .
Let {{formula:37d9482b-d195-414c-bd4f-0eb5fa12edfc}} be a 2-form such that {{formula:e70c9946-9126-401b-adcd-715b1e7e37b2}} is a symplectic fibration. A {{formula:1740ffa7-f572-4935-a750-7338ee86ab19}} -fiberwise trivialization (REF ) is called {{formula:95915703-d251-40ce-9837-34d37fb12dc1}} -fiberwise symplectic if for any {{formula:6920f6f9-e74b-40a6-a0fd-640602dc7a08}} we have
{{formula:f57ddc46-3d0a-44ad-bef1-5df8ba00e5ea}}
where {{formula:13093e98-4f26-4ba5-a86a-efedb773195b}} is the restriction of {{formula:9bc9ff13-1b5e-49ab-8de1-b520c0d13df9}} to the fiber {{formula:71b112fc-2090-40d6-8d45-08932b37f71a}} .
Let {{formula:f21e6125-70d3-4c38-8f49-da0adba4f94c}} as in Definition REF and {{formula:3df3103e-3f4a-4420-b163-7d8b0789b02c}} be such that {{formula:203aeb1c-5cff-494f-b4af-d9b9ff91c3db}} is a Liouville fibration. Assume that we are given a {{formula:fa764f24-1880-4b34-80a7-66f549fa9696}} -fiberwise collar trivialization
{{formula:4e353294-7407-48d3-961e-521c68a14595}}
that is {{formula:88dc9735-6f46-4fc9-834f-eb8f44cc5f99}} -fiberwise symplectic, i.e. satisfies (REF ). For any {{formula:16317e30-c563-4d50-93c2-18de859af7ce}} choose an open neighborhood {{formula:8bb12cb6-cb1f-43c8-9c07-3eb97be1e6c1}} of {{formula:b01665c8-6177-45b3-8cbe-ba0b5a8aba36}} in {{formula:ec98e9e4-a3d3-42fb-b367-d3abe35143bf}} and a {{formula:21de6735-da4b-4d7a-9492-993bac1b5e64}} -fiberwise local trivialization
{{formula:7da9a048-09fd-4e1a-b1a0-2c9253b79d7c}}
which is induced by {{formula:97cd2399-6cdf-4f86-8c6a-6a328f2c6edf}} on the collar {{formula:d9b8067c-cb45-4707-a7f6-4c490cf8725a}} , that is, such that for any {{formula:0790f13b-5e16-40ca-98e6-bc2dd24b78fd}} and any {{formula:2a17c6db-1f7f-463b-8f1d-0ec41bfa25e1}} we have
{{formula:ca0efd87-5c88-468d-90a9-4cbeefba565a}}
where {{formula:81734274-f956-425d-abdd-703c125f8102}} , and {{formula:8ceaebff-56ec-4cfe-8fe5-64b581ae2111}} is the projection, see Notation REF .
For every {{formula:6c74dd1b-832f-4ea3-a7aa-e7d6da248f46}} the form {{formula:4fce6a5e-7f43-486d-a8b1-fc3c90e9a797}} can be interpreted as a family of symplectic forms on the fiber {{formula:1c2d2fe8-43ce-4533-b311-06f71cbf2076}} parametrized by {{formula:ec86aa9c-d51a-4b07-a695-fa410efe159f}} ; for any {{formula:6d621589-3824-4c26-9eda-9baadfa6f64c}} , denote by {{formula:a724fa5a-ecbe-4af0-8466-a5db109487ea}} the corresponding form.
Since the collar trivialization {{formula:ddb53eba-a276-4e1c-9e10-8d98951fa735}} is {{formula:dc6e1a4f-4132-4562-82a5-1b984b6afeb8}} -fiberwise symplectic, for every {{formula:50fb80c1-5e97-4c84-a259-9b968770f2d6}} we have the equality of restrictions
{{formula:1f225509-5fd1-4718-a524-7fa6e3f08fba}} ,
hence the difference
{{formula:251f0a60-72c3-468d-ab2a-2e20c57bc895}}
vanishes in a neighborhood of the boundary {{formula:fe9363ce-8e2a-4598-b49a-91c3fa9c58f3}} . Thus we obtain a relative cohomology class {{formula:feeb4b86-92c6-421b-91b9-80a619585a9f}} . We claim that this cohomology class does not depend on the choice of the choice of the trivialization {{formula:24c0cc8c-6677-4c3a-8c3e-3fe65fb929c6}} , and hence denote it by {{formula:5d5eab6f-6e1b-4a16-bbc7-2e88d0414169}} .
To prove the claim, we choose any class {{formula:60c2fff8-80f6-44f2-8b0e-1ca212d1168c}} . We have {{formula:2bf318df-9fe6-4c04-a652-f74b321c2db5}} . Hence, if {{formula:9cc0c8ac-e898-44a3-8597-36fd9faa0ef9}} is a different trivialization we have
{{formula:c43d2cdf-95ff-45c8-9b5b-8aa98fa5cbdb}}
Since {{formula:461137ae-0463-43ee-8d2b-ab8eb117b33b}} and {{formula:6810be2d-273b-49b5-9daf-c1580ac0523f}} coincide close to {{formula:82c2f529-4857-4b0c-8d1e-f519d51e8c0e}} , the chain {{formula:f0608200-a8ca-40ab-b437-929760b95295}} is closed. Therefore, since {{formula:9c40774a-4067-48f6-ab6c-e7865df67625}} is exact we have the vanishing {{formula:13f76b95-a72b-4f15-8cc3-5a3687d20872}} . The second summand vanishes for the same reasons. This proves the claim.
Definition 5.4
Fix a Liouville fibration {{formula:9e828dc7-b212-47f9-b03b-3f77b9bd7d11}} and a {{formula:be0ecc22-f72d-46ff-a29c-a99a119df425}} -fiberwise symplectic collar trivialization {{formula:2977c0ed-fa26-46f8-a940-b9b4e9ed7924}} as in the discussion above. Fix {{formula:73014cb3-13a3-425e-8794-55fdd1e6bc99}} . We say that {{formula:db6566e0-3cee-42c1-a96e-bab6d313e065}} is cohomologically trivial with respect to {{formula:169b4b32-d53f-4413-9da7-53e50c443d7c}} at {{formula:e36de84f-95ab-4ef5-abb2-0e8cad7c958e}} if for every {{formula:880632e3-f468-4556-8912-f89fca63adb8}} and a trivialization {{formula:e859dbbf-e42f-4aa7-a5a6-02473a5be091}} as above, the cohomology class {{formula:8434a19f-b925-4dc7-9d53-71ff19cbe15d}} vanishes in {{formula:214911a0-f079-4223-84fc-2dd3c73df00e}} for any {{formula:c61723be-0792-4aba-bdb0-e1d72a0cc7dc}} .
Definition 5.5
Let {{formula:9069aeca-11d0-4cea-a711-1d4cc51cbb3b}} and {{formula:0aa848f8-fb7c-4389-9498-1defadff9484}} be such that {{formula:e6dfe7fa-c075-4ad8-b91d-b3715e8baff1}} is a symplectic fibration. A symplectic monodromy trivialization is a smooth map {{formula:61a3f3fe-929c-43f5-90f0-b33167417d29}} such that {{formula:a4086838-99c2-4199-bdbb-3c581114f555}} for all {{formula:ae888ee6-c160-4eb2-9b96-d181b4e625f6}} , and such that for all {{formula:eb3c38bd-f70d-496b-a710-adb2e1c04e0d}} , the restriction {{formula:aacd04c1-2301-422a-b7c1-bb65642a272b}} defines a symplectomorphism from {{formula:97fb0883-b5b4-4e01-b5e0-908c737cc731}} to {{formula:ca961ed5-7eeb-446e-acd6-49bc3002d6cf}} , where {{formula:f68ac932-a177-4031-ba3e-02acd7bf8a1e}} . A symplectic monodromy is the symplectomorphism {{formula:669c5cb2-c120-4760-94db-4ab2f9d6f21d}} for a symplectic monodromy trivialization.
Assume that {{formula:ba9dbf63-3d82-44fb-8140-1bead474f821}} , and that {{formula:ab89ecd9-d9a2-4fbc-a352-063e53d3f96c}} is a Liouville fibration. Fix an {{formula:c3cbfa3d-77e7-4e94-b771-c265380c7a80}} -fiberwise symplectic collar trivialization {{formula:8c84a9e2-0ba7-48a5-8476-5c5631f99711}} of {{formula:73f2bc3f-ea0b-4784-ab74-73c1a0261b29}} . A symplectic monodromy trivialization relative to {{formula:24e9ef7d-eecb-4a19-abea-c4cdeb489812}} is a symplectic monodromy trivialization {{formula:86521962-affe-4a3d-9065-aa18c46af9fc}} satisfying {{formula:6061fd09-bcab-4d4a-9861-242a6a5db8d6}} for all {{formula:c175ef9e-7b4f-495d-80a9-267e8344c0ed}} . A symplectic monodromy relative to {{formula:28d03a45-3c01-4e88-bc0a-e80ab93332ae}} is the symplectomorphism {{formula:45638202-bc8d-4940-9841-af5dfa09c587}} for a symplectic monodromy trivialization relative to {{formula:6e63d9bd-b65a-46f8-a0fc-3dbac33e1b23}} . Observe that in this case {{formula:ec7a4ac7-59a2-4da8-9536-c9f5d0d8de1e}} is the identity close to the boundary {{formula:1063ba22-6541-4a9b-a618-b991c9dcd257}} , so if {{formula:3e5450cd-cc3d-43f9-85d5-67759654f5a3}} is exact, then {{formula:facb464c-1132-41c2-b59f-917efa1a47fd}} is an abstract contact open book.
Remark 5.6
Fix {{formula:1ee4ca66-89f8-4374-9d4b-0da38d09428b}} and {{formula:b0ed16c1-a8f5-4de4-a3f9-411cb0a1dd94}} such that {{formula:846bbee9-7683-4d88-8102-ac91880f3e83}} is a symplectic fibration. If a symplectic monodromy exits, then it is unique as an element of the group of symplectomorphisms up to a symplectic isotopy. Indeed,
given two symplectic monodromy trivializations {{formula:4166171a-2324-4d39-8826-009ac7b89574}} and {{formula:a872b02e-8447-45ef-8c68-fa87c92b7414}} , the needed isotopy is {{formula:6f6ca141-5c44-4b40-82fa-0d81fd352273}} .
Fix {{formula:9aef490a-6be7-418d-9a8c-b617929c9c43}} and {{formula:5d124419-f78e-4466-8167-6fbb22e0dcdc}} such that {{formula:93663107-df0a-4849-86d5-9996d5f5dd94}} is a Liouville fibration. Fix an {{formula:2186a795-7b90-4b42-a6f9-7d39b9b3fb0e}} -fiberwise symplectic collar trivialization {{formula:1ffabbc3-9e28-4e2b-b91b-eb3e4bdb1a99}} of {{formula:36a06bce-9300-4876-9135-a3c90a2ed61e}} . Then as before, if a symplectic monodromy relative to {{formula:33b82f14-0d35-4b76-bf26-7c30f3654bea}} exists, it is unique in the group of compactly supported symplectomorphisms of {{formula:54efe8cc-3645-47cb-b21f-c44c740171b5}} , up to a compactly supported symplectic isotopy. In particular, if any symplectic monodromy relative to {{formula:bc233cd6-6a8f-4ae2-af2c-43df637d9e62}} is exact, then the isotopy class of abstract contact open books {{formula:ec3a5fd7-7520-4cd8-a6cb-9c4700ec9bdf}} does not depend on the choice of the symplectic monodromy relative to {{formula:42434b00-1473-419a-ae41-b1974ca2623b}} .
The next proposition provides sufficient conditions for constructing a symplectic monodromy.
Proposition 5.7
Let {{formula:dfda0c0c-eb52-4ccf-bf9e-a887f3e8a2c7}} be as in Definition REF and let {{formula:d7a9e3a3-ccdd-4032-ab41-da7c476b0036}} be such that {{formula:3deb1c25-0e12-4c6c-8a87-c32e8b136b85}} is a Liouville fibration. Let {{formula:1fccd664-7dd1-4255-ac4c-38a99b143af4}} be a {{formula:687adc24-7afe-4a7d-9d73-469883573623}} -fiberwise collar trivialization that is {{formula:3cc8de55-e7a3-4b64-9ee5-a054e29c016f}} -fiberwise symplectic.
Assume that {{formula:52c34c91-da03-4ad4-96b2-f9a5f4764d16}} is cohomologically trivial with respect to {{formula:d3e55812-e104-47ad-9190-15349e63010c}} at every {{formula:d2dfe330-64f0-4e0b-bd2d-fad7f27e046f}} , see Definition REF .
There exists an open covering {{formula:340591f2-4d4c-4162-b376-cfec772fd0a7}} of {{formula:31c2b399-19c1-4017-8b55-4a1e93d90aba}} , and, for each {{formula:fd6dc4e7-f6f8-4094-baa3-eabd89d5c57e}} , a smooth map
{{formula:17521c98-2dc2-465b-a592-2be79d0389d7}}
such that for every {{formula:dbdbea1a-a931-4938-8761-2a48eead58fd}} we have {{formula:7975bf39-9a75-46e0-b487-333e7eeaa97c}} , and for every {{formula:f6a55194-3e5b-49f2-977b-2627c84454ea}} the restriction {{formula:3af55a0e-b705-4235-ac85-e29ead502576}} is a symplectic monodromy trivialization relative to the {{formula:8438b6cf-b6ef-46ca-9887-96e5f29a23ca}} -fiberwise symplectic collar trivialization {{formula:f94bd30b-5597-40e3-91d4-d4cc8255c2bb}} .
For {{formula:cf3cfb31-2f76-4bf4-ae71-8a2ab250a85e}} and {{formula:da8466f3-868a-4f53-a4df-95d86404d389}} , denote by {{formula:0a3acbe1-83e9-4572-91a6-f51e586c4e2b}} the associated symplectic monodromy. If the group {{formula:ef581e45-0e43-4b82-925d-628b6fe8e37a}} vanishes then each {{formula:a9634f5d-d499-4196-b134-e7b3a50162f7}} is an abstract contact open book. If {{formula:c09fc823-a68e-48ba-925f-8a2e970cc9a9}} is connected then {{formula:bab0b8ce-93b0-4932-953e-2d75dbdf79fb}} for any {{formula:77a90712-4e76-4c9e-80cb-05f03df2e4b9}} , {{formula:01d31c53-6a0c-4c8e-a486-6565409578e2}} .
{{formula:01725211-2ec0-4b98-ba67-9944b015d384}}
.
For the first assertion, we will use a parametrized Moser argument.
Since {{formula:503cb36f-4790-486b-9f87-736dea947a10}} is a fibration and admits a {{formula:48ca6738-d9db-49ef-97ea-478b47794e0a}} -fiberwise collar trivialization {{formula:7239bc31-e7b6-431d-9553-c350071da4df}} , there exists a smooth map
{{formula:402e9bec-c73f-488f-8e77-3d55e7fef95b}}
such that we have {{formula:e639e019-2779-4abd-b62e-8c5e3f5f24fe}} for any {{formula:7f830950-b2e1-4913-9233-a8aa38cfb332}} , and {{formula:3ae95fba-b3bb-4ee8-a07b-0360f808c14e}} for any {{formula:b8a77756-9bb6-4382-9b96-d8882953a65b}} . We view the pullback {{formula:4a7d2b85-c778-4a37-bdb6-96e46b846c09}} as a family of closed 2-forms in {{formula:1eb87033-7bc1-402e-9f4c-3b5649b3841c}} parametrized by {{formula:db66b574-5e56-4991-b4a9-30f780ece66c}} which are symplectic fiberwise over {{formula:edd20e78-7ac9-4891-9b59-b1cc9aa82b64}} . We denote by {{formula:7bc61ed8-5c07-450c-bdf2-c8ce46937215}} the restriction of {{formula:f39ad290-594a-4249-8973-04ffa941ba41}} to {{formula:cd4bfbf4-20cc-4c29-8eeb-472cd378d692}} . Since {{formula:c6007a71-6faf-4837-9569-4a81b6674565}} is {{formula:2086050a-43d2-4ab4-902f-294a102262eb}} -fiberwise symplectic, {{formula:ac32293f-f0bc-4990-89d4-c0d0e5e89cd9}} can be chosen so that the restriction {{formula:8a4008ac-7114-41b9-bc2b-43c56dc8c491}} is independent of {{formula:4a697690-32a5-4651-b8de-6d27003b6093}} for any {{formula:45746f1b-804e-4833-90be-6a0cc9308c2a}} . We follow the strategy of the proof of {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}, taking care of the fact that our construction is parametrized over {{formula:3622fdac-29b9-46e8-be26-0cd00b65b404}} and compatible with the collar trivialization.
Since the assertion is local in the base {{formula:73051883-4a45-4210-8fda-bb3cfab72f8c}} , we may assume that {{formula:52e0cb39-bf23-4688-9fff-232b5606b0ad}} is of the form {{formula:d9473009-f644-4534-85ae-206e2443b162}} . Then we can view {{formula:09d7bb75-1a51-4ab0-8b8d-7eb4d1ff1ee5}} as a family of closed forms {{formula:208d10f5-9eba-41a2-bd4e-0dc3b51b1c4b}} which vanish near {{formula:abc7825d-5716-4043-bba8-b5946892501d}} . Since {{formula:6dd71667-131b-457e-b580-470e6e2a70a8}} is cohomologically trivial with respect to {{formula:73ca8e49-796a-4250-af5d-d1613ac71bb1}} , the relative cohomology class {{formula:adeb9d3d-d423-4034-a27f-e4126df27cee}} vanishes in {{formula:4bdcd578-8f6e-4d63-9a16-31b487a83db2}} for every fixed {{formula:7df1452d-b3bb-4af9-bf78-c088e8b79273}} , and every {{formula:feffb939-9a75-40f3-8215-88ca4ce9bd45}} . Therefore, the class {{formula:72a8c90c-28d4-4bd8-b316-ba8b4b0afbfa}} vanishes in the same group. This means that for any {{formula:4d92e21b-435f-43bd-a1b1-a15fe00a4745}} and any {{formula:4168d42d-cfa2-4c5d-8394-bc396df4b3ff}} there exists {{formula:df2bd1e2-ed4e-4e27-a3c0-7fd7379b48ab}} such that {{formula:237a0009-dec5-45a2-bdbb-213bf1e69c9a}} and such that {{formula:79ea4ad6-a334-4db1-9f0c-41889db092f6}} which vanishes near {{formula:8f7ef448-9f47-4bc8-af00-3f787b80ff6c}} .
We will produce such a family of {{formula:54397186-ed48-4e7c-9a4d-e4908eff6726}} smoothly dependeding on {{formula:242089f7-0e35-45dd-bf99-e6c13a944e2d}} and {{formula:02dec72e-06c6-42b3-802e-e3f28c55cd42}} . For this, out of the two methods suggested in the proof of {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}} to construct {{formula:f569f72b-a3c9-4328-ac9c-4b2746596008}} , we adapt the first one, mimicking the proof of the compactly supported Mayer-Vietoris sequence in de Rham cohomology.
To be precise: denote by {{formula:fa16b671-27bf-4701-973c-3d1ab4c094a4}} the chain complex formed by families {{formula:977cc447-f771-4b66-96cd-83b34e7f2acf}} of compactly supported forms in {{formula:4d1075c2-5d6d-468b-8228-f95df6bf872a}} smoothly depending on {{formula:fb0c7d05-78a6-4abd-8cfc-ce4adcf99705}} . Let {{formula:9599199c-ccef-405f-9424-417528e66788}} be its cohomology. Define a homomorphism
{{formula:7bbace1e-c0b4-4c99-927f-86fc24f1e354}}
as follows: choose a basis {{formula:178961d2-a093-47ad-bea2-b3e8278dee9b}} of {{formula:5a921cf3-7809-41e4-9e11-1fdf4f69790c}} and define
{{formula:87cd917e-450e-4a41-a6ea-93747471c66d}}
where {{formula:8422446d-385c-4ada-bedc-0c997df34d04}} is the basis of {{formula:7d1271ec-926d-4d64-aa69-5024f41e3d69}} dual to {{formula:c3d19364-ff0d-45d4-bc43-f30c6aded058}} .
In order to produce the needed family it is enough to prove that {{formula:b057715e-7c1e-42e7-b904-1f98cf8ed9e0}} is an isomorphism. This is proved by induction on the minimal number of open subsets of a good cover of {{formula:2a642ddb-f2cd-42d6-81b1-09d1cdf276c7}} , where a good cover is a cover by contractible open subsets such that any finite intersection is contractible. Existence of good covers is established in {{cite:a10b521733a67e3443ab5c671580eb940ea4e2f9}}. The initial step of the induction follows the proof of the compactly supported Poincaré Lemma {{cite:a10b521733a67e3443ab5c671580eb940ea4e2f9}}. The inductive step follows the Mayer-Vietoris argument of {{cite:a10b521733a67e3443ab5c671580eb940ea4e2f9}}.
After the smooth family {{formula:b98892d1-5a41-48c7-bb9d-38539d41d80d}} has been constructed, applying Moser trick to the family {{formula:83e3c3ba-eec2-4a18-9dff-04d0fcdc42c1}} as in {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}} we produce a family of diffeomorphisms {{formula:04643f7f-4ccf-45c7-b398-e0a31805abc9}} that are equal to the identity at near {{formula:a41f11ce-2349-4c3b-b180-266d55c029db}} and such that {{formula:7745f9e1-1161-4bf9-bca7-eb58beb51acb}} for any {{formula:c4ff6b38-653f-42d3-aa93-0288b0c5e692}} and {{formula:f967b1a1-74d3-485c-a3bf-352e482fed62}} . Then the required smooth map {{formula:e4cfa2d2-0ecd-49b7-92ae-368b76cc69dc}} is defined by {{formula:c062b816-59e8-45f3-9ea0-76bb1d8b40fc}} .
For the second assertion, we note that for any {{formula:9744a4b1-ab67-4c9b-9730-a478dc4aec6a}} the monodromy diffeomorphism {{formula:d1b3c899-afa2-4b9b-9467-42504367b4af}} induced by {{formula:df3984a4-ecd3-4e9e-b112-7c23f3dfc6f2}} is compactly supported, so if {{formula:2a7a23f1-d4a0-4015-9733-f39c43f80324}} vanishes then {{formula:9dbb565c-16fe-4868-b8ff-fef7cf3868fa}} is exact. Therefore, {{formula:a7b599ab-ff27-41ff-a3e4-7a08bdbfcb8e}} is an abstract contact open book. Clearly, {{formula:e6ae8f6e-4c33-40e8-9dca-0a8b5eb0ed84}} if {{formula:a42c2faf-8f47-498b-b6ca-94789e39b041}} and {{formula:b1ba9b68-1b76-46aa-a79c-369a909d8ce1}} is connected. If {{formula:da8c20d1-11a9-478e-9a7c-413ae8978c3f}} then since {{formula:339af424-30ad-4e9c-8602-fe5614f16dec}} and {{formula:1c3c1a48-512b-4899-b733-ffb465401ee0}} are both symplectic monodromies relative to the same collar trivialization, we have {{formula:36f93638-c5c9-4e7b-af74-abd2186d8aa4}} by Remark REF .
Definition 5.8
Let {{formula:2342d87b-4e0a-4000-a392-e4aee97f50ad}} be a Liouville fibration, with a connected {{formula:ba48af74-4d7e-486e-960a-07ee23d6fbf1}} , and let {{formula:80f1f277-6abb-4186-b104-9209b39bbee8}} be its fiberwise symplectic collar trivialization.
The abstract contact open book from Proposition REF will be called the monodromy abstract contact open book of {{formula:2605138b-d545-4ac2-a358-8c5ab8d23f72}} . It is unique up to an isotopy, and exists if {{formula:db09c876-f60b-49db-a06a-9a64ebcb35dd}} for {{formula:aa57de11-5e65-4ffa-8703-c9aa8805c138}} . Indeed, the vanishing of {{formula:4cf2f308-fc10-45a6-9788-451142db27b0}} guarantees that {{formula:4584e4e6-2373-492c-8f92-55b68f45ad57}} is cohomologically trivial.
subsection2-.5plus-.7.5Grading and Conley-Zehnder index
In order to define a grading of the Floer complex, we need to consider graded abstract contact open books. We will now recall basic notions leading to their definition, following {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
subsubsection3.5plus.7-.5Graded principal bundles Let {{formula:8a0e71b7-cb5c-4d39-a3b8-4a6d477e12fd}} be a Lie group with universal cover {{formula:8a85c425-77d3-44c0-8b2a-e43aa26d51c9}} , and let {{formula:7fe37f3a-e170-4d43-8463-eac1e3f8f339}} be a principal {{formula:33370cb5-57cb-4fec-bc2d-1e75f199602e}} -bundle. A grading on {{formula:800253e5-6706-4d1f-9706-7582367b85a8}} is a principal {{formula:c31e400e-fbdc-4404-8fd3-15de23351dfc}} bundle {{formula:f36a0cb4-5933-4717-a485-350a628f1895}} together with an isomorphism {{formula:e67d8a05-c9b0-4955-ab3f-b8dc9596121f}} . Thus there is a natural covering map {{formula:4458136d-e0cd-4d08-a432-ca62634b61b6}} . An isotopy between two gradings {{formula:25adb8cc-44f0-4516-a2e2-c040e0704201}} , {{formula:bf276a02-3d59-416a-abeb-5bcde2ff6aec}} is a principal {{formula:edd10d37-439d-4d48-ab7f-f537af84a759}} -bundle over {{formula:9f89ce3b-9a16-4b57-9f03-41bbb4c017dd}} , together with a grading which restricts to {{formula:172a4829-71fb-4598-816a-899cfabb4504}} over {{formula:311b8fc7-88c5-4917-9a3e-cf3f34204a6e}} for {{formula:c8ec74a3-2e21-4572-bc76-6fcd96fd36fa}} . Choose a base point {{formula:6694752f-bfda-4993-9c05-adb2eaa27768}} . Then by {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, there is a one-to-one correspondence
{{formula:66e1f760-2501-42e4-927f-eb386148cf90}}
sending a grading {{formula:9689b014-ad96-4543-8500-128c0f9763c4}} to {{formula:508c7efe-ad9f-4659-bd47-dcb5af5f1ae1}} , where {{formula:ca9d8492-743c-432f-9bb3-0e8e0092749e}} .
subsubsection3.5plus.7-.5Graded symplectic manifolds
Let {{formula:c01bd3b7-5269-4831-8eee-926509000757}} be a symplectic manifold of dimension {{formula:65b9b8ba-7697-4095-9d74-67803eb469b8}} . A grading on {{formula:60100eb1-81fc-4c01-9376-64ae8edae82a}} is a grading on its symplectic frame bundle {{formula:6e7906d6-c6ba-4c14-827f-74758bae13be}} , which is a principal {{formula:6c6411ed-4c50-457b-9752-1076d31da137}} -bundle, see {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. Choose an {{formula:38e08adb-08b9-4080-8bfb-d3185f43ec36}} -compatible almost complex structure {{formula:d5f273cb-e641-40de-95e9-e7f28fbb38ea}} , so that {{formula:f79d6fac-9648-4d37-b171-eccd28197488}} becomes a complex vector bundle of rank {{formula:256cc649-8b95-4d33-9e6d-3ff0b517e995}} . Let {{formula:4243f674-b6d4-4691-9cb7-f7778664d383}} be the corresponding anti-canonical bundle. The natural maps
{{formula:72162a69-76d5-4bc9-8d7a-5d88eef4bd49}}
induce isomorphisms between fundamental groups, hence (REF ) gives a one-to-one correspondence between gradings on {{formula:6accfbc5-04bb-4af2-ad63-ef857c962f33}} and gradings on the {{formula:67e8d633-7b56-4ebe-8f3c-1ce0d1ebae55}} -bundle of {{formula:9e202175-b1db-47bf-96ec-2a37f5822344}} . By {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, the latter are in one-to-one correspondence with homotopy classes of trivializations of {{formula:2dc71491-6d61-42e8-86af-0783f5cdb1cf}} , or, equivalently, of {{formula:fb94b4e9-a403-462f-b865-ea870f07185e}} . Recall that {{formula:6223fde4-5c8d-4654-ba35-855c798ab0d8}} is a complex line bundle, so its trivializations are given by nonvanishing sections. To summarize, we have a one-to-one correspondence
{{formula:976b042a-09ea-46c9-874e-8cbec8a43fba}}
Likewise, given {{formula:948b6184-0fa0-45ca-a151-05890acbab1f}} a symplectic fibration with fiberwise symplectic form {{formula:9347c5b3-5d3a-4753-9fa4-154fb60d6a57}} and {{formula:ac8ffd0c-feea-49c2-a611-d58021d87e02}} a {{formula:ae24913a-2a23-4793-9f74-d81e627d5324}} -compatible almost complex structure, let {{formula:c45ddc24-f203-44ee-bee8-82899568c5e5}} be the vertical symplectic frame bundle, and {{formula:9d92d285-6dcc-47f2-8215-9d98a3d07370}} the relative canonical bundle. We have a one-to-one correspondence
{{formula:f5782eda-7da3-4861-96e8-aeb5e17bf740}}
subsubsection3.5plus.7-.5Graded symplectomorphisms and graded abstract contact open books
Let {{formula:6b6d4660-41c4-4b09-8a23-29f66f9ef7a9}} be a symplectomorphism. Let {{formula:70f6055f-de84-44a0-8f82-b2c21f099673}} be the induced automorphism of the symplectic frame bundle. A grading on {{formula:664696b2-7672-4717-bde1-c91cedeff142}} is an isomorphism {{formula:c2b0a8e9-b35b-4b41-a421-caa2d4b3e4da}} which lifts {{formula:8967af88-5fe1-4cf7-8962-b7c38f760ad9}} via the grading isomorphism {{formula:690a146a-fecc-455e-8579-fcc0893a4a26}} , i.e. satisfies {{formula:3e9a70ab-3709-46df-8c13-5ab90f849b2c}} .
A grading on an abstract contact open book {{formula:68291b23-33fe-4583-85ed-960e158f7f1f}} is a grading on the symplectic manifold {{formula:30509170-9364-4852-8e87-4565e74f3504}} and a compatible grading on the symplectomorphism {{formula:eb60ce6c-9764-481b-9f9d-1c97a26461c2}} . An isotopy of abstract contact open books {{formula:2bb7b4c6-619e-442d-8887-cd2d548ca400}} is graded if there is a family {{formula:0dddb207-e908-4f5d-8ea1-1074cb72a93a}} of gradings of {{formula:e674d125-0264-43f6-9d37-ba2e977cec7d}} depending smoothly on {{formula:373a5b1b-f7df-4ab9-8cf4-4978e26fe6e8}} .
Lemma 5.9
Let {{formula:e6e04990-acad-4603-8cc7-5cbe6a1ed977}} be a symplectic fibration with closed fiberwise symplectic form {{formula:150dfdff-991a-44be-851b-48344f7c48e7}} , and let {{formula:96634e46-7be5-4748-bb05-21d531a2060d}} be the vertical symplectic frame bundle. Let {{formula:b30ab9d7-2703-4364-9784-0e00c6561c7e}} be a map that preserves fibers over {{formula:841b92de-36e8-4e63-9731-85dc75eeb404}} , and is a symplectic monodromy trivialization fiberwise over {{formula:ca9daca8-68ae-4f7b-b089-32919a5c5a80}} . Any grading {{formula:4bac0658-69df-4bf8-8091-64340464ea7b}} of {{formula:e3d10322-ca33-46b9-95ae-79290d7dbf7f}} induces a grading {{formula:4a6bf293-e731-44e0-af1d-b69d70d48a04}} of the symplectic monodromy {{formula:5e8876f6-cdfc-452d-8ad6-eca6a9b99561}} defined by {{formula:100ca818-79ce-436a-bf50-fc5096411721}} , which depends smoothly on {{formula:2eac98ed-4ff7-4f0c-92f7-5282baa3b16a}} .
{{formula:188d8d05-e878-4113-9bc6-d97bdfed63f4}}
.
By the symplectic monodromy construction, given any symplectic monodromy {{formula:09d06f31-39c3-4452-ae2c-232635e14415}} there exists a closed fiberwise symplectic form {{formula:2c507434-e50c-4217-af9e-31454f753b96}} such that {{formula:0a90fd68-7ff4-4cde-91ae-f69982c92484}} for all {{formula:63e0fb62-ac37-49ce-9e17-f83043360767}} and {{formula:1b7a88f0-96a5-4027-a3ce-b45fc09e4350}} is the time 1 flow of the lifting of the unit vector field on {{formula:2c3ac162-8a0a-481d-b038-f254c8f78f16}} on {{formula:f5fbb7b8-03c7-4717-abbe-da4ec4cb8e5a}} by the symplectic connection associated with {{formula:c60ff060-a44c-4a14-97df-738a0c0668d2}} . Indeed, we define {{formula:af061f5a-e127-4c47-9dc8-ad9d3bacc05d}} . Since the statement only concerns fibers we may assume {{formula:6cc4bfe9-d29b-492b-a5b0-28befd88f274}} for the rest of the proof. The symplectic connection defines a {{formula:9b3d6c07-fe40-42d5-8624-828703ef0ecc}} -invariant connection {{formula:8a260992-f249-4acc-8a1e-0bf5fb4f022d}} in the principal bundle {{formula:8a2d0a18-7485-4346-8bd9-3cd2c7ec1c1d}} . We lift it by the covering map {{formula:de284574-b9af-425d-bcac-cbf0fb58e519}} and obtain a {{formula:333033e8-924e-45e7-9963-d22d4792f945}} -invariant connection {{formula:bf09ad15-df45-4aac-a376-c65585528dcd}} , on {{formula:5301ed6c-84d7-47db-91ec-b38724d98561}} . The symplectic connection, together with {{formula:694f51c5-d4e4-4291-9c56-1fdb435c8486}} induce a connection {{formula:e33d0f2e-6376-42ed-ad69-fa820f57a6cc}} on the fibration {{formula:d1f9555d-488a-431c-9d34-4da8886079b4}} obtained by composition. The required mapping {{formula:6efa85bb-7218-4560-879b-7bfb73eec348}} is the time 1 flow of the lifting by {{formula:8755aa64-0e80-41be-a2e2-d6397a0cf6f5}} of {{formula:e4f718fb-eeb7-47ab-a208-9c1fb842bf5f}} .
Corollary 5.10
Given a symplectic fibration {{formula:fe9979d9-8e75-4d21-b90e-9d961aca838a}} and a grading {{formula:1a11e69e-340f-4135-8328-df66b3862290}} of the vertical symplectic frame bundle, any symplectic monodromy inherits a grading, and the graded symplectic monodromy is unique up to graded isotopy. Furthermore, in the situation of Proposition REF , a grading in the vertical sympectic frame bundle of {{formula:e96404f7-45e0-438a-bd46-3ba63b84a946}} induces a grading in each of the symplectic monodromies {{formula:ca5ba682-b689-417c-9b44-c493efbe0eab}} , and in the isotopies connecting them.
subsubsection3.5plus.7-.5Conley–Zehnder index
Let {{formula:2f8725d6-8870-4cd9-8240-32803e24a6c4}} be a graded symplectomorphism. Let {{formula:6d708f0d-17be-4e35-a8b1-64006e86d63e}} be a fixed point of {{formula:75b3dd22-bca6-415a-bde8-74ea3145811b}} . Then the grading induces {{formula:84b9e6c5-4cc1-44eb-b451-eb2f26fb454b}} sends the fiber {{formula:b7c672fe-7ab9-437a-a05b-075ccf442df0}} over {{formula:89738694-3002-4db8-a3a3-de76964f71a2}} to itself. So its restriction {{formula:abda7156-1d99-4919-9782-9924a1ff8564}} is identified with an element of {{formula:6a5989a1-5a4c-48a9-94c4-192b45972bcf}} which can be viewed as a homotopy class (relative to the ends) of a path {{formula:aea5375e-c1ac-46d9-b5f3-4066f2fd069e}} of automorphisms of {{formula:2b6c1f97-251c-4961-828c-c3138b133f10}} , starting from {{formula:45dcd97b-377d-4d73-9843-237f24648e14}} . The Conley–Zehnder index of {{formula:8ff25470-99a4-4fea-afc2-927f31cc5920}}, denoted by {{formula:951b7ada-ccbc-4770-a6d3-0d89eefc0d49}} , is the Maslov index of {{formula:d5627cba-f5bb-45fb-b9d5-e79e78d5be38}} . we refer to {{cite:025c28c491ae8ba339649b2b63ff3d4a3dae68c0}} for its definition and {{cite:7aec0adf2b1ebf6cf76a67be588ba68490afb6a8}} for basic properties.
We will now give a formula for a Conley–Zehnder index of a symplectic monodromy with good properties. Let {{formula:184e7d65-0f11-4962-bca4-b441397703e8}} be a symplectic fibration admitting a symplectic monodromy trivialization {{formula:d2c53a83-9023-4133-9736-5dbf7c257180}} . Let {{formula:ed0352e0-be5f-46a0-9c83-b19ba2d2f353}} be the symplectic monodromy, see Definition REF . Let {{formula:57411e37-b793-4c12-a8b6-3a61f7541021}} be
an almost complex structure on the vertical tangent bundle compatible with {{formula:bef3a1eb-652f-4d90-8b7d-eb3c915cfabf}} . Then {{formula:80425481-ce05-4f10-babb-bda455c5e95a}} is a complex vector bundle of rank {{formula:8b78597a-03f1-4d03-9c28-cf71a5dc6530}} . Let {{formula:fc72ff43-9889-49de-bbe2-67badf79b23f}} be the relative canonical bundle. Assume that {{formula:27f29636-a7f5-4a34-8b28-4224209edf22}} admits a nonvanishing section {{formula:3110ab8e-fdde-4472-9017-4d3912e4fc56}} . By (REF ) it determines a grading on the vertical symplectic frame bundle {{formula:6cea7dad-095d-40b9-ac10-3dbf05cb453e}} . As in Section REF , we get this way a grading on {{formula:21e6960f-baa3-41b9-96b1-98ada65456e3}} , too. Let {{formula:8f49f6d5-15e8-4aa7-8464-1890a28ea52e}} be a fixed point of {{formula:08917e49-6bae-478a-b773-ba32b4953505}} . Assume that {{formula:b5f4f97b-ed8e-4b71-9989-6c1c604118c5}} . Let {{formula:2901b5b8-f275-4f13-b30e-7efcf8105ae2}} be the winding number of the path {{formula:153448ff-db7d-43f2-9971-8331d84a5dea}} . Now by {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, the Conley–Zehnder index of {{formula:5b84260b-fdf9-47fc-a631-82075cb9cab2}} with respect to the above grading equals
{{formula:e1274be8-f78d-4c88-8945-85c7e9b11986}}
We will also need to see how the Conley–Zehnder index changes under a small Hamiltonian deformation. Let again {{formula:632915f8-978a-420b-85d6-d13e714fb9ae}} be a graded symplectic manifold of dimension {{formula:79def850-f0b8-4526-b278-a65f04eb457a}} . Let {{formula:913416fa-24e0-4bbb-9591-82e36978b88b}} be a time-independent Hamiltonian, and let {{formula:69ff5689-8839-4692-b3d7-046e3cb4dbc4}} be its time-one flow, see Section .
Assume that {{formula:8c5d797e-fa31-4421-8ef0-1856e42c0e67}} is a Morse point of {{formula:4108b5b1-30fc-46dc-bf40-b1dc98cad232}} , of Morse index {{formula:fec8a58e-be11-4ccd-9584-6404aa90f532}} . Then {{formula:267010de-d5ec-439e-8aca-4e273992117e}} is a fixed point of {{formula:f639b7b5-408f-490e-8d88-af25a196111b}} . Assume furthermore that {{formula:094ecd25-a518-4375-89d2-0788ab57ba0a}} is {{formula:49b4b8f9-85b8-4d7d-8e2c-09e06ed56964}} -small at {{formula:d792d8a1-1edb-43a7-9bca-3519cc0c7ca1}} , more explicitly, {{formula:15f00f58-c4a3-4295-9b94-de2e7bcb9fb1}} . Now by {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, the Conley–Zehnder index of {{formula:a9328756-e0d0-4e7c-bd9b-972349143dcb}} equals
{{formula:9d046479-ea06-4dd7-8ac6-1351338e7081}}
Note that the formula in loc. cit. differs from the above one by a sign. As explained in Section , this is because we use the opposite sign convention for definition of the Hamiltonian vector field. Therefore, to get the same {{formula:db393c80-2ded-4b93-8cba-7284bbfa0f61}} we need to apply loc. cit. to {{formula:f4ae5e1b-84d5-47ef-a774-cdfd868c8027}} , and the Morse index of {{formula:5fcbcfb1-2652-4b85-bee6-27b2507c8787}} at {{formula:bcec3eaa-4374-4936-9247-aee2ec9e41cc}} is {{formula:d6b5fd09-02ba-4c93-afec-31cdc91eae6d}} .
Now, let {{formula:7cfdb63e-d84f-40eb-95c7-a18442447201}} be a graded symplectomorphism. Assume that {{formula:35bb5995-5e9a-4cea-a422-5cbef4ceb7e9}} is a fixed point of {{formula:6e805e93-f399-4286-b744-0020222918dd}} , with Conley–Zehnder index {{formula:d604b94b-0372-4c4d-a50f-e3861b57b288}} . Put {{formula:ccc70c2f-aacc-4a37-b0cd-54610f24e8fc}} , so {{formula:9bed9fd1-9e15-45ed-9979-525b915ca267}} , too. Since the Maslov index of a concatenation of paths is a sum of their Maslov indices, see {{cite:025c28c491ae8ba339649b2b63ff3d4a3dae68c0}}, the Conley–Zehnder index {{formula:a429e15f-acc5-41e1-9227-ff0f2c22d03f}} of {{formula:bd28840d-110d-4770-9e36-1ccd1cab86fa}} as a fixed point of {{formula:d1443d64-c9bc-4325-bf7e-40c3406e3a53}} is the sum of the Conley–Zehnder indices of {{formula:2f2e1fc3-cabd-4bd4-87bb-8d1d7497b52f}} as a fixed point of {{formula:993e5666-e46f-4928-a232-8faa0dbdb1d0}} and {{formula:89234e26-0282-467e-9e43-ea1f01383631}} , that is,
{{formula:92ac4ea2-87b3-47a6-a31c-9a5e2ae460b4}}
subsection2-.5plus-.7.5Constructing symplectic monodromies
Setting 5.11
Let {{formula:1198fe66-41bc-48a7-aa70-a0e64a8e512d}} be either a complex variety (possibly singular) or a smooth manifold (possibly with boundary). Let {{formula:906b9616-a072-481f-9d13-46b3ed97a401}} be a compact domain in a manifold, and let
{{formula:bb5c20d4-3def-4015-ac35-d2c530f74ad8}}
be a smooth family of regular maps {{formula:a99c9874-ff5c-4968-9513-6a9c0f292fb6}} in the case that {{formula:efdcace9-b89e-480f-9100-fa16f6ea5af9}} is a complex variety, or smooth maps if {{formula:0804fdf3-f874-4149-a307-612e9b4862b7}} is a smooth manifold. Let {{formula:5cf02dbb-a8c3-4fe8-ae39-92d05009714a}} be an open subset whose closure {{formula:f1eb35b4-5b85-42ab-9a33-9df3969cecaf}} is compact and whose boundary is a manifold, and such that for every {{formula:ea66d021-080e-4c76-b3a2-09a6bb83fbe3}} , we have {{formula:6005f7ca-2dd6-4a02-a123-94232869dd1a}} . Assume that we have open subsets {{formula:dfd2637d-b9d6-411e-803b-ef41c032014e}} with the same transversality property and such that {{formula:3a45ba9f-38cf-417b-a730-52842c0cad9a}} . Assume that there is a {{formula:da21a032-8a33-4fe6-adb7-e42b2a6eebcd}} such that for any {{formula:ff468256-b9a2-46f4-8b94-e55d102813f3}} , the fiber {{formula:a0d3a06f-39da-4fa3-84a9-23fd1bec8cca}} is smooth and contained in the smooth part {{formula:60821f60-7507-4b3c-a6e8-80b46ecdf8d8}} of {{formula:6d65c883-a9c9-4acf-ab59-f90bab22b5be}} . Then, by Ehressmann Theorem there exists {{formula:01d85bb0-9632-4e7b-a404-6d16ee1e5980}} such that
{{formula:80d5974a-a330-40fe-996e-daeaf04c1a59}}
is a locally trivial fibration of pairs. We put {{formula:b148cc49-0f3b-4c98-8f0b-c16f3c1addf9}} . For each {{formula:31ded077-706c-4aa4-b3a5-b6c9ac4ab3c2}} the connected components of {{formula:8d768ce4-95e4-4e1d-bdb0-d70bfecdb415}} cannot meet {{formula:b732ab5e-7716-4b70-b80e-fe81be4974bc}} and {{formula:d5e3bca8-503a-4d1b-a2fc-87ceefe04cc6}} at the same time. We call the components meeting {{formula:3aa5227d-b1ea-485e-bdf0-2f50a054837c}} the inner components of the fibration (REF ) over {{formula:e49de282-d3b0-4c44-8627-4c51267be18f}}.
The reader may have in mind the following example: the function {{formula:446250cc-6985-4db9-8f4d-254f148ba793}} is {{formula:a4a28c98-031b-4d38-ac09-a96ec46f5904}} , a family of holomorphic functions with isolated singularities, and {{formula:7abe17d1-093d-4cae-a0e2-31a4aa63b2c8}} , {{formula:fc48d377-4440-4003-8332-d790d3e43f93}} and {{formula:99fbf14a-afb5-49c4-b9da-82b3431777e9}} are products of concentric balls in {{formula:085cd22e-d8a2-4a07-9a5f-811eecde63d1}} with {{formula:36cd842d-7ccb-433b-b404-4b2c9e146c3a}} , such that the only critical value of {{formula:1551ab71-fade-4a7f-b283-f979bd0a25f8}} is 0 and {{formula:96c8478e-8293-4e00-bb6b-a9d61f11f0b2}} is smooth in {{formula:b4c6c16b-b39a-441d-8be3-fe0050b99484}} , for instance see Example REF below.
Let {{formula:c79babc1-3f28-4edb-9e05-c09085ee18da}} be a compact sub-domain. Let {{formula:325e6f70-4672-42ab-8cca-a3b807861005}} be a 1-form whose restrictions to {{formula:730655b2-6b85-45a5-a6a8-f7cc1d41d33c}} and {{formula:089a23e7-9145-491b-aeb2-d564fdd3b18b}} for {{formula:c3e10119-3093-4f96-b32a-6f03c4bc2a47}} and any {{formula:41887277-9b73-4c52-8161-9926e5bb0380}} are Liouville. Assume that the Liouville vector field of {{formula:5a22e000-10da-49ff-816e-2c46261ad0e2}} points outwards {{formula:1d7010ce-7313-4a0e-b132-6a77855791f9}} . After shrinking {{formula:e91a03af-1470-4d5c-bfde-e0c355cc165c}} around {{formula:4c95d6c1-629f-4fb8-8d69-ad6a050d713e}} and {{formula:f4b444b9-5b54-4bde-8f07-3bb290cf3620}} , if needed, we can assume that the same holds for every {{formula:3d965ed2-fa55-4282-a388-4076c1a6991a}} , and that the fibrewise Liouville vector field on {{formula:c7440646-1740-4a6e-9308-a3f87b6ed9c0}} points outwards {{formula:43ea3865-a330-4046-81f2-975f5c42f47c}} for any {{formula:49d14ff2-3303-45cc-89b1-1ca39b409b8b}} .
Then
{{formula:1e6b3b73-9773-4703-805b-411fd38a4fe3}}
together with {{formula:917d4839-6a32-4786-aa3d-5584fdfe068a}} forms a Liouville fibration.
{{figure:4d8f1e21-b7c2-4bfb-bc76-1b5baab25f2b}}Let {{formula:9ed9757e-69ed-4356-9125-bf81e46cc4ee}} be the polar coordinates in the disc {{formula:c98256e1-1455-4afe-8bb3-d0d93386b2f5}} . The radial vector field {{formula:6e00b1b6-5996-40c0-86b4-3e8d8c22ef21}} in {{formula:0108fa35-90a2-445a-af1c-900956a9b4c7}} does not extend to a smooth vector field at the origin, but it does extend if restricted to a ray {{formula:4756db31-26f6-462b-966d-5ee5a68693b9}} of constant argument {{formula:77e7cf34-078b-4022-8cf2-e62b0926eb02}} . Let {{formula:a59c27f9-71b0-4a3b-8858-43574a2375ca}} be the symplectic lift of {{formula:e3379b71-ae54-49c3-a229-0366b8140a5f}} in {{formula:bf26aec4-60fc-4389-8bac-ac0b7f121148}} . Like {{formula:002f181f-961f-4ca6-bae7-d36432a9e79a}} , the vector field {{formula:fc7d3150-8b2f-4004-a1c7-76a5c8b18e63}} is not defined in the collar {{formula:fabb84e5-321a-4edd-92c3-a7ec590d77b0}} , unless we restrict it to {{formula:4aaa3833-19ff-4c2d-9819-5e42cc3a35e1}} . Denote this restriction by {{formula:fcd46958-3a07-4a93-9932-d94695959c7f}} .
Let {{formula:2b007318-a083-446a-85d2-6036eed7328b}} be a compact fiberwise collar neighborhood of {{formula:e26aa043-a38b-4b92-9922-76c94e6d25a9}} in {{formula:027743d6-e1e9-455c-a36b-1dc1952f1f28}} . By compactness of {{formula:74b60452-5e87-4dc6-949f-04fc894aa1a8}} , there exists, {{formula:84d7794a-873c-42a0-a38f-6636afe93b08}} is small enough, so that the integral curves of {{formula:15f7fa07-6353-48f3-858e-472018ca87ca}} with initial point in any point of {{formula:611f2f4f-58de-4d97-8195-ce31806c71e1}} exist for all times {{formula:2605280b-cf62-48a0-aff0-d2647b22baec}} and are contained in {{formula:50829226-c4a5-4071-a295-4952218b94f4}} .
So we obtain a smooth flow
{{formula:28d1b4aa-5cbc-471d-98b2-b484ffd9e2ab}}
such that {{formula:c664c702-14c6-4eb2-bd84-239602e59c36}} is a symplectomorphism onto its image for any {{formula:5971e53d-8945-42ef-a48f-aafe8e2a606b}} and {{formula:c7a51cd6-3468-4031-b7b3-60521628b4e0}} . Put
{{formula:8c063afd-b690-44ef-80dc-0952cc96d234}}
Then, for a fixed {{formula:a483e75c-feca-47da-ae90-c1f2769e2cc3}} , the map
{{formula:183e4fc4-edbe-47a8-8710-b9c5dfee5754}}
is its {{formula:0b02b872-3b24-4a84-9030-ab92cd852d50}} -fiberwise collar trivialization which is {{formula:10f2e20b-8b39-4078-9557-fd33c18d9d7f}} -fiberwise symplectic. On the other hand, after possibly shrinking {{formula:7cccb758-6b46-4eb1-8d20-b01adaf9905f}} the map
{{formula:7f6b11dd-8aa8-4017-8c92-18665f858cc4}}
is a locally trivial fibration of pairs, that coincides with (REF ) over {{formula:43180628-7acd-4ab6-b1f9-47d6bfe68cf5}} . Thus, we still have that for each {{formula:fbaac232-6bb3-4e89-8e03-37873d37dbd0}} the connected components of {{formula:981c0049-4f8f-4810-ac96-59828ae5a83e}} can not meet {{formula:3adc57ca-8211-4995-bcd6-c31b47a9b012}} and {{formula:b1d6f866-62e3-4fac-813e-3bfd361d2916}} at the same time. We call the components meeting {{formula:7ff149fe-01f7-4e70-ace6-999832f40010}} the inner components of the fibration (REF ) over {{formula:35f4cae7-b02e-43f5-9511-1d682e5c788f}}.
Define
{{formula:acd785f4-ae41-48f2-8058-7ce0bd73c82d}}
as the locally trivial fibration whose fiber over {{formula:7da75d9f-85ee-490f-ae24-02cf2ca7616a}} is the union of {{formula:aa8d8cdf-ee76-4b96-82f8-07d1d412edcb}} and the inner components of the fibration (REF ) over {{formula:1ce95256-1427-4954-bf6c-53611b0bd7bb}} . Since the fiberwise Liouville vector field points outwards {{formula:33a91cba-3584-4436-be6e-b9874655ac8e}} , shrinking {{formula:990b7646-5ce0-4fdd-9c72-ebda4cd9bd49}} if needed, we can assume that it points outwards {{formula:a30efdee-bd39-43aa-924e-a67aaf903289}} for any {{formula:1d189c8a-f1d8-4756-9d98-94925683a6a8}} . Since {{formula:ecb13ee9-789e-4750-9b88-d40a3fbf159b}} we have that {{formula:e141f247-6613-47f9-835f-9d652226fe5e}} is a Liouville fibration.
Remark 5.12
Note that the above construction of the collar trivialization works equally well if the target {{formula:3964b4b4-f09a-44bb-b815-2eb3b5d6c906}} =[0,)S1{{formula:66801b35-d1f4-4bd7-bdbb-4610f64a6d3c}} r{{formula:2a0da0d4-a8b7-4755-b33b-4c6608d1440c}} X{{formula:28bc006e-0140-48c3-9ac2-bc4b4ee81dd0}} R{{formula:c4ff39ed-44eb-4256-87da-5b05161e5594}}
It follows from the Ehresmann Lemma that the fibers {{formula:d11b81a0-0474-45ac-8109-9751872da78c}} and {{formula:c710ac21-a159-4d3a-98e6-0b3952ac5cca}} are diffeomorphic, and their diffeomorphism type does not depend on {{formula:e716b261-3d4a-4629-a9be-562a75e7a05d}} . Assume that for some {{formula:a6ab096a-b88d-4bf3-8044-e0cddadcdcd0}} we have the vanishings
{{formula:bb1e784b-3827-4459-8ad4-41d2cb8e345d}}
Then by Definition REF , we have monodromy abstract contact open books
{{formula:278e417c-37d3-4e33-b4dc-111ba6fb8c02}}
whose isotopy type does not depend on the choice of {{formula:91cf92e9-346e-4fda-a9c6-965e634fd103}} .
Fix {{formula:959da5d5-7d06-485d-be6e-3b0ee4a1ee8c}} . Assume that {{formula:0b12973c-d700-4ca4-8efd-c88a4dd6f677}} is a complex manifold such that
{{formula:5aa4f6d7-c40c-4c08-84f8-6378b7777092}}
Then the abstract contact open books (REF ) are graded, as follows. Since {{formula:bcb10382-ce0e-4b4b-a362-4d4ed154ab9a}} , the canonical bundle {{formula:5a4f3e4c-201a-4550-a45e-0d6798109512}} admits a smooth, nonvanishing section {{formula:e51bfbac-d21d-439c-9c1b-301d81ed5980}} . Let {{formula:19a2b912-5de7-48bc-91ea-cb4fd420caf6}} be the vertical tangent bundle, and let {{formula:0bbc230f-30ee-45ff-bb62-ce5d137d5af4}} be its top exterior power. Then we have an isomorphism
{{formula:a4fd4171-d205-4f05-89cc-49fc58fd47fb}}
Under this isomorphism, {{formula:9f1a075f-05d4-4e94-b1c7-e540404c1a01}} is mapped to {{formula:081849f9-3846-4dde-817a-f32e671f3a46}} for some smooth, nonvanishing section {{formula:b3a0b3a7-59f2-4e32-b8c0-6021785feb0a}} of {{formula:c3987a31-82f7-4c3d-8d5d-3c218a70e057}} . Now {{formula:d9fd9cad-4bdf-4680-ba21-5e047bc95a9c}} trivializes the canonical bundle of each fiber, so by the correspondence (REF ) and Corollary REF , it makes each {{formula:594ff988-5567-46ff-9505-1aed3ae54ee5}} a graded abstract contact open book, whose graded isotopy class does not depend on the choice of {{formula:bb06dfe4-f5c0-4d3e-931c-cb02f57a9fa6}} .
To summarize, if in Setting REF the vanishings (REF ) and (REF ) hold, then the above procedure defines monodromy graded abstract open books (REF ), all graded isotopic to each other.
Remark 5.13
In terms of {{cite:d20ff9e1ba10f8d7a97dd150ef78cb99d18bc788}}, the restriction {{formula:a0ca55fd-f947-46f2-ba06-19d0616a938b}} is an exact symplectic fibrations with singularities. In turn, the restriction of {{formula:8ae19af8-9c01-412a-9a60-fc7313004797}} to {{formula:eb698f0d-4cf8-4710-bee4-d56c90502698}} is a Liouville fibration admitting a collar trivialization: such fibrations are called exact symplectic fibrations in {{cite:3417f296cf2b82b4ff749de60ebc59d1eab13d83}}, cf. {{cite:d20ff9e1ba10f8d7a97dd150ef78cb99d18bc788}}.
A typical situation covered by Setting REF is the following:
Example 5.14
Let {{formula:26fe042c-4b9a-469e-bc5b-964ecd1fb3de}} be a strictly plurisubharmonic function. Fix {{formula:5a052fc4-5e66-439c-b0bf-2321fbaef846}} and consider a closed analytic subset {{formula:5772aff7-7eaa-45d9-8ae6-5fe1cce99015}} . Let {{formula:fd46ea76-4cdf-4681-bf0f-b5ec4958b00d}} be a family of holomorphic functions with isolated singularities in {{formula:195a88da-d437-46fb-890b-5803be553376}} , holomorphically depending on a parameter {{formula:94d91ad5-90be-434d-8edc-07baf1c06f00}} . Assume that
{{formula:7d4bb2c9-bd97-4408-925a-0994a212bd69}}
Let {{formula:eb21cb06-60e3-4633-acf3-920c476d7de5}} be a family of 1-forms smoothly depending on a parameter {{formula:36d0a667-9786-43a3-93be-c53da308d2bf}} such that
{{formula:c663b668-daff-4b49-93ea-5c7cc24696b6}} . Put {{formula:90aec1a4-2f09-4123-b22c-45d6ad432fcc}} and define
{{formula:9f383ca8-3ac7-4128-baa5-f03f571f30a3}} by {{formula:e796f4f6-18d9-4b6b-be4c-fc64c0d4720f}} . Let {{formula:6dc34ec1-4858-466f-b782-a7cb65aae449}} be regular values of {{formula:2e12540f-d8ea-45c3-8e0d-4efe87b984db}} , and let {{formula:8557cb00-8ac6-4a02-b78b-0d3d14bed538}} , {{formula:d88bf68a-fdb8-4e9e-8be7-d3eed282d095}} and {{formula:6c849f4f-f43f-471a-a7d7-dddb30d22ef2}} . Then if {{formula:94aba8d9-c8a7-4705-8ce6-f427c7205085}} and {{formula:9010dcb8-3632-4200-8424-30a00ebcc1a4}} are chosen small enough we are in the Setting REF .
If {{formula:761aa605-3dc3-496d-8a5a-a3cfdedc45b0}} then the smooth fibers of {{formula:1861ec72-6958-408c-98a4-dad05c8b50a3}} are Stein manifolds of dimension {{formula:9092b327-b28f-4d8c-94cd-eeb4b8e1cf91}} , so the vanishing (REF ) holds and we get abstract contact open books (REF ). If {{formula:35c2611a-6c23-4922-ad44-c6503e5be62a}} then these abstract contact open books are graded.
Remark 5.15
In the situation of Example REF , if {{formula:810ded5f-97b7-4fb1-827f-fa68ab8784bc}} is smooth and near each critical point of {{formula:c6a6bea0-9ce9-4bb6-8626-11772003433a}} and the family {{formula:8dfb1687-381d-42c6-bccb-9e6061a4014f}} has constant Milnor number, then condition (REF ) is satisfied.
Recall {{cite:edcbc0062181eb9053bb57ecfae662692e9a247b}} that given a holomorphic function germ {{formula:457439ba-1dfa-403b-b9ec-de03091f8cf0}} there exists a radius {{formula:6d0f0d01-289c-4cd2-8732-09e547f28334}} such that the ordinary sphere of radius {{formula:7af05d7e-4021-4e5d-93bd-47f5bd09fdf9}} centered at 0 is transversal to {{formula:b1eebd70-0917-4370-9f08-981deece658c}} for any {{formula:93a48961-f893-41eb-829f-6f9423679adb}} . Such a radius {{formula:7c9f4858-2b7e-4817-8f12-ec588a50d169}} is called a Milnor radius, and the balls {{formula:c61230a7-c357-43bf-98a6-ca22d4e525d3}} for {{formula:8bb5d628-14e0-4ec3-8723-3de0cfb40ea0}} are called Milnor balls. Fix a Milnor radius {{formula:9f073f13-39c5-4b96-94b1-44f82138ed14}} . There exists {{formula:696473db-c9c4-4130-9508-edc55b65edf0}} such that the restriction {{formula:ffbd84d6-5c9a-4399-8d5e-5f2016951eb6}} is locally trivial. The restriction of this fibration over {{formula:51c6e502-34eb-44c7-bf46-bf12dcb8f902}} is, up to a diffeomorphism, independent on the choices {{cite:edcbc0062181eb9053bb57ecfae662692e9a247b}} and is called the Milnor fibration of {{formula:d94ad02c-2157-4a9d-8e8f-713c480f7c80}} .
Example 5.16
Let {{formula:ce9ee7de-69ac-4144-8a21-9a4d7344454b}} be a germ of isolated hypersurface singularity. Let {{formula:f0c90a02-c735-4b25-9872-3d73b6a1ddde}} be a Milnor ball, and let {{formula:bd3f154e-ecdb-4374-a558-3281baf5bc01}} be the standard Liouville form, see Example REF . Then the Liouville vector field points outwards {{formula:21eaad1e-9429-49ef-9b24-8b5b31094acf}} for every {{formula:06d3b4c1-3907-4904-8816-149e306de7e7}} .
Consider the Milnor fibration {{formula:1e4b251c-e401-4ba8-a44f-5c25b0528ac2}} , where {{formula:52aefdac-3118-480a-9258-ac2cefce21cd}} is small enough. In Setting REF , take for {{formula:8c429fce-5dfc-42e0-b6eb-52449b223a06}} balls of decreasing radii, smaller than {{formula:7280665f-1ab6-4260-969f-620cc524277b}} . Then, the above procedure provides a symplectic collar trivialization for the Liouville fibration {{formula:7b9e598d-9288-49f8-af66-6be32dcee07d}} for some {{formula:9ee98f36-5ba0-4bf9-b660-63751bdf1480}} , whose fibers {{formula:62374a5a-2164-43b9-9647-f818a57adf20}} are diffeomorphic to the Milnor fiber {{formula:d7ee3c9d-7411-4737-bf4e-637de7e264a8}}
Assume {{formula:9ac4f949-ed43-4334-a0b1-0bbb9e366880}} . Then {{formula:eaa46793-56c8-4269-b13c-b7dcbcadc411}} for {{formula:8444ae2b-31a3-4297-8d5c-98abc3f5ed64}} , so above the collar trivialization is cohomologically trivial, and the symplectic monodromy gives an abstract contact open book {{formula:f48a543b-754b-401b-887f-843b37d91967}} , see Definition REF . Moreover, since the canonical bundle {{formula:bdd8d3be-69ac-41e0-a06d-8176af32800e}} is trivial, a choice of its section defines a grading on {{formula:be8c9c0e-8738-4eee-90fe-975927cd8901}} .
In fact by {{cite:edcbc0062181eb9053bb57ecfae662692e9a247b}} we have {{formula:e187e7ea-2df3-417d-adf9-1e8c2142797d}} , so by the correspondence (REF ), the above grading on {{formula:bc05efbc-0878-4908-95bb-2ea6eccaed46}} is the unique one.
Example 5.17
More generally, let {{formula:10e07fc9-8818-484a-95fb-81db836b2e24}}{{formula:91991a5a-7f26-45fc-9769-367c123dda95}} n4{{formula:d580c821-a96d-4854-b5af-bf5f2950ca01}} (x)||x||2{{formula:67b754bb-8731-4ce1-8a13-0c19afd2ba43}} -dc{{formula:1021cc9e-eb07-448b-8b44-2ed11eb26a7f}} bV<bW<bU<c{{formula:b0eddb84-8d64-40e9-8e76-37fa1b1e9653}} f0{{formula:cf6e09ab-4bc4-430d-9689-98d179925695}} >0{{formula:76c0f1c4-c045-4bb3-9d62-af7dab7f8c11}} (Fs,s,s){{formula:bee12ed2-ac7a-4972-9c20-25c9a8a3ff5e}} s{{formula:1f4158b7-6f29-46a6-a7e1-2dfc8de4461f}}
Note that the Milnor radii chosen in Example REF for {{formula:01c96fef-2acc-4620-a0b0-ffee8c3d6783}} are not necessarily Milnor radii for {{formula:ff445eb5-1260-4844-ac3a-980bebf54bb4}} , {{formula:d9209522-d066-446a-982f-493317a75f94}} . In fact, it is a long-standing, open question if a Milnor radius for {{formula:b9bab26e-14e6-4986-95e1-441dc6f1bee9}} can be chosen independently of {{formula:b1a7beac-a072-4775-bbf7-5cad49dc483d}} ; even modifying the distance function {{formula:3251ea71-6360-4905-87cb-9f0965dd0705}} , see {{cite:1a7a0600b04c00025ac436bd59ea3f5c0c1b05bd}}.
In particular, the graded abstract contact open books produced in Example REF for {{formula:b7b21d8a-400f-4c06-a4a7-eaa01bb6c122}} ; and the analogous ones produced using a Milnor radius for {{formula:771d5808-9ee0-43c7-b6c7-f7562a869c11}} , might not be isotopic. To prove Theorem REF , we will compare their Floer homology. A crucial ingredient of this comparison will be the fact that they differ by a trivial cobordism. More precisely, the following result is known.
Proposition 5.18 ({{cite:2becdf6e265a68c3177b7f1043cb33dc4350d607}})
Let {{formula:dbee9c0b-8c74-4d3e-a840-e804b4782b08}}{{formula:26aa58f5-ed9b-47a8-bc2e-a7c6178e3755}} 0{{formula:8af41559-3278-4936-8483-13aa9601af62}} f0{{formula:beea73b1-0143-4f77-8e34-f46940fb58d2}} s{{formula:f1351b62-1e12-4a0c-9972-1173edb12bb2}} s<0{{formula:0c8f0997-add0-4669-868c-389e6af8f72b}} fs{{formula:ddcec055-b793-4b8d-8519-cb5776e220f4}}fs-1(0)B0Bs{{formula:e6366f46-79b1-4267-bee3-c1d1875b9a68}}
{{figure:77654d6a-4c7b-42b4-abd4-76632634e701}}subsection2-.5plus-.7.5Codimension zero families of fixed points
We end this section by introducing the notion of codimension zero families of fixed points, introduced by McLean in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. In Section , we will see that if all fixed points of {{formula:a84c4259-ba28-4e84-8ff2-019361ffa64c}} come in such families, then the Floer homology {{formula:792f7746-5711-4e29-9de2-01a806a5a59a}} can be controlled using just their topology. In Section we will see that this is the case for radius zero monodromy on the boundary of the A'Campo space.
For definition of manifolds with corners see Section or {{cite:f24d18a69e9c818fe069ba78d0f67630c8ad8ab2}}.
Definition 5.19
Let {{formula:9be9e243-0d89-4909-a22d-9d6b321d9c8e}} be an abstract contact open book.
A codimension zero family of fixed points {{formula:addd9281-5cd2-471a-805b-1e4b459a44a4}} is a compact, connected codimension zero submanifold of {{formula:4530f978-577d-4433-a20e-e87fd8872e36}} with corners, such that there exists an open neighborhood {{formula:2abe8465-1be9-4bc5-a960-3768a9c79945}} of {{formula:0707da7e-09b1-4967-a258-3ecaf33a2fcd}} in {{formula:3bfd3db5-f583-4368-9c34-135bbd0ceba5}} , and a time-independent Hamiltonian {{formula:ef94c071-462c-4745-94ec-ff3469ba0fac}} satisfying
{{formula:44721511-82b9-4fb8-bb4b-a0723754093b}}
The action function {{formula:8b6a130b-073d-4198-be23-b3d20f0fa329}} associated with {{formula:657ad869-7906-4163-a6a3-bbc96ab04f62}} is constant in {{formula:78010e96-0434-4eed-9da2-4c01cf5f1279}} , and we denote its value by {{formula:2c4c2047-9037-466a-8f5f-a98291ed3f5d}} . If {{formula:e69ea5d8-5c33-4c77-9deb-80553529798c}} is graded then the function {{formula:6f017277-81a2-40d7-b6f0-3cc9dcaa6686}} is also constant in {{formula:48b04974-0075-4a95-958d-8a5a9f9cad5f}} , and we write {{formula:5ba0bf80-2eab-4ad4-98aa-40c906dd323e}} for its value.
Put {{formula:20b89df5-18c3-4f3d-a42e-30057bc3325a}} . Since {{formula:fbee5a80-c749-44c4-937a-df936fdc1aea}} , and {{formula:db835f75-df42-4993-86f9-097c7a4d75f0}} is a codimension zero submanifold with corners, at every point {{formula:44e02e40-c88b-4c36-9226-7486466b87f4}} there is a chart {{formula:21c4e058-d4b3-4f1b-94be-04c065f610c8}} such that {{formula:d6e5244c-5fbb-44d2-84c0-8a90c006a702}} is either positive, or negative. Denote by {{formula:6f51729e-8954-4227-bb56-095d7947a9d8}} (resp. {{formula:f51f2840-e078-4528-b826-f192d831584d}} ) the set of those points {{formula:17ba71b0-d963-43e9-a077-8b88e1ff42bb}} such that {{formula:49cc9dcb-a91c-4563-a533-01349231e0ec}} (resp. {{formula:d805af66-7939-42f5-86c1-8836c9cd87fd}} ). This way, we get a decomposition
{{formula:3ec2b29f-06e6-49c2-aeb8-e80c95c16c16}}
where each factor is a union of connected components of {{formula:32ae4690-e844-48da-af54-2fbd4144fafe}} .
Remark 5.20
The codimension zero families considered in {{cite:97005650a8ac11991dc2e980883b47d01674846e}} are assumed to have {{formula:331ab1db-664c-4d89-876b-b4aed399bdfb}} (or rather {{formula:3c661991-1fa8-47b4-9b2b-8fb34ec8c44e}} if we take into account the opposite sign convention for the Hamiltonian, see Section ). We will need to work in this more general setting, which is why our version of McLean's axiom (HF3), stated in Proposition REF , involves relative homology,
In practice, codimension zero families of fixed points will have corners. For example, in Figure REF they are the quadrants {{formula:bdea9a12-fd54-422d-9a10-f14903099290}} , see Proposition REF . The following lemma allows to get rid of them.
Lemma 5.21
Let {{formula:a87ea9ab-c166-4cb3-8b3a-7f85481099be}} be an abstract contact open book, and let {{formula:e0006afb-67c1-4678-a2d2-0f17bb81ed1c}} be a codimension zero family of fixed points. Then there is an abstract contact open book {{formula:2423a83c-bfe9-44de-a5d3-0d26135a53cf}} such that {{formula:0ee2ecd2-ffd6-4a2a-866a-6e908e81d96b}} , where {{formula:b5d6410d-0cd5-4d6b-bdf7-6c341d2192df}} is a codimension zero family of fixed points, which is a manifold with boundary (but no corners), and the triple {{formula:7afe940a-c169-4e48-bfea-2ae2fa3109b8}} is homeomorphic to {{formula:61872392-2bec-4631-a92a-fb1aeb9831d5}} .
{{formula:8c110390-4a54-4eca-b996-af4410e5798b}}
.
Let {{formula:4b571fa3-3d65-436d-97a2-5f057e6a8a2c}} be the time independent Hamiltonian from Definition REF . Then {{formula:868919eb-1cdb-419f-aba7-bc4cd0dbdf39}} has no critical points, so {{formula:8823f5d0-6740-499d-b815-293557f25c53}} is a submersion for some {{formula:205d9c3f-0bb8-4370-8e40-38ea3fa03928}} . By assumption, the Hamiltonian flow {{formula:2cbeaf5e-71e0-4d0f-b884-830f058db725}} has no 1-periodic orbits in {{formula:ac9fa77a-430e-4324-b80b-a95b5825d5a4}} . Reducing {{formula:87cea47d-7519-486a-a6f1-eb385a4e1dc1}} (hence shrinking {{formula:3d4a4266-ca9e-4f1c-a8ef-4100d450ed69}} ) we can assume that it has no {{formula:5fa96511-a56a-42a6-8ec5-68624e7f8dd8}} -periodic orbits for any {{formula:199aba28-d731-4af3-ac74-4aadb19cae01}} .
Define a smooth family of functions {{formula:c897b615-4479-4906-a9cc-4306846d536f}} , parametrized by {{formula:3283113a-7020-4508-93d0-5612fb97c7af}} , such that {{formula:31841f6f-d7bf-4a13-a170-8607abf34f5f}} vanishes identically in {{formula:a531d7fc-9cac-400b-84cd-9aa7640dd1f1}} , equals the identity outside {{formula:ea42a0db-5c0f-4c64-94dc-6e48d24d5bcb}} and is strictly increasing away from {{formula:46c2125f-8e0a-47c2-a908-75146ad7699c}} with derivative bounded by 4. For each fixed {{formula:c786f8fa-444b-4774-8ae3-3341be5b87a5}} define the time independent Hamiltonian {{formula:1adf656e-8f88-4262-9760-108a60dec18a}} .
Let {{formula:2338ed9e-6802-4efb-a472-6d4a00430fad}} be its time one flow. Recall that {{formula:4d94be0d-27c2-448d-9f6e-a659f440e1e7}} is the time one flow of the Hamiltonian {{formula:a4d7ee5d-9282-4051-9879-3828aace3fa3}} , so {{formula:aaacaaee-1ea9-45f2-874d-42f192fe7a27}} , and {{formula:ee85bfa8-1d1e-477f-967d-6d390cf9183a}} agrees with {{formula:f7383b99-5f60-4fd7-83a7-a9977bfcc34a}} away from {{formula:60892664-dd00-49a7-9f0b-1a7ea3b169fb}} . Therefore, we can extend {{formula:9d3ca0ea-bf84-42f2-9b63-122353defcc4}} to an exact symplectomorphism {{formula:769a68aa-85b8-413e-8e45-e8b2a4771fa7}} , equal to {{formula:01aae81c-1ec8-45c9-aa29-9eebc573f698}} away from {{formula:1467dd92-8041-4f57-9c41-dbb7f0503385}} . In particular, {{formula:9aabff7b-8ea7-4a27-b425-a144ab89ebe3}} and near {{formula:bcf87d85-b932-442b-aaf7-7b54c692e3fa}} . Thus {{formula:02bee61c-0c9d-46d1-92bf-95de372169c0}} is an isotopy of abstract contact open books, and {{formula:9a3208c2-07b5-485b-a7e1-728952accce4}} .
Put {{formula:1fa9ddb5-3d04-4f13-96d5-04faa5e00bd1}} . Because {{formula:80faeb2c-8629-4bb5-ab66-0741aa732d2a}} are regular values of {{formula:7b640965-94b1-43ad-a383-19ef17152edb}} , the subset {{formula:e03a8b4c-db24-462f-982f-b9284981e393}} is a submanifold with boundary, of codimension zero. We claim that {{formula:5e171d2c-4c95-4dba-8977-babecc056235}} . Clearly, {{formula:12d61453-7a7a-4dad-b067-043aa4294ebb}} . Outside {{formula:8b770992-05d8-4676-b328-b9a5114ce60f}} , the Hamiltonian vector field associated with {{formula:37271473-ec73-46d6-9a99-871d51a2a5f7}} coincides with the Hamiltonian vector field of {{formula:b6845dd6-4e4c-4ccf-a938-3b3a120e0b41}} , multiplied by the derivative of {{formula:0406633c-d1c6-44b4-bdc1-41dff728de88}} , which is bounded by 4. The claim follows because, by assumption, the time {{formula:dae8a695-24d7-4bf0-b64c-bb426d17c751}} Hamiltonian flow {{formula:42cf73df-ff61-4229-86a4-fdcd3814b798}} has no fixed points in {{formula:fb348ef3-ff95-4d57-b235-8b0aa36f52d0}} for {{formula:7b1cc5f7-6482-4bce-a1db-0fdd5d12f738}} .
Therefore, {{formula:2f1fe889-8ab3-457b-903e-0d58d15fb3b8}} , and {{formula:76f01525-cf87-4870-b5d8-b0abb78686a8}} is a codimension zero family of fixed points, without corners. It remains to see that {{formula:865dd0da-f757-4578-b45a-6b526e637471}} is homeomorphic to {{formula:6f989da9-4f10-4bb1-bafb-946221d2d3d7}} , for {{formula:a7c40b9d-847e-4517-8edd-b74815597c98}} small enough. Write {{formula:82318d0f-7c91-4cad-84d5-6e0ab307c41f}} , where {{formula:e8ba3bd1-d981-42e3-8bcb-2b8f47b9c28f}} on {{formula:c43f6d21-fbb0-45c6-bd04-4514456ca438}} and {{formula:c7459af7-8f6a-4d62-95d3-79551eea6a54}} on {{formula:33e5aff2-14f0-43fa-97ea-9f2e40986592}} . Define a smooth vector field {{formula:e4a37f63-fe95-4c07-8bcf-03cdb1be8936}} as {{formula:8c741421-8986-497b-8bd0-f350085f241b}} on {{formula:663a54f6-9219-4508-b755-20ccfd0349a7}} , {{formula:8619eca8-d473-4659-a7d5-401084645707}} on {{formula:65f7da8a-3f56-49ec-8152-8762fa6de908}} , and 0 on {{formula:5778fe15-b5e1-4837-a938-531441dbf3fa}} . Fix a vector field {{formula:9d6ab66d-1549-4cc9-8c7b-02809c3e9d67}} on {{formula:5a54075d-efea-4d77-b96e-34ec211bc1a3}} , pointing inwards {{formula:5c88cf44-cb9f-47ee-b87d-25d092be99ec}} , see {{cite:f24d18a69e9c818fe069ba78d0f67630c8ad8ab2}}. Since {{formula:224ec1e4-3de0-49eb-b2c4-056c85a9f39b}} vanishes on {{formula:d031dabc-209a-429b-a374-b53c4927120e}} , we can choose {{formula:b899cfc0-90b9-40ca-86d8-006f73911e8c}} small enough so that {{formula:f99cf5fb-3c52-4ebc-8fac-a4a80490b876}} does not vanish on a neighborhood {{formula:c5582078-da68-47e8-a442-799c52904986}} of {{formula:bc675210-39b3-4a97-8219-7ae9c416b9ee}} , and points inwards both {{formula:28676a9a-c58a-4ec7-a99f-f13f6c75bdf3}} and {{formula:8bbe1275-c4e6-4398-a3bd-c8da6c6f8c94}} . As in {{cite:f24d18a69e9c818fe069ba78d0f67630c8ad8ab2}}, we conclude that both {{formula:4918fe7e-eb74-48f4-8adf-62d1e234597f}} and {{formula:82d7e486-5bfb-4041-bd8e-76bb549bebd5}} are homeomorphic to {{formula:74d32071-c9e3-4df5-9eb2-91be770705cc}} , and we can extend this homeomorphism to the required homeomorphism of triples.
The McLean spectral sequence in fixed point Floer homology
In this section, we introduce fixed point Floer homology {{formula:09935dd0-67dd-4f7f-b8bb-08840f28d55e}} of an abstract contact open book {{formula:28c4a0d4-11fd-4627-adbc-72e936a51fde}} , following McLean and Uljarevic {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. This definition extends the one from {{cite:d5bc3ba72058ac6304d8c4621b90572df69b7cd1}}, and was first introduced in a slightly different way in Seidel's thesis {{cite:e86f6801593e482c23ac7cc24e84e7c0073c645d}}, see {{cite:c580055fb2a31bbd41b397a216821656bfb54c49}}.
After some preparations in Section , we introduce the Floer complex in Section . In Proposition REF , we construct a spectral sequence converging to {{formula:37559c92-6ecb-4605-a1b8-b9810ba233a9}} in case {{formula:7db96da5-4f08-4c29-bb0e-12e5ebf008c4}} is a disjoint union of codimension zero families of fixed points. This spectral sequence is a slight generalization of the axiom (HF3) of {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, proved in Appendix C of loc. cit. Our arguments are essentially the same, but we give a less compressed presentation for the convenience of readers which are not experts in Floer theory.
subsection2-.5plus-.7.5Completions of Liouville domains
A contact manifold {{formula:98c98189-05da-4ea9-b8af-759d817081b2}} is a compact manifold {{formula:8139ea91-05cf-4f6d-8ecc-185bef2d7b7d}} of dimension {{formula:4bcbee70-ca6b-493d-a933-78e13a48c1e7}} , together with a 1-form {{formula:628a2dbf-e715-4059-a2fb-7fb9da64dfdc}} such that {{formula:0c0095f5-a9cb-4d7a-bd6e-fbb004fdb464}} . The Reeb vector field {{formula:f9548d0c-28e6-47e2-97a5-470867440478}} is defined by {{formula:dc98ff11-beab-4490-a95a-a7a9b5c23d40}} and {{formula:508fee8b-a8ff-4a2c-9280-70036649c3b4}} .
Let {{formula:d6cac8c6-06d1-4dfb-bb65-bf76b515a181}} be a Liouville domain. Then {{formula:41ff7e02-6a19-4fb2-bebd-138116181589}} is a contact manifold. The backwards flow of the Liouville vector field {{formula:504df797-f798-49b8-bf8f-0c75d000aa19}} defines a diffeomorphism onto its image
{{formula:cc769199-0268-4a89-90b4-b606e45e7884}}
such that {{formula:e4dd572b-32ed-4ccb-b518-878209ef3bc9}} , where {{formula:3d621f1e-9795-4411-a84a-6e45961f5e51}} is a coordinate on {{formula:dc8f5ddc-1cb8-4476-8f16-e423e753b2e4}} . The completion of {{formula:f25e1a05-faf6-4b9d-b5ef-dbc353d8cc98}} , see {{cite:b1204f911b5b321f5d286fa4927a3bf3bb9cf2d1}} or {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, is a manifold {{formula:417474fd-87e9-41bf-874b-8d4bb3491519}} , where {{formula:4b332621-2a6d-48ee-85eb-77af6cd87e43}} identifies {{formula:3c1ce74d-58cb-483f-9b25-d649914c4b78}} with its image under {{formula:851ad6a4-051e-4598-90dd-a58b6b29374d}} , together with a Liouville form defined as {{formula:de813bbe-25ac-46de-9934-af033d595ff1}} in {{formula:a44d6159-4b4e-4fdc-a436-cfd70f026437}} and {{formula:ab9059f1-08a2-4992-b7ea-b6a85ad5c28c}} in the cylindrical end {{formula:61153d23-e539-4f4f-b760-5ee24c18f809}} . Abusing notation, we denote this 1-form by {{formula:5bd48343-fcfe-4dd3-a808-6b2bf1f24d50}} , too.
Let {{formula:98994823-855a-4006-9f2f-941e053efaa1}} be the vector field corresponding to the coordinate {{formula:5bc46b65-ca97-4070-90f7-2ea3e4479947}} of the cylindrical end {{formula:91752910-7721-4ac0-9f9b-bd0019245f1e}} . The restriction {{formula:ce209955-c961-4bca-ae4d-a70b11125c89}} is a contact form, and we denote by {{formula:e58ebe9f-e6e4-47cf-93ba-28a3cd9d98c1}} its Reeb vector field. Therefore, {{formula:36e6b359-398b-4375-b981-4501762969e7}} is a vector field on {{formula:892cc880-3930-4b56-b101-339ed7587cc7}} , which is tangent to the level sets of the coordinate {{formula:471f516a-2c53-4fa1-bb56-e5e73ca720e3}} .
An almost complex structure {{formula:d3d356de-4863-4ba4-892d-6d2137970221}} on {{formula:3772d0cd-3ec5-4432-9ec6-73494a52423f}} or {{formula:f5ed4e15-17c4-4963-9d5f-207b0296580c}} is cylindrical if there is {{formula:b13e880b-5dcb-4c01-997a-478a1700eb4c}} such that {{formula:0540c185-d424-489b-b837-7a6ee003314a}} leaves invariant the contact distribution {{formula:de8c5895-2ac0-4978-8243-61a25ed9f421}} and {{formula:9c4038a8-4430-4813-a151-88ec0e6c9136}} for any {{formula:f67b9bfa-dddf-4102-a6fe-5b0416a56876}} . This notion coincides with {{cite:97005650a8ac11991dc2e980883b47d01674846e}} and {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}.
Fix {{formula:85faa92a-2778-4472-93cc-0828c970c9f9}} . A (time-dependent) Hamiltonian {{formula:c12c8be9-70c0-45fc-bd9a-a3a84c6e0a7b}} is of slope {{formula:9b9e5cff-81e0-4adb-ba43-8b84d68baec9}} if there exists {{formula:71437940-a3af-4032-b6d1-470351018e0e}} such that {{formula:a89d0482-bc67-44db-8c2b-410dd3742a11}} for {{formula:74b0e3a7-47a1-4699-b658-fc95e29d1bb4}} , where {{formula:41501823-9eff-4b7d-8a7e-494d62400708}} is the trivialization (REF ), see {{cite:97005650a8ac11991dc2e980883b47d01674846e}} or {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. A Hamiltonian of slope {{formula:27a52378-2ec1-4588-8ac5-e63f205f0440}} is extended to the completion {{formula:27c6fc16-35a2-47c8-b7cb-9eaf73949a69}} by the formula {{formula:3e0945d3-d4e5-4edd-bdba-5aa00b61225d}} at {{formula:ea776b75-cfbc-47d5-8488-95b1c459132f}} . At the locus where {{formula:38624efa-3588-4585-ae14-bb74f4b9df88}} the Hamiltonian flow at time 1 coincides with the Reeb flow of {{formula:dabbfdb9-f20a-45cb-8015-a691f7e0f060}} at time {{formula:ac6c3bd0-793d-48e5-ad37-361b7aca75ba}} . A nonzero slope {{formula:570ed527-c90e-4505-974e-b78243d09ad9}} is small if {{formula:4ea72a86-71d9-4faf-afcb-8cd9964f39d3}} is strictly smaller than the minimal period of a Reeb orbit. In this case, {{formula:b8dd5b12-e04b-42c2-9bdd-82df2a399224}} has no fixed points in the cylindrical end of {{formula:60257c3f-d98c-45a3-b3de-190001ca3303}} .
subsection2-.5plus-.7.5The Floer complex
Let {{formula:013277b3-de8d-4576-bf86-1ff373647516}} be a graded abstract contact open book, see Definition REF and Section REF . We will now give two, equivalent definitions of the Floer complex {{formula:255dfd29-bafe-4b72-b1f3-b82ed53b914e}} , using the perturbed and unperturbed Floer equations, see (REF ) and (REF ). The first one is adapted from {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, and mirrors {{cite:d5bc3ba72058ac6304d8c4621b90572df69b7cd1}} in the compact case. The second one is used in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. It is known how to pass from one setting to the other: we outline the method in Section . Nonetheless, to derive the spectral sequence in Proposition REF , both the perturbed and the unperturbed settings are most appropriate at different parts of the argument. This is the reason why we explain both approaches.
subsubsection3.5plus.7-.5Periodic almost complex structures and Hamiltonians
Let {{formula:16a5319d-08ec-4596-9ef1-1b51cb28bd4d}} be a graded abstract contact open book. Extending {{formula:1d407b4f-d66d-4799-aa91-031955d062ca}} to the identity in {{formula:3daeabea-8e7a-4130-a607-4e88abcfe9c2}} we obtain a compactly supported exact symplectomorphism {{formula:16b4211f-e436-468b-9c7b-74c4ed415d89}} , which we denote by {{formula:d3fc0ade-c380-4577-abbe-c040aee13e5e}} , too.
A family of almost complex structures {{formula:885455d0-29ac-4eb5-be01-72acf966d7a9}} on {{formula:ffcdfb58-e9b8-40d6-9f5f-45c62e6aa16b}} or {{formula:b3475698-c8af-421f-8576-4d7b1948ae11}} , smoothly parametrized by {{formula:98caf15a-bbe6-42d8-a96c-da25689846d2}} , is {{formula:d9ea0e31-3850-4a66-bfb1-fc3edb5e786e}} -periodic if {{formula:08b1dc3b-b80c-4be3-9463-425041a229b1}} , see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. A (time dependent) Hamiltonian {{formula:79cb022d-977e-4454-9c1a-96456da5e230}} is {{formula:ff8b55a6-8b1d-48f1-9a81-0d8cba9afde2}} -periodic if it satisfies the equality {{formula:0c9a7e19-dd8f-4bc9-a3f8-4e53d04a5c1c}} , see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. A {{formula:6929afc7-d61a-4136-acf4-c4450e2b1e0e}} -twisted loop in {{formula:5b1ff509-4dec-465b-a03c-e55b7f1201f3}} is a path {{formula:4b82522e-25de-4a8c-8bea-43988dbf1fef}} such that {{formula:cedfe102-a0da-410f-87f5-96465b255b8c}} , see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}.
We warn the reader that in {{cite:d5bc3ba72058ac6304d8c4621b90572df69b7cd1}} the periodicity conditions are defined in an opposite way, i.e. {{formula:e5234033-f467-4697-ad2d-bcab77407675}} , etc. Also, the symbols {{formula:869a0d40-1a7b-4b78-8241-0094ae81bffc}} play the opposite roles in {{cite:d5bc3ba72058ac6304d8c4621b90572df69b7cd1}} than in the discussion below, which follows {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}.
Let {{formula:543d7f10-40cc-4830-9627-21d0bd4f9105}} be a {{formula:13ca677d-94a4-438c-90dd-40d6228f1d69}} -periodic Hamiltonian. It satisfies a basic equality
{{formula:7180d25a-da44-48ab-a482-3c9171b0ec92}}
cf. {{cite:d5bc3ba72058ac6304d8c4621b90572df69b7cd1}}.
To prove (REF ), note that {{formula:982485e6-786d-4e16-be4e-0c5e7e89efd4}} , where {{formula:b1a88221-94d5-4bea-9145-259f22250aed}} . By {{formula:6eaa2062-5708-42eb-82e0-681948e4db1e}} -periodicity, {{formula:3d96a19c-3c8e-45e4-b4d1-2d3f50974020}} , so {{formula:7f8f3758-56cc-4c18-afc7-38193e326d0b}} , see {{cite:340e109d9e1838dc962e6460f23b38ad1ff82b9c}}. This proves (REF ).
We denote by {{formula:fafbc47a-ac24-47d3-8070-02184c919196}} the space of {{formula:d746b5c4-e3e8-4656-8f68-b4a2eb4782f9}} -twisted loops which are Hamiltonian, i.e. of the form {{formula:3b68e96c-4166-4940-bd77-16dff0b797fb}} for some {{formula:1595fd97-e377-4f42-8797-d88f63f87502}} . We have a one-to-one correspondence
{{formula:b344c33e-baac-48bb-b96e-6b047d616d16}}
Indeed, if {{formula:a8485e9b-f45d-491f-b371-a85bb0d4c359}} then {{formula:3ef260fd-904c-4e5f-b010-1ef514dd14fd}} , so {{formula:b4656390-48f8-4bb5-9a41-afda5c72aa10}} . For the converse, let {{formula:cebaa2b7-80db-45b7-a679-5ac1cb150593}} and let {{formula:607bcd0f-41c8-4859-84f7-0f94de58b390}} be the Hamiltonian {{formula:021def67-b4d1-4f5c-b9c7-f4ad35dec093}} -twisted loop starting at {{formula:4e38ea3f-0ec7-41f8-9137-1742605f1e51}} . We have
{{formula:e60f37a6-0cab-4917-ace2-6df13daf5034}}
where the first and last equality follow from the definition of {{formula:28665a10-35ba-4a3f-99ad-3fda9b4a8558}} , and the second-to-last holds because {{formula:ee8ae9b4-64d3-40ed-8b46-46c515eb663c}} . Thus {{formula:74d75288-0632-4da4-b6d5-7e3152255d7f}} is {{formula:acbab973-20f5-4b4b-9db4-1c5236b6c1a5}} -twisted, as needed.
Note that if a {{formula:3b219d1f-be21-42a9-9857-185d0b771bdf}} -periodic Hamiltonian {{formula:ec1ae25b-2b91-459b-976e-e67f617727e6}} has small negative (or positive) slope, then all fixed points of {{formula:bbc15003-ec79-4394-8c7e-ffe299d18224}} are away from {{formula:45344996-c269-410a-be9f-47164402dd26}} , and away from the cylindrical end of of {{formula:132f5219-150d-4d04-aaff-5b7980caa175}} .
subsubsection3.5plus.7-.5The Conley-Zehnder index of a Hamiltonian twisted loop
Fix a Hamiltonian {{formula:0479e19b-769b-4479-990b-275b771d065e}} -twisted loop {{formula:1def1b2e-5b75-4a79-bf5d-05cf33693c97}} . By (REF ), {{formula:30f8e60e-6958-4a9c-83b8-2411d4e5bacd}} is a fixed point of a symplectomorphism {{formula:2bbc7a03-8efe-4dc6-9f2a-4f271ae45a9f}} . The latter has a unique grading, induced by the grading on {{formula:94e7a0d3-ece7-4a41-adb6-a8ffdde3af27}} by the isotopy {{formula:dec630d5-b099-47d3-bd79-15a0be99dc19}} . We define {{formula:62dfcbd0-548f-4380-ae44-dfd5cc118b21}} as the Conley-Zehnder index of {{formula:39e96180-09f9-4ffe-9c19-2c6181b04d9e}} , see Section REF .
This definition lifts the relative Conley-Zehnder index introduced in {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} to an absolute grading. To see this, let us recall how {{formula:759ef007-f4fd-4588-a1d6-2be933c200d6}} was defined in Section REF . Since {{formula:622285b9-589e-4ec3-936f-d47cde72adf6}} is an integral curve of the Hamiltonian flow {{formula:b134c177-aca3-4d0c-b292-e964949b324f}} , for any time {{formula:c2db5f63-8dbd-4228-8907-fd40d0e50787}} the differential of {{formula:64324efe-e395-471a-8462-61334282b675}} at time {{formula:f6b78b63-cd04-43a6-a431-5c27cdfbe905}} induces a mapping {{formula:0d40bfff-823b-4d51-b5c7-e4afcfe7c86e}} , which lifts to a mapping {{formula:7752e02d-6908-4c39-8548-428ce19ecca0}} . The grading on {{formula:83d4658e-fcfe-4bc2-a479-abe4d6ef44d3}} defines a mapping {{formula:3328f42f-2f7c-4eca-a9f7-be727e118d07}} . So, composing we obtain a mapping
{{formula:1696507a-17e2-42e3-b7e4-ed9edf5ba967}}
which gives rise to a homotopy class of paths in the symplectic group relative to its ends as in Section REF . The Conley-Zehnder index {{formula:aa0af4fc-434b-4a57-9882-9f788ff9769d}} is now the Maslov index of this path.
Given two frames {{formula:e07a267f-c71d-42e0-9ac1-f699e4c9a0a6}} and {{formula:3d7bdc0f-43e4-4957-9e48-6b5ddf7eca79}} such that {{formula:528eacbc-dd4c-48db-ab8e-9697aa117b12}} , any homotopy class (relative to its ends) of frames {{formula:ee6cc5a1-f853-477f-9842-c4a6e1ffe05c}} connecting {{formula:e7038f21-581e-4149-b7ee-3e00dd38f17f}} and {{formula:b04130ab-4c6b-46c7-85c4-c55df19f5ba8}} determines a map {{formula:a403a827-5621-48cd-bc33-adda5e6a385b}} . Let us call the homotopy class admissible if the map equals {{formula:cc97b344-a99e-4e1d-9fdb-0b13885c2a99}} .
Fix {{formula:db5e4b1b-0db5-4b3e-88a4-7d48b680d776}} , {{formula:60e32c8b-f45b-48bb-a78f-f4a6e94b4615}} for two Hamiltonians {{formula:e2794837-c505-44ce-812c-3ce316e2e5dc}} , {{formula:a39e1307-072b-4251-85d4-c80e6a3a1370}} , and fix {{formula:315415c4-42c4-4bec-9693-b7b4bc21c8df}} satisfying {{formula:ad00eb39-752a-4fbc-b755-b1393303aefc}} , {{formula:77a3fc39-5bde-4788-9aeb-a606bedbaeeb}} for {{formula:2c11f730-d587-4c6d-aa0a-8a70022a5097}} and {{formula:1985e954-d5ee-406e-bab2-68fbd4a8bbd9}} for {{formula:54566c90-1d07-4d57-b39c-df070c4aaef6}} . Then the grading induces, via the choice of a homotopy of admissible classes of framings along {{formula:3f1e61d9-2be3-40c7-bf75-4870985c0346}} for any {{formula:29845f33-bcae-454c-bf69-38495fbe1c16}} , a symplectic trivialization {{formula:845d76c2-fdb5-4d59-adc9-e95469d328d0}} of {{formula:77adf1a7-de31-4d38-b849-982c98f9ba77}} such that {{formula:34064a88-5e76-4b2a-9731-4de7296e7d7a}} . Now, the relative grading of {{formula:7b93b6de-6e3f-4035-a830-77bae4feb1ff}} is defined in {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} as the difference between the Maslov indices {{formula:b5b9abf8-e5ad-40fd-a39e-ecc69fecad3c}} , where {{formula:c517b8a8-17df-4734-aabe-330f29b12345}} . The path {{formula:35217664-b1a1-4470-9384-981c8cfb9966}} represents the homotopy class used above to define {{formula:fca40ce2-f030-48b4-a270-78f0a6913435}} , as claimed.
subsubsection3.5plus.7-.5Perturbed Floer equation Now, we introduce fixed point Floer homology following {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}.
Recall from Definition REF that associated with an abstract contact open book {{formula:0b9675e0-04b6-43bc-bfa6-4276b430a98e}} we have an action function {{formula:21a53dea-94e2-4a03-a69e-0e9c487b9cc4}} , defined (up to a constant) by {{formula:893a19bd-bdac-49af-8de3-a344b1258c14}} . Fix a {{formula:4778df80-6987-4534-9bb6-f567ec02ea70}} -periodic Hamiltonian {{formula:de6172cc-f920-4730-aae3-701a3a94cf39}} of small negative (or positive) slope.
The action functional on the space of {{formula:5309f2d9-434f-409f-b5ed-5d60302a39c9}} -twisted loops is given in {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} by the formula
{{formula:208d055a-f603-4673-8982-fe15d7baec29}}
Note that if {{formula:6097fd5e-a159-4916-890f-204a7613e030}} is a constant path, then its image is a fixed point of {{formula:1c50636d-264a-4138-b871-07910db733dc}} , say {{formula:ce599d46-84cc-414d-8bc1-cd6bdea6a689}} , and {{formula:f34b2bca-2719-4735-9e05-39aa5c2a0f70}} .
The set of critical points of {{formula:d18d6230-96b6-47a9-b999-565f8bdd9d08}} equals exactly {{formula:af3e277f-c8c1-4441-a5b1-b7f1a7680d0d}} , see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. A {{formula:2e135bc2-9b94-4f00-8bb7-084b219ec778}} -periodic Hamiltonian {{formula:5f3b532e-ef7b-45be-88f5-6ebfb51243ef}} is nondegenerate if {{formula:2b3e945f-c341-4dfa-958f-4cd525aff2e5}} at any fixed point of {{formula:09f9f3c3-6e16-4bcf-b001-4a0992768be7}} , see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. If a {{formula:fae419ff-67ec-4007-80ca-17899a152178}} -periodic Hamiltonian {{formula:ae43d84f-1f78-4a74-85ca-d67e89612f45}} is nondegenerate then the set {{formula:33d7bb45-ef7c-420f-aba4-de523b37ff27}} is finite.
Let {{formula:7f5d5ca8-e81d-41cb-94d8-db3ad7d1df82}} be a family of cylindrical, {{formula:2cc350fb-1c11-42c2-867e-5dad478df7c6}} -compatible and {{formula:bc9ba51e-296f-4f84-966c-7d7a19357cab}} -periodic almost complex structures on {{formula:e5319771-39de-436c-8ce2-bd49f555a7d3}} . Fix {{formula:36fce340-7c18-47da-86e9-752c73c68c31}} . A Floer trajectory between {{formula:f522fbf2-b82a-4ae4-b900-e0c8abed9263}} and {{formula:9c43b7c4-2bcf-4a92-ab22-0e2b65d29a2a}} is a smooth map {{formula:35763535-cf6a-4a9b-a87f-be6e8d650386}} satisfying
{{formula:1c8ad7b6-1d25-4273-a196-cfe98349f4a5}}
and the perturbed, {{formula:09f958ef-3ec7-4f35-bd70-52787c0ae645}} -twisted Floer equation:
{{formula:2bfcf090-d45c-4f85-9bc7-e3418f307da6}}
see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}. Note that if {{formula:b8f81514-25da-4a91-b803-03678c1c20fd}} is a Floer trajectory between {{formula:76d089c3-a0b9-43be-98e0-fae55e65b2df}} and {{formula:7ac744a4-9f4c-4ff7-8bf8-e04806f2dff7}} then so is {{formula:c64bea6f-087c-4fbe-81d8-3a37484aee55}} for any {{formula:b49fa424-52b1-4f01-9b38-2974a22c733d}} . So, the additive group {{formula:80b99dac-9b3a-4655-9b95-b333e2f969ec}} acts on the Floer trajectories.
Definition 6.1 ({{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}})
A pair {{formula:f711b6d0-0d9c-4da6-8d4b-558e9fb8120a}} is regular for the perturbed {{formula:d3bb5a98-f418-4ac4-8d24-d4708195384a}} -twisted Floer equation if {{formula:d290e998-4390-409d-97ad-7819c6175274}} is nondegenerate and the linearization of the operator (REF ) for every Floer trajectory is surjective.
The Floer complex associated with a graded abstract compact open book {{formula:39f54c35-7a7c-4634-b17e-f699e009214a}} and a regular pair {{formula:dde0890f-b4a0-4e97-b7e8-b19b8a375dd5}} is the {{formula:05967a5e-7206-4815-8cf6-137ee144b3f8}} -vector space {{formula:1460cc40-55c9-40cc-8677-b4e68584fac2}} generated by the Hamiltonian {{formula:e1333fdf-bfbe-4f92-8195-4b815efef91f}} -twisted loops, graded by {{formula:973a7134-ea85-4b2d-a9af-81e75b5f6c24}} , whose differential is defined as follows. Given {{formula:4a2a2f10-b868-42e0-829d-a4a4e63683db}} such that {{formula:863fec9e-7388-4b6a-86ef-ec231368bb1e}} , the coefficient of {{formula:fc6d5cef-2c8f-4e60-a9bd-90d7090393c9}} in {{formula:f50b45d9-e2be-4af0-8ec0-4bca4cf81c2f}} is the number (mod 2) of Floer trajectories between {{formula:7d877446-b559-47aa-bcca-044f17186f33}} and {{formula:7146db2f-3efc-45ea-8275-61eaffa9454b}} , up to the additive action of {{formula:e3c61712-e74b-4644-a137-bf294b03eda2}} . By {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, this complex is well defined. By {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, the homology groups of this complex are independent of the choice of a regular pair {{formula:e07f1dc0-f00d-4fd8-ad40-a669c040568d}} , up to unique isomorphism, as long as {{formula:943b3dd9-fee6-478e-887f-b8669d33b36f}} remains of negative (or positive) slope. The reader should note that we have upgraded the relative grading used by Uljarevic to the absolute grading defined in Section using the grading on {{formula:ba29c6d8-2523-47c7-9fab-7812b6b0dbfe}} as in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}; the proofs of {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} work in this setting.
As a consequence of the above independence, we can associate to each graded abstract contact open book {{formula:73025728-0a30-4437-8076-d327e1b47c82}} two {{formula:68f81e3d-d97e-411d-a3e3-a56afb2a4372}} -graded Floer homology groups {{formula:ad4f44bc-326c-437f-adf3-abbed7234ccd}} and {{formula:ba6fa7ef-5684-4b7c-9ce2-1d1e0eecfca6}} , given by {{formula:7c4cbbe4-66e6-42e4-bb40-528a4712e389}} of small negative and positive slope, respectively: here we use the notation of {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, so the signs get switched as explained in Section . The relation between {{formula:7e61cc26-ce0d-4441-a226-b33b8c450b72}} and {{formula:aae000d9-1c68-4791-90c7-ef6f63b6b363}} is outlined in {{cite:c580055fb2a31bbd41b397a216821656bfb54c49}}.
subsubsection3.5plus.7-.5The action filtration
Let {{formula:cd417fbf-c24a-4370-8b27-b0c6ac5f03f9}} be a regular pair in the sense of Definition REF , and let {{formula:0704a1c9-183f-469f-81bb-ec92b4795fca}} be the Floer complex introduced in Section . For any {{formula:8c36883e-7793-4e59-9d9a-1ca308344a2f}} , let {{formula:6162e1d6-ba2f-45ee-a38e-3c3e83871180}} be the subspace of {{formula:2c4ef005-ad06-4783-8361-f857cf33c24a}} spanned by those Hamiltonian {{formula:96ef4d3a-b76f-431f-9cf7-cd7ebd2f6b59}} -twisted loops {{formula:6b39c56f-d658-4235-bdb6-f0a4ed8b9b20}} whose action satisfies {{formula:557488c3-a065-4c96-87f9-51c43ac872b2}} . It is a subcomplex of {{formula:d962679a-15a5-48aa-b1fd-cb30d7becf07}} . To see this, recall that for {{formula:79770080-8b68-4c28-8881-8b6bab2a3c76}} , the coefficient of {{formula:dee8a26e-d5d3-4e86-ae62-5d290ac2762f}} in the expression of {{formula:fa1df93d-0668-46ff-9858-b8103bbc7290}} is the count (mod 2) of the Floer trajectories connecting {{formula:88f16e0b-d4ee-4dba-86f8-9c853a44b27a}} with {{formula:4a9195d3-3abb-42a3-9140-4a69d1677b1b}} . By {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, these trajectories are anti-gradient flow lines of the action functional {{formula:6edffb5d-62c3-4774-851b-117350634933}} . In particular, the differential {{formula:c11479db-7112-4f6a-9b81-b19df5eeb086}} decreases the value of the action, so it maps {{formula:1e49d2e2-0977-469d-ac29-428a94199122}} to itself, as claimed.
subsubsection3.5plus.7-.5Unperturbed Floer equation
Now we review the definition of {{formula:78ee8c8a-4375-4917-b1f7-675d868e858f}} given in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. In the subsequent section , we will see that this definition is equivalent to the above one.
Let again {{formula:75ee6168-b4c7-431b-b261-1585eff24d4d}} be a graded abstract compact open book. Consider a small {{formula:f0a5eb7f-861f-4b51-a23e-1bec0f41f371}} -periodic Hamiltonian {{formula:8ece0382-f5c4-437f-88b1-99b5cfd8a7b1}} which is of small negative (or positive) slope. Assume that all fixed points of {{formula:6b4a0d4e-93f9-4e18-a233-3778ed11742c}} are nondegenerate (hence there are finitely many). Now, we define the Floer complex {{formula:6120afc1-5e71-4340-833a-5a432b2dfdf6}} as the {{formula:886c7c4b-bade-4ebc-a9ef-1e6afbb875e4}} -vector space freely generated by the fixed points of {{formula:0a88bfa8-e92a-4cac-aa24-0e9f06c3478a}} , and graded by the minus Conley-Zehnder index, cf. {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. The correspondence (REF ), together with our definition of {{formula:a05f7c96-b213-41ee-9fc3-0085f6fd8c2c}} , allow to identify it as a graded vector space with the Floer complex considered in Section above.
Let {{formula:905be757-a520-4596-8618-f1e39446ae48}} be a family of cylindrical, {{formula:f21c7402-ecfa-4c1c-bbf6-5d6765f45a40}} -compatible and {{formula:659e64fb-fb12-432b-95eb-d60f822f1aec}} -periodic almost complex structures on {{formula:ca88678a-7d1f-453d-85eb-0b4b21832a85}} . Let {{formula:f15798e8-26df-4652-a9b8-d6ba1f3f4895}} . A Floer trajectory between {{formula:8035e3be-2766-41a6-b0ec-abeddfbe53eb}} and {{formula:3102b9ad-43dd-41e9-854d-d00323be4303}} is a smooth map {{formula:ad55622c-a981-47f1-8c07-298632c96842}} satisfying
{{formula:6acbadc1-69b6-4f2c-96e4-bb5695c906d4}}
and the unperturbed Floer equation:
{{formula:9129c721-aec9-4782-9e33-9bcd3cc2dcdc}}
see {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
Given {{formula:84677385-579e-4d1e-826a-e0387cbaa8e6}} such that {{formula:d57d920a-5eca-4b40-b6b1-4a3cfbfa3c5f}} , the coefficient of {{formula:b4c55129-a5c8-48f0-a0ca-feb5e84a550b}} in {{formula:eb225dc2-5f73-4873-968f-a84ba2dc33e3}} is the number of Floer trajectories between {{formula:f4669776-dc14-4fef-aecf-5d29aac26f36}} and {{formula:f9385432-9756-4497-b912-093e540b5e67}} , up to the additive action of {{formula:7db33818-4b42-4726-a93d-66f04806e4b8}} in the {{formula:c1f11320-b985-4836-96bc-0edf613919b3}} variable of {{formula:5ee3bcb8-e0df-4561-8d7a-673530c04f75}} . As we will see in Section below, under the correspondence (REF ) this differential coincides with the one introduced in Section above. As a consequence, {{formula:185e4f73-7768-4dba-9a8b-f6a7275e60b9}} is well defined whenever {{formula:53d50969-d72f-48db-875f-c494dc5e5b56}} has the property that for every Floer trajectory connecting fixed points the linearization of the Floer equation (REF ) is surjective. Moreover, in this case the corresponding homology groups coincide with {{formula:c6fc2ea8-eb18-49d1-b4ea-7cdc0e9f0404}} defined above. In particular, they are independent of the choice of the almost complex structure {{formula:0207fc91-faa2-4104-913b-8044fd1d75dd}} , and of the Hamiltonian {{formula:72c361c4-d1fb-414b-8b61-e1df8f1e7d85}} , as long as the latter is of small negative (resp. positive) slope.
subsubsection3.5plus.7-.5Coincidence of homology groups Here we explain the equivalence of McLean's and Uljarevic's definitions of {{formula:1c1d11f4-e046-495c-8f9f-b63341050379}} , introduced in Sections and above.
Let {{formula:80f8319c-e5be-4986-bd36-66e41deb0348}} be a {{formula:41f86058-7fb2-4b19-ac90-d8d07a97f9d3}} -periodic Hamiltonian and let {{formula:908e6d1c-0a0e-4348-a313-d37b9e8bdd7e}} be a family of cylindrical, {{formula:02426e97-64c8-4923-89e0-4d9362c4231d}} -compatible, {{formula:1c017ff6-b0ab-4d20-b6e4-77370085dd48}} -periodic almost complex structures such that {{formula:a53b2c80-9efe-44f4-a50a-43626929ef06}} forms regular pair in the sense of Definition REF . As in Section put {{formula:06d81e6b-e0e1-4c46-b4e9-b6bd94753cb4}} . We have already seen that the correspondence (REF ) identifies as graded vector spaces the Floer complexes defined in the perturbed and unperturbed settings of Sections and , respectively. For the equality of differential we use the following, known argument from {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}.
Let {{formula:e6c14d39-f716-4b1b-928c-70639fd28706}} , and let {{formula:50a331f1-8c82-4376-80b1-8a0066ed9516}} be a Floer trajectory between {{formula:3c94eb2c-4d65-451c-8454-4c0d0612b2bf}} and {{formula:32f8979c-b210-4fcc-acbd-4ecc28107e21}} . Let {{formula:8c63e34f-f942-46fd-818b-2a2eab97c8db}} be the corresponding fixed points of {{formula:a9db7732-77e1-4cdf-8459-e81d66b3dac4}} . By {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, the image of {{formula:6a7b35ec-48cd-4131-a7b6-d2f834170e86}} is contained in {{formula:b34b23af-29f0-4311-8d94-c0e7d9125e55}} .
Define {{formula:0b3bc225-8053-41d1-b44e-adddc3fbca33}} and
{{formula:cc1c7dc1-b5cc-4c6c-897c-a7c061d824cb}}
The proof of {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}} precisely shows that {{formula:91ac8a01-44b2-4f08-836f-502ef84c5aaa}} satisfies the unperturbed Floer equation (REF ), with {{formula:7caef110-89ad-4cd5-bd0f-3752412dd1f6}} replaced by {{formula:34aeff1b-7c2b-474d-aec1-9024fa975fae}} . We have clearly {{formula:e7cfa774-05cb-47bd-ad08-e4bae93d47f8}} . We claim that we have the equality {{formula:7a894404-e2a3-4f01-b413-73336c932970}} and that {{formula:3848a322-8294-4da2-bbe3-1a53f679dc93}} is a {{formula:f98f2d79-1e2b-4690-b806-4a00c1e05794}} -periodic family of cylindrical almost complex structures. Hence {{formula:d93cd042-fd9b-4958-a8c0-ae3b0d906bf2}} is a unperturbed Floer trajectory between {{formula:6338a34b-b06d-4d24-9231-8ac181fca0d4}} and {{formula:9a425995-f85a-4cbb-8f1c-f738146268c2}} for the family of almost complex structures {{formula:c8f3232b-1419-4c02-acec-2d680ca43319}} . The claim follows by elementary computation, using a basic equality (REF ). Indeed, (REF ) and definition of {{formula:9aaf9dfd-116d-4ba6-8a07-449a1d8c3c89}} give
{{formula:ed454517-01d2-448b-8e33-020b12d7d057}}
Hence by definition of {{formula:3c18c6dd-d316-4c0a-8e6d-c569fa9127bc}} , we have
{{formula:aae69c61-253e-48d7-a84e-64235a2a742a}}
Which proves the periodicity condition (REF ). To see that {{formula:965ea0e1-4900-49ca-a0d3-186d33b213b7}} is {{formula:5066140c-a823-4eb5-a380-9244e8d49726}} -periodic, we write
{{formula:e593d2b9-f2c0-4fba-8778-264bde13637b}}
as needed. The equality {{formula:33d6f86d-72f2-48ea-b6a4-f458d0ca07fa}} above holds by the {{formula:8afcc400-02b8-4fdb-b792-cbed5428623f}} -periodicity condition {{formula:058cc318-49ff-4bcf-a891-aed723c35fab}} .
Since the pair {{formula:902f3adc-9f4a-4e3f-a729-4c07889aa399}} is regular, the linearized Floer equation for {{formula:6f9a194e-ce0b-4e9d-ad20-9a77b1815159}} is surjective, and hence the same happens for {{formula:91608d66-affd-4067-a7a9-3e581d6c8778}} and the family of almost complex structures {{formula:2dd094b7-8b90-4667-9c6e-fbaab3c59724}} . This way we get a one to one correspondence between the Uljarevic Floer trajectories {{formula:513a9c72-381c-4005-8eb4-cddade7dfc82}} and McLean Floer trajectories {{formula:4e9d6221-2d67-4a9a-a015-3e9e4d2c0533}} , modulo the {{formula:2c69f5e4-5986-4712-8cf9-5e49d113ff74}} -action. Therefore the differentials in both Floer complexes coincide: in particular, the results of {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} show that the McLean differential is well defined. We conclude that both constructions in Section and give the same Floer homology groups {{formula:74bf9697-62e3-4673-a4e9-fba06f34fd70}} and {{formula:555e5286-c638-43e6-b5a1-882814837e0d}} .
subsubsection3.5plus.7-.5Invariance by isotopy of abstract contact open books
The following result, crucial for us, was proved in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. We include a proof for the convenience of the reader.
Proposition 6.2
Let {{formula:87978bbd-548f-430c-bc04-c706ebaf15f0}} be a graded isotopy of abstract contact open books. Then the groups {{formula:4168090f-3d7c-4305-a98d-6223db3a6346}} and {{formula:11eaa4f2-7510-44d7-b623-64cd54c3ad3f}} are independent of {{formula:2d1b8422-380b-424d-bb23-2606d15062ab}} .
{{formula:5d62dee3-929d-44fc-aaaf-88e1cd98f520}}
.
Let {{formula:19ab7905-c8ef-4513-b34e-3e1b7f723bc0}} be the total space of the isotopy, and let {{formula:9e7d1e70-daed-459d-852b-781ccb63f261}} be the diffeomorphism which restricts to {{formula:0d9d5b3c-d100-4031-b149-ff83450bec99}} on each fiber, see Definition REF .
Given such an isotopy {{formula:c59b2be5-0927-4379-b100-0e3e51da2d74}} of Liouville domains, in {{cite:79d04bdbf72ad1ca2a8ea83646d1cfd16b340e58}}, an isotopy of completions {{formula:eed3e04e-5b89-48fe-9f82-945895420e95}} is constructed. Since {{formula:5e46b4a0-e3ff-4883-94bf-16f63aaa90b1}} is compactly supported fiberwise, it extends to a diffeomorphism {{formula:e51bbc64-4349-4c14-85fb-589c667ce83c}} which respects fibers and at each fiber is a compactly supported exact symplectomorphism. By {{cite:79d04bdbf72ad1ca2a8ea83646d1cfd16b340e58}}, {{formula:3bdd644f-db0d-4a72-9736-74128ea59ae2}} admits a trivialization by an exact symplectomorphism, i.e. there is a diffeomorphism {{formula:5779493a-b452-4f78-9705-62c64d601157}} such that {{formula:6ecb53cd-15ad-48d0-8aa3-d8fd6aa474c1}} , where {{formula:5353d260-6165-48a0-adbf-319b77d3e2f7}} is the restriction to {{formula:2bed98ab-fed0-4b5c-99ba-9bccac1493e8}} of a smooth function {{formula:2d6f7232-10d9-4cdf-97c7-c03fd7b7b153}} . So, given an isotopy of abstract contact open books {{formula:8d5ebdbf-34be-4ebb-ab3c-a71d42813db5}} , we have an isotopy {{formula:bb9aa9fd-74d4-4dd6-bcb5-57df7ddd9831}} of their completions, such that {{formula:180c452a-f591-4bca-b71f-18f4814b310d}} , and {{formula:7c9f1378-ce22-41e2-b171-7165a1745c10}} is a fiberwise an exact compactly supported symplectomorphism. If the isotopy {{formula:72059a2f-f0d7-4bfd-8c1b-98843942f948}} is graded, then this grading is inherited to {{formula:d463067b-202c-41ec-9307-c86cdbf7e987}} .
By construction, the Floer homology of {{formula:76ba19c5-7073-44b2-a185-1368b07c0240}} coincides with the Floer homology of {{formula:cfebd90d-9569-49fd-80f7-6c4700a0817b}} . Then, the result now follows from {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}.
subsection2-.5plus-.7.5The McLean spectral sequence
The remaining part of this section is devoted to the proof of the following result, which is a (slightly more general) version of the axiom (HF3) from {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, proved in Appendix C of loc. cit; see Remark REF for comparison. Our proof follows the same path, but we provide a more detailed exposition for the convenience of the reader.
Proposition 6.3
Let {{formula:f1174688-14d8-47b8-a2b9-46d0a4c9be54}} be a graded abstract contact open book such that {{formula:738ce3e3-85be-4a85-9274-cd0d370b3443}} and {{formula:f507327e-2b61-4b38-8d23-01f5385e56ee}} . Assume that
{{formula:6adf91ff-7aa1-4092-bda5-7a838eb6a5e0}}
where each {{formula:69c0b257-ae3b-4869-89df-9d61048a7544}} is a codimension zero families of fixed points, see Definition REF . Choose an action {{formula:812c1bca-3dc2-4d7d-994c-bcc350d1cf90}} , see (REF ), and let {{formula:5d163356-fb6e-4870-8f5d-7e609801cf89}} be the (constant) value of {{formula:a2e15c27-1342-4e86-ac2a-bf095add506e}} in {{formula:6dbc21a3-ca80-4b7e-b20d-b92aea8703d3}} . Fix a function {{formula:86c559ae-9e8e-4fc0-a241-b1e11f6360da}} such that {{formula:6d01f77e-857a-4b07-aec0-546f35375ab4}} (resp. {{formula:fd40590c-68f8-4a7b-967a-528d3f4dbe14}} ) if and only if {{formula:aa76b318-26e1-4e26-95a6-5581a52c49e7}} (resp. {{formula:430ee3d1-5578-4a05-afb7-e16edb1c16a8}} ). Then there is a spectral sequence {{formula:69a3e0d8-8e7c-4600-aa2e-b05a40828bfb}} converging to {{formula:5f061b83-f524-4f63-b92c-7dc629ea0bba}} , whose first page equals
{{formula:919b2aa0-d252-4747-8763-d3324a26c69c}}
where {{formula:f65cb64b-b9c6-4048-b3b4-9580a2e54051}} is the splitting (REF ). Moreover, there is an analogous spectral sequence converging to the Floer homology {{formula:8f2c82b5-f834-46cc-8f19-d9ac68f88774}} , whose first page is as above with {{formula:fa5d3c0a-17e7-488b-b705-17e823db45b7}} replaced by {{formula:fd929f21-393d-4bf5-b77a-2b21c9d993ee}} .
The assumption {{formula:5238d228-c18a-4d29-a037-a98a4923b92a}} is a technical assumption appearing in our construction, and probably it is not necessary.
Remark 6.4
The original McLean spectral sequence introduced in {{cite:97005650a8ac11991dc2e980883b47d01674846e}} is converges to Floer cohomology. Its first page is {{formula:90404019-7c57-4a11-8c8b-b9bd0bf4961c}} . Let us explain the difference with respect to (REF ).
Most importantly, in loc. cit. the definition of codimension zero family of fixed points given in {{cite:97005650a8ac11991dc2e980883b47d01674846e}} requires {{formula:6bdbba40-02a7-49c9-a532-5fc8c024d01d}} , so the homology groups in loc. cit. are simply those of {{formula:0891dac9-8d0d-47d3-a6a8-d4a0f07e1378}} . This way, Proposition REF generalizes loc. cit. so that it can be applied in the proof of Theorem REF , see Proposition REF .
The remaining differences are: the opposite degree of homology groups, which comes from the fact that the sequence in loc. cit. is cohomological and converges to {{formula:0e8da5eb-0053-4beb-95b8-29247dd5db56}} ; and the more general coefficients {{formula:1e7cbc9e-d18f-456a-882f-834d4e8f4098}} . We have chosen the coefficients {{formula:cbe805cd-6b86-45d4-8d50-80715202b7b5}} for convenience, since the same choice is made in our references {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}, {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}. This coefficient ring is enough for most applications, including the proof of Theorem REF .
For Proposition REF the general idea is the following: choose a Hamiltonian perturbation of {{formula:c1f2ec44-2876-46b0-ad75-8b7aab8d5042}} such that the fixed points become non-degenerate and contained in the union {{formula:95faff33-c891-4902-babb-600dabd2146e}} . Prove that any Floer trajectory connecting two fixed points contained in the same component {{formula:159f4e36-197c-4d6b-839a-ac828b2d4c04}} is contained in {{formula:9e4c091e-893b-4d01-9184-fac16d080082}} too. Then, since {{formula:15443a16-d791-497d-a521-46416219a173}} equals the identity, following the technique of the classical proof that Hamiltonian Floer Homology coincides with ordinary homology, it is possible to identify the pieces of the first page of the spectral sequence. For the precise implementation of this scheme a couple of technicalities need to be taken into account.
A confinement lemma {{cite:97005650a8ac11991dc2e980883b47d01674846e}} of pseudo-holomorphic curves due to McLean is used: it allows to confine Floer trajectories in a small neighborhood {{formula:72e2194a-831b-4230-9822-1f4a770d8dc0}} of {{formula:6e229f20-1878-46c1-804d-500948e89fe1}} rather than in {{formula:512d4748-dd73-46bd-82d6-212a0320b0cf}} . Since the almost complex structure {{formula:bcb26d37-f3a2-4273-8d17-2c1caace3bbb}} needs to be {{formula:640c4cab-2040-4d61-af9d-a4791c8b1b48}} -periodic, and {{formula:9e60223a-032b-40c7-a7c9-4c3c44ebd210}} is not the identity, it is not possible to choose {{formula:97cc369e-2cac-42ed-8385-8a1257e27408}} independent of time in {{formula:3eac895f-a4db-41ea-8b7a-511eccd79aa8}} . On the other hand, for the proof of the classical fact that Hamiltonian Floer homology coincides with ordinary homology, an almost complex structure independent of time is used in order to prove that Floer trajectories associated with small time independent Hamiltonians are in fact Morse trajectories. In order to deal with this we will produce first a Hamiltonian deformation of {{formula:b212ff3a-6bbf-4e77-ae33-4baf56a92a27}} whose fixed points is a disjoint union of codimension zero families of fixed points {{formula:d2834202-ba31-4995-8def-6a95b1f3c2fd}} , and each {{formula:b93e8c01-9be1-4f33-8712-1b36d18623f3}} is strictly contained in the interior of {{formula:21ec31bb-ecff-42ea-8cab-855c3784f4d5}} . Then a comparably smaller second deformation will produce non degenerate fixed points inside {{formula:4e8cc2ff-3a2f-4e72-8019-b19b4b04769b}} , and such that any Floer trajectory connecting two fixed points in the same {{formula:07c81999-526c-4b1b-b556-9f50a1c42849}} is contained in {{formula:9b60eaac-7895-41f1-985e-276658cec122}} . This way, a {{formula:b52e83d1-f8a4-4339-9a36-20635e7308a4}} -periodic complex structure independent on {{formula:bbb16caf-de1d-4be0-912a-97be94ba911b}} in {{formula:e5f3e4a8-266a-4883-9798-f661a3199fd3}} will suffice to identify the groups appearing in the fist page of the spectral sequence. The plan of the rest of this section is to carry out the proof following the program just hinted.
subsubsection3.5plus.7-.5Technical preparations Our first aim is to provide small Hamiltonian perturbation of {{formula:8636eb1b-29e6-421a-86a4-4b2145ee228e}} , which is well suited to apply the confinement lemma {{cite:97005650a8ac11991dc2e980883b47d01674846e}}; and a similar perturbation of the identity, which we will later use to compare the Floer complex with pieces of the Morse complex.
paragraph4.5plus.7-.5Step 1
By Lemma REF , we can and do assume that each {{formula:b2c90841-6b95-49a7-86d1-db791e254e13}} is a connected codimension zero submanifold of {{formula:f4602131-358a-4b06-a1eb-e2153d7d1a8f}} with boundary, i.e. it does not have corners.
Choose a collar structure {{formula:75dc4d3d-67c4-43c2-af46-8b42d3be0bcd}} near the boundary of {{formula:825eb722-2718-484d-ad42-cf51a40d39db}} in which {{formula:a03e44dc-32b5-445d-8a7e-1c34831fb4e7}} , see (REF ), and {{formula:dbaeba40-5309-4bb5-b554-ac47c8009418}} . Choose a small negative slope {{formula:3034c930-d672-448c-bcea-af627ad344ec}} and consider a smooth decreasing function {{formula:31664913-d086-49bd-a8b3-f150ebd4fbe6}} that is constant near {{formula:35516953-78b2-4298-9296-03e7af918aa6}} and equal to the linear function {{formula:42969140-7d7c-4d8b-a12c-a3b871c946a2}} near 0. Then the time-independent Hamiltonian {{formula:0c712e7f-9133-4d40-9a68-07574a78344e}} has small negative slope; and is {{formula:1358aad5-0ac5-4748-a83c-ecbaefd078ff}} -periodic since it is constant whenever {{formula:11dfa23a-19d2-4c9d-85cb-2d1e8aa92386}} . Let {{formula:9783a730-c25b-4e4e-b656-f67af1054938}} be the corresponding small negative slope deformation {{formula:4d98420c-48a2-4bc6-b71e-e1970e6fb018}} of {{formula:9ca6028f-b0ad-43c5-b967-b193ad5a10c6}} . Now {{formula:cbea3e4c-ac13-46e1-a78a-8acead41889e}} , where each {{formula:fc648ee1-2289-4082-8b9f-a33caea49ce2}} is a codimension zero family of fixed points, contained in {{formula:8a69a8cc-15f8-4434-bf82-8c78a0d9b971}} in such way that {{formula:731eb449-b759-4a5a-b5fd-dfcbf3fb7bb0}} does not intersect {{formula:0cd1397b-d0d6-449c-8259-42b5769a3aff}} anymore. In terms of the decomposition (REF ), we have
{{formula:998712c9-dcf0-41eb-8148-7ebe5ebaf9e0}}
{{formula:507062ad-16a6-48b2-a6e0-b68ee27d0aa3}} , and the triple {{formula:86964e7d-bdcc-4f45-9e22-f3b75d3d1563}} is homeomorphic to {{formula:afc7be91-a98b-403b-8865-f7c5a7f9b9e5}} .
Notation 6.5
For the rest of the proof we will abuse notation and rename {{formula:c495ac6d-7bba-401d-a6ca-70baacaafc9a}} . This way, {{formula:c719a8bc-5845-465b-b430-25a91ef350e4}} .
For the next steps we need the following notion:
Definition 6.6
Let {{formula:2b5bbb1f-b0e3-4ff1-851c-11c91e209150}} smooth functions, and let {{formula:20970472-d443-4a85-8c86-c7851737cecb}} be an open subset. We say that {{formula:daea8773-2f13-486a-ad27-209343ed0ff4}} and {{formula:05a4f6a2-5802-4630-8c76-7e3319092356}} grow compatibly in {{formula:804374b9-12d2-4ec4-9ff9-c58358d3393e}} if the level sets of {{formula:416c44d0-ee4f-4940-852b-fd501b1fd6ee}} and the level sets of {{formula:b5ce26c1-dc65-40fa-be37-c3023bb19105}} coincide, and there is a smooth function {{formula:76086946-3cdf-49b0-9631-19bbd0d7d9a1}} , constant on each of those level sets, such that {{formula:cd45cde7-f747-4923-a211-a52cd8d042ab}} .
paragraph4.5plus.7-.5Step 2
Let {{formula:437affd4-7236-4a57-acf2-389959e73db1}} be the time independent Hamiltonian associated to a codimension zero family {{formula:99047dcc-3bdd-4a52-8b79-89f2d62cff90}} of fixed points of {{formula:d4a6a274-0191-4886-8521-0a04ea373de5}} . Choose a compact collar neighborhood {{formula:114a70a2-7e3b-42df-95e3-1168edaf238f}} of {{formula:6416c557-3a06-4151-9c8b-9e95c71ee332}} contained in {{formula:3741dc10-86c4-46bc-a956-98d17129bb60}} such that the restriction
{{formula:9560425c-449b-42dd-b474-51f4577473eb}}
is topologically locally trivial. We claim that there is a smooth function {{formula:a4858a41-d36d-4fbb-a885-3b161bc1a195}} such that
it is topologically locally trivial and a submersion at the interior of {{formula:22b34f0a-9ee0-45e4-8f98-90eb718b7b72}} ,
the functions {{formula:b6ae0b38-7cf9-489d-96c2-9d8b1c2bae88}} and {{formula:4fa99f44-6831-42ef-85c1-c950617865a3}} grow compatibly in {{formula:305e7949-90f7-46eb-8915-85e1fe649609}} , see Definition REF .
the unique extension of {{formula:3f457a58-536b-4301-8a04-4177f1d339ad}} of {{formula:25b395ec-1e40-40b2-a0c0-563bc373555c}} by constant functions in the connected components of {{formula:b96e46f9-bc8b-4801-85c4-9b0c422c8cc7}} is smooth.
The function {{formula:38cd1fb4-8b98-4ae9-9d3f-7231bc2574cd}} is constant in {{formula:35f1b31c-9cd3-46a8-ab42-9718d176721e}} and in each connected component of {{formula:bed1e833-0754-4abd-bd85-003b27122518}} . Furthermore we have {{formula:46e24836-c355-441e-bc11-b788e99fba12}} .
It is easy to produce the restriction of the function {{formula:e4925ca5-06ee-49d3-817f-30ffe34b0194}} in a small neighborhood {{formula:73b3efe2-dee5-4450-b7f4-08b49f37ad1c}} of {{formula:f03db9ed-86fd-412b-9e99-d6985a378230}} . Its differential is a compactly supported 1-form in {{formula:ffedc6f5-2697-45b7-a037-94f80d48596a}} , which extends by 0 to a compactly supported 1-form in {{formula:6775d538-b49c-4f83-ac45-09f0a1682872}} . Then a function {{formula:1b22ccb5-5317-4809-a49c-c937f26bb9a6}} satisfying REF exists because of the assumed vanishing of {{formula:897ba3a1-b5e8-449b-884f-1f339e6a3a58}} . Property REF follows from Properties REF –REF .
Since
the restrictions of {{formula:790fd08a-26dc-4017-b433-636f4e8084f0}} and {{formula:510ae501-a9de-4278-8d83-fc3cdfcf5278}} to {{formula:b79e4b16-0ec4-4387-bf16-bf6618898326}} grow compatibly, there is a function {{formula:a305a82c-06b5-4821-8956-66ff9975c99a}} , constant on the level sets of {{formula:3394b91f-0d49-4d43-be7a-b40261decd0e}} , such that {{formula:7f349322-684b-4272-8144-079154e35b70}} . Since {{formula:256db54b-5b3a-4153-85db-b946c31244d2}} is zero on {{formula:5a22a175-1088-4fef-afbc-5ac3c553a37a}} , and {{formula:905a13d5-7201-48d2-83d5-40f9da786382}} does not vanish on {{formula:e456d576-b857-4f01-a5a3-fe724927f4e7}} , the function {{formula:4d61c959-6788-4093-a1a3-68a45b18ae21}} extends by zero to a smooth function {{formula:f30a1cbb-2be2-49f2-b8ea-74fb20b5dc20}} .
Fix {{formula:0e9e1b98-669f-4c85-9cf6-b387d17c7350}} and put {{formula:13324f0a-43cc-4982-bee8-dbecd0567362}} . Then {{formula:956d2d44-5e79-4019-ac1a-f8da2339fde8}} is a time independent Hamiltonian such that we have the equality {{formula:513a601b-dee5-4c47-8ca9-77587bddd524}} . Indeed, at a level set of {{formula:ca0c0226-2ff7-4a68-8853-563e8b95f508}} the composition on the right hand side moves a point first time {{formula:2cd355b3-2dd7-4ce3-bae5-8a2477ced7e1}} along an integral curve of {{formula:e53d518e-caac-4bd5-914e-ea7b16d0183f}} , and then time 1 along an integral curve of {{formula:ec65860b-1624-409f-a016-d0801088656d}} . Moving time 1 along an integral curve of {{formula:1a201eb2-7b47-4518-bd64-d93f2479083d}} is the same than moving time {{formula:f4f40ed7-a8b6-4c53-9502-879989e7feb2}} along an integral curve of {{formula:04e9b828-8c40-4146-a116-d4c76b423674}} (recall that {{formula:0d2b1860-557d-424b-af36-7f4eea8908ae}} is constant on those integral curves), so the total effect is moving time {{formula:8950daca-eddb-4a72-93f3-c40d28d9db61}} along an integral curve of {{formula:c824fd2e-a62e-43c7-91be-4559179c06f2}} .
Define {{formula:ef789b51-9e54-4aa8-b377-8f739017640f}} . We claim that if {{formula:3ce34f79-cf2a-470b-a842-dfb1b78c4575}} is positive and small enough the set of fixed points of {{formula:90a3bff0-fd22-432e-b221-fe250adec774}} is equal to {{formula:91543e5d-f077-47c2-93d9-80563358e3ef}} . Indeed, {{formula:f400449a-3543-4acc-989a-0e452757425f}} is obviously contained in the fixed point set of {{formula:97c676ad-6b76-4ef0-8edf-6d79711a9e61}} . To check the opposite inclusion we notice that, since the time 1 flow of {{formula:0c02b0f8-97f5-416d-bd53-4adb4ee876b4}} coincides with the time 1 flow of {{formula:60101fce-56ee-4f64-a030-0f57aec7b4e0}} , and has no fixed points in {{formula:396da0b1-3f20-4f27-a314-af1db3d68378}} , there exists a positive {{formula:dcccb0c3-6e00-4ea4-9192-1c59c63b893c}} such that the time 1 flow of {{formula:7b4655be-925a-4251-8515-75901720c721}} has no fixed points in {{formula:a8fea9b8-6a90-445e-974c-42e57f866f66}} , either. We fix such {{formula:4dd10a73-5dc2-414a-a660-4dd55ff430d8}} for the rest of the section.
Let {{formula:d2f122be-8bd0-47af-87fd-891efd4e32e7}} , ..., {{formula:fe2fb19d-3a6d-4d2a-86f9-0d56474b24a2}} be the collection of codimension zero family of fixed points of {{formula:83c1b6de-d3d7-4aca-83c8-4ecb70d5e723}} (see Step 1). Define {{formula:b7d83b64-b759-47bc-852b-4ec6c1746110}} . The composition {{formula:2e5470e6-2739-4299-a40c-888cdeee9f87}} is an exact symplectomorphism whose fixed point set is the disjoint union of codimension zero families of fixed points {{formula:c711d49a-ca26-442a-8a9d-1356c38fa5f5}} . For every {{formula:92baa236-8a37-4e6a-84c0-cb150115746e}} we have constructed a Hamiltonian {{formula:a08c47c0-d4c7-457e-9d5e-fd0f7d2e6c42}} such that {{formula:e1053500-f5b1-40a4-a96d-b6a7aa7ac671}} is equal to the time 1 flow {{formula:ddea4d41-e0c4-4622-96dc-16c25e302dd9}} .
paragraph4.5plus.7-.5Step 3
For each {{formula:a24962b1-122e-4e29-b4bd-a65232ae5d09}} choose a function {{formula:5cbbbcae-8353-4d18-af3e-95b5e27ebb8b}} such that:
{{formula:29dddff8-df48-4ee0-8350-f7a8b759e73a}} admits a smooth extension {{formula:b8b5f2c8-d22b-49dc-b848-c70984f66d02}} by a constant function equal to 0 in {{formula:5a6b85da-35ea-49b6-a409-aa365f5293df}} .
the restriction of {{formula:8dc27959-7ff5-47b8-9fcd-72e5517501db}} to {{formula:ff8da972-0c9d-4e2b-8178-f332157d23b5}} has no critical points.
the functions {{formula:89223776-db21-4407-bea2-a93fe3e74317}} and {{formula:974aa32a-3313-48cb-8fbc-38b929e235b8}} grow compatibly in {{formula:d9fa842b-df33-452b-8acb-09b0e6ca63df}} .
the critical points of {{formula:906a0147-bdbb-4af6-9783-da286d3eff02}} are of Morse type
for every {{formula:3f33c3eb-b9fc-41d2-ac0a-2a7975cbb9d5}} , the only {{formula:5958bd53-6f6a-4b61-afd4-10bb857a320f}} -periodic orbits of the Hamiltonian flow of {{formula:4d430bec-df92-48e4-9b65-510ac55d34c0}} at the interior of {{formula:a265c44f-16d0-4743-af18-4b4cd8e8de12}} are constant and equal to the Morse points.
It is easy to produce a function with properties REF –REF . A small generic perturbation of such function which does not modify it at {{formula:03cb3d5d-5416-4a71-853a-2ba6119baf52}} achieves property REF . Property REF is achieved multiplying any function with properties REF –REF by a sufficiently small positive constant, as a consequence of {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}.
Define {{formula:b1183b0b-81a6-4e13-9818-355b26946b31}} . For {{formula:8289cb70-2555-477d-b371-b87d96151188}} small enough define {{formula:29eeac1a-6658-43ff-9838-8a600e3fc5c1}} . In the same way that we proved above the existence of a time independent Hamiltonian {{formula:0d0c75c2-245b-4d14-aafa-afa7be40f08c}} satisfying {{formula:f0c8b970-43b9-4aff-b470-24cebf08e0c7}} , here we prove that there exists a time independent Hamiltonian {{formula:e93d14af-066e-4c54-a482-1a34388b4656}} such that {{formula:445061df-d8b4-4d3a-9780-547bce71c61a}} . So, we have that {{formula:b416e09a-9720-4443-93ae-ec7594991b44}} is a negative slope Hamiltonian perturbation of {{formula:ea3198fa-f589-42eb-97d9-0e76480bad49}} . Moreover, if {{formula:ce5282de-816b-47cb-8704-37cda6e9167b}} is small enough, that slope is small, too, and the fixed point set of {{formula:e3d5c595-3584-462a-901c-fea26f394933}} is the disjoint union of the Morse points of {{formula:c1169e16-698d-474c-8006-50d826bba16c}} at the interior of {{formula:4e0b0dcb-6042-4289-82e8-b3cda8855fda}} . Indeed, by Property REF of {{formula:29fc387f-e8ba-4ad4-9e19-41a2001dcbfd}} the set of fixed points of {{formula:fd916031-7162-45c1-baad-30072043d99d}} contained in {{formula:4f12e31e-59c7-4d34-96b6-d7071b243175}} coincides with the set of Morse points of {{formula:906edcd4-d3f3-4617-a589-1e72571f5b52}} . Outside {{formula:508adb53-d12b-463c-812f-7e4709b3255d}} , {{formula:83b51e06-7ed2-431b-ae2f-122dba3044e6}} coincides with {{formula:04b31062-8648-4b82-8b4c-a04fe7b58513}} , so it has no fixed points there. It remains to show that {{formula:50075971-d2d4-4fb2-8707-721b13d5d73c}} has no fixed points in {{formula:9f6a6d95-3aa7-4123-8e21-3abcc7011a86}} . This follows, if {{formula:ae762b87-0887-44c7-85ad-ed3861729b16}} is small enough, by the same arguments that proved above that if {{formula:75cdeff9-4854-4330-a999-1560957afbac}} is positive and small enough the set of fixed points of {{formula:979ee87b-ec3b-4496-837f-65dcfe53cd43}} is equal to {{formula:ad338504-ce86-4ef3-8269-e766202249f4}} .
paragraph4.5plus.7-.5Step 4
In the three steps described up to now we have provided a Hamiltonian perturbation of {{formula:6bd20639-c64f-4173-ae05-5887d8f228ec}} that is well suited to analyze McLean's spectral sequence. In the next one we run a similar procedure for the identity symplectomorphism in {{formula:22f56217-e6a2-4fe3-9cbb-3bd76cef208f}} , with the later aim of producing a spectral sequence similar to McLean's. Then a comparison of the first pages of both spectral sequences will lead eventually the proof of Proposition REF .
Consider a smooth function {{formula:fb6cc2cc-71cc-4c5c-a128-1fee49c4040a}} with the following properties:
we have {{formula:f28d1fff-610b-41fa-85c8-0d85038357d1}} ,
the restriction of {{formula:4512b7cc-6dd5-417d-b5b2-1114c6f505cb}} to {{formula:87591f55-1a03-435f-a521-6a7cba468522}} is a Morse function, and moreover the set of values of {{formula:3d094b37-e809-4a00-af11-9f564397a71f}} at the Morse points of {{formula:ac8d724a-8a8f-4281-9379-38cd50ac962d}} is disjoint from the finite set {{formula:6b072c92-a30c-420e-afcf-3733a58ec8cd}} .
We consider the exact symplectomorphism {{formula:cf50dd0c-0068-4a77-9221-d28b4b38b6e9}} for {{formula:923e32fb-bed2-489f-9727-4369b2659e3b}} . As before, we prove that there is a time independent Hamiltonian {{formula:535ed26e-0ffc-4836-86c3-4428fa2553b0}} such that {{formula:e3c0459f-0dfd-4211-8f38-63998d2c126e}} , so that {{formula:1da895b9-4f0f-46b5-8cd8-e054f586f3f8}} is a small negative slope Hamiltonian deformation of {{formula:ca88de89-8cbc-4f52-8f45-5c9ab6b821ec}} . Moreover, {{formula:88409a1b-c988-4d8f-9309-909838b69891}} is a Morse function in {{formula:17cea246-7ea3-462c-8567-2aebcd092164}} which satisfies {{formula:26265d4b-9444-4eca-a4fa-466e8dcf35e6}} . Therefore, by choosing {{formula:e1a9448e-c1f3-4157-9fb6-a9fe8dfd3ef1}} small enough we may assume that the fixed points {{formula:67145ea6-0031-4bf4-ab0a-db7c936070c0}} coincide with the Morse points of {{formula:e319d2be-c953-4a0c-b4b3-442672a6c2ac}} {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}.
The following two properties are satisfied by {{formula:4e199394-1799-4652-8bb8-e3b49bbf579b}} :
{{formula:22eb5cdd-1071-4acd-8375-e81162dd4133}} .
The functions {{formula:401e369b-3bab-4e10-83c9-e460631a9a85}} and {{formula:672f3e06-79bf-4150-a226-0161e5c8a11d}} grow compatibly in {{formula:c8d361df-addd-4311-a4c1-050ab96d74d4}} .
For REF notice that since {{formula:99c4896c-6a88-48ae-9db7-f7835f5e4971}} , we have {{formula:34f41704-6b1a-492c-9cb9-b4c4554afa34}} . Therefore Property REF of Step 2 implies REF . Property REF is a consequence of Property REF of Step 3.
paragraph4.5plus.7-.5Step 5
Let {{formula:962d26ae-a049-4800-a194-63ec27428153}} be cylindrical {{formula:86012061-23e2-4e41-9304-0dc4a8d0b939}} -compatible almost complex structures such that {{formula:2e2a9207-395e-4e5b-89b7-69aec352e7aa}} is {{formula:1b54ec5a-6aa2-48e2-b48e-0904273a1a4f}} -periodic, {{formula:eb2f9721-2a25-40e8-8508-8c63df596112}} is time-independent and they have the following additional properties
{{formula:dc9c5ad7-9005-4f97-ae92-eaf186d8a577}} is independent of {{formula:19fa4e9f-4767-4994-bed5-a36723b1e226}} in {{formula:959c4687-e70a-434c-b80f-451232cd7a43}} and we have the equality
{{formula:0322bf33-7ac1-4516-a885-3ff19d6fcad1}}
the pair {{formula:58be9876-9726-48af-a725-3c12c77795e5}} is Morse-Smale for any {{formula:adcf6674-803c-4ce7-b312-b44ab10eb1e4}} , {{formula:0a24e7e5-410f-4bf2-bfbd-69007e388a59}} (that is, the linearized Morse operator at each gradient line connecting two Morse points of {{formula:1704b766-fd13-4cac-9d22-fba89db81b62}} is surjective, see {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}} for more details),
for each {{formula:9ce8ab80-b336-4ae6-aff9-94893544e490}} the pair {{formula:227db692-a7b3-49dd-83b0-90078dcc1e2c}} is Morse-Smale at the interior of {{formula:1606fbee-2116-4773-b756-fc3ea285b0bc}}
In order to construct {{formula:388f752c-2482-487c-96b7-a50789062254}} and {{formula:b891da72-74bf-4ca7-bd73-ae42cc4f1437}} we choose a time independent {{formula:0f7f0c99-4fed-4263-b482-ab7ce7905370}} satisfying property REF , following {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, and then choose any cylindrical {{formula:78530974-6437-4326-b494-1266b16a772b}} -compatible and {{formula:0817237b-27fc-4878-a5ee-e55fe8144504}} -periodic almost complex structure {{formula:954903df-1747-4162-93c8-5e6b78e380b0}} satisfying property REF . Property REF is satisfied because {{formula:e03c8685-192f-425b-ace3-e53a97abf02d}} is proportional by a non-zero constant factor to {{formula:91270e3f-4d0b-4f20-b9f8-34bcd71b93c4}} .
subsubsection3.5plus.7-.5The two Floer complexes
Having fixed {{formula:0b17fe45-94b6-4af8-b256-f827c24a0365}} (for {{formula:8a5a0aa7-119a-4e2d-ba0b-5a9e8e1af19d}} small enough) and {{formula:2fd9dab3-2f61-45f8-8bd6-9717fdf096c5}} with the properties of the previous section, following {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}} we find that there exists a time dependent perturbation {{formula:678aeb74-8c8e-4c89-b471-0cf9b197ada7}} of the Hamiltonian {{formula:0eee617b-8e67-4e7c-8821-85c95b1f7235}} , which is as close as we wish to {{formula:8a1710fb-7bac-49f6-aca2-0162a0cc330f}} , does not modify {{formula:664a99d3-902d-48c4-ad2e-e0c707a3408f}} in a neighborhood of the fixed points of {{formula:b7f94aec-50e4-48d2-84f8-913be344c201}} and in a neighborhood of {{formula:76b54850-2b42-40e1-bff6-db0ecead453c}} , and such that {{formula:cc94189c-e6ef-437e-9cca-e28832d26ad5}} is a regular pair for the perturbed {{formula:3c48beef-25b3-4d84-9715-9e9f20664ecf}} -twisted Floer equation in the sense of Definition REF . That is, the {{formula:5cdc161e-7dab-4935-a0e9-bafd5f282fab}} -twisted perturbed Floer complex of {{formula:b354e129-d00e-4329-b7ec-234297bafa49}} , denoted by {{formula:1842f850-bef0-4e48-8b17-79d32122ee31}} , is well defined and by Proposition REF , it computes {{formula:1f7b17e9-de71-4c83-a07c-a7ebf00433e0}} .
We define {{formula:403ce71a-973a-4143-8ce6-3f41c6a2f223}} and {{formula:6d862dd5-6b04-4244-a4e8-c0282d66dc19}} , see Formula (REF ). Then the unperturbed Floer complex {{formula:4bfb89af-a874-408c-97bd-447ca44f30e6}} for the pair {{formula:6d778459-c65d-48b4-b6d4-fb260a5ab56b}} , introduced in Section , is well defined and, as we have seen in Section , it computes the same homology as {{formula:736168f6-47e8-4445-af33-a47c1ee88d21}}
Similarly, there exists a time dependent perturbation {{formula:b870b2fb-8ded-457a-9686-eec05298f731}} of {{formula:e14800e9-3f54-4bd5-8179-0b00b5bd7e5a}} which does not modify {{formula:c17fb6df-1ef8-4cb0-9a47-cb7c61fd7b38}} near the critical points of {{formula:ff9f466f-6d87-455a-b359-154e2af0a739}} , such that {{formula:b38f16de-c644-4814-8dfe-52c23745036b}} is a regular pair for the perturbed twisted Floer equation. That is, the untwisted perturbed Floer complex of {{formula:c553ef97-a080-494b-9b9e-12c55dd5e78d}} , denoted by {{formula:4cb57cf2-a99a-4e41-b71f-ae2ff21012a1}} is well defined, and it computes {{formula:cedf9dd6-90bd-4aae-a334-e74e865df399}} , see {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}}.
As before, putting {{formula:acebacf6-66a5-4954-ab6f-432b90e1621f}} , we see that the unperturbed Floer complex {{formula:3936fd43-7e3b-4935-96c0-7a9eec34972a}} is well defined and computes the same homology as {{formula:a76827d6-970b-427c-b1b7-7ab87365dfa1}} .
The perturbation procedure described in {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}} makes clear that the perturbations {{formula:2c26a3f8-2837-46e2-bc2b-4a3d34f4ac51}} and {{formula:c0378ff6-f0d5-479f-9769-769e512df6c2}} can be chosen so that they coincide in {{formula:38e9669d-545c-44e8-a6ce-345826841f41}} , that is,
{{formula:8bdd8beb-289f-4554-a008-13babab64100}}
The idea is to replace the space of perturbations of the Hamiltonian, denoted by {{formula:48bdb02f-bb0c-477e-98b9-c427567beaf8}} in {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, by a Banach space of perturbations {{formula:6ddb7cea-b46d-483a-9f2d-b037dcccbc78}} whose elements are pairs {{formula:911168db-993f-46b2-84b2-043965db2959}} where {{formula:7fc1b580-57ef-4519-b69d-9c8babfb9bd1}} is a {{formula:af9bdaf1-29d8-403c-9ab3-7283627a2192}} -periodic time dependent Hamiltonian, {{formula:23d55ff1-c47c-4a96-8ef7-d4511082b8a4}} is a periodic time dependent Hamiltonian and {{formula:268b5a8e-d5e9-4942-9005-447db26788c3}} and {{formula:cc0b25db-7d20-407f-b4a1-bde770e43586}} coincide in {{formula:80dfb7a2-f350-4dc1-a243-2db65657e63d}} . Then it is possible to prove like in {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}} that the locus of {{formula:ef4acfea-6590-4bbb-ad0b-86b95cf169ea}} such that {{formula:c311134e-c812-4985-8daf-08a94913361e}} and {{formula:2c40b009-8a3a-4357-9530-1c25b9783b9c}} are both regular pairs is dense in {{formula:acbd4f0c-1b5c-4da7-a56e-5698d00bf348}} .
subsubsection3.5plus.7-.5The McLean spectral sequence We will now use the action filtration, introduced in Section , to define the McLean spectral sequence.
Recall that the action function {{formula:9d587a76-d946-4afa-b68a-b38afc22a869}} of {{formula:df2503ab-a63d-4e1c-baa9-85525d0ae844}} is constant in each codimension zero family {{formula:1af5b06b-4735-4833-ab2d-30f22fb8d06d}} of fixed points of {{formula:1880f762-af85-4f37-8e98-0aaaa6feeb5e}} . We have denoted its value by {{formula:0b046663-de41-4c03-a197-aa2d2a30ccab}} . Choose a positive number {{formula:e6dedead-1b30-4176-8b28-0a03e26100a0}} such that {{formula:c89015e1-7c74-4433-a7d5-9c2180c514a0}} .
Let {{formula:d67824a0-8d13-47ad-bc82-6974ca54d3dc}} be the indexing function from the statement of Proposition REF . For an integer {{formula:2f37dc68-ef9d-4fee-b77d-0fabe6b43ee3}} put {{formula:cb6c103d-4764-44ad-aa52-a8988cc3b6e9}} and
{{formula:a14e33ce-0d61-4f5c-a582-72be8edcf1ab}} . Then for every {{formula:c0d88568-5285-48ca-b337-5b2fa69a7541}} we have {{formula:67db5c52-02ac-42ea-b5a7-a485bcb9b88e}} . Put {{formula:b9122c28-f842-4dd6-8e8b-cd01bbab14a0}} . Clearly, {{formula:6262dec5-62cb-4184-beb7-204cba405adb}} if {{formula:5144a947-7d6b-4c27-961b-6f94f1bf97d3}} .
Put {{formula:ad18d1cd-385c-4960-b323-26f419fb45a0}} , where the right hand side is defined in Section . This is an increasing filtration by subcomplexes of {{formula:03a23380-6890-4aed-b60e-ba01559f91e9}} , see Section . The McLean spectral sequence {{formula:7a9fb9cc-b5de-41bb-9a40-2a1dbe54289a}} is the spectral sequence associated with this filtration. We have {{formula:4055f1c7-bb66-4bab-8de2-370bfdbcdbf1}} .
The {{formula:3318e8f5-7f9b-46c6-8ac0-ea2cfe4a741f}} -twisted perturbed Floer complex of {{formula:6d5e2d92-0eda-4337-a462-2760aa2a21fb}} is generated as a graded vector space by the set {{formula:c259b44e-954c-4ea8-87a9-3c251b067a55}} of Hamiltonian {{formula:2e7aa37d-1c2c-47b2-af7d-18d08ac7f19c}} -twisted loops. Since we have chosen {{formula:4e7d84ac-1e6f-421f-97e0-dd73bc1a852b}} and {{formula:6825932f-a289-4ce5-9a6b-105ce729efea}} small, the image of every {{formula:4b3a36a8-a164-4816-9bc3-7a2391bc2d10}} is contained in some {{formula:ce615367-bd2e-4494-9848-b45b4a1817c5}} .
We have that {{formula:fe045460-895d-4e09-b400-f45d6ecd8ed8}} is equal to the time 1 Hamiltonian flow of {{formula:c0fed333-9059-4ae0-b28d-8a6ce47b3d1d}} , which is equal to {{formula:c95e38da-eb0e-4a4a-bd5d-93cdaf899c5f}} . Then, by property REF of {{formula:da00e13d-d806-418d-941b-0215e03e2668}} , the only 1-periodic orbits of the Hamiltonian flow of {{formula:ab0f2f6c-58d8-4e05-9c5a-91480b588cfa}} are constant and equal to the Morse points of {{formula:4789c708-8a1b-4052-8a36-b2d9a853fe32}} . According to the procedure used to define the small perturbation {{formula:a1c45c21-65bb-4c4b-8179-4a1fd5c3c760}} following {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, we have that the only {{formula:3df42a19-ab62-4f61-9eef-2c33cee3a2a0}} -twisted Hamiltonian loops for the Hamiltonian {{formula:ef5711a9-ba57-4d67-be55-76037461def1}} which are contained in {{formula:f0007629-879c-49e3-a519-191e8790d00d}} are the same than those of {{formula:e7775289-fef1-48e1-9a73-fb4d2e018c18}} . Hence each {{formula:d3673280-07e5-425a-93f2-dab59dc662dd}} is constant and equal to a Morse point of {{formula:0dceea4b-24cf-4a81-86c0-1de9b69d43b7}} in {{formula:e13b05cf-7d3b-4e87-a4a5-63dd6452196e}} , for some index {{formula:37ff3d7e-3893-4b82-979a-ad0d86df1695}} . Formula (REF ) shows that the action {{formula:a738c22b-837c-4ac3-9461-85e6caba534b}} of such point belongs then to {{formula:b4b965a1-9deb-426e-a6ec-1f32c968ddd8}} . We conclude that
{{formula:172f2612-1ed8-442b-b235-7640648b0d64}}
where {{formula:d3541e56-e9e6-40d8-87a1-4dc4eae6b928}} is a subspace generated by the Morse points of {{formula:9e720051-5b44-43cf-af8c-31fc7056ca1f}} . Lemma REF below implies that each {{formula:92668bd4-5869-4908-8cc0-67dfcc3a2e21}} is in fact a subcomplex of {{formula:96936bf7-feed-48c1-b87e-92c9302edb96}} .
In order to compute the first page of McLean's spectral sequence we will compare it with the first page of a similar spectral sequence associated with a filtration in the Floer complex {{formula:06f4153d-50e8-4a00-b288-91e9c57efc40}} which we introduce now. Let {{formula:efb187d6-5bfd-4cf1-8126-ca3a1ab03839}} be the critical values of {{formula:d1a58071-571a-4044-936a-a02237f0a921}} ; notice that some of these values are equal to {{formula:527cf21b-aae9-4360-8f33-756bd21298ce}} . Let {{formula:2f6ed34f-344a-4a53-bd41-713079783027}} be the set of those indices {{formula:72c7005e-6c9f-4091-8ce3-47a667896d52}} such that {{formula:32635651-a83f-4ddc-9c89-8eae148c2797}} .
Since {{formula:cb57c6d3-19de-4291-b739-3726cd917d60}} is as small as we wish, we have that {{formula:b776a1cb-4bab-4897-aba5-5c10b4fe5854}} and {{formula:bc61b897-5364-4a79-9665-df42810bdba5}} are as close as we wish in the maximum norm. Therefore we may assume that the critical values of {{formula:83c21c90-def5-4cd8-b282-6cfed45e0162}} belong to {{formula:b78564ec-6eb2-4b7f-a335-cdb5973ca65f}} , for a fixed {{formula:63228d72-e2d6-4bbf-8dd0-577b2e5e1f9e}} such that {{formula:26dc05cb-8b72-4b13-83b6-2917e85670ea}} .
Let {{formula:0317284a-1dc5-4b47-b2ee-d12b1ef94168}} such that {{formula:5412fa4d-3782-47ff-a72c-34e84de53d21}} and {{formula:bf2404c6-fddb-49f0-9f5e-67595d8a71c3}} for {{formula:6df462ed-0806-4d50-bb6b-33ca0a839dca}} . We define the increasing filtration by subcomplexes of {{formula:b8991d2a-2e35-498b-be30-564d47c003a5}} by {{formula:f458458d-0a00-432b-a605-37c95531c155}} , where as in Section , {{formula:64aa735e-fdc2-47f2-b9d3-c4f189184431}} is the subcomplex generated by Hamiltonian {{formula:2e652a97-e4bc-4599-ac5b-f8145c059288}} -twisted loops whose action is upper bounded by {{formula:11e6093c-9219-4b04-b2dc-557292dbe05e}} .
As before, since the perturbations {{formula:64236f05-7708-470b-8127-81c1cd7b0355}} and {{formula:bebeb069-1820-4528-b24d-a1481b8a568d}} are small, we conclude that the set of Hamiltonian untwisted loops {{formula:09c6f0dc-f907-4dd1-89d6-410c4582979b}} coincides with the set of Morse points of {{formula:bb2c083c-b870-4f99-8edd-120a58288bfb}} . Since the action function for the identity symplectomorphism can be taken vanishing identically, Formula (REF ) shows that the action {{formula:f01c0a2f-cbf0-4a44-86ba-77b44d47e7bc}} of such a point {{formula:c7c4de2b-6fca-486f-938b-cc4604614f9b}} belongs to {{formula:fac93349-457e-4748-8abc-59a763669d06}} for a certain index {{formula:fe127f59-0c69-4779-b36c-34384e94c593}} . If {{formula:3322675f-7b37-4a1f-960e-3e48fda51b29}} , property REF of {{formula:ebe7c3ea-0524-40ac-9cca-6d3c06a68f1e}} shows that {{formula:eb7a960a-2a42-465b-b377-eec09e7c8e27}} is a critical point of {{formula:91150177-b952-475c-b109-2302e327246f}} for some {{formula:d455e664-fc8e-4d92-a548-ec9f7a007b20}} . We conclude that, whenever {{formula:0e8fb488-d72b-4d69-84fa-f7282be03d17}} , we have
{{formula:a521973f-f445-40c0-80c4-969d7a00380f}}
where {{formula:82c0ef14-28c3-4f1a-b272-8ce0ee554b75}} is generated by the Morse points of {{formula:0e56329d-9388-4d65-a8a6-8beebee4b694}} . As before, Lemma REF shows that {{formula:059fc7c3-e3d1-469e-9a78-c93c0e4baaa0}} is a subcomplex of {{formula:71ac69b8-dce2-4bf8-82ff-6e61e5bd4dbf}} .
We claim that we have an isomorphism of complexes
{{formula:afbd773e-bcd1-4562-90d8-123bb20fa5ee}}
By definition, the underlying {{formula:af2aa333-71c8-41bf-be8e-04ae55c278f4}} -vector spaces are both spanned by the Morse points of {{formula:fc3d528a-fc7c-409d-b246-f6f0bfb43613}} , so they coincide. The grading of such a point in {{formula:13ce1dfe-f8b2-4151-989a-d290b4ff4488}} and {{formula:40988a17-4a11-49dd-99ff-30e830527cd3}} is its minus Conley-Zehnder index, seen as a fixed point of {{formula:aac430ef-b68d-4163-93b9-2a04a56e3ba1}} and {{formula:15e41023-b29b-4ba3-81bc-7aca3b0941b2}} , respectively. By construction, the flows {{formula:76423eff-e6a9-479b-a8be-e41f7e951d00}} and {{formula:4a9b2941-ba9e-4fff-b97e-a9524baac372}} coincide in {{formula:d8e274a7-8fc9-43aa-af70-58c88c8ebe31}} , see (REF ). Denote this common flow by {{formula:ac924ab5-3d7a-4eaf-b53e-9b6c816cbd68}} . Formula (REF ) shows that {{formula:1e9669d9-cf48-48ff-8660-d5e5ec6f1353}} . Thus the grading of {{formula:af62155d-b900-4b84-9299-f44e9ffa8037}} in {{formula:a8a90ecb-c292-4226-a01b-5b43b26dd2fa}} equals the grading of {{formula:89af9d8c-a5c8-444e-9ee7-b8adeefd4161}} in {{formula:2eb058dc-7fdb-4dd7-82fc-ba4dc9c3574a}} minus {{formula:dfc497b0-5a15-4e33-b810-11d7423e5eec}} , which gives the required degree shift.
It remains to show that the differentials in these two complexes are equal. They are computed by counting Floer trajectories connecting the critical points. By the confinement Lemma REF below, these trajectories are contained in {{formula:52202f33-af2b-4527-9691-9ea6364f81e8}} . There, we have the equalities {{formula:0ea211b5-8c7a-4cef-91e1-e285f148c3d1}} , {{formula:449fe67f-0abf-4cf5-a956-dd585d6f4421}} , see (REF ), and {{formula:9db1d565-757c-4aed-9bee-94b3c341eb16}} , see (REF ). Hence the Floer equations (REF ) defining the Floer trajectories counted by both differentials coincide. Thus we get the isomorphism (REF ).
In the above proof, we have used the following confinement lemma for pseudo-holomorphic curves due to McLean, see {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
Lemma 6.7
Fix any index {{formula:95d2d3fd-eda2-4518-8c73-749347de2f75}} . If {{formula:cb773444-2231-4eb1-9dc8-c9aaebc85a7f}} and the perturbation {{formula:a5d46651-e70d-4000-9dee-8eeaf421d791}} are chosen small enough, for any two Morse points {{formula:1e9707b2-a9f6-4292-8821-df4bf7c846e7}} of {{formula:84c15b98-843d-47b7-bde0-cc6ed4c8b6d9}} which are contained in {{formula:ac7ce1fa-3348-4a89-8993-8f73f3f8d60d}} , we have that any Floer trajectory connecting {{formula:eb2a8273-2006-4f60-a4f0-3eacf31fa487}} and {{formula:3e333796-f2aa-464c-8ef0-a5d236a22401}} for the pair {{formula:52569c96-2f5b-443d-83ef-9dbfab198a28}} has image completely contained in {{formula:83e8093d-a12d-4af4-9a9b-93fa2c8e6d68}} .
Similarly, if {{formula:dd84a920-112c-4b4b-92b2-e89b4eecdf76}} and the perturbation {{formula:33a01c8b-c8ba-4ce0-873d-86d539ab8b46}} are chosen small enough, for any two Morse points {{formula:5f2d1e87-ba50-4854-93e5-765ba973dc9b}} of {{formula:b4ab56a0-0e2f-458e-9a09-e232a345f28d}} which are contained in {{formula:fda95b4c-ebcf-4016-b64b-99f9683a290f}} , any Floer trajectory connecting {{formula:bf464b4b-5aa1-4a85-a765-a116dca67f90}} and {{formula:1fa1c818-3c49-4b87-8dfd-52109be7df1d}} for the pair {{formula:6aa1f641-1a8a-4b64-a953-4db77d8da561}} has image completely contained in {{formula:35caf5c1-4075-4290-bc6e-681f9ea8bdcb}} .
{{formula:7260361f-47ca-4cb2-a2c2-9f9d238eda12}}
.
For the first case the Lemma coincides exactly with {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, applied to {{formula:43c8aaca-99af-4a59-8db6-1866d17076fb}} , where {{formula:c2542498-d0fd-4c1e-907a-dd2a52e5800b}} plays the role of codimension zero family of fixed points and {{formula:8cac795e-1b9f-4dbd-bc3a-5596ffe1693d}} is the neighborhood where {{formula:9096ccf9-34d0-4c71-b1c8-e99dfc3d350e}} is Hamiltonian, except that in loc. cit. the definition of Floer Homology uses unperturbed Floer trajectories. The case of perturbed Floer trajectories is reduced to McLean's setting by the procedure described in .
For the second case notice that by {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, if {{formula:76d348c4-a8a0-4e7f-849a-14b09d02d68d}} is small enough the solutions of the Floer equation for {{formula:040da360-0b9d-4326-9f01-f849d84550c2}} are independent of {{formula:e632f4cc-d8ba-4f6c-b17c-424d49469c4d}} , and hence equal to gradient trajectories for {{formula:d573c748-8341-4352-9489-d74e2e5f3b2d}} . By Property REF of Step 4 a gradient trajectory starting and ending in {{formula:288bc0d5-47c4-4db7-b427-df2e69cc4521}} can not escape {{formula:35007e9c-7fb0-4ad0-a4c6-6a35386a8784}} . Moreover, by Property REF of Step 4, a gradient trajectory can not escape {{formula:eea02b73-69e1-45b6-99f8-21a14fc1bba7}} either, and since the critical points of {{formula:e2992f05-7295-4aac-8af4-084c50f90ee7}} are contained at the interior of {{formula:66ecfd44-87be-4661-ad6b-815475c16190}} we conclude that all gradient trajectories are contained at the interior of {{formula:13737e21-2f41-430c-9bdf-6fe741bfee4c}} . This proves the confinement for the solutions of the Floer equation for {{formula:4d8907c5-ff56-4e16-969e-f58af51629b8}} . Then, by Gromov compactness the confinement follows for solutions of the Floer equation for {{formula:f5551517-c150-4b80-b228-8e8f82c20016}} if the perturbation {{formula:5a51318b-ebb5-460c-bdf1-10b34cf78565}} is chosen small enough.
subsubsection3.5plus.7-.5Computation of the first page To prove Proposition REF , we need to compute the homology of the complex {{formula:78c9ce26-a461-45b1-bf0f-672c154f7f3f}} appearing in the formula (REF ). Using the isomorphism (REF ), it is enough to compute the homology of the complex {{formula:37e15e36-5aca-4623-b892-012f8f61c00d}} , defined in (REF ).
We need to show that the homology of {{formula:65596693-3e20-4083-bb33-13a2a0808d9b}} is the relative homology of {{formula:8e279267-f16d-4419-b26c-39f9c5cfa01e}} , up to a degree shift. For this we follow the proof that {{formula:e2265db8-f73c-4bd4-a8af-c795674cf56a}} via continuation maps in Floer theory. In {{cite:687f659b6ad85164acd4322b4b39e7c12fc6d193}} it is remarked that our case (in which {{formula:f0fb7202-4d75-4b3e-80b8-fdacb4f19f00}} is a Liouville domain) is treated like in the classical proof that Hamiltonian Floer Homology for compact {{formula:11e55337-02ff-4171-afbe-423a04fbac48}} with vanishing second homotopy group, equals ordinary homology shifted by the dimension. This proof is fully explained in {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}.
We recall the needed details here. In {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}} it is proved that since the pair {{formula:d1a43f7b-54f3-46b7-a4f4-a0fd01591734}} is Morse-Smale, the Morse complex of {{formula:82a3d062-58fe-49ea-beb8-f1e6d944e33d}} is well defined. Since we have {{formula:7123262b-feec-4f58-85e7-e27d93a1e41d}} , if {{formula:5c917707-ec56-4833-8f18-89f7a166eceb}} is chosen small enough, the Floer trajectories for the pair {{formula:39620f9b-77b6-497f-934d-2185e8fdb68b}} are independent of time (see Proposition 10.1.9 of {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}). Hence {{formula:3f285c25-fed0-49e5-8fb6-c2dfda3f08b1}} is a regular pair in the sense of Floer theory. The Floer complex {{formula:3e15c8a6-a72a-4611-8af9-bd8275da4cf6}} is well defined and coincides with the Morse complex of {{formula:704c85bc-9704-49e1-9d04-4d95654b4057}} .
In {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}, using Floer's continuation maps, a quasi-isomorphism
{{formula:43dc71be-6317-4b35-8a1f-c30e60ed0ae5}}
is defined. The action filtration {{formula:3fcab8f0-615d-40ef-a4e7-549c6f6516ce}} is defined for the complex {{formula:fea23a0d-cd4b-48ee-8700-da78e3d0f1a8}} in a similar way as above. We claim that the quasi-isomorphism (REF ) can be defined in such a way that it respects the action filtration in both sides. To prove the claim we recall some ingredients involved in the construction of the continuation quasi-isomorphism. First we define continuation data {{formula:46dd625f-8d99-4ee1-9185-1ba05a772426}} , where {{formula:66f065e6-9973-47f5-820c-5c03f7cdcc57}} is a function such that {{formula:2d19a3c2-fb4d-481a-80de-e1d1f7ee0433}} for {{formula:130caff2-8301-4734-bdb3-068f799157e8}} and {{formula:aefd02f6-2a44-4fd7-8beb-b645a68b8cc8}} for {{formula:fdc9d1a3-c1dd-4b52-94da-f2ec28393e63}} . Then a small generic perturbation {{formula:dc0d0b28-552a-4321-87a6-e90dbea679eb}} is constructed, without modifying outside {{formula:2cf83523-2dcb-4dd7-bdbf-a5f10b6c4202}} , and from it {{formula:6fb6e20c-8c5e-4ef9-a4d2-30fc6afac174}} is constructed by counting pseudo-holomorphic curves {{formula:2cfe85cc-0bc2-4cc9-b8e3-87d1182d11f2}} . The point is that the pseudo-holomorphic curves satisfy the energy bound provided in {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}. If a pseudo-holomorphic curve {{formula:5d44144c-2886-4e2e-a602-b9398a2a6d85}} connects {{formula:188e612b-7788-4175-b863-5d57e1e2b7de}} and {{formula:970ce6f3-63d0-4642-a452-d37d54939466}} then its energy is bounded by
{{formula:9d100ce2-1752-4a37-a3c8-8e686cd8e234}}
where {{formula:2c9783fb-8736-409c-8fb8-13b1aa9a57ad}} is a constant determined in {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}: it is enough that {{formula:279f5b6e-53a9-41e3-927f-3bd6bc2ba1bb}} bounds {{formula:db1e509d-e698-487d-9777-9992caf16b57}} . Choosing {{formula:32430321-b88e-4fcf-b4c5-91a90fcd7fbe}} and its perturbation {{formula:19521086-447f-4347-b479-76315bb490ff}} small, the perturbation {{formula:5c306347-4f95-4a22-a2bb-8617a1881315}} can be chosen small, too. So we can set {{formula:02d60d92-6506-4788-955f-021692b21c0f}} as small as we wish, and doing so it is clear that {{formula:3f489f42-92b6-4c2f-a671-a915375c0cc5}} preserves the action filtration.
As a consequence we obtain a quasi-isomorphism
{{formula:9541ca41-d538-4ee3-8709-7f431fadbb8a}}
Assume {{formula:4200d2d2-8e88-4e2c-aae3-caa0edfed11c}} . By definition of {{formula:bd79b81d-fb32-4267-946c-e09c2dd98cf3}} , see (REF ), the right-hand side of (REF ) equals {{formula:c719f880-f081-4fdb-9747-c406391ecc31}} . The left-hand side is defined analogously, as a vector space spanned by those critical points of {{formula:5ec9613d-0cc4-4805-9817-e328cd9b3127}} whose action lies in the interval {{formula:6c2b518f-d36a-43dc-8390-95853daeb3d9}} . They are precisely the Morse points of {{formula:7ae1fd72-774e-4400-90a4-0b2bcf6f2d67}} contained in {{formula:440dad66-85ed-439a-8852-50fc252868fe}} . Since {{formula:c33b40b2-1fc7-4e15-aef2-488c571daf65}} are independent of time, the Floer equation (REF ) used to define the differential in {{formula:f760f9ca-373a-45be-a9df-89ff7db36f3e}} reduces to the Morse equation used to define the differential in the Morse complex, cf. {{cite:c0b9236abfb1f289f1b60cff899b7f43e01970f3}}. Therefore, {{formula:b712187f-19a4-455f-860a-1f7a19d97b58}} corresponds to a piece of the Morse complex for {{formula:9da58487-7b22-4d04-9019-378b47637f1b}} generated by the Morse points contained in {{formula:b8a542d1-1982-4db2-b0bd-953d38f69c03}} . This piece computes the relative homology of {{formula:494661c7-13e2-429f-b03f-c48f74b42324}} . Indeed, in the decomposition (REF ), the gradient of {{formula:7ea5becc-a46f-49d5-969f-0b22f4440d45}} points outwards {{formula:41d9514f-1c28-40d8-b13c-43541c9a81ba}} , inwards {{formula:91e7ed1d-fb9a-45ad-9316-08bc9dccd1a0}} , and inwards {{formula:8fd6685e-8af0-4b95-b44d-94bae78ca43e}} since {{formula:2c514c81-63b8-4f39-b72d-e1ec7c8cc1f0}} has negative slope: thus the claim follows from {{cite:15573b4b8b622b43c4a9b6f916b04c8da970c617}}. Using the formula (REF ) for the grading, we conclude that
{{formula:1d9b54e1-e39c-4c09-8b72-c527cf017516}}
Together with the formulas (REF ) and (REF ), this gives
{{formula:2a764d60-75c6-42ca-8a96-b54a2dbb91d1}}
as claimed in Proposition REF .
To get the analogous spectral sequence converging to {{formula:40c4ec97-2d92-487c-a5b5-8ed307c446e0}} , we need to replace the small negative slope by a small positive one. This way, the gradient of {{formula:c97fa246-7a9f-4097-a746-351dd80c9e91}} points outwards {{formula:a972ab87-6480-4a74-998b-8b685a6f6b69}} , and at the end of the day we get relative homology of {{formula:c609d6df-c740-4a16-864f-9773dbebb5c4}} modulo {{formula:68355939-00c4-484b-9051-4fe7c159216b}} , as claimed.
The A'Campo abstract contact open book
In this section, we will phrase Proposition REF in the language of graded abstract contact open books. This way, in Proposition REF we will get an isotopy between the natural abstract contact open book, constructed at (any small) positive radius in Setting REF , and the one at radius zero, given by the flow of (REF ), which has good dynamical properties just like the topological A'Campo model {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}}. This model will satisfy the assumptions of Proposition REF , so it will give a spectral sequence converging to Floer cohomology of the monodromy. In Proposition REF we will state this spectral sequence in a setting which is convenient for the proof of Theorem REF .
subsection2-.5plus-.7.5The dynamics at radius zero
Let {{formula:1d10cada-7ac3-4713-ac3e-d41bd690b39b}} be as in Section . That is, {{formula:fa6adc2f-ed7e-40cd-89a2-0680820a0f08}} is a complex manifold of dimension {{formula:230c4cd1-3f93-42db-a1fc-949bb7c6c47f}} , {{formula:6d509df5-124d-4975-a921-c1307d4c36b0}} is a holomorphic function such that {{formula:4bad683e-59c4-4f54-b159-c7544a092791}} is a submersion, and {{formula:6d76386c-39a1-435d-bc38-5878352fb474}} is snc. Fix an integer {{formula:2b0e62b5-0fd7-4068-8513-9c0150cb87c6}} . We assume that {{formula:0025ee5e-fff9-4ac7-85b2-400a6ffac005}} is {{formula:45f4fc3b-194f-41e4-932f-5d01e87c667c}} -separating, that is,
{{formula:81fe7793-fdfc-4cc7-940c-0247dc2757e7}}
Let {{formula:dcf5fd8e-2d8c-4bb3-97b9-504ba6e61bd3}} be a 1-form such that {{formula:a5633b01-7d23-452c-88f9-b4025d54fba0}} is Kähler. Let {{formula:b304f817-6b69-4495-be38-3ffa65f27f6d}} be a function such that {{formula:9979e7a0-ec2e-491d-8ba4-83229f8d65f2}} is the residue of {{formula:ac3ec5b2-7a41-472b-bf4b-08dec0e9a7dd}} along {{formula:89e9a5b9-91c9-4ea4-808b-c63be2a9eace}} .
Assume furthermore that the canonical bundle of {{formula:584b5c6b-972c-4f4f-b2c3-5f5509664613}} is trivial. Then, the canonical bundle of {{formula:271b6be6-cc9a-4fc1-9830-7f90f7632b90}} can be written as
{{formula:bcb37cf2-3b5e-4ad6-9352-147dedb511d8}}
i.e. it admits a meromorphic section {{formula:aab1c96e-9a4c-429f-8d45-ac87650c1d61}} with a zero of order {{formula:54f81345-68b4-4053-aa89-c42fc6954934}} along {{formula:598e5889-60f1-43a0-9643-95702196d8f8}} , and no zeros or poles on {{formula:9a694ab8-6d3e-43aa-a42a-0f6d9ee18ba1}} .
In order to apply Setting REF , we fix open sets {{formula:a056f357-563b-496f-92d4-01bd261a70b8}} with compact closures such that {{formula:3a8f8d68-7089-4bfe-9cbd-3835af89b525}} and {{formula:7702669d-f068-432b-8024-3ed7d6457eff}} , {{formula:4b8d3e43-1670-4899-9032-558ee041ae7a}} , {{formula:c4471e7a-ceec-430d-af53-1c9b416afc45}} are manifolds, meeting transversally all the fibers {{formula:092279c1-ac1e-4b48-bac9-5f2a2fa1a0c2}} for {{formula:66469bbc-2d09-4a45-97b5-70fba94522b9}} . We assume that {{formula:e8c96afa-e548-4318-bead-b740eaf1f578}} is smooth, and that {{formula:d8589fde-c069-47ad-889a-9857675bc891}} does not have poles along {{formula:da2f402f-ef40-42e2-a1f8-ca629a86cebb}} . Then since {{formula:d3ef1a92-4d42-46aa-964c-561ffda5b615}} is Kähler, and {{formula:1fa63e6e-13c8-432c-a43b-d6a407ae33d6}} is a holomorphic function, {{formula:a7a424a6-6bc8-4c95-add5-3c15fe1be5bf}} restricts to a Kähler form on the fibers {{formula:301f82d5-5f2f-4f10-991f-bd9dc80d022d}} and {{formula:29969479-c833-40c2-98e8-2d68db916c4c}} for {{formula:2bbb0e6b-e054-44bc-87a1-a3b9c4795893}} . We assume that the Liouville vector field of {{formula:8e5df3f7-29ca-49fd-8cf9-5dff17372acc}} points outwards {{formula:533cfda4-9a98-4a07-a653-10326c713d69}} for {{formula:e6a7341e-cee1-4145-bd79-e2e5ce214fe6}} . Then we are in Setting REF , with {{formula:1e5a842f-92d2-4094-9036-60676c396406}} . The construction in REF gives a {{formula:fbe80cd4-fe64-460c-9ce3-817e875e5dbf}} -fiberwise symplectic collar trivialization of {{formula:877e8afa-c7ce-420a-8e73-c3ff4bc9609a}} , for a subset {{formula:f3797db6-e494-4d7e-9aa7-0fa386c0a949}} such that the fiber {{formula:8070c75f-44bd-4eeb-a28f-fe75d450c110}} is diffeomorphic to {{formula:8bd3bf2f-0524-4351-bff5-c67623fe705f}} , for any {{formula:1d396b0f-f347-46b5-a038-e594b4a1ed0e}} . Shrinking {{formula:551c4f3c-56ff-47db-b118-18814551aca6}} we can assume that the Liouville vector field points outwards {{formula:620cdad7-3c00-4cc4-84b6-ec3d5cc295b0}} , so {{formula:b94b0408-c02c-4261-a522-5dfa7516c5c9}} is a Liouville fibration. Assume that for some {{formula:b01c4522-0646-4d7f-a474-21f5b0ef3e84}} we have
{{formula:8ab4dbda-dc02-41ab-ac5b-247e1ad130f5}}
Then Definition REF gives symplectic monodromy abstract contact open books
{{formula:837492e6-8dff-40ae-9d4c-42c8fa8d58ef}}
where {{formula:d715bf78-ddf2-42ba-8979-7acd4d120474}} . As explained in Section REF , a nonvanishing section {{formula:7e8e109c-8495-43d7-aa4f-ef2e4e7146f0}} of {{formula:4d75fd84-b751-47bf-a77e-5858d5d45958}} induces a grading on each {{formula:839894f7-f84a-4db4-bf34-809c4149d2d1}} , and we have a graded isotopy {{formula:ba3b5d37-e746-4113-aec0-3b58dc9e513b}} for all {{formula:217454dd-f664-44c9-9461-95c97b86d612}} .
Let {{formula:bf2b115e-496b-4cd7-953c-f1acd6ac01c1}} be the A'Campo space for {{formula:a76c87fd-4fb2-4c54-bd0e-18719977e70e}} , equipped with a smooth structure constructed in Section REF . Let {{formula:6b206f3c-c496-43a4-936f-413b7d6aaf34}} and {{formula:2767ec78-1e18-455e-ab36-c8b2427b3306}} be as in Definition REF . Put {{formula:a01c090d-9d14-45e3-8dcc-572ce396b3ab}} , {{formula:f3914ef8-cd58-4dec-9e43-3fb0cc416c0f}} , {{formula:0c295d31-2886-4621-b898-b8093004ff2e}} .
Choose {{formula:07538599-d102-41d3-b112-ac34f0624e8d}} satisfying the statement of Proposition REFREF . Then in some neighborhood {{formula:c26abdbd-d247-4f01-bf7f-c032efeafee9}} of {{formula:107462e0-e0c5-488d-9365-2ddc92ecb5ab}} all forms {{formula:0e90f271-b43b-4887-897f-699ff637140b}} are fiberwise symplectic, for every {{formula:ac62a2af-7cdc-416b-9619-240065e270eb}} and any {{formula:6a6d1e7b-765e-4fb3-b8bf-b3acee7aa100}} . We shrink {{formula:73b98fe1-c0f1-4b78-8c5e-ec91c1967861}} so that {{formula:6faddb54-2210-428a-bfac-e7604470e0b6}} and fix {{formula:3aa8da4b-80c5-43d6-a09c-e161552ad390}} . Let {{formula:75dcdd28-6482-48de-b180-42c9359f9a05}} be the image of the collar trivialization constructed above, and let {{formula:dedaea76-590a-4c56-a6ef-10d46d34f265}} . Recall that {{formula:a940887c-fd8d-4cb5-86db-63f1b0fa3671}} is fiberwise symplectic. Since {{formula:33f24059-f815-4e67-8b09-c903eca994dc}} is fiberwise a diffeomorphism, the form {{formula:b4b994d4-8e7d-4941-a251-f9a702a6417d}} is fiberwise symplectic, too.
The fiberwise Liouville vector field of {{formula:5f707fa0-b700-413f-804a-8583af0dfcc9}} points outwards {{formula:cf7b9383-13b4-4153-a805-554164d321a8}} because {{formula:c20799ec-ead3-4be9-a42a-da6f117d72b8}} is a fiberwise diffeomorphism in a neighborhood of those sets and the same property is satisfied in {{formula:b3ab4195-7d3a-403f-8d53-22e38750afb2}} . Since pointing outwards is an open property, we can choose {{formula:0fb096b5-d92f-47b1-97b5-4f0690026907}} small enough so that the Liouville vector field for {{formula:6e6a51ea-609f-4c4f-9bf7-a93b14642523}} points outwards {{formula:65ec8868-65d3-4583-95aa-648bff1b1b52}} .
Therefore, the map {{formula:19adb78b-5fdb-4af3-a24e-a48940aa2a5d}} fits into Setting REF , with {{formula:4ede104b-9b46-49d0-9fe7-024e028a807b}}{{formula:a8e249de-d3cc-4cb5-a0c0-98fd335df189}} P=[0,t0]{{formula:9ec195b7-6cba-41a5-a92a-ec7ae7c03202}} t0{{formula:c15f1e6c-5611-4c6e-99ea-a7e578e627b3}} ”*{{formula:eb900233-ce83-4883-b2e3-684543ed8512}} ”*[0,t0]{{formula:063bc815-a32a-40a9-bb1b-c9507907ffda}} '<”{{formula:6bd15297-3969-4357-a783-9b13d0556df9}} 0{{formula:b6f06cd4-c4d4-4e72-89b8-c52b7a072f3f}} ”{{formula:f46020fe-4ae8-458e-8667-47aad93f8572}} (Fz,t,z,t,z,t){{formula:6fc28299-6edd-4bb3-bab3-66157c715031}} t[0,t0]{{formula:edbb453b-2b76-4ae0-b9dd-badf9c4576a6}} z”*{{formula:192743de-cf9c-4861-a07a-14934fa61b74}} '<”{{formula:9652a44f-368c-4505-ae68-cda339c80caf}} A',t{{formula:adf5b5fc-0a32-4edd-b9c5-a1291d18c07e}}{{formula:11b3bb04-87bf-4fb1-9981-e2915574fec1}} fA-1('){{formula:a5c05ecd-584e-414c-9a68-53defdee71a0}} (Fz,t,z,t,z,t){{formula:c90cbd03-002a-43ce-9db7-b0852b478186}} (Fz,z,z){{formula:6cba3218-d9fd-4186-b640-2470673bf062}} Now, since {{formula:d67d8241-1245-4504-8795-4664844e9625}} , the form {{formula:09c19906-25f2-44f5-bee5-6e32f6af1046}} is fiberwise symplectic at {{formula:983702c4-3d9c-475f-9457-9b2bbc59e8c1}} , too. By Remark REF it is exact and we have seen that its fiberwise Liouville vector field points outwards {{formula:1b568008-58aa-4846-9e17-50cf534f5077}} , and we have constructed a fiberwise symplectic collar trivialization of the Liouville fibration {{formula:af357298-02b9-4b26-b963-1674b5e8ed82}} . Let {{formula:1c1e2689-e105-4d30-8df4-1fa44036810c}} be its fiber at radius zero, and let {{formula:488f3136-348d-4898-9a6a-f0f4605bbd96}} be the associated monodromy abstract contact open book. Since the symplectic monodromy is unique up to an isotopy, see Remark REF , we infer that {{formula:1a634ee6-e50a-4b5c-9c3c-da2c802e9969}} is isotopic to {{formula:138f16ed-4659-4a80-b45a-5d08a6a1843f}} for {{formula:386bddad-5274-4e0f-ae85-00cf51349790}} , hence to {{formula:ebd5cedf-a8b1-48c2-9270-ee055d54dbbd}} .
As explained in Section REF , this isotopy defines a grading on {{formula:17d82cb4-d3df-43a7-a1d6-ea0ced01abd3}} . Therefore, the Floer homology groups {{formula:65dd8a3f-0c91-4503-957b-6d6ff0616457}} are defined and isomorphic to {{formula:c6e430a3-a001-4142-9bfc-cf4ca93c2c6d}} .
By Proposition REFREF , the symplectomorphism {{formula:91d8a05c-ff3e-4648-b2e6-f0de7306bbd4}} is the time one flow of the vector field given locally by (REF ). In Proposition REF , we will study {{formula:8067c81c-7b1e-431f-a271-6c9558fe253c}} , using dynamical properties of {{formula:e6455899-cca7-4c31-9a7b-e33e8d8f4ed3}} summarized in Proposition REF below.
To state our result, we need one additional piece of notation. Recall from (REF ) that for {{formula:3e84476f-4c33-4dea-b215-2e6a8fe5aeb4}} , we put {{formula:a374ab35-f0ab-4a39-87e5-ac6c653ed282}} . Write {{formula:216f3613-794d-4f3d-b832-48f6b59f084e}} . Then {{formula:60548c0c-beb0-4fcf-8919-0ab2074a82cc}} if {{formula:90c5e03a-ad5f-45fb-808f-d11969755240}} , but if {{formula:e4a8c1da-5fe5-418d-8c11-2b072bcd8ac6}} meets {{formula:29c9ce7c-b767-4bf2-9808-05991d71d7b9}} then {{formula:5423c796-dfac-4e88-9ecc-f0b7ef6e5932}} is a submanifold of {{formula:214a4561-4623-4066-92c6-076966d3c6e3}} of codimension zero, with boundary {{formula:eb276969-1e47-48ed-9047-8bbbdbc1de13}} . As in {{cite:3a14ff98229456cb788161a4b4eb6b3a0b5c13fd}}, we introduce an unramified {{formula:dbb90cd3-b574-4422-ad3f-7348d6e6951f}} -fold covering
{{formula:6b41daef-1665-4d96-8412-10d0ac154cba}}
in the following way. Fix any holomorphic chart {{formula:1f50c49a-9efe-4b39-af33-a4574a5c069f}} around a point of {{formula:5eec8490-8de9-45e4-a9cb-31a2eeb86b3e}} , and let {{formula:ec0a96a1-25d5-4f9a-861a-5ba4da8c88ce}} be a local equation of {{formula:89cedcd6-9f38-4e4c-945c-379c55114ee6}} in {{formula:18c4ea98-0fcc-4904-8d63-bcb6ee970a8b}} . Then {{formula:d10a6cf9-8884-425e-9055-87a2d577bf4c}} for some {{formula:591fc02b-5f5d-4288-ae0f-7d981b20f000}} . Define {{formula:65241184-b867-471e-b14d-ea6bde026296}} . Gluing these charts, one gets a topological covering {{formula:fe959dcd-7a81-4582-95e0-3222cbe2279a}} with Galois group {{formula:7626fdbe-bb5f-4485-87dd-a93ea9b567b9}} , as needed. Note that if {{formula:f6499583-e3fb-4803-bd3a-2a3825efb2cd}} then {{formula:15cb1c19-5c4f-4742-82ac-482219187c37}} is exactly the covering considered in {{cite:3a14ff98229456cb788161a4b4eb6b3a0b5c13fd}} or {{cite:97005650a8ac11991dc2e980883b47d01674846e}}; and if {{formula:0af4d4ff-cbca-4ffe-a3e0-2fce5bce51b5}} meets {{formula:d06b1cb8-9842-4074-94ab-7c102536d416}} then {{formula:7360a406-cb4c-4646-8dcc-fb78b632f966}} is a restriction of that covering to {{formula:4cb39d7d-c6f8-49c0-8181-a6913f632945}} , which is a manifold with boundary.
The first result of this section mirrors {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, see Remark REF .
Proposition 7.1
Let {{formula:cb4a49fc-5d82-4148-8ed3-b96e3373287c}} be the graded abstract contact open book constructed above. Then the fixed point set of {{formula:a4a23750-c9e2-4c98-b54e-82708ee5f85d}} is
{{formula:92cc704d-ed8f-447e-b6b7-7fe134037863}}
see Section for definition of {{formula:66c5d329-21ca-471a-b492-e34ecbf59330}} . Each {{formula:2e59738e-de4b-45a1-b6c4-f7ed77ad30a2}} is a codimension zero family of fixed points of {{formula:d4f96001-9103-479b-94e1-3de9b16b3b55}} , see Definition REF , which satisfies the following properties.
In the decomposition (REF ), we have {{formula:4ef508c5-5446-413e-a2c3-dbba275dc482}} and {{formula:28f15959-3cf2-41f1-8b7d-1187822d3796}} .
There is a diffeomorphism {{formula:d4164432-0c2a-4c18-84d2-8ee0e62aabd8}} such that {{formula:3b57b773-cf7f-4be5-9727-fba2e8fdcfe6}} .
The Conley-Zehnder index of each point in {{formula:18324541-7d21-4522-b108-3fb619467474}} equals {{formula:dc7a1efa-6d81-4483-9c63-c46874941599}}
We can choose the action {{formula:7c16050e-0938-4fd6-88f6-bd83611fefef}} so that {{formula:4b26def2-9a38-45b7-a593-030bf207c8fe}} , where {{formula:9f850675-6582-41e3-8edb-b5dce132dc7a}} is the residue of {{formula:9a4bde8b-b2dc-46ee-b76c-340a28dbca5e}} along {{formula:5e5424a5-6c21-4cb2-8f31-f460a35cafab}} (recall that if {{formula:774a5e7d-d52f-44b0-8b93-15d6ade6bfc3}} meets {{formula:1a461968-b711-44b7-b33a-f96b7e67e830}} then {{formula:cbc880ec-58fa-4e9c-8c57-8b5115c1059b}} .)
{{formula:a7bfe15b-7761-4251-b340-aa28035ae2a8}}
.
Recall that {{formula:1b0dc859-9e5f-4c7f-b058-7debb9e5659f}} is the time {{formula:31c1c251-6b15-4d8f-a6df-9aeff4772053}} flow of the symplectic lift of the unit angular vector field on {{formula:8f576096-3558-4b54-a111-2c6fa894bfe3}} . Fix a fine chart {{formula:717ce36c-5d6f-444a-8bfb-48fa19da8458}} , i.e. one satisfying the statement of Lemma REF , let {{formula:88829862-2a27-46a0-a2db-87808636d709}} , and let {{formula:4c4d7972-1985-4c1f-98c5-c44f3ddbcbd3}} be the associated index set (REF ). In local coordinates (REF ) on {{formula:b7c10060-381c-42f2-8dab-41cb5923aaa1}} , this lift is given by the formula (REF ). Therefore, {{formula:f113586f-4fda-4255-b703-b30c2f9edaef}} is given by
{{formula:7238a16e-b875-440b-a791-39d1b1ebfd6e}}
In other words, {{formula:c35a3d0b-18c8-490a-8ced-7c3ca5fcafe4}} is a rotation by an angle {{formula:d33aa762-fd49-4488-8abb-d8643d14965b}} about the {{formula:5ffb57c7-ea20-4835-9029-3b29c14ed830}} -th axis, see Notation REF . Note that by Lemma REFREF , the values of {{formula:d63c9fe6-e21b-470d-80bb-6046208d98a7}} do not depend on the choice of the chart.
To prove the equality (REF ), we use the {{formula:a32aef40-6dda-4fb9-8585-c9b677fbe37e}} -separatedness assumption (REF ), cf. {{cite:1ebc1139fe2d6fb512a4508d6af9f68e1135a9c5}}.
First, we claim that {{formula:88053e87-aa53-4118-bb66-9e66d105d8eb}} . To prove this, fix {{formula:813b4bf2-a5a3-4305-a35d-a4bda3022e04}} such that {{formula:43ee807b-998d-4749-8b17-50eb9bf0a79a}} for any {{formula:11ffdc2d-248a-44be-9570-308e164e4a1a}} . Since {{formula:3fabd142-7d4e-4ec4-9331-87288895c5a5}} , we have {{formula:1f544172-d060-4ecf-9b45-cd1de811ffad}} for some {{formula:bfd78e27-7749-43e3-857e-f416a49e1700}} such that {{formula:c4b64dcb-03f0-4d6b-b1c4-cfaeaeac1bf2}} . Then by Proposition REFREF , we have {{formula:f7e102fe-f39c-4c17-b1b9-fbd72ad29b7e}} if {{formula:bf5648d7-1c11-41be-9f6f-22175ebc8dd5}} and {{formula:3460b58a-27cb-44ca-89ea-9fc267d7d197}} if {{formula:365e4431-77e9-4920-85eb-06710027835b}} . Choose {{formula:a68248fc-7143-4f59-b501-1c8de87cafcd}} such that {{formula:bb55d9d5-4ae4-4843-a884-db0975e1ef40}} for all {{formula:75513b85-b4f5-4f2b-acb9-733cf2a750a3}} .
Since both functions {{formula:3f58f437-1b31-431f-a2fc-54808d2942c4}} and {{formula:c26f78b3-2d6c-4ee2-ba55-9df1858d3ed5}} are increasing, we infer that {{formula:1830e6d5-f557-47b1-99ba-50cb8ec730e2}} for all {{formula:91e052a2-8afe-4a64-978b-c826c8da2370}} , so {{formula:03397bef-2669-46bb-874a-aca9c714a032}} if {{formula:bd07de9b-c2ae-496f-bacd-dd8814507832}} and {{formula:09e7d587-584b-4a2f-a15b-926c33b10203}} if {{formula:6f55d723-c310-4eb3-8420-ec445d6704b3}} . Hence
{{formula:226edf34-2d2e-4237-bcec-087dc5c05bfc}}
Thus {{formula:57ec081a-b9cc-468a-8587-cef612264989}} . Formula (REF ) shows that {{formula:8613cd8f-d31e-4d1a-a796-56eff7014bff}} , as claimed.
Thus {{formula:8120a06c-5119-4719-a106-88997cd49037}} . Take {{formula:62955298-2bb5-436c-93d6-a4b1bb4829b4}} . By Proposition REFREF we have {{formula:dbaf84a9-d715-4102-aa81-4d15ec14d49c}} and {{formula:19dab298-5668-4a13-8a0b-0de2087b713d}} for all {{formula:cc9240ec-60a5-44c8-9b42-1530b52a7912}} . Thus {{formula:daaa4c83-1676-4fa4-8296-8b34ed1a59e8}} and {{formula:69ec7ede-5ca9-4bd3-9bab-d50d53e16517}} for {{formula:84f0279a-1007-43cd-a2ee-9d9923707517}} , so formula (REF ) shows that {{formula:1b198286-8a11-4e51-a4a9-547dd8db859c}} if and only if {{formula:3e9bc499-c3f5-401b-95d4-3902cd2b48a1}} . This ends the proof of (REF ).
Now, we claim that each {{formula:1f835fc4-2cb1-4cfb-a489-002dfa07896a}} is a codimension zero family of fixed points, see Definition REF . Say that {{formula:5ecf2348-e233-424b-8627-b05e8e4448b6}} . First, we claim that {{formula:e1b2743e-4da5-44ca-962e-5d31f40996a2}} is a codimension zero submanifold with corners.
Fix a point {{formula:3986efa8-1837-4570-a600-d93246475bf4}} , let {{formula:d1cc90c7-9ac0-4e0a-a5e4-167f5f77c29a}} be a fine chart around {{formula:be6b92fc-d1b5-4877-8d24-622be60fe2af}} , let {{formula:c85a14a5-7e34-40c0-ad01-c084b63e9d1c}} be its index set (REF ), and let {{formula:b5b8a767-30fa-4433-bff9-7fbcc3b02176}} . Recall that by Lemma REFREF , we have {{formula:fb9e03df-ae9b-40f8-bace-466206fec6b0}} and therefore {{formula:5c78b143-b782-434a-b3c5-31af470292f4}} for all {{formula:880a4691-2791-4d91-a4a3-1dc3751c89c9}} , where {{formula:004737ab-b261-4f7f-9a02-6fc9266bd516}} , {{formula:5445cfc5-e74a-43ff-854c-398f7a015327}} are global functions introduced in (REF ), and {{formula:31c7def7-845f-44d6-b542-224675da4a50}} are the local ones, introduced in (REF ), (REF ). By Proposition REFREF , we have {{formula:cfcb0467-f2f0-4a47-913c-b8d92a0d141c}} .
In particular, {{formula:8da3a5c5-e42a-4f55-95d9-ce8158cead57}} is contained in the open subset {{formula:ae7383ac-67f1-4b72-9758-da40e25aee84}} introduced in (REF ). There, we have a smooth coordinate system {{formula:31ac6650-ad2f-408b-adf4-ac8365b29320}} , see (REF ), where {{formula:389dec78-95fc-4052-81bc-becc72266d61}} . The fiber {{formula:b4e508fe-9d4c-4dc0-85e6-6aa7d521410d}} is given in {{formula:497fe9c3-ebc2-43d9-8ad9-3716d67befb2}} by {{formula:28877db4-7966-466f-8ae4-0f398c366916}} . By Lemma REFREF , the subset {{formula:3f77a7b7-1a32-4d41-9a25-285a8088cc8f}} is given by the inequalities {{formula:b672b7e9-0187-47f3-9905-703e843f1aed}} , for {{formula:c359d844-95bc-4963-b493-61fdcb3a812f}} , see Figure REF . By Lemma REFREF , on {{formula:74cb43a3-b58a-4bd9-945b-44a11bc58810}} we have {{formula:1386ce3b-1163-493c-afbd-1573b5e8fbf9}} . We conclude that the subset {{formula:5742e9a8-637e-47c7-848d-1dffa5c11305}} is given by inequalities {{formula:82d24137-a678-4c8d-92e4-6abc0af00fe9}} , for all {{formula:3107c921-87cb-4978-a3ca-74a5ac5fb5e9}} . Therefore, {{formula:785f94db-4768-465b-b59f-da9a6c368687}} is a codimension zero submanifold with corners, as claimed.
Now, according to Definition REF , we need to find a neighborhood {{formula:c66ab025-cba1-45b5-aba4-678141200ca3}} of {{formula:f381edb7-6288-49ef-a76a-5e13d51974b9}} , and a time independent Hamiltonian {{formula:b9becb72-af04-402c-9bd8-157a966eb6a5}} such that {{formula:f7bf7212-216e-4a89-aab4-663b238fa7f4}} . Define {{formula:0f1d6df8-321e-4c3d-a0a7-30e5fd39f9e7}} as the union of all the sets {{formula:a1121004-f942-4291-8662-7d57ae602288}} , where {{formula:30637913-479f-4183-8749-bd08c42f6861}} is the set (REF ) defined for an adapted chart {{formula:22b5ad38-c37d-43c1-a4c9-ec1d6b5e8104}} , and {{formula:f5c11938-bac1-4af6-bd5b-46830ec9566d}} ranges through all fine charts. We have seen that {{formula:85c52ba9-d3e9-4661-8c68-19b0ce3fd637}} is a neighborhood of {{formula:51b36cc7-32f7-4994-a29e-1cf949543c74}} in {{formula:637399e4-8d55-4dde-87a5-fffd4d9b1a9b}} . We claim that {{formula:07ce0205-bee9-4229-931d-157af6cb8406}} is the required time-independent Hamiltonian.
To prove this, once again we restrict to a chart {{formula:4a832b86-1453-4dd1-87b6-e68310cf5e6a}} with coordinates (REF ). There, the monodromy {{formula:4161bc13-c7a6-4e19-8c85-ba64bca9bea7}} is the time one-flow of a vector field {{formula:b3b7a019-d321-4933-854c-2f336323559d}} , where {{formula:d0756e06-831b-44eb-b833-2c2e59e7600c}} is the symplectic lift of {{formula:48436cb1-c06e-4e67-9d4c-724bf26f9dc0}} , given by the formula (REF ). Recall that {{formula:0452e14c-3acc-4c47-b1da-988935664bca}} are coordinate vector fields introduced in (REF ). Put
{{formula:a2f2e3a9-69f5-4f13-ab00-a8f1a75095b8}}
We claim that {{formula:4a033e42-fb65-4d5d-855a-ad4b6f63cc81}} is the Hamiltonian vector field {{formula:b7973ad2-8b51-4ee0-82df-224852d5609c}} . Since {{formula:5e3de1ad-ca7e-4b13-9f72-8790672a84ff}} by definition of {{formula:ed1b29ee-1f9e-4d8c-ab0e-30df7ee40999}} , we have {{formula:c74bc523-519e-4294-b2f0-a1a9411fa967}} , so {{formula:3cc98cbc-8c16-4814-9409-a987aeda2067}} is tangent to the fiber {{formula:dc0c938d-38f4-498e-80b7-c201bd058827}} . Because {{formula:8515475e-85be-45e5-a6cb-2f6e8d2ffcc1}} is orthogonal to {{formula:da2c3874-71aa-4ec5-af59-9fa520b60e01}} , in the tangent space to {{formula:95ffd165-abc6-40a7-a40b-7e15fb8a0412}} we have
{{formula:504d9260-7fb5-4b8b-a8a2-9b5e47e2b0a3}}
where the second equality holds by Lemma REFREF . Thus {{formula:87408ea5-b3cc-46b4-b19e-4e7e9a32826b}} , as claimed.
To verify the conditions of Definition REF , we need to check that {{formula:249f49b1-4256-4e22-a1f6-01339fd46441}} . Recall that {{formula:6c80902d-7946-4811-9951-7a303b7cef4f}} is the time one flow of {{formula:1a63568e-a184-4525-a8c0-09bad1bf84ff}} , so it differs from {{formula:98005358-cf22-4fdf-9655-8c578184f833}} by a translation by {{formula:d147ed12-c6f5-4357-85b6-ab2d14b918d7}} in the {{formula:dded13c1-44a8-4db5-a0eb-d9d9cf955127}} -direction. Since {{formula:3fbd86f3-30d4-4669-be6c-f2b07b9a54f7}} , these time one flows are equal, as needed.
It remains to verify the properties REF –REF . Part REF is clear since {{formula:c3b53f8f-f805-47ab-a09c-08eafc0dc175}} .
REF By Proposition REFREF , {{formula:c732bce6-6301-4622-a32c-cc07eec77e9b}} is an {{formula:53106014-ae2e-4739-9f69-44dafadb5a77}} -bundle. The fiber {{formula:dbd13963-46d4-4f41-be8b-bac0da8134f8}} is locally given by the equation {{formula:30a17fdb-bfc2-480c-a518-db0e9930e4b0}} , which yields the required covering.
REF Recall that the Conley-Zehnder index is constant in a codimension zero family of fixed points, so it is enough to compute it for {{formula:8a126799-298b-4d0d-bae9-52d86e983641}} . Choose a fine a chart {{formula:050642df-5f25-4f97-bfab-98aa650634f7}} around {{formula:664f105d-a2ea-4b27-b4a2-168d4f39e43c}} , adapted to {{formula:f992e3af-de9c-4eba-b964-05656ab0a0ee}} , whose associated index set is {{formula:46d2683a-0047-4ffc-b4de-7c365c36df17}} , i.e. {{formula:1e734370-3fba-4265-85d8-e50e54a16ef1}} does not meet {{formula:47856c5d-e117-4454-981f-633d754a53f5}} . The smooth coordinates (REF ) on {{formula:90b93ddf-de4a-436d-a602-4cf17ef00e28}} are {{formula:d966d611-60c3-4d26-9c26-8a54635edd51}} : we have skipped {{formula:c96409de-5cc7-4a80-813a-307f73593842}} since it cuts out {{formula:3620cb90-8cc5-489b-83c9-0310518ab2c8}} . We will compute {{formula:e23816ed-1141-4e1d-9913-550f846f5095}} using the formula (REF ). To do this, recall that the grading on {{formula:1cf69b21-8f7f-4805-84bc-f9f1158eb0ba}} is induced by isotopy from the one on {{formula:14386f12-4632-4f3a-b45c-fd2b1d6bfb3c}} . The latter is given by a section {{formula:9ac650b6-7245-4c1c-8d6c-7dfa179fa64f}} of the relative canonical bundle {{formula:8385b7c7-ab9d-4dd8-938a-6118a9638bef}} , such that {{formula:ec58da37-5bc4-4b73-ae33-7ed7c42f196d}} maps under (REF ) to the chosen section {{formula:efd99634-6454-47c0-89ea-5a9a8d61ea35}} of {{formula:4285376b-f2b9-4586-ba5a-46a9e8129274}} . Since {{formula:e6f91b55-75d2-4c6b-b028-bbf2dbb61e5a}} has zero of order {{formula:f19cb4f5-2f5d-4f71-8994-d0b73860e62e}} along {{formula:fe1b4c11-b1c5-4276-891a-487f8e5572d1}} , we can choose the coordinates {{formula:81643c29-7165-43aa-86ca-487c34e717a2}} of our adapted chart {{formula:7e212086-084d-4da8-b78f-4bae4f6d66c0}} so that {{formula:e17163b9-f814-4bf6-9d82-c5ca10ebc49e}} . Since {{formula:b441a675-347a-48c6-b0f4-686bdc54e807}} , we get {{formula:c3702f85-134b-4473-9896-e11a4131bbba}} . Therefore, the induced grading on {{formula:9bf7cbe1-60ba-44b9-a9e1-d59b0e26fb2b}} corresponds under (REF ) to {{formula:5119429b-90fa-4d33-8e85-776f45101fee}} . By (REF ), {{formula:7dbe0719-2f50-41dd-90bb-d88fa55ab8ea}} is given by {{formula:f9d13ccc-e9ef-4c1c-9873-a871431da19d}} , where {{formula:1c342e9e-d828-40ff-ad7f-98ca9c2b4763}} . It follows that the winding number used in the formula (REF ) equals {{formula:2c459fe8-df81-4c52-a0d8-4aaf075579f0}} . Thus {{formula:a57a069d-5bca-433e-a884-9d0c51fbdfec}} , as needed.
REF
Note first that the functions {{formula:eae6b13c-24e3-49c0-b267-2c29c5c5b763}} are constant. Indeed, if {{formula:73e2130a-70ce-4ec8-94fb-6ba279724d2a}} is not constant then {{formula:0bdb99ca-ddf1-4856-8257-ae7d362e1849}} is not compact, but then {{formula:e00bc148-3202-489d-9123-c2c73f2f513c}} meets {{formula:2fe369d0-01a4-47c3-8123-2013cc022418}} , so {{formula:24128883-d465-446d-ac2a-3f1d521a0fdd}} by assumption. Choose a point {{formula:1fca3671-28cc-42da-8748-b4ddfcf676a4}} for every {{formula:b8075f28-702a-4682-a94b-ec311c6832a2}} . It is sufficient to prove that for every {{formula:981eee88-79e0-4935-b031-fa9027d1f8a7}} we have
{{formula:bb74c3cc-504b-41af-9d40-e90c4cd7654c}}
To see this, recall that the function {{formula:297048b7-ca5d-495e-919d-bb5d512fe207}} is defined up to an additive constant. The above equality allows to choose it so that {{formula:73e326ee-5c88-445e-822f-f25b5c914da3}} . Since {{formula:cd0c0a99-8058-4c80-82a7-113f6bc3cf96}} is constant in each {{formula:b1b38f40-7f6f-4587-bd11-5139d604547d}} , this implies that REF holds.
By definition, {{formula:3312e9cb-e5b5-49ef-ba2f-ef9ac9cd7511}} satisfies {{formula:3d8f14eb-dee5-47f9-9679-bb21b571daa1}} . Therefore, it is enough to prove that for some path {{formula:83d7aaa8-3713-4d09-94ae-e6191b6dfd4e}} such that {{formula:64097d91-39d3-4a0c-9802-883d9dc271a8}} , {{formula:31aa36e5-9407-42a5-abb4-4b1898375372}} , we have
{{formula:0ec8af3c-d22e-436f-aefc-cd473ec0754f}}
We can choose {{formula:779f8f75-228e-45ca-a8c8-ac10d38a3c2a}} in such a way that it avoids triple intersections, i.e. {{formula:25190d1a-79b6-4427-8777-844f881f96c1}} . Then {{formula:c1e11610-ceeb-40dc-a554-1457042ce1e0}} can be split into segments with images lying over adapted charts whose associated index sets (REF ) have at most two elements. Thus considering each piece of {{formula:0d59706e-3d45-4a72-a1cf-c3fc0121ba7a}} separately, it is sufficient to prove (REF ) for {{formula:10c9ea92-2468-4690-933e-8871a2971018}} , where {{formula:93df33ef-74af-456e-b58a-5c599e1fb495}} is the preimage of an adapted chart {{formula:e9504d14-04ec-4810-ae1b-9b0caf34ff01}} with index set {{formula:4ece062d-bcf3-4b7d-9bc1-a22b5259f5f0}} .
Since {{formula:0b3f2362-f836-4fc7-ba2b-a55b616a4391}} has a logarithmic pole with residue {{formula:8aa2e08d-1761-45a5-adaf-fd3a87284c37}} along each {{formula:18c00706-8a68-4471-a876-7ec90bac6dc5}} , we can write {{formula:620297b7-5e84-447c-856c-19f8c09878a1}} for some {{formula:d4e2723b-53b8-477e-a5c9-9ce1c4c13883}} . Using Lemma REF , we get {{formula:75e15422-9922-49aa-90a5-1e1d9306a3e7}} for some {{formula:f870d92d-022a-4796-9efe-4341174b2568}} . The formula (REF ) shows that {{formula:c4b48896-5fc6-43b7-86e6-56e01ba0abee}} , so {{formula:28581416-f2dd-4f09-8468-c1c5e7c3aa5a}} .
Now by (REF ) we have
{{formula:a49d738b-7f3e-446e-add2-ff7ed27bee54}}
If {{formula:67e5a2f3-2cf1-41fe-af5d-13fb2acc4a35}} then the image of {{formula:dd0831c4-f58d-4a2c-9157-fc307674ec11}} is contained in {{formula:93629e87-4da3-4308-a752-39c3d60c932f}} , where by (REF ) the function {{formula:6ee22abd-ef96-4ec8-a3d0-6bce8c1af39a}} is constant. Hence the above integral is zero, which proves (REF ). Assume that {{formula:208fc2c4-72aa-4865-8170-739bb97e765b}} . We claim that, along {{formula:656d4a16-07e8-44d3-b30f-561ead711148}} , we have
{{formula:8fcbd964-8839-4228-853c-f89b32d62e03}}
Inside {{formula:9545858f-df67-4e5d-85fe-17629170a9de}} , the functions {{formula:64aad551-dce4-4e14-bc2c-21a68470ab63}} , {{formula:d201490c-9339-41d4-a7a9-08de681eb6c7}} are constant, and {{formula:9b059369-da91-4609-92db-e2d25bde474d}} , so {{formula:defb5cc9-1a0d-4580-ab3d-1946c0b901e7}} is constant, too. Hence both forms are zero in {{formula:78645c55-9e5c-412d-afd3-167a81325cdb}} , and by symmetry the same holds in {{formula:c26f8848-252a-41fc-acd3-e9118f68f222}} . It remains to prove the claim in {{formula:3a013e11-b159-48a0-aecb-4150656dfe3e}} .
Fix {{formula:e684f4ff-6129-4cc1-8154-8aac6d9e144d}} . By Lemma REFREF , in {{formula:170164bf-bf83-4e2a-8e21-90c6e88b6115}} we have {{formula:67db460b-ddc0-4496-b15d-87f20ac66375}} . Recall that, by definition (REF ) of {{formula:784060d6-8980-40dd-a53b-ff1f619368b1}} we have {{formula:e734fc50-f770-4d03-ba8d-6b240c10b82d}} , so {{formula:87476a99-bbe5-4afd-be94-0b1e31b34de3}} by definition of {{formula:8af7efe4-7932-41d1-a688-fb99023cec83}} . We conclude that {{formula:6548f72b-0db9-4fb9-9474-1f6fc1a4ffb0}} . By Lemma REFREF we have {{formula:6a20fdf5-ed6e-4af0-9929-69191a354308}} , so {{formula:0e3a6e82-1913-4f27-b106-ece6bc858b2c}} . Hence {{formula:24af7709-1a68-45e2-99fa-2bea9fb590ec}} by definition (REF ) of {{formula:c4d3abc7-be73-4e8a-b409-fa982acc3f83}} , {{formula:7045e614-ac9b-4d23-b838-621071d7535e}} . This proves the claim.
Now, the above claim, together with the fact that {{formula:27b0f52b-4800-44d7-a13e-21fc597f2f1f}} , {{formula:9f115d3b-30a1-4f5c-b2ab-3ba0b4e4ff16}} are constant, imply that the above integrals equal to the difference of values of {{formula:82f29189-9ac2-462f-9caf-7dcb4c788c98}} at the endpoints {{formula:4fe34525-dbdb-4fbd-972a-e6dfe99144c0}} , {{formula:b2c6e7c8-f55a-4735-a45b-9d29b7058f57}} of {{formula:3c9d8354-e26c-42d8-b353-d0672675c0c6}} . Since {{formula:3c914df3-6c36-4fe6-8bfd-6eff4409ebed}} , we have {{formula:24165fd6-0ddd-448e-9d10-e1c36514a19b}} , and similarly {{formula:a05eaa7a-3bf8-4716-b3bd-b3629abbe6f7}} . Thus
{{formula:4baa3448-cec9-489d-854c-bba5f8a81698}}
as claimed in the formula (REF ).
Remarks REF , REF and REF below compare our results with the ones of {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
Remark 7.2
Assume that {{formula:fd4da1e0-2cf0-42d4-b23e-17981b4bb5ed}} is an {{formula:db66ead4-fd16-4ae6-9416-e7bddae5b231}} -separating resolution of an isolated singularity {{formula:83ecb08f-4155-4cbf-8139-2404c5b06bbb}} , and {{formula:b1ec53f7-1b78-43d6-93ed-2fd533558855}} is a Milnor ball. Then, Proposition REF mirrors {{cite:97005650a8ac11991dc2e980883b47d01674846e}}.
The key difference is that our monodromy has codimension zero family of fixed points corresponding to the proper transform of {{formula:120f0a5f-36c9-4aa6-87e7-3c7b6a2fb4bb}} . If {{formula:f121fb81-bc0b-4508-b223-077ebcaedd4f}} is a Milnor ball, they can be avoided by a small perturbation near the boundary, as done in loc. cit. However, to prove Theorem REF we will need to apply Proposition REF in more generality (namely, keeping the radius of the ball fixed within the family, see Example REF ).
Note that in Proposition REF we allow arbitrary snc fiber, not necessarily one coming from a resolution of a single isolated singularity: for example, {{formula:9eb3f1a4-a23e-4a39-b0e1-10bbfc41b40c}} could be an embedded resolution of a family of Stein spaces be degenerating to more than one singularity in the same central fiber, or, with minor modifications of the proof, be a family of projective Calabi-Yau varieties.
Moreover, the abstract contact open book in Proposition REF is actually isotopic to the symplectic monodromy of {{formula:53a89cfd-a161-4381-925c-5fc57971e7d8}} (defined in a natural way in Example REF ). Meanwhile, the model resolution constructed in {{cite:97005650a8ac11991dc2e980883b47d01674846e}} just shares the same associated graded contact pair, which in this case is the embedded contact type of the link.
Remark 7.3
The formula for Conley-Zehnder index in Proposition REFREF differs from the one in {{cite:97005650a8ac11991dc2e980883b47d01674846e}} by {{formula:5348f45c-d5a2-4af5-80c2-a9f63e0e6200}} .
This shift most likely arises due to a mistake in the computation carried out in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. Let us indicate a possible source of this mistake. The Conley-Zehnder index is computed in loc. cit. from the winding number associated to a trivialization of the vertical canonical bundle {{formula:9338841e-c01f-4121-8c89-79a4b909a8d3}} , using the formula (REF ). At the bottom of {{cite:97005650a8ac11991dc2e980883b47d01674846e}}, it is claimed that this winding number is the same as the one for the trivialization of {{formula:af59d0b8-8996-4d5d-b0e1-c598661002a7}} . To be more precise, loc. cit. uses an analogue of canonical bundles on the model resolutions, but these two coincide locally, when {{formula:7cb71e24-2be9-4f62-8541-55ee5b10dad9}} is a chart adapted to {{formula:793c60a9-6df5-4676-a2b0-5bfdf0d28288}} .
The claimed equality is false. To see this, take {{formula:ddbb5c12-cc83-4f4b-999a-7624a5db9120}} with coordinates {{formula:96a8e64c-1666-4587-9f62-fcf5a33f59fc}} , let {{formula:52674e16-28b4-41d9-aa62-a9b64d3e2edb}} , and let {{formula:8700d47a-09d6-4f3d-9fe0-18dd45240883}} be a trivialization of {{formula:feb93e5d-4cf6-45ab-b8e5-daf260eeea59}} . As in the formula (REF ), write {{formula:2dfee6c0-399b-41f8-a41f-4c32cac5860e}} . The trivialization of {{formula:38592cd6-33cf-4699-aa71-b26e466158b4}} is given by {{formula:431dd3f1-1305-40ff-aae6-fff58af15e3f}} , so the compatible trivialization of {{formula:dc6d51e6-dbe2-42e1-965d-2674b5d67bb7}} is {{formula:abb24403-83c1-4122-8c79-c57612148e75}} . To compute the winding number, as in the proof Proposition REFREF , we pull back these forms by {{formula:01d9cf1f-90d0-4160-9044-fc9d84019adc}} , and trace the path in {{formula:632f3fb0-0636-4614-985b-31587b07908a}} made by the leading coefficient. For {{formula:284f7b17-049d-495c-8bab-c1f931236ed6}} , the pullback is {{formula:ee7935ff-72e8-467d-8960-7a83479075f8}} , so the winding number is {{formula:6e426542-8b51-475d-a17b-f09367cc52fd}} . In turn, for {{formula:29ffeaf9-1f20-4c0f-9fbb-c162f6b24bd4}} , the pullback is {{formula:f4a40139-a4ac-4c00-b3cf-ffbf2ac37b41}} , so the winding number is {{formula:d2484ff8-bf40-4ed2-8422-da227ad8a620}} , exactly {{formula:d484ce3e-745e-47b9-a792-e46e31c025b9}} less than the previous number. Now the formula (REF ) shows that to get the correct Conley-Zehnder index, one needs to add {{formula:e6f3ad93-fd67-4992-a044-7fdff7bed632}} to the one computed in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. This explains the {{formula:91b34df3-4af5-4a55-89bd-dc50e6ec2900}} shift.
subsection2-.5plus-.7.5The generalized McLean spectral sequence for isolated singularities
We will now apply the McLean spectral sequence from Proposition REF to the abstract contact open book constructed in Proposition REF . We will work in the following setting, which is well suited for the proof of Theorem REF .
Let {{formula:67798dad-b136-4858-8821-380a5c674e1c}} be a Stein manifold and let {{formula:44624897-d065-42bf-bb71-7c34dd3cfee4}} be an exhaustive strictly plurisubharmonic function. Consider the Liouville form {{formula:636cff64-1eda-4fd3-9e3f-0c0b0fbb5e43}} . Let {{formula:7ac96794-4f2f-4de0-a404-c56d78269d57}} be a closed analytic subset of dimension {{formula:fc9011cc-5fb8-44de-af43-23ecb4e21c60}} . Let {{formula:d00a9af1-8627-4e50-8932-55dfbb595884}} >0{{formula:ab8b1880-8893-41ce-b87a-0259ceb7b819}} f|f-1(*){{formula:902e8bc2-2a01-4d72-8d17-ce29f286665d}} f-1(0){{formula:897a217f-14b2-4124-9e67-bade3d885a11}} Let {{formula:d4506470-82d6-495b-a8ea-45dbc9547107}} be a regular value of {{formula:66466b43-5cd7-4ab4-aa3d-4c03b96b473a}} . Put {{formula:84c5130b-9a4c-4d7d-b65e-f81443d7bf6a}} and {{formula:6af8e3ef-19f6-46f5-abac-f156be70ab6b}} for {{formula:fb0a033a-5667-4f51-908e-ddeb4c3e9ccb}} . Then the Liouville vector field of {{formula:c0e53e97-03d5-444b-8d95-df89bfd68b81}} points outwards {{formula:e7cc3cce-27ee-4462-8310-e3f2fd61ef64}} . We are now in Setting REF , where {{formula:36f6801f-1c5c-4e7d-ba99-8513fa7d74a8}} plays the role of the space {{formula:78fa1a5f-19ba-4353-995d-eb26d7887389}} , and {{formula:78059acf-440c-4058-bd47-644d879cf145}} . We choose {{formula:b8e3948c-860b-448a-bf25-b5692a8500c3}} and {{formula:cedc957c-9728-4461-9c69-402b2de87574}} for any regular values {{formula:72a67321-9640-4945-ad77-88d061d55e56}} of {{formula:5bf4a059-6b6a-43bb-a5f3-8cade4abb990}} , such that {{formula:cab419ee-5d98-4e21-b2a0-81d25b295096}} is smooth and contained in {{formula:2471d2ad-51a4-4e4e-9770-e3ac3dead3f9}} . Then, after shrinking {{formula:209f1801-1f70-4d72-9b37-70dc5c726151}} if needed, the formula (REF ) provides a {{formula:e7aa1e2d-9865-4d55-905a-54ccc86eac88}} -symplectic collar trivialization of {{formula:e4ddf075-f04e-40c9-bc62-dbf7ee26dc7f}} .
Note that for every {{formula:b2e890c4-7816-403e-ba53-1a1276c8d727}} , the fiber {{formula:fd324343-9d84-40c7-b098-cd8f83ecc944}} is a Stein manifold of dimension {{formula:e5f4814a-9b3b-4147-96aa-8e48060f6d2a}} . Since {{formula:f329fce9-cee2-420f-b6c0-ca81033a7565}} , we have {{formula:6d25fa1b-20e4-4403-897f-31012a7652ea}} for {{formula:47ef4231-1bd0-4e8d-941d-54ab8bc2df80}} . Assume furthermore that
{{formula:5b7dd877-7e13-461a-9785-c52b2d1773b4}}
Under these conditions, in (REF ) we have constructed monodromy graded abstract contact open books
{{formula:63eeddec-6ebb-4b0b-b499-b0b2372aef29}}
where {{formula:f0a4ef22-0293-487b-8094-3663ef6c18b8}} , {{formula:937d2f0c-69a8-43ee-a471-66c84d5c95e9}} , and {{formula:6df21c67-8628-4508-aeed-313f6c331b39}} is the symplectic monodromy of {{formula:db56f56e-1224-49f4-b05e-8d43466473c5}} . Its graded isotopy class does not depend on the choice of {{formula:f62c6846-5bc3-4be7-9a64-77e3b2a16470}} . We will now provide a spectral sequence converging to its Floer homology {{formula:c953c34c-f7d0-45ad-b759-b166d6786334}} .
Denote by {{formula:783bfa1b-72c1-4d9f-94e9-3db889d2b4a7}} the irreducible components of the fiber {{formula:937f94ef-e006-469f-8833-099ec4fbed5e}} .
Fix an integer {{formula:ae69ca40-9599-43c4-856a-cb2e442fd67c}} . Let {{formula:41c105cf-a3d7-403e-af9f-9a44679777a2}} be an {{formula:6c194baa-a558-4bff-b4db-0f2c74907ac9}} -separating log resolution of {{formula:299c0669-e92c-4f53-89df-33d08c45dbb5}} . That is, {{formula:48d1b5a5-c492-4c67-ae47-792198bf3e27}} is a proper modification such that, if {{formula:b4ac39c2-a143-4846-9071-5252ba619e4b}} denotes the proper transform of {{formula:39d71222-bf02-4313-92a5-7e84cfce41d7}} , then {{formula:167fef65-96d3-4229-bbb6-a7bfb9ef2b39}} is an {{formula:e0538f62-bf32-445a-96f5-78a57915084e}} -separating resolution of {{formula:8ae5fd6f-06b2-4ba2-b66f-6b2e7cba6421}} . Put {{formula:75a9c241-c10f-472b-9a0f-fe0721a6a12d}} . Then {{formula:12741dbb-196f-49b3-85a1-d037a1570672}} is a smooth complex manifold, and {{formula:e5d8f8fd-dbb1-4229-ad17-f773f828f259}} is snc. We assume that {{formula:0c92383a-8bbc-4be2-a043-cad4841bd689}} is an isomorphism away from {{formula:8c8d2597-5194-4bfe-baf2-7803e7284362}} and near the preimage of the collar constructed above. We write the irreducible decomposition of {{formula:34341c3b-dd09-4a12-b286-c3a1724c8c59}} as
{{formula:c1a0ef38-f12a-4771-b826-726c33314be4}}
where for {{formula:ccee9ee7-9a54-45b4-b8ce-f16ba750846a}} , {{formula:7171929c-0200-423e-a313-9a1f57462013}} is the proper transform of {{formula:8c2e0df1-ec8e-4f72-a42b-0b41b869f33f}} , and {{formula:fc72491c-2dd2-4f19-a4c5-130f08468948}} . For {{formula:24475b85-1c80-44c1-a658-ecf54aa9d76f}} put {{formula:a6c5e60b-b9a9-4c39-a41f-259bce930505}} . The {{formula:47687d81-a212-428b-96b2-8b614849636a}} -separating condition means that {{formula:b59d4012-3a09-4043-ae96-75844a4b51f9}} whenever {{formula:68144c99-c34b-4467-80ff-470d129880d7}} , see (REF ).
We have assumed that the canonical divisor {{formula:bfb6a8f6-5090-47f7-8f3b-4d1958ea9309}} is trivial. Therefore, we can write
{{formula:ee41efac-1d7d-4471-a153-210e69756a2d}}
for some discrepancies {{formula:5bd3e277-a6e6-463a-a570-0556dbca4483}} . Since {{formula:916ccc4e-6a58-41bc-84d6-be78fa9dc6bc}} is Stein, we have a very ample divisor {{formula:6f73e39a-c64c-46f8-9f79-88621e5a352d}} on {{formula:2e9b560e-950e-4a77-8959-fa95120e32e8}} such that
{{formula:2103030a-bccd-4916-a6e7-e10c05e236e3}}
for some {{formula:60f494dd-8077-4c41-b027-2914aff0f99a}} . For {{formula:2e3ed1c7-5073-4533-af80-3a875854bad7}} we denote by {{formula:7d05660f-a15c-4ffe-95d0-48ba100fdcc9}} the {{formula:f4a3263c-0b02-4f8f-940d-09130022e1ad}} -fold covering of {{formula:8cb0fb4f-ef18-4f14-b5f2-353de557a6cb}} constructed in {{cite:3a14ff98229456cb788161a4b4eb6b3a0b5c13fd}} and recalled in Section above. For {{formula:f5021a0f-c217-42c2-aad8-76fbd9ffc2ea}} we put
{{formula:9423bea6-9798-461e-a38c-dc2c4bd851d6}}
Where {{formula:9086a8fb-c059-4c8b-9e69-cdd104fc6e3c}} is the locus where {{formula:8a26b034-150e-41c9-947e-a9ed7fa946ab}} is not an isomorphism.
Note that {{formula:927535ac-513a-421d-a77e-5661240c0b5b}} is a manifold with boundary {{formula:fce9fc9c-e85e-4484-9f35-18db08f198f1}} , but {{formula:562bfa49-eb18-4dc4-8990-69acea82e8c4}} is not compact unless {{formula:529715fc-339e-4962-adc2-e26c7a54149e}} , {{formula:0ba9bd17-086c-4ce9-91b7-249169f020d4}} . Eventually, we put
{{formula:afa6ca80-5269-4d84-9c82-34df5c4341fc}}
so {{formula:eb0eb08d-c8d3-4346-8e28-d1c244b25c14}} is the set of those indices of exceptional divisors whose multiplicities divide the fixed number {{formula:d005335e-167b-4adf-a77d-f81240ffbb56}} , and {{formula:13d12af6-757e-4e3d-a8af-a09388aafaa5}} . With this notation at hand, we can formulate our next result. We denote by {{formula:6f10596f-df11-4338-9693-36a5f8740e7c}} the Borel-Moore homology with coefficients in {{formula:36f89f21-7c87-4815-ac42-b5c9a307cdb0}} .
Proposition 7.4
There is a spectral sequence {{formula:97007a8a-11c4-4cd0-8fd2-3ec14951eb8a}} converging to {{formula:64293b05-12c9-49c5-96bc-64792d56f0e9}} whose first page equals
{{formula:49525916-fa20-4eab-ba5a-0ea4db2a84ec}}
{{formula:8f630e8c-1d60-49d9-a392-5bcaee67202a}}
.
Put {{formula:8ae962f5-9316-4806-a386-ee6b613571a1}} . Since {{formula:fc5af503-827e-4c81-9515-d6b23b9faf3b}} is an isomorphism, the form {{formula:b5746768-b49d-4f27-a12e-0ce07d5299e1}} makes {{formula:19f94eff-0f0a-45dc-80bc-3285f8bc1175}} a Liouville fibration. Moreover, since {{formula:fad2bb08-cc40-4dbe-83dc-90f1438d470b}} is an isomorphism near the collar, the chosen collar trivialization lifts to {{formula:98437f05-db45-4a94-afc1-576b786767ef}} . The associated monodromy abstract contact open book is the same as {{formula:ebd3cd32-8511-46fe-a2e3-6b9184648eed}} . We will now modify the form {{formula:1d70e14f-d72a-440b-91df-cff0d53cedf3}} to a form {{formula:8fb10375-23a2-49df-85c6-683b62f2b1fc}} such that {{formula:ec2fa6a7-e78a-4121-8ba9-3af9faed5a62}} is Kähler.
Let {{formula:0855ee33-d1b3-42b6-95b6-b167f9615159}} be a section of {{formula:797235a7-dfc3-4c4f-9e94-04ed5f14f28a}} with zeros of order {{formula:a2471eef-8156-402b-8a12-70a25217348d}} along {{formula:e393e488-cf13-451f-93e9-31ee7dc56727}} , and let {{formula:babfa994-d39d-4514-ab74-3de90f887670}} be the embedding given by {{formula:e4f98c60-68c9-419e-997a-c8c4bcb50ed0}} . Let {{formula:e29332e6-b0e3-4592-89df-7b1b913515c3}} be the pullback of the Fubini-Study form through {{formula:91237448-e22e-4262-b600-d1e966c428a7}} . The basis of {{formula:f4f641b5-5cc8-4f26-b0a5-6400fc0e65b0}} used to define {{formula:153240ef-609c-4a47-b198-fd7de7fc78b2}} gives a hermitian metric on {{formula:ae65e294-b0b4-4b72-ada4-3004ae304c10}} , see {{cite:33bc970490d96eb88a26a85b4c569281965aba7f}}. Let {{formula:496f562f-8cb9-4c5b-84a4-abef04eafc89}} be the corresponding norm. Then by {{cite:fdc0ac8128f8d39a8b52a80e254a84a0f9afc49e}} we have
{{formula:0c802b6e-c239-48a0-addd-c52ecc4936c3}}
For {{formula:f2581be3-4b1a-4b2c-a857-8d97a04edad6}} put {{formula:7e4d62f5-292a-4daf-a75a-7f82bd236772}} , and {{formula:1ddb4b00-d161-4e62-ad7e-85b2c24a9789}} . Then {{formula:95786ef4-8283-434c-9352-1afb14bccfe7}} is a symplectic form on {{formula:9c2bab60-c33f-41df-b1df-eec5fab41518}} for every {{formula:23af8fa7-9835-474c-8ffa-37af87035d2b}} , and on {{formula:f881305e-3915-4e5c-a3e8-f34656997747}} for every {{formula:06275740-3107-4792-9966-3f2e2d2cd6b9}} . We claim that the residue of {{formula:9762b0bb-ea3b-4091-adbc-0f6cd610ec0a}} along {{formula:0daa55e8-97e3-48a3-9c1d-ecd12e8d03f8}} equals {{formula:6dd4141f-ca0b-4bcc-a7c7-8e492602310e}} . To see this, fix a chart {{formula:9cb5dc35-6a58-418c-b43a-3fc75eb6ba53}} adapted to {{formula:fc157928-2161-4a11-9a42-8dc36e96466e}} , and let {{formula:d7c37182-64d1-4d3f-8c4f-bfd6bb475af3}} be its associated index set, see Definition REF . Let {{formula:17e5dbdb-5ac8-4234-9913-c113456ed07b}} be polar coordinates corresponding to {{formula:174d7f47-57fc-46f4-8188-d802a2692992}} . Then {{formula:6387f825-8b00-4f2a-9779-3d091ec5a3ab}} for some smooth, nonvanishing {{formula:969cf630-b722-4d9c-b929-0425c318884e}} . Hence {{formula:92e40768-0a19-4750-aa65-10cea2a7c0f1}} for some smooth 1-form {{formula:933fd127-5eaf-47f8-81e7-6791d5821d3e}} . Recall that, in our notation REF we have {{formula:9d3cc8a5-03a7-45db-9960-0a2868a4d309}} , so the residue of {{formula:dcbdc9b5-a941-4fad-b10c-068e846ec98c}} along {{formula:98042f22-a1ec-4756-955a-4255626e556b}} equals {{formula:6afff572-79f7-4193-9d68-237eb5129ceb}} , as needed.
Put {{formula:cf6842c5-1e5a-4979-be45-ffe6f20d3747}} , for {{formula:486135a2-c0b9-4e51-90ef-ef416a811ec0}} . since each {{formula:f6450fc7-baae-4c54-98b4-2a08ed32f59d}} is Kähler away from the exceptional locus of {{formula:0fe54c56-5165-42b0-91f8-fc7040224080}} , it restricts to a symplectic form {{formula:70c8e5ab-141c-4b35-86a7-365ef10d6f9d}} for {{formula:46a65884-4c6a-4f4e-ba1a-af57b6f766d8}} , and on {{formula:4364c9ab-4744-4f71-8fe7-8c76f086bc8f}} . Since the Liouville vector field of {{formula:72b12f72-ea51-4292-b85e-e01751f83bf1}} points outwards {{formula:7cb58882-b36c-4f6f-8ed1-6258e54fba74}} , there is {{formula:5e68bcbe-bfb5-480d-8fe5-b0c90fa1a862}} such that the same is true for {{formula:15e6e070-306c-4e77-a960-726805f4b2ca}} , {{formula:e2bd2224-137e-45dd-bb5c-7bc847e8363a}} . Therefore, the map {{formula:efc837e6-00ef-419b-a4ab-796c94f7588b}} fits into Setting REF . This way, after possibly shrinking {{formula:1d629d90-cb02-4eb5-94b3-e1bc2e857663}} , we get {{formula:513632c0-0e90-4170-9aec-1f6794418732}} such that the above map restricted to {{formula:6851cee4-7807-4f37-9a5a-4ca089ad366f}} admits a {{formula:91c23337-b65a-4dbe-b7a1-fe49b22aaeaa}} -fiberwise collar trivialization which is {{formula:0520a1f7-b33c-4a87-9d69-6f3de3db9084}} -fiberwise symplectic. As before, shrinking {{formula:84d941b5-f0ca-447a-a2b5-6643244f69c0}} if needed, we can assume that the Liouville vector field points outwards {{formula:0c75ee24-d076-45b9-9867-60b2d4d53521}} , so we get a family of Liouville fibrations with a collar trivialization. Put {{formula:e6471c89-97f1-4b37-9b52-7f9adf79e36b}} , {{formula:47d3d397-c4b7-4a0f-af61-31cb452c9afd}} .
Using the vanishing of {{formula:f47e72fd-3e31-4cb2-b2e1-64affac0eab3}} for {{formula:a2b699df-a2fc-43c5-9e8e-5da033ce80a8}} , as in (REF ) we get monodromy abstract contact open book {{formula:a16fadf3-e57d-43ea-a097-229e1b90d5e2}} , isotopic to {{formula:957d8153-db00-4dc8-91f4-c76d6a90cc79}} , which by construction equals {{formula:23b3d9a9-f531-4b1c-a5ee-3bd4fa9d7f1e}} . We endow this abstract contact open book with the same grading, induced by the chosen section of {{formula:496d164e-5a79-40ae-ac3e-84b22ab0e5a2}} .
Since for {{formula:e8e19db0-a246-4210-bb70-d986f785f66a}} the form {{formula:b62cf1ae-d3a8-4872-91df-0a0354874e0e}} is Kähler, it satisfies the assumption of Proposition REF . Therefore, our abstract contact open books {{formula:46183219-86e4-4ab6-bd48-d8d4bd3635ed}} is graded isotopic to {{formula:bbc5b399-46e1-4f7e-90bb-3665597b3e7c}} such that {{formula:08786fac-93c0-4b39-957a-8027bb54b3dd}} is a disjoint union of codimension zero families {{formula:e9934461-abde-4d65-935a-1ba170a8fa95}} , {{formula:1c54eda3-607b-4e4e-a499-1d58116fde51}} , with properties listed in Proposition REF .
By Proposition REFREF , we can choose the action function such that {{formula:b03f2904-4330-4cde-b3e4-0ac795fc04a1}} if {{formula:f026722f-2ebf-4d0a-8a0b-1ea780289c03}} , and {{formula:0632e125-962c-4aa3-ae40-bfee8e8a61ee}} if {{formula:9c2f865a-f1f4-4c74-a0ba-afaf4c7cca22}} , where {{formula:43278a2d-f567-439e-afee-aef51a0f10e6}} can be chosen arbitrarily small compared to {{formula:ecd1ea7d-5501-46f2-889b-c1f1aea08ec8}} . Therefore, an indexing function {{formula:5cb846cf-4419-4dd1-b85e-ff1bc602a482}} defined by {{formula:cf18726f-ae01-467a-a290-de4c5f410236}} if {{formula:aed0c0c7-8058-4e65-9079-582451798c54}} , and {{formula:c3d1163f-013f-4321-9abc-8a5ab5766960}} if {{formula:911295b8-6b77-4782-bfab-8c110391ab0e}} , satisfies the assumption of Proposition REF . Thus we get a spectral sequence converging to {{formula:50acb00a-4cb0-4042-81ae-9c1727f6cf6c}} , whose first page equals
{{formula:97705cec-fe9f-4be0-af71-9dbac2703d43}}
where {{formula:956b8314-2940-42aa-ad9c-211f07df9f20}} if {{formula:6d5482ea-5aa7-496e-855e-cd7fb8c3ff26}} and {{formula:5309c585-e4bf-4125-8a4f-3407e114c1dc}} . By Proposition REFREF , for every {{formula:e0e1b238-8476-4dc1-ae4e-aed80790d29a}} we have {{formula:bd928544-f6a1-4548-aff6-d4e96938e99b}} . In terms of the splitting (REF ), this means that {{formula:7b9a651b-dca2-407f-a87f-38d0ee5b950c}} , where {{formula:c9dcd532-9fd6-4250-a2fa-79005b430fbb}} .
Assume {{formula:27181371-5127-4628-8a7e-d0e11a7e05a2}} , i.e. {{formula:1d8a4dd5-5b69-4a92-b2a1-55f568dc0237}} is an exceptional divisor of {{formula:4c514fe9-b3ae-47c3-8fda-57827596c718}} . Then {{formula:a6c834a4-1a62-4850-973b-f5c908641815}} , so {{formula:3960f0fb-a8cf-4afc-8f6d-9cda72586527}} . By Proposition REFREF , we have {{formula:b1b6e819-2aec-4055-8925-b176c2483ce1}} , as required by the formula (REF ).
Assume now {{formula:2d0094f6-f24a-40f4-99ab-608fcece7981}} , i.e. {{formula:9d5677ed-82fb-449b-b2a2-ac0d92c3eb4e}} is the proper transform of the component {{formula:8fa2ea37-e3fe-45b5-bb69-cb21e31dd6de}} of the fiber. Then {{formula:bd0a12bf-ff2f-46c7-82ae-d1d5fe482d19}} is the manifold {{formula:2c2a8d2b-eaab-4d1e-9d0d-0ac96f7d2910}} . Hence the terms {{formula:5975aed6-1088-4a5b-94ef-bb1a03f61dbc}} agree with {{formula:5ef80c16-9833-4c61-b8b3-e38239579f71}} in the formula (REF ).
Remark 7.5
Let {{formula:a6e6990c-760f-4b20-9764-ee0339df4d96}} , let {{formula:fe44badd-d65d-46a7-b757-dd8a78e80af7}} 0n{{formula:4a4e751c-b340-4986-86ae-6a4240570ed3}}U=B'{{formula:74ee7eca-fb22-4414-bcc1-d76f62ff629e}}W=B{{formula:75731242-5906-4a1a-ae70-cdccd7a5539a}} In this case, terms {{formula:8451754b-9f53-4989-8129-65250fdb119c}} of (REF ) corresponding to the proper transform of {{formula:4ee713f4-052d-4f51-8bab-9f11c11d30d8}} do not appear. Indeed, in this case {{formula:befb2be2-72cc-48f1-be88-fecfb8dc4e4c}} . By the conical structure theorem {{cite:edcbc0062181eb9053bb57ecfae662692e9a247b}}, {{formula:4e6b60fb-8289-4607-a652-cf784afc14c9}} is a cone over {{formula:19334202-7b04-42f1-9cc6-6df7b13d04a4}} , so {{formula:86ef9aec-6495-448c-a0bf-99ae211ca7f7}} .
Therefore, in this case the spectral sequence (REF ) is similar to the spectral sequence in {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. The latter converges to the Floer cohomology of the symplectic monodromy of the Milnor fibration {{formula:860f4a66-b2f2-4e87-958d-bddd49e1fcc6}} .
Remark 7.6 The exact values of the action and Conley Zehnder indices, computed in Proposition REFREF ,REF were used to write down the precise formula for the degree shift in (REF ). However, this formula will not be used in the proof of Theorem REF , where we will only be interested in vanishing of {{formula:e3439b04-cce0-43ab-b275-da7c69a43cd1}} .
Proof of Theorem REF
In this section, we apply the techniques developed above to prove Theorem REF , which asserts that any {{formula:91549956-3d1c-47a2-b427-070e83e7fd13}} -constant family of isolated hypersurface singularities is equimultiple.
Let us recall the definition of a {{formula:ec82e65c-089d-4070-82df-44ddb5216818}} -constant family. Consider a family of power series {{formula:cc0fc9d9-cee2-43b8-b25d-48560a3bbb5d}} , where {{formula:4fc9057f-e6f4-4d22-95ae-6f8a8a00170b}} t[0,1]{{formula:4c72af1f-8968-46b6-a814-6f983ff4e0b4}} ft=Nna(t)z[z1,...,zn]]{{formula:18802a9a-cd72-4a2c-9335-03d1db561158}} (ft){{formula:d3459cd2-8ab1-4d1c-a853-ec332600408c}} ft{{formula:c18f7280-e91c-4bee-ab84-7a26edfb8979}} f{{formula:27c29b90-e433-4ed9-adbd-c3e02276f6fd}}{{formula:f37fcd98-5255-4045-976b-6e79b4328795}} (ft){{formula:5262e1f0-f389-4dea-adf4-59a58e8843bc}} f{{formula:fb9ec081-556e-44bf-b4fe-6b34828f970d}} (ft){{formula:52068de4-a657-4d5f-ab6e-79b848d902c0}} ft{{formula:2de2cff8-e5ac-49c1-b76f-1376a3a607a9}} t{{formula:9379a309-177f-4a6d-9b43-797cefcdfc53}} Note that the assumption {{formula:e0da0cf3-4c4f-4b0c-88d2-3334e83cffd1}} , which we include in the definition for convenience, ensures that the singular germ {{formula:2dd3d7dc-b097-40c6-accd-c459246a53f3}} is isolated.
We begin the proof by recalling a known reduction.
Lemma 8.1
Assume that every {{formula:bba4580c-0218-435d-b185-448dce2fc224}} -constant family {{formula:7058b5e1-5da8-4394-952f-75de5773b38e}} satisfying the conditions REF –REF below, is equimultiple. Then every {{formula:be5340cc-b910-4f56-b5f0-de94bb7249b8}} -constant family is equimultiple.
There is {{formula:7f25d1a0-6a27-4518-97ba-a46f31c7df4e}} such that {{formula:7a95f8e8-90ca-4685-9800-2a5e8fbffc45}} for {{formula:9bf06126-dd16-4d73-9693-3725c7913edb}} , i.e. each {{formula:815ec65e-cece-4670-9281-156db466f89e}} is a polynomial of degree at most {{formula:a69656d8-907e-4540-bc05-6145e888f9b6}} .
The functions {{formula:5058d822-a45a-4cc5-8c3f-b6ccd6e927fa}} extend to holomorphic maps {{formula:1d6edc0a-8a88-47be-a002-179a643f6687}}
{{formula:aed2ba55-2277-49ea-895c-0f252a209df0}} .
{{formula:262733b7-1ca6-480c-9daf-241506e37a90}}
.
REF Let {{formula:9053c60a-0b48-4789-883b-5081935f9a3b}} . If {{formula:c670e994-3e71-4a87-ba4e-bc8053860189}} is finite then {{formula:5d903106-41d0-4388-b942-93cb210cfd55}} is {{formula:1417f66e-37bf-4842-b225-43c9b11bebd3}} -determined {{cite:0561cdd946f67939e77edce69565b1e24e17ff7b}}. This means the following: for any {{formula:b635f5dc-1402-4430-8465-5f9c959c19ad}} there exists a formal change of variables {{formula:9673f067-0b0c-4cc5-a73f-4214fbdd3a3d}} such that {{formula:15550193-2d6e-41f4-8da3-cb39b27152b3}} ; if moreover {{formula:0366bc6a-b95e-43e0-b775-eae632287b5b}} and {{formula:63996fa7-9377-4c2d-9715-b8b1d13b3b76}} are holomorphic then {{formula:f582b221-9a3f-4fbd-ba42-27932399827b}} can be chosen to be holomorphic. As a consequence, {{formula:bb97a514-58ff-4256-9f3a-a243a01b5bf3}} and {{formula:d970d504-577a-45b6-9aae-66f05bd8b98a}} .
Now if {{formula:569bffcc-c2b7-43a9-ad38-922d524a7c2b}} is a continuous family with constant Milnor number {{formula:00fb8943-7dff-4e60-b714-f25cfe431147}} , then for {{formula:7cc3cfc4-5afd-45a2-bddb-1b9e6be46cdb}} we have {{formula:62b5ff0c-c414-4adb-826c-09c7342270b9}} and {{formula:bb62da9d-8435-4069-86e1-f160a8e37f82}} , so to prove Theorem REF for {{formula:086e851b-11bd-42d3-8d5f-d11e80ac4b0d}} it suffices to show it for {{formula:13b10c37-af9c-4b95-8576-16345d983590}} such that each {{formula:07bcb9d6-97c1-4213-8128-5291d26a582e}} is a polynomial.
REF Polynomials whose germ at the origin has Milnor number precisely {{formula:050f5019-9c68-4794-a8f1-eb0ef34cc777}} form a Zariski-locally-closed subset of the space of polynomials of degree {{formula:257821b5-1f16-4175-96b7-b0acd56676db}} , called the {{formula:e1d6a73a-ae7a-421b-926c-8605f900a2e1}} -constant stratum. Denote it by {{formula:b57c061b-578d-4e9e-8580-97ed821bd53c}} . The family {{formula:b9cd6bd6-ec4e-4190-8a24-dd35e5bcb22f}} describes a continuous map {{formula:d9119566-3688-456c-be27-720d79a3dbbc}} . Choose a resolution of singularities {{formula:cc4571bb-fe18-4a8e-9b6a-9cadede2d002}} , and let {{formula:0501056a-c4aa-4c6d-b0dc-7e654fa40515}} , {{formula:beb72d48-f961-498f-9dde-b44a0d8e8b80}} be points which are in the same irreducible component of {{formula:a4f8a786-071e-410f-a15c-f37634a4b3d8}} and such that {{formula:bf11aa92-e50a-4b29-9b5c-7c40510dbcc1}} and {{formula:8f98251b-45eb-4920-a173-dda0e8a9e0f1}} . Then there exists a continuous path joining {{formula:caaf50ef-8e7e-4f00-9b6e-a699f5156a66}} and {{formula:009fd93c-6920-4b39-8656-88c65e5b6ebb}} . Because {{formula:6c3b616a-2a36-4936-a226-5bac1c096b9f}} is smooth, there are points {{formula:5c412e69-f54c-4983-bac2-ac1eb999032e}} and holomorphic parametrizations {{formula:38570ce8-6852-4542-9e8d-de8bd317d40a}} such that {{formula:8433fcdf-b7f9-438d-8d42-78a4f0419c22}} and {{formula:d15fde45-7aff-40f1-b5fa-71303ae0b936}} are contained in {{formula:0e3a022e-e3a4-4dc4-9add-fd9a01c1fb77}} for any {{formula:a0bb668d-4947-417f-8c9b-35a99492b1ed}} . The result follows after replacing {{formula:285ce411-7784-4c85-9c6c-7c7c5d5b736a}} by each {{formula:e3b9fab6-81dd-4a42-b126-4ef97a42be18}} .
REF Write {{formula:25ee1d00-aeef-48e1-92a1-215783fdb0e3}} , where {{formula:db39a006-4ac5-4f8e-8c13-8871b7162d97}} . Then {{formula:f1189534-a5ee-4431-b11f-f99dce63d465}} , so {{formula:8eac364f-abd4-426a-b3fa-c32f21118c0d}} is {{formula:798db9c1-f929-47df-a40b-2fd60ba59c55}} -constant, too; and {{formula:82f5f3a8-20eb-4e4c-8d9a-b1d3990a053e}} .
{{formula:a453a948-c68e-4c38-8d58-d0ac1ef86b42}}
Proof of Theorem REF .
By Lemma REF , we can assume that {{formula:11d5a05d-7a2a-44f2-83ac-c2ba5da71054}} , and our {{formula:9bf2b317-5ef1-481a-ab99-a4c22688e98e}} -constant family is holomorphically parametrized by a disk {{formula:320f7d7d-643c-4011-9580-70149d3ae7a7}} . Clearly, it is enough to show that the multiplicity remains constant for {{formula:747709e7-c1cb-4ffc-93d7-ac0acca83148}} sufficiently small. In Example REF , we have chosen three Milnor radii for {{formula:b0d1f938-fbe8-4b5b-9702-35460344218a}} , say {{formula:02e5a11d-f29f-4292-b661-5348eb306b4f}} , and constructed an isotopy of graded abstract contact open books {{formula:de2b4cab-e1db-4763-abda-29056ed8e760}} , parametrized by {{formula:d76ecbb5-be8c-4c87-a960-2c9051792de8}} , such that {{formula:55487f44-bcd2-45e7-a037-a0924be5be52}} is diffeomorphic to {{formula:16ce4888-e325-4af5-98aa-ffb2028ad391}} for small {{formula:dbbc6d23-e6f0-4596-a0d1-4176d19324e6}} .
By Proposition REF , for each {{formula:96188606-8ba0-4436-a77e-c58391d2261f}} , the Floer homology groups {{formula:e9eddf80-e04d-406c-92d9-99531ef62aa8}} do not depend on {{formula:84b79295-13cb-40d7-8882-c9785aee3c11}} . Fix {{formula:46225bf4-5c0c-49a3-ad0d-f330cd59c27c}} , a number {{formula:7c8da9e8-4446-43e8-8f07-982ed5a399c0}} , and a minimal {{formula:5eb25de5-5cfa-414c-a306-ccde46fa6f82}} -separating resolution of {{formula:438202ca-e693-4148-94c9-ba0df567c69b}} . We will analyze the first page (REF ) of the McLean spectral sequence converging to {{formula:f7287a54-8c69-49ca-8abd-c0ab14d7bffb}} .
We claim that the terms {{formula:63e993a3-22c8-4505-821a-c0fe34cd315d}} of (REF ) corresponding to the proper transform of {{formula:3aa15bda-0d46-4cde-8bb8-14c528c5a551}} are zero. By definition, {{formula:34122386-3f5a-4906-b553-ac9bbbbd3326}} for {{formula:51963eb4-6e72-418d-b084-2f1c5d228dbb}} , and {{formula:eb7b601c-d8de-432d-9637-0b96b18a2a98}} , see (REF ). In our case, {{formula:c4fc7245-6326-4758-9192-ae09d871723c}} is the Milnor ball {{formula:8bcfede4-a875-4ead-92bd-57a4fcc93149}} chosen in Example REF . Thus {{formula:be1ff564-9271-4ea3-b416-aa8872baa29c}} .
The cobordism {{formula:dcfac0a7-784a-44e9-a807-5eceb47f4280}} is homologically trivial by Proposition REF , and therefore we have the vanishing
{{formula:24a1e111-9220-4e45-8d25-27f1d71290f0}}
This implies that {{formula:c8e07da8-05bd-4c27-b1d3-4a37e8a26410}} vanishes, since {{formula:f13203f2-d00c-4b3b-9e91-6b165276a6cf}} is a Milnor radius for {{formula:aabaacb6-beb3-45b7-835b-44cd13e7dc0b}} , and by the Conical Structure Theorem we have that {{formula:c3613b95-a3b8-48e0-b047-497dc8951a41}} is the cone over {{formula:5db8776e-5c09-40ce-9d76-c7529fa33414}} . Notice that when {{formula:7b4b095e-7e2d-4224-a598-f17838f00140}} the homologically trivial cobordism does not appear and in that case the argument reduces to Remark REF .
Let {{formula:134eb159-dd04-4da4-90eb-98bfad47c36c}} be the multiplicity of {{formula:8d371fc9-8c8d-4924-9652-0dceb8d1f86c}} . Since the {{formula:0bbfb762-1d93-4ed8-97b6-fe886929c0d9}} -separating resolution {{formula:701eeaf7-6554-495c-901b-839115f4648c}} is minimal, we have {{formula:509372c3-98ab-477b-899f-7bc1bf75e5a1}} , with equality if and only if {{formula:374c81e0-58dd-405c-8e46-e7735a36919e}} corresponds to the exceptional divisor of the first blowup at the origin. Therefore, if {{formula:77f74c94-5822-47b2-9e54-d2a79815d37b}} the index set {{formula:d6906c2a-6470-4e44-a8f9-30e2ed186a4f}} in (REF ) is empty for all {{formula:57cbcc28-5416-406b-9af9-c713ee163123}} . If {{formula:622a3704-b51b-40af-901c-ee5c8a8bb4ba}} , then it is nonempty for exactly one {{formula:88b6a6e2-33e5-430c-a271-e45ccb03f2c9}} . Thus the first page (REF ) is zero for {{formula:127577e4-39c3-410d-b965-f1fc39de0820}} , and has exactly one nonzero column if {{formula:c9795b54-c460-4d7d-8dfc-39d4a31ffa96}} . We conclude that
{{formula:5cf0f643-ef16-4bce-b0bd-bcad24c0c503}}
which is independent of {{formula:5a03e7ce-85a7-4b7b-9f6b-7bee36988a4b}} since {{formula:d131a057-acac-44a2-a2ca-06ec39e7c039}} is independent of {{formula:6d1f9dd6-ef4a-4887-b754-27fdbf6c272d}} .
Remark 8.2 (cf. Remark REF )
When {{formula:ab96db82-fbec-4981-a93b-da4b50c0f5e7}} , Formula (REF ) is a version of {{cite:97005650a8ac11991dc2e980883b47d01674846e}}. While McLean uses the graded abstract contact open book arising from {{formula:75d5d617-7d8f-4c54-9039-64efd7536793}} restricted to a Milnor ball, that is, the Milnor fibration in the ball, we use the graded abstract contact open book associated with the Milnor fibration in the tube, constructed in Example REF .
Remark 8.3 Let {{formula:bbf872ae-1e31-4dbf-a4d2-69244b161cfc}} be a smooth, {{formula:71505918-a489-45ce-92ae-0528c547841d}} -constant family. Theorem REF asserts that the initial forms {{formula:606809f6-ee50-4d5f-9135-ca96eca597dd}} are of the same degree, so they form a smooth family, too. Since {{formula:9d7dfaf7-9e71-48fb-a2a0-489e5194a3c8}} is homogeneous, its Milnor fiber is {{formula:7077e4f8-b94e-4e3f-b6e7-018b9cc6022c}} . Therefore, the Milnor fibers {{formula:51af4ffb-241d-4379-b62d-6bc953a08da9}} form a smooth family, too. In particular, by Ehressmann lemma, they are diffeomorphic to each other. This gives a positive answer to {{cite:5a1a6037ae4af47b859ad972de595c63e02816f2}}.
On the other hand, we recall that, by {{cite:fc21421237ff2e50dfc85104de1a7a9584e364c5}}, the projectivized tangent cones {{formula:697f0965-2e40-49a2-8685-06ba4252666c}} may not be homotopically equivalent.
Remark 8.4
Theorem REF implies an analogous statement in the purely algebraic setting. More precisely, let {{formula:af69dfcc-0d3e-47dd-a272-2ade9f2261e4}} be an algebraically closed field of characteristic zero. Fix {{formula:f49c6ffe-8b77-4a3b-af38-8a9c3bbf12c7}} and write {{formula:6c2a6afa-5ba2-4d31-bb52-4d87c68e87ad}} ; so {{formula:59fcd137-d599-45ab-8d7d-c6f66ff9e193}} is an algebraic family of power series. Assume that the Milnor number {{formula:fb6f2394-ac82-4a2d-94c5-8d773a7c5f2e}} of {{formula:17435c0d-2b32-4b08-bda7-ebec9460484d}} is finite and independent of {{formula:4e271818-196f-41c9-a6b8-667a08a9b725}} . Then, Theorem REF shows that the multiplicity of {{formula:a61e5eae-95d1-44e0-a092-2ebb872de407}} is independent of {{formula:98736f44-8e0c-4dd0-8b22-21e45a5dcca1}} , too. Indeed, as in the proof of Lemma REFREF , using finite determinacy we can assume that each {{formula:4e60a569-36de-4325-af46-3370c2bfc8ad}} is a polynomial. Now, Lefschetz principle reduces the claim to the case {{formula:c37b7528-242d-4ff3-b282-5310c9fb247a}}
| i | b9ce0ded4341544fe068d01d09f23e9c |
Label-only Attacks.
Label-only attacks {{cite:b42e2e97074bbcd87d2414d024d68c0c26a8d215}}, {{cite:a610faba2a54ad94a6570adc14bf3ab6c18c2e39}} consider a more restricted scenario where the target model only exposes the predicted label instead of intermediate features or gradients, or even output scores.
Thus, label-only attacks solely rely on the target model's predicated label as their attack model’s input.
Similar to previous attacks, this attack requires the adversary to train a shadow model.
The adversary queries the target model on a data sample and perturbs it to change the model’s predicted labels.
Then, the adversary measures the magnitude of the perturbation and considers the data samples as members if their magnitude is greater than a predefined threshold, which can be derived by perturbing the shadow dataset on the shadow model.
| m | 3b36ab8ed333e5a1a195cc59673d981c |
In this section, we introduce a better choice of {{formula:37d24246-ae51-4206-882f-002ee272be9c}} and compare it to {{cite:fc27f29ebef93da171ee7047f5bed4c2154e8f2f}}. We also extend the computational framework to some widely used network structures.
| r | 949a578fb92edeb643596af4345d62a1 |
Textbook examples of this subject are usually presented in terms of
continuous variables, typically position and momentum. However, there
are many quantum systems that can be appropriately described in a
finite-dimensional Hilbert space. These include, among other, spins,
multilevel atoms, optical fields with a fixed number of photons,
electrons or molecules with a finite number of sites, etc. An elegant
way of approaching these systems was proposed by Weyl {{cite:e6ad3daf6a79deb9318413137b99a147928ef7c9}}
in his description of quantum kinematics as an Abelian group of ray
rotations. Similar results were also obtained by
Schwinger {{cite:8a3e6c1c07ab90228f9fa570c406e2e87c302e58}}, {{cite:d94aa8ac306ab11673102d9e5e9435c0d1a2fca5}}, {{cite:ccc6e0644076e0d8e0005e6d1679ca8ceb854ba2}}, who
showed that a set of unitary operators (defined through cyclic
permutations of state vectors) can be constructed such that they are
the generators of a complete operator basis, in terms of which all
possible quantities related to the physical system can be built.
| i | 17714431f158c8265f4c7b559372908e |
The optical-UV and X-ray Swift data of OJ 287 taken in the course of the MOMO project
{{cite:e0eb8a7ab77d5ae4fc55545b1c571a59bc3dee2a}}, {{cite:b21d3f1074649d90a969cc8eabfd66b6fd55c380}}, {{cite:5e5553ebcfc3e2ee09a403babb2ff8981f52c323}}, {{cite:f15444f89ae0a82a3ff11e2c5f1eac8d54600791}}, {{cite:80d4ee4fbad3e28a45c8ac633bd0481feb1a825e}} have already been published until May 2022 {{cite:0a59184997f644f985940207ca801a4c67d56c43}}, {{cite:f62bcc330319c64b1ac6eff12c4cb66489bb96cf}}. Data beyond May 2022 will be reported elsewhere.
The Swift data is also consistent with the idea that the source of the flaring in the early part of 2022 was in the main jet {{cite:0a59184997f644f985940207ca801a4c67d56c43}}.
| d | 16c4b20f9491d992a000d174ae744530 |
Our luminosity functions reach similar absolute magnitude limits to
the luminosity functions constructed by {{cite:1a81cbbbc087976edb7fdc7c03e7df237c744e66}} in the same
redshift ranges, despite their Swift UVOT data having a much
longer UVW1 exposure than our XMM-OM data. {{cite:1a81cbbbc087976edb7fdc7c03e7df237c744e66}} based their
study, which had somewhat broader goals than ours, on a master sample
which was selected in the UVOT u filter, and as a result the
faint UV absolute magnitude limits were set by the onset of
colour-dependent incompleteness. In their Schechter-function model
fits, {{cite:1a81cbbbc087976edb7fdc7c03e7df237c744e66}} fixed the faint end slope {{formula:ea9f14a2-75d2-4ab3-96ff-21e147736ced}} to the best
fit values obtained by {{cite:6e631beb328acfb7846c13fa98b38194280f160d}}; given the covariance between
{{formula:7c627e43-28ef-4cdf-b9f4-6ab92bad0c7e}} and {{formula:26873daf-6e7b-402e-8ae5-6ef9412726af}} , their measurements of {{formula:aee59948-d28f-4cb0-a9b8-eedc7ae6b0bb}} are therefore not
strictly independent from those of {{cite:6e631beb328acfb7846c13fa98b38194280f160d}}. Nonetheless,
their fits support the picture implied by our study that {{formula:0d182562-b23d-4516-bcdb-66881bc130a4}}
evolves such that it is brighter at {{formula:212ff09b-b785-4826-973a-51d4747672c6}} than at
{{formula:0c2eec21-47ca-43b0-9138-2c8f1c21dcfa}} . Visual inspection of the binned luminosity functions of
{{cite:1a81cbbbc087976edb7fdc7c03e7df237c744e66}} also suggests that {{formula:7d9d8244-7196-4e50-a569-cf42b83a088d}} evolves in this fashion
between the two redshift ranges.
{{figure:d8be5742-3bb5-460c-a42c-e64445200f25}} | d | 40f583f56523a978248b5660c6c9536a |
In matrix models, there are phase transitions in which distributions of dynamical variables change topologically,
like in Gross-Witten-Wadia transition {{cite:4e8bf59d49123d83273f6ea2d6a11af7e15657d5}}, {{cite:cf5e32f0b61db4d919d8c0c4cde08533a47918c6}}
and in the transitions among the large-{{formula:67b6e5dc-435e-4a1d-962b-c40d17bc4f66}} limit multi-cut solutions {{cite:5a7cf99769091bb24985ddbc553189d1b75f62fe}}.
In a recent study {{cite:d58abfaad175a704d881cdf76bb52a171b965c3d}},
similar splitting-merging behavior of dynamical variables
was observed in a tensor-vectors system by numerical simulations, but
the results were not convincing enough to characterize it and conclude whether this is a phase transition
or just a crossover.
In this paper, we have studied the system exactly in some large-{{formula:b27b4bdc-8b1e-4532-8555-f4ae1952fd19}} limits,
and have found cascades of first-order phase transitions for fixed tensors, and a second- or first- order phase transition
for random tensors, applying the replica trick for the random cases.
These phases can be distinguished by breaking patterns of real replica symmetries
for fixed tensors, and those of replica symmetries for random tensors, respectively.
We have also performed some numerical simulations to compare with the exact results:
We have found consistent results, which support the assumptions made in the derivation of the exact results;
We have also found rather slow convergence of numerical data toward sharp transitions of the exact results,
which implies the necessity of our cautious attitude toward numerical simulations of our system in future study.
| d | 1744c2a05b043ea38b962ecf10ac4b5d |
Nonetheless, we compare our proposed approach to nine other baseline systems. The first four baselines are representative of the state-of-the-art in image retrieval in the computer vision community. These systems were developed for the Oxford 5k {{cite:43cd34938c8872c86b7a7725549564edbefdfa61}} and Paris 6k {{cite:4b70c0f6572b1e784c183bf5c7806a7f61458cd1}} benchmarks, where the goal is to identify a famous landmark in a query image given a database of known images. All four systems are built on top of pretrained ImageNet classifiers like VGG {{cite:20784629a89a9a275aba689021b09a7d059f3d72}} and ResNet {{cite:6a01908bcc1669e41f282d2e041efe02c99bc333}}, but they differ in the method by which they convert model activations into a final feature representation. The first baseline (MAC {{cite:1c56df5aa3606dd3675254d8f22496c4f309e81e}}) takes the {{formula:d2b1c4b8-cc21-4df5-a240-ca0535e3c223}} tensor of activations at the last convolutional layer and computes the maximum activation within each feature map. This yields a fixed-size {{formula:1453b8ce-170f-45fc-a547-c09074ea7b9b}} -dimensional feature representation regardless of the image size. The second baseline (SPoC {{cite:2d728d968949b5317815dcdf3a8c2ac07e98daab}}) adopts a similar approach, but uses average pooling rather than max pooling. The third baseline (GeM {{cite:cb419400b1f7f964c1fcaa60dae22aac72951faa}}) uses generalized mean pooling, which is a generalization of both average and max pooling where the type of pooling is specified by a single, trainable parameter. The fourth baseline (R-MAC {{cite:76d7ad62379ec146d86a886f587510f07bfe8af2}}) applies max pooling over different regions of the image at various scales and combines the results through another pooling stage. All four baselines also apply various forms of post-processing, such as dimensionality reduction through principal component analysis, whitening, and L2 normalization. Given a query feature representation, similarity with database images is computed with a simple inner product. In our experiments, we compute piece similarity as the maximum similarity with any page in any of the piece's constituent PDFs. We evaluate the baseline systems with their provided pretrained models.Training the baseline systems from scratch would require a large amount of labeled data (to retrain the ImageNet classifier) and would constitute a significant research project on its own. In this work, we simply evaluate the baseline systems out-of-the-box using the provided pretrained models. The last five baselines are equivalent to our proposed system but using a fixed n-gram fingerprint for {{formula:ff6b5d77-f0f9-4049-a026-f94e14f24c97}} . The 1-gram system corresponds to the approach proposed in {{cite:effb91c70dbdbdd5b362ac8fc1fbe456e2e3b791}}.
{{table:1a180a90-45ca-43a2-91a1-117fed5267a4}} | r | 43f2ecac439de265e2d9b4a69221c821 |
In this paper, we propose to look at the role of stochastic resetting
in the so-called encounter-based approach to diffusion-mediated
surface phenomena {{cite:e2ac746266c8c0ff0209ce06dd9857144e285572}}. This approach is based on the
concept of the boundary local time {{formula:4e281eec-f404-4cd0-818d-788d663ab636}} , which quantifies
the number of encounters between the diffusing particle and the
boundary up to time {{formula:86c293f8-7402-42a8-a9fe-aa4cc5623941}} . The diffusive dynamics is entirely
characterized by the full propagator {{formula:9c3f05f5-790c-43e0-998f-a7d50d199c7b}} – the joint
probability density of the particle position {{formula:bee29eaa-794e-43dd-a688-bfb55d2c2e62}} and its boundary
local time {{formula:521907ef-5a25-443c-8871-ebb8cf1347dc}} at time {{formula:fcd25df7-d8a3-494c-9ba9-a459a416649d}} , given that the particle started from
a point {{formula:a1c8b0c9-3df7-4558-ae22-02483dd048d8}} at time 0. Once the full propagator is determined
for a passive (non-reactive) boundary, different surface reaction
mechanisms can be implemented (see Sec. REF ). In this
way, one can retrieve the conventional constant reactivity described
by the Robin boundary condition as a specific model, one among many
others. Several extensions and applications of the encounter-based
approach have been recently discussed
{{cite:e2ac746266c8c0ff0209ce06dd9857144e285572}}, {{cite:d5e780f41bbb46af1877fe2cb4dfd0df6a8286c7}}, {{cite:0d1632c7d2365519f5373f2fa9f938a8bbc5f0b8}}, {{cite:c660fbb94b8cac47bf4b4537484939c8ab282c79}}, {{cite:dce1fc8dad3270482e889917847ada40075c1418}}, {{cite:a9643cb5b0810db41e3a2500e3b9932ebdb0546e}}, {{cite:ac3c3c1b91398188d9044d9d44b49f229798960b}}, {{cite:451b3c534c0015ab8171355044f3dd2c014d9b50}}, {{cite:96a4052c054d04dc96e07bb9367a3d8e577e9765}}.
Here, we aim at investigating the role of resetting within this
paradigm
At the submission, we discovered a recently published paper
{{cite:96a4052c054d04dc96e07bb9367a3d8e577e9765}}, which undertakes a similar study in the case of
Poissonian resetting. Even though all our results were obtained
independently, we systematically outline eventual overlaps with
Ref. {{cite:96a4052c054d04dc96e07bb9367a3d8e577e9765}}; in addition, a comparison between two
approaches is given in Sec. ..
| i | 3e8963d49808bcd3643bd36e6aa474b3 |
The discovery of the accelerated expansion of the universe made about two decades ago {{cite:ddb1bb004c567319839b1cda3e1defa84ddff74d}}, {{cite:994dc25cc386879af6e3ba1a5214f50a84963797}} established {{formula:1c0aeea0-161d-4fcd-87a7-a91c382a329f}} as the standard model in cosmology. This paradigm relies on three pillars: that general relativity correctly describes gravitational interactions at cosmological scales; that at those scales the Universe appears homogeneous, isotropic and spatially flat; and that the Universe's content at late times is dominated by non-relativistic, pressureless cold dark matter (CDM), and the cosmological constant term {{formula:13a06ef9-7170-444a-9171-b9d511b22e57}} . The resulting {{formula:9c0f473d-d7b6-4e36-b538-a857b94a842c}} CDM model is in good agreement with cosmological observations from a wide range of temporal and spatial scales {{cite:b36345526a690c97f3b053f8b368ef28072eacc4}}, {{cite:f865b83092ef6e5295d0ba4d3ac478c6c28ee497}}, {{cite:284a836fdb84661b27d90e6ab9bfa124352c81af}}, {{cite:55d3e9b5228bbb8fd59f9de68c34d12de68cefc0}}, {{cite:a0053d7cf20e918f84c0089998a8333725e24efe}}, {{cite:3ffe6b6101abd0ba70dac2b84914fe122694b950}}, {{cite:d69f381daf13349d4aa54389c51bb92fd6129377}}, {{cite:1d91757725077747a00d5c57d13fe0ec3adffa4d}}, {{cite:cedca2bfa246a68c37c6cec6d69ec1e738ba0bcd}}, {{cite:f2163c916bff49b5d7cbb3690ca072458fdf17b9}}, {{cite:7ca8960f1419ea56e53c6eece2497f08e2c5468b}}, {{cite:ed4b67bb1822c89126c70d49d3f5a28879ed690c}}, {{cite:6e5b3684aeb183388b25d0fc76e1d2d2bc29d8b1}}, {{cite:9e84445618959125299ebfcd3867d2d6c2cc350f}}, {{cite:8f359432009497d2f8813aeb7718198881180265}}, {{cite:05e44d30ffa7b47473635b0704443baa7c386a2d}}, {{cite:66bd9e1cf3aa4e8ba9d919e50430cdd4e57ed361}}, {{cite:71f756517383dd198e51b53cb864288f329de120}}, {{cite:522a15a03d73a752c402a3f23a06b0b2d0ce11a3}}, {{cite:e9fd67a60a4bc84c12f7b124e8a2e317e69dee5f}}.
| i | 6d2778db0cad64bcc3e4c60dc7734734 |
The success of recent papers {{cite:8f5551ade48ccb773adc561f7db341cac95e72e6}}, {{cite:63f4f98fe3cf019bb26513571ee4728c3e40cfc5}} in supervised RE is fueled by advances in deep learning, but also, crucially, by the availability of a large training set such as TACRED {{cite:f38ebf7feb0b7232aed62506c04a7816ef588bd4}}, containing tens of thousands of training examples.
For most relations of interest, such training data is not available.
| i | 66d5405aa1359376f3c96f0fe29723ae |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.