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c5f107cc19855eb76b50e01207bcf806919ccfe0 | subsection | 21 | 57 | Crystals on shifted tableaux | Blocking will depend upon a parameter i, for 1 \le i < n, and for i-blocking, only entries \overline{i},i,\overline{i+1},i+1 will be considered.Definition 3.6 Let i \ge 1 be an index and T a semistandard shifted tableau. A pair of entries y,x of T, with y \in \lbrace \overline{i+1},i+1\rbrace and x \in \lbrace \overlin... | {
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866ecb40c7be706f18781a25a1dfc1a400b285b4 | subsection | 22 | 57 | Crystals on shifted tableaux | Thus no i-free entry \overline{i+1},i+1 may precede an i-free entry \overline{i},i in w(T), establishing the claim.Recall that m_i(w(T)) = \max _r(m_i(w(T),r)) is positive if and only if \overline{f}_i(T) \ne 0, and in this case the smallest index p for which m_i(w(T),p) = m_i(w(T)) occurs at an entry \overline{i} or i... | {
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78fae61ae6bce96e4c9f09ed40985d7691ffc622 | subsection | 23 | 57 | Crystals on shifted tableaux | As z=\overline{i+1}, it needs to be i-paired with a cell labeled \overline{i}/i after it and before x in the reading word, which can only happen in the column of x (Figure REF , left). Assume there are t cells labeled \overline{i} in the column of x. The cells left adjacent to those can only be labeled i or \overline{i... | {
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176b163e8a10113cf953a14c9a96add4d4a32938 | subsection | 24 | 57 | Crystals on shifted tableaux | This leads to a contradiction as there is no way to fill the cell to the right of \overline{i} (see Figure REF , right).
[Figure: Case L1(a), showing the cell above z can not be i+1 or \overline{i+1}.]Case L1(b): In this case we assume z \ge i+1 (since we are not in case L1(a)) and either y does not exist or y > i+1, s... | {
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dd37a2323bc2cd7ebfa3bb3b947076fcec289f51 | subsection | 25 | 57 | Crystals on shifted tableaux | We have two cases based on the possible values of z as i or \overline{i+1} (since we are not in case L2(b)). We first consider the case z=i. If the rightmost i in this row is not followed by \overline{i+1}, we are done. If it is, then it needs to be i-paired with a cell marked \overline{i} between x and itself in the r... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
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] | [
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5d7a47edfafe6f457459eb05064627a856f3b32b | subsection | 26 | 57 | Crystals on shifted tableaux | As above, the bottom \overline{i} in the column must be followed by an i, and we can continue until we find an i not followed by \overline{i+1} (Figure REF , right).In order to prove that the shifted lowering operators are invertible when the image is nonzero, we offer the following explicit rule for their partial inve... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
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c0e2cdb761595836c96f5d353a2fba23c4bae1d6 | subsection | 27 | 57 | Verification of local axioms | To prove that our operators define a normal crystal, we begin by showing that they satisfy the conditions required to be a crystal.Theorem 3.10 For any strict partition \gamma , the shifted raising and lowering operators \overline{e}_i, \overline{f}_i for i=1,2,\ldots ,r together with the usual weight map define a crys... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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8fca85dd02672beefa9ebfe2d96d3e0afa980a9c | subsection | 28 | 57 | Verification of local axioms | Furthermore, the result is that the rightmost i-free entry i becomes the leftmost i-free entry i+1, landing us in case R1(a) for the shifted raising operator, which will precisely undo the action.
[Figure: Example of i-blocked/free entries when \overline{f}_i acts by case L1(a).]Case L2(a): We assume x=\overline{i} and... | {
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5b06cc35fff0453c88baebffae3b8a3a6d1b80b2 | subsection | 29 | 57 | Verification of local axioms | Furthermore, the result is that the rightmost i-free entry \overline{i} becomes the leftmost i-free entry \overline{i+1}, landing us in case R2(a) for the shifted raising operator, which will precisely undo the action.
[Figure: Example of i-blocked/free entries when \overline{f}_i acts by case L2(a).]Case L1(b): We hav... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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4e9135fdf4690e587b0f3eaee3f69c52ed8dea5d | subsection | 30 | 57 | Verification of local axioms | Second, comparing the position of u=\overline{i+1} in w(T) with that of x=\overline{i+1} in w(\overline{f}_i(T)), u moves left past any marked entries that lie weakly between u and x in the column reading word (bottom to top along columns, from right to left). By the same analysis of ribbons, where now we count columns... | {
"cite_spans": []
} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
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ed998d8bedef718cae74c872e09ce3a483e4ef09 | subsection | 31 | 57 | Verification of local axioms | Therefore all entries i+1 remain i-blocked in \overline{f}_i(T).
[Figure: Example of i-blocked/free entries when \overline{f}_i acts by case L2(c).]Case L1(d): We assume x=i, y = \overline{i+1} or i+1, and z > \overline{i+1} in T, and, letting u denote the northeastern-most cell of the (i+1)-ribbon containing y. Note t... | {
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d49d404f2fd96828e7fdd1cf6893add52bbed420 | subsection | 32 | 57 | Verification of local axioms | To begin with, we show that the notation used in Definition REF is well-defined for the graph on semistandard shifted tableaux with edges given by the shifted crystal operators.Lemma 3.12 The graph on \mathrm {SSHT}_n(\gamma ) with edges x {\stackrel{i}{\longleftarrow }} y whenever \overline{e}_i(x) = y and x {\stackre... | {
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1f87899ce455c829ed9af49d881365feaa63cf41 | subsection | 33 | 57 | Verification of local axioms | Therefore we need only show that the traveling entries in the cases of L1(a),(c),(d) and L2(a),(c) of Definition REF do not otherwise change the number of i\pm 1-free entries. This is a case by case analysis similar to that in the proof of Theorem REF .Theorem 3.14 The shifted raising and lowering operators \overline{e... | {
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36d2cd2399c08db880885acddce95f780ddecf1c | subsection | 34 | 57 | Verification of local axioms | Finally, consider the case |i-j| = 1.For axiom A5, note that by Lemma REF , we have\nabla _i \varphi _j(T) = 0 \ \Rightarrow \ \varphi _j(\overline{f}_i(T)) = \varphi _j(T) \ \Rightarrow \ \left\lbrace
\begin{array}{rl}
\overline{f}_i \mbox{ removes a $j$-free entry $i$} & \text{if} \ j=i-1 , \\
\overline{f}_i \mbox{ ... | {
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932b07d9fa6556576dceb3cb746d5b413dc719df | subsection | 35 | 57 | Verification of local axioms | Applying \overline{f}_{i}^2 changes the newly created i-free entry i of \overline{f}_{i-1}(T) and the original rightmost i-free entry i of T to i+1 so that a final application of \overline{f}_{i-1} yields the same result, and we have \overline{f}_{i} \overline{f}_{i-1}^2 \overline{f}_{i} (T) = \overline{f}_{i-1} \overl... | {
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aa82d1997f0f5645c6dc5b76ca8483b9c2532d6e | subsection | 36 | 57 | Verification of local axioms | By Corollary REF , this gives an explicit characterization of the Schur expansion of a Schur P-polynomial.
[Figure: The Yamanouchi shifted tableaux of strict shape (4,3,1).]Corollary 3.16 For \gamma a strict partition, we haveP_{\gamma }(x_1,\ldots ,x_{n}) = \sum _{T \in \mathrm {Yam}(\gamma )} s_{\mathrm {wt}(T)}(x_1,... | {
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b701aaa16819dc47954ef8666a96f144c211603a | subsection | 37 | 57 | Verification of local axioms | Moreover, if \gamma is a strict partition such that \gamma \ne \delta _k for any k, then P_{\delta _k}(x_1,\ldots ,x_n) has more than one term in its Schur expansion.Let T \in \mathrm {Yam}(\delta _n). Note that by Lemma REF the highest row of T contains n, so all the entries on T are bounded by n. Also, as the leftmos... | {
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005cea4304678ace7825e0c8b88a083760040d47 | subsection | 38 | 57 | Verification of local axioms | In the discussion to follow, we alter the presentation in only in switching their notation from English to French to coincide with ours.The HPS reading word (p.13) is different from our hook reading word (Definition REF ). The HPS reading word of a semistandard shifted tableau first reads all marked entries up columns ... | {
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b5612ee1ae2cf55869765b894826ebcf49dd7d58 | subsection | 39 | 57 | Crystals for the quantum queer Lie superalgebra | Recently, Grantcharov, Jung, Kang, Kashiwara, and Kim developed crystal bases for the quantum queer superalgebra. In this section, we review the queer crystal theory arising from U_q(\mathfrak {q}(n)) from the combinatorial viewpoint. In §REF , we review queer crystal bases and define normal queer crystals as those ari... | {
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9eeb602386efddcb6e94c2cfd91f8c7ec92e674d | subsection | 40 | 57 | Queer crystals | Using notation and terminology from § REF , the dominant weights \Gamma ^{+} \subset \Lambda are those \lambda \in \Lambda such that \lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _{r+1} \ge 0 and \lambda _i = \lambda _{i+1} implies \lambda _i = \cdots = \lambda _{r+1} = 0. In other words, \Lambda ^{+} is to partitio... | {
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"Sami Assaf",
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05a5992236b8a7f5ed817b3f8e0fac0d815758eb | subsection | 41 | 57 | Queer crystals | Given a queer crystal \mathcal {Q} of dimension r+1, we define automorphisms S_i, for i = 1,2,\ldots ,r byS_i = \left\lbrace \begin{array}{rl}
f_i^{\mathrm {wt}(b)_i - \mathrm {wt}(b)_{i+1}} (b) & \text{if } \mathrm {wt}(b)_{i} \ge \mathrm {wt}(b)_{i+1}, \\
e_i^{\mathrm {wt}(b)_{i+1} - \mathrm {wt}(b)_i} (b) & \text{if... | {
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e888e8c246bddff04e3ee3ac48c02c7b9f0621a6 | subsection | 42 | 57 | Queer crystals | As in the classical case, we have the remarkable fact that the following combinatorial procedure on queer crystals corresponds to the tensor product of the corresponding representations.Definition 4.3 Given two queer crystals \mathcal {Q}_1 and \mathcal {Q}_2, the tensor product \mathcal {Q}_1 \otimes \mathcal {Q}_2 is... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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cbf610b6b3fef04ce79123abfd31d863e7ed8278 | subsection | 43 | 57 | Queer crystals | For example, we havef_{\overline{2}}\left(\left(\raisebox {-0.3}{\vtop {
{&\hbox{t}o 0pt{\usebox 2\hss }\vbox to 12{
\vss \hbox{t}o 12{\hss #\hss }
\vss }\cr 1\crcr }}} \otimes \raisebox {-0.3}{\vtop {
{&\hbox{t}o 0pt{\usebox 2\hss }\vbox to 12{
\vss \hbox{t}o 12{\hss #\hss }
\vss }\cr 3\crcr }}}\right)\otimes \rai... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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b0550b5305eb8d26199c0585cdf7299a3d11931d | subsection | 44 | 57 | Queer crystals for shifted tableaux | Sergeev established that the characters of irreducible tensor representations for the queer superalgebra are Schur P-functions. Grantcharov, Jung, Kang, Kashiwara, and Kim developed crystal bases for the quantum queer superalgebra and gave an explicit construction of the queer crystal on semistandard decomposition tabl... | {
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42e9bd6696765cd64d42801c4a9929b9c61cb45e | subsection | 45 | 57 | Queer crystals for shifted tableaux | There are two disjoint cases: either (i) T has a \overline{2} in its first row, or, since \overline{e}_0(T) \ne 0, (ii) the leftmost entry in the first row is 2. For case (i), \overline{e}_0 changes the \overline{2} in T to become the rightmost 1 in T^{\prime }, therefore \overline{f}_0 will act non-trivially on T^{\pr... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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77fb0e000467c281cc6a881b1e32c4e4a15d0acf | subsection | 46 | 57 | Queer crystals for shifted tableaux | Then, T^{\prime } has weight (a_k,a_1,a_2,\ldots a_{k-1},a_{k+1},a_{k+2},\ldots a_n) and outside \theta it matches T exactly. On \theta , the ith row has a_{k} cells labeled i, one cell labeled \overline{i+1} and a_i-a_{k}-1 cells labeled {i+1} for all i<k, and the kth row contains a_k cells labeled k. In particular, t... | {
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"Sami Assaf",
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fd0aef2f03c9d3a37b79dbbf19fd07aa597335af | subsection | 47 | 57 | Queer crystals for shifted tableaux | Consider, then, a shifted tableau T of degree n-1 \ge 1. By Definition REF , we havef_0 \left( \vtop {
{&\hbox{t}o 0pt{\usebox 2\hss }\vbox to 12{
\vss \hbox{t}o 12{\hss #\hss }
\vss }\cr i\crcr }} \otimes T \right) = \left\lbrace \begin{array}{ll}
\vtop {
{&\hbox{t}o 0pt{\usebox 2\hss }\vbox to 12{
\vss \hbox{t}o ... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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7a82cb24cc057d7f15fce174953237e724cd1dd2 | subsection | 48 | 57 | Queer crystals for shifted tableaux | For T \leftarrow 1, the 1 will either bump the \overline{2}, if it exists, or will bump a 2 in the first row, if it doesn't, with the result that T \leftarrow 1 will have a \overline{2} in the first row. Therefore, in all cases,f_0(T \leftarrow i) = 0 \ \mbox{whenever} \ f_0(T) = 0 \mbox{ and either } \mathrm {wt}(T)_1... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
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1f6320ce3ec1bb00d0944dc895b1e66f76a562db | subsection | 49 | 57 | Queer crystals for shifted tableaux | For example, removing the \overline{f}_0 edges and inserting edges \overline{f}_{\overline{1}} = \overline{f}_{0} and \overline{f}_{\overline{1}} = S_{1} S_{2} \overline{f}_{0} S_{2} S_{1} for the queer crystal for \mathrm {SSHT}_3(3,1) results in the crystal shown in Figure REF , which clearly has a unique highest wei... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
"Sami Assaf",
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125a1e21111efdd8974c3a735892bfb87f4812e7 | subsection | 50 | 57 | Local characterization for queer crystals | Following Stembridge , we desire a local characterization of normal queer crystals to aide in proving that a given queer crystal is, in fact, normal.To this end, a queer graph of dimension r+1 will mean a directed, colored graph \mathcal {Y} with directed edges e_i(x) {\stackrel{i}{\longrightarrow }} x {\stackrel{i}{\l... | {
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b8fd5be3ec5a5d3405bf0b83d8d03b0b6e4dc5c5 | subsection | 51 | 57 | Local characterization for queer crystals | (B1)
all 0 paths have length 1, and \varepsilon _0(x)+\varphi _0(x)=1 if and only if wt_1(x)+wt_2(x)>0;
(B2)
for every vertex x, there is at most one edge x {\color {ForestGreen} {\stackrel{0}{\longleftarrow }}} y and at most one edge x {\color {ForestGreen} {\stackrel{0}{\longrightarrow }}} z;
(B3)
assuming e_0 x is d... | {
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} | 1803.06317 | A local characterization of crystals for the quantum queer superalgebra | [
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8f14398306b47e23d610ba2739ad4a530ffb959f | subsection | 52 | 57 | Local characterization for queer crystals | This is enough to establish the statements for i\ge 3, therefore we need only look at how 0 moves interact with 1 and 2 paths.For axiom B1, assume f_0(x)=y. Then either y contains a \overline{2} or the leftmost box on its first row is labeled 2. In either case we have f_0y=0, so all 0 strings have length 1, and \vareps... | {
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b49fb50323071c9bb3bd0727e1bb7e3d861de0b7 | subsection | 53 | 57 | Local characterization for queer crystals | Otherwise, x has no cells labeled 1, so the second row contains no 2, and the leftmost 2 on the first row comes before all other 2 on the reading word. In both cases, the 2 that turns in to 1 with the e_0 move has an index \le q, so the e_0 move increases m_2 by 1, and does not change \varepsilon _2. We have \Delta _0 ... | {
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"Sami Assaf",
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b0417923b6e5545f5c1d6641258dcc2b66bae3fd | subsection | 54 | 57 | Local characterization for queer crystals | If k=1, we are done. If k>1, consider z=e_1 (e_1 (x)). By (B1), either e_0(z) or f_0(z) exists. If e_0(z) existed, by (B6) we would have \Delta _0 \varepsilon _1(e_0(x))=2, \Delta _0 \varphi _1(e_0(x))=0, contradicting the maximality of the 1 string. Then f_0(z) exists and is not equal to f_1(x) as 0 strings have lengt... | {
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"Sami Assaf",
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9aa013e1b698b7dae27d3e6ec3fffac199e4ca39 | subsection | 55 | 57 | Local characterization for queer crystals | As every regular queer graph is a regular graph when f_0 is ignored, we must have f_1 and f_2 as shown on the left side of Figure REF . Note that e_0(v)=0 as it is a highest weight, and \varphi _1(v)=1, so we must be in case k=1 of Figure REF . Therefore f_0(v) = f_1(v). Similarly, as an e_0 move from a vertex of weigh... | {
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66fe82466e713149b3f122bdb4b83328cb032bb1 | subsection | 56 | 57 | Local characterization for queer crystals | However, this is not the case. The primary difference between our characterization and that of Stembridge , is that we do not give explicit conditions for when the potential 0-edge (dashed in Figure REF ) is present or not for the f_0, f_2 components of the queer crystal. We believe that making this precise will lead t... | {
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dcf63e275de72a77aced278332dcec8e0199f94f | abstract | 0 | 14 | Abstract | Structural planning is important for producing long sentences, which is a
missing part in current language generation models. In this work, we add a
planning phase in neural machine translation to control the coarse structure of
output sentences. The model first generates some planner codes, then predicts
real output w... | {
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27248057dd2f168d19d29f24b24d7245653cc7ae | subsection | 1 | 14 | Introduction | When human speaks, it is difficult to ensure the grammatical or logical correctness without any form of planning. Linguists have found evidence through speech errors or particular behaviors that indicate speakers are planning ahead . Such planning can happen in discourse or sentence level, and sometimes we may notice i... | {
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a47e22fc51e780aab70bb5d8990acbc20e5f22bb | subsection | 2 | 14 | Learning Structural Planners | In this section, we first extract the structural annotation S_Y by simplifying the POS tags. Then we explain the code learning model for obtaining the planner codes. | {
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16c7b342bb0a501ac4b897661558c4d2df836ef8 | subsection | 3 | 14 | Structural Annotation with POS Tags | To reduce uncertainty in the decoding phase, we want a structural annotation that describes the “big picture” of the sentence. For instance, the annotation can tell whether the sentence to generate is in a “NP VP” order. The uncertainty of local structures can be efficiently solved by beam search or the NMT model itsel... | {
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a23e6cc261b15b5039b6ff80507984b51c664dbf | subsection | 4 | 14 | Code Learning | Next, we learn the planner codes C_Y to remove the uncertainty of the sentence structure S_Y when producing a translation.
For simplicity, we use the notion S and C to replace S_Y and C_Y in this section.We first compute the discrete codes C_1,..,C_N based on simplified POS tags S_1,...,S_T:\bar{h}_{\mathrm {t}} &= \ma... | {
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ac05918b803e2dc3b6e6ffb663ade1286a4f0d3a | subsection | 5 | 14 | NMT with Structural Planning | The training data of machine translation dataset is composed of (X, Y) sentence pairs. With the planner codes C_Y we obtained, our training data now becomes a list of (X, C_Y;Y) pairs. As shown in Fig. REF , we connect the planner codes and target sentence with a “\langle \text{eoc}\rangle ” token.With the modified dat... | {
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c822164205d9be5b34cb6df2a9a82bc08bfff481 | subsection | 6 | 14 | Related Work | Recently, some methods are proposed to improve the syntactic correctness of the translations. restricts the search space of the NMT decoder using the lattice produced by a Statistical Machine Translation system. takes a multi-task approach, letting the NMT model to parse a dependency tree and combine the parsing loss w... | {
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d3632444efb233b982e5ad94da134230d6000dc3 | subsection | 7 | 14 | Experiments | We evaluate our models on IWSLT 2014 German-to-English task and ASPEC Japanese-to-English task , containing 178K and 3M bilingual pairs respectively. We use Kytea to tokenize Japanese texts and moses toolkit for other languages. Using byte-pair encoding , we force the vocabulary size of each language to be 20K for IWSL... | {
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4bd235d1940bf7a9b5152491546bd6e4fb87a961 | subsection | 8 | 14 | Evaluation of Planner Codes | In the code learning model, all hidden layers have 256 hidden units. The model is trained using Nesterov's accelerated gradient (NAG) for maximum 50 epochs with a learning rate of 0.25. We test different settings of code length N and the number of code types K. The information capacity of the codes will be N \log K bit... | {
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61f2ec2d6a02acd1b627e7e3f9522dca3e46dff5 | subsection | 9 | 14 | Evaluation of NMT Models | To make a strong baseline, we use 2 layers of bi-directional LSTM encoders with 2 layers of LSTM decoders in the NMT model. The hidden layers have 256 units for IWSLT De-En task and 1000 units for ASPEC Ja-En task. We apply Key-Value Attention in the first decoder layer. Residual connection is used to combine the hidde... | {
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2a60cd7fce1f7a736bdb201c4938b7b86c9beb9d | subsection | 10 | 14 | Qualitative Analysis | Instead of letting the beam search to decide the planner codes, we can also choose the codes manually. Table REF gives an example of the candidate translations produced by the model when conditioning on different planner codes.
[Table: Example of translation results conditioned on different planner codes in Ja-En task]... | {
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62eb6add53fde3acc046d935dee9f758e40a0500 | subsection | 11 | 14 | Discussion | Instead of learning discrete codes, we can also directly predict the structural annotations (e.g. POS tags), then translate based on the predicted structure. However, as the simplified POS tags are also long sequences, the error of predicting the tags will be propagated to word generation. In our experiments, doing so ... | {
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9e0dd63ef138e01a5e5348b8fd40ffe7db950cd9 | subsection | 12 | 14 | Conclusion | In this paper, we add a planning phase in neural machine translation, which generates some planner codes to control the structure of the output sentence.
To learn the codes, we design an end-to-end neural network with a discretization bottleneck to predict the simplified POS tags of target sentences. Experiments show t... | {
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b083b067bda8616fae2017f2cf4c7d79869242d5 | subsection | 13 | 14 | Examples of Generated Translations | We show some random translation examples in ASPEC Ja-En task. The length of input sentence is limited below 10 words. The second code tends to be “\langle c1\rangle ” because it may learns to capture information for long sentences.\textbf{Input: saigo ni， shorai tenbo ni tsu ite kijutsu .}<c3> <c2> <eoc> finally , the ... | {
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d5d0db6c02cd06c21ba6e145be19d123ae770d0f | abstract | 0 | 48 | Abstract | We consider the residual B-model variation of Hodge structure of Iritani
defined by a family of toric Calabi--Yau hypersurfaces over a punctured disk $D
\setminus \{ 0\}$. It is naturally extended to a logarithmic variation of
polarized Hodge structure of Kato--Usui on the whole disk $D$. By restricting
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dfd9502a5f2b41d188b83e7403135e41c20b4e32 | subsection | 1 | 48 | Introduction | Let d be a positive integer.
Let further M be a free -module of rank d+1 and N:=(M, ) be the dual lattice.
We set M_:=M \otimes _ and N_:=N \otimes _=(M, ).
Let \Sigma \subset N_, \subset M_ be unimodular fans whose fan polytopes \subset N_, \Delta \subset M_ (i.e., the convex hulls of primitive generators of one-dimen... | {
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3c37d83d5012fff91ea399ac098827572897a38f | subsection | 2 | 48 | Introduction | We show the following in this paper.There is an injective graded ring homomorphism
\psi \colon H^\bullet _{\mathrm {amb}}Y, \hookrightarrow H^\bullet B, \iota _\ast \bigwedge ^\bullet _.
The radiance obstruction c_B \in H^1 B, \iota _\ast _ of B is given by
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2499181712be1ea717867a105439defa0a46ada8 | subsection | 3 | 48 | Introduction | We define H^\mathrm {amb}_{A, , 0} \subset H^\bullet _\mathrm {amb}(Y, ) byH^\mathrm {amb}_{A, , 0}:= 2 \pi \sqrt{-1} ^{-d} \widehat{\Gamma }_Y
\cup 2 \pi \sqrt{-1} ^{\frac{\mathrm {deg}}{2}} \mathrm {ch}(\iota ^\ast ) \in K(X_) ,where \widehat{\Gamma }_Y denotes the Gamma class of Y, and consider the lattice structure... | {
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fe052fcb1f3b6a5f8cc1ff1d94bfb05eb8efb506 | subsection | 4 | 48 | Introduction | See sc:lhs for the definition of PLH.The following triple H_^\mathrm {trop}, Q_\mathrm {trop}, _ defines a polarized logarithmic Hodge structure on the standard log point 0 :the locally constant sheaf H_^\mathrm {trop} on 0 ^\mathrm {log} whose stalk is isomorphic to H^\bullet _{\psi , } B, \iota _\ast \bigwedge ^\bull... | {
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b10102853545202bea816ce5c4ad78678caaf34d | subsection | 5 | 48 | Introduction | It is known that the valuation of the j-invariant of an elliptic curve over a non-archimedean valuation field coincides with the cycle length of the tropical elliptic curve obtained by tropicalization , .
The definition of periods for general tropical curves was given in .
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6912a9ffff6d3a45dd3d30fbfb89eeed530bfa41 | subsection | 6 | 48 | Integral affine structures with singularities | Let N be a free -module of rank d, and set N_:=N \otimes _ and \mathrm {Aff}(N_):= N_\rtimes (N).An integral affine manifold is a real topological manifold B with an atlas of coordinate charts \psi _i \colon U_i \rightarrow N_ such that all transition functions \psi _i \circ \psi _j^{-1} are contained in \mathrm {Aff}(... | {
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... | |
47d479620876ec0353be1cecaa5e860dcf8e96d6 | subsection | 7 | 48 | Integral affine structures with singularities | We call c_B \in H^1(B, \iota _\ast _) the radiance obstruction of B.The inclusion \iota \colon B_0 \hookrightarrow B induces a map \iota ^\ast \colon H^1(B, \iota _\ast _) \hookrightarrow H^1(B_0, _).
Then we can see \iota ^\ast c_B = c_{B_0} from the definitions. | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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f87e73415b02a972624dd7d58147189ccd7e67bb | subsection | 8 | 48 | Constructions of integral affine spheres | Let M be a free -module of rank d+1 and N:=(M, ) be the dual lattice.
We set M_:=M \otimes _ and N_:=N \otimes _=(M, ).
Let \check{h} \colon M_\rightarrow be a piecewise linear function that is strictly convex on a fan \subset M_ whose fan polytope \Delta \subset M_ is a reflexive polytope.
Let further \subset N_ be th... | {
"cite_spans": [
{
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"doi": "10.1007/s00208-005-0686-7",
"end": 1604,
"openalex_id": "https://openalex.org/W2102777516",
"raw": "Mark Gross, Toric degenerations and Batyrev-Borisov duality, Math. Ann. 333 (2005), no. 3, 645–688. MR 2198802",
"source_ref_id": "c9d... | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0e4ca87233a7472869811821f931538314c63710 | subsection | 9 | 48 | Constructions of integral affine spheres | Cones of the second type is called relevant.
We set(\tilde{\Sigma })&:=(\mu , \nu ) C(\mu ) \times 0 + C(\nu ) \times 1 \mathrm {\ is\ a\ relevant\ cone\ of\ } \tilde{\Sigma } ,\\
(\tilde{\Sigma })&:=\mu + \nu (\mu , \nu ) \in (\tilde{\Sigma }) .Then one hasB^{\check{h}}=\bigcup _{(\mu , \nu ) \in (\tilde{\Sigma })} \m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00208-005-0686-7",
"end": 397,
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"raw": "Mark Gross, Toric degenerations and Batyrev-Borisov duality, Math. Ann. 333 (2005), no. 3, 645–688. MR 2198802",
"source_ref_id": "c9dd... | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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fe1f279eab521cfd122d79c42bc47168beee235b | subsection | 10 | 48 | Constructions of integral affine spheres | Let further _ be the sheaf on B^{\check{h}} \setminus \Gamma (\tilde{\Sigma }) of integral tangent vectors, and _{, v_0} be its stalk at v_0.()
The parallel transport T_\gamma \colon _{, v_0} \rightarrow _{, v_0} along the loop \gamma is given byT_\gamma (n)=n+m_{\tau _1}-m_{\tau _0}, n (n_{\mu _1}-n_{\mu _0}),where we... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s1056-3911-2010-00555-3",
"end": 827,
"openalex_id": "https://openalex.org/W2049640390",
"raw": ", Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010), no. 4, 679–780. MR 2669728",
"source_ref_i... | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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2b0d1dfb994b061f386f67a80c1d02b92e42cec1 | subsection | 11 | 48 | Constructions of integral affine spheres | From the assumption, we can see that _e(\tau ) is an elementary polytope.Similarly, fix a vertex v_0 =\mu _0+\nu _0 \in (\tilde{\Sigma }) contained in \tau .
For f \in R(\tau ), the polytope \Delta _f(\tau ) of becomes the convex hull ofn_{\mu ^{\prime }}-n_{\mu _0} \in N_\mu ^{\prime }+\nu ^{\prime }=v^{\prime } \prec... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4310/jdg/1143593211",
"end": 349,
"openalex_id": "https://openalex.org/W2963639110",
"raw": "Mark Gross and Bernd Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), no. 2, 169–338. MR 2213573"... | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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da81026997825a700df410739c48cae80d723507 | subsection | 12 | 48 | Proof of th:1 | Let M be a free -module of rank d+1 and N:=(M, ) be the dual lattice.
Let further \Sigma \subset N_, \subset M_ be unimodular fans whose fan polytopes \subset N_, \Delta \subset M_ are polar dual to each other.
Consider a piecewise linear function \check{h} \colon M_\rightarrow that is strictly convex on the fan and sa... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1070/rm1978v033n02abeh002305",
"end": 1535,
"openalex_id": "https://openalex.org/W2017700406",
"raw": "V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247. MR 495499",
"source_ref_id... | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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e147675fb1275938cccf8f354b8e064884bdfd60 | subsection | 13 | 48 | Proof of th:1 | For each v \in (0) and \rho \in (1), we define an element n(v, \rho ) \in N as follows:
Let \rho _i _{i=1, \cdots , d+1} be the set of cones in (1) such that v=\bigcap _{i=1}^{d+1} \sigma (\rho _i).
We define n(v, \rho ) \in N by the following d+1 equations:m_{\rho _i}, n(v, \rho ) :=\left\lbrace \begin{array}{ll}
-1 &... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-0.05403191223740578,
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... | |
fdec089d04a32e91ca6658e8eb491d6bfa1b9f03 | subsection | 14 | 48 | Proof of th:1 | Since we have \pi (\tau _0) \prec \pi (\tau _1),
when the face \pi (\tau _1) is the intersection of facets \sigma (\rho _i) _{i=1}^{l}, the vertices \xi _1(\pi (\tau _0)), \xi _1( \pi (\tau _1)) are contained in \bigcap _{i=1}^l \sigma (\rho _i).
Hence, we havem_{\rho _i}, n(\xi _1(\pi (\tau _0)), \rho ) =m_{\rho _i}, ... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03237002342939377,
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0.03761756792664528,
0.0... | |
ac711b7aa41000329419a03ec1b1f441802b0b36 | subsection | 15 | 48 | Proof of th:1 | We will show that the coboundary of \phi (x_\rho ) coincides with \psi (x_\rho )-\psi ^{\prime }(x_\rho ).First, we check that \phi (x_\rho ) is certainly an element of \check{C}^0(, \iota _\ast _).
When the face \pi (\tau ) is the intersection of facets \sigma (\rho _i) _{i=1}^{l}, the vertices \xi _1(\pi (\tau )), \x... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.018437283113598824,
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0.0... | |
269c4d2e6b622c70f35b497cad8272dea75512b7 | subsection | 16 | 48 | Proof of th:1 | Via the map \bigwedge ^k \iota _\ast _\hookrightarrow \iota _\ast \bigwedge ^k _, the element \psi (x_{\rho _1} \cdots x_{\rho _k}) defines an element of \check{H}^k(, \iota _\ast \bigwedge ^k _).
Since \psi (x_{\rho _i}) does not depend on the choice of the map \xi _i: \rightarrow (0), the element \psi (x_{\rho _1} \c... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.02455713599920273,
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-0.03852219507098198,
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0.00947... | |
5ce80470f3291a2219fb633b4bb32df7bb0065f9 | subsection | 17 | 48 | Proof of th:1 | If \pi (\tau _{k}) \prec \sigma (\rho _k), the relation \pi (\tau _{k-1}) \prec \sigma (\rho _k) is obvious since \pi (\tau _{k-1}) \prec \pi (\tau _{k}).Next, we show that for any i_0 \in 1, \cdots , k-1, if the statement holds for i=i_0, then it also holds for i=i_0-1.
From the assumption \psi (x_{\rho _1} \cdots x_{... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.030397137627005577,
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0.005199711304157972,
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-0... | |
39dce5d63fc81766f6242bd619dde8b9e6556566 | subsection | 18 | 48 | Proof of th:1 | When \psi (x_{\rho _1} \cdots x_{\rho _k}) \ne 0, there should be a k-simplex U_{\tau _0}, \cdots , U_{\tau _k} of such that\psi (x_{\rho _1} \cdots x_{\rho _k})U_{\tau _0}, \cdots , U_{\tau _k} \ne 0.From lm:2 for i=0, i.e., \pi (\tau _0) \prec \bigcap _{j =1}^k \sigma (\rho _j), we get \bigcap _{j = 1}^k \sigma (\rho... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.006676471326500177,
0.0366252139210701,
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0.0023367649409919977,
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0... | |
684c39448be7ec8692b2df72b2f24289df1be070 | subsection | 19 | 48 | Proof of th:1 | This map will also be denoted by \psi ,\psi \colon H^\bullet X_, \rightarrow H^\bullet B, \iota _\ast \bigwedge ^\bullet _.The cohomology group H^1(B, \iota _\ast _) has the d-point function induced by the wedge product\bigwedge \colon H^1 B, \iota _\ast _^{\otimes ^d} \rightarrow H^d B, \iota _\ast \bigwedge ^d _\cong... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.010065826587378979,
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-0... | |
84151d513610e449f75dc43dc74b8fde47a5e522 | subsection | 20 | 48 | Proof of th:1 | Since the fan is unimodular, such facets \sigma (\rho _0), \sigma (\rho _{d+1}) uniquely exist.
We define e_i \in N\ (0 \le i \le d) and e_i^{\prime } \in N\ (1 \le i \le d+1) bym_{\rho _j}, e_i = \delta _{i, j}\ (0 \le j \le d), \quad m_{\rho _j}, e_i^{\prime } = \delta _{i, j}\ (1 \le j \le d+1),respectively.For the ... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04357358440756798,
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-0.015455196611583233... | |
945b96c8a5d8f45846265e265f88875285747918 | subsection | 21 | 48 | Proof of th:1 | Note that \pi (\tau _d) is a facet of B, since the dimension of \tau _d is d.First, consider the case where \pi (\tau _d)=\sigma (\rho _0) or \pi (\tau _d)=\sigma (\rho _{d+1}).
For any i, we have \pi (\tau _i) \prec \pi (\tau _d), and \pi (\tau _i) \prec \pi (\tau _d) \cap \bigcap _{j \ge i+1}^d \sigma (\rho _j) by lm... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-0.... | |
23bbde22efc444299d306807e66d9ad6d5a44a7e | subsection | 22 | 48 | Proof of th:1 | \\
\end{array}For i \ge i_0, this can be shown by lm:2 and comparing the dimensions as we did in order to see (REF ).
For i \le i_0-1, we can check this as follows:
Since (REF ) is not zero and v_1 \prec \pi (\tau _{i_0}), we haven(\xi _{i_0} \pi (\tau _{i_0}) , \rho _{i_0}) - n(\xi _{i_0} \pi (\tau _{i_0-1}) , \rho _{... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.041789788752794266,
0.008783792145550251,
-0.017262326553463936,
0.003687895368784666,
-0.0058342390693724155,
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-0.01578182727098465,
-0.03391413763165474,
... | |
e192c41d82d64bca3bedafd91987abc504a12268 | subsection | 23 | 48 | Proof of th:1 | Therefore, (REF ) is equal to\bigwedge _{i=1}^{i_0-1} -n(v_0, \rho _i) \wedge n(v_1, \rho _{i_0}) - n(v_0, \rho _{i_0}) \wedge \bigwedge _{i=i_0+1}^d -n(v_1, \rho _i) \\
=\bigwedge _{i=1}^{i_0-1} e_i \wedge e_{i_0} - e_{i_0}^{\prime } \wedge \bigwedge _{i=i_0+1}^{d} e_i^{\prime } ,where e_i, e_i^{\prime } \in N are th... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.0007281119469553232,
0.022689245641231537,
-0.032164715230464935,
-0.03539949655532837,
-0.0006799526163376868,
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0.03250040113925934,
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0.029265617951750755,
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-0.011222555302083492,
-0.021468574181199074... | |
544f3bd3dc234df9dcf6da4ae8a01d3b02def156 | subsection | 24 | 48 | Proof of th:1 | Therefore, on U_{v_0}, (REF ) is equal to\bigwedge _{i=1}^{i_0-1} e_i \wedge -s_{i_0} e_0 \wedge \bigwedge _{i=i_0+1}^{d} e_i+s_i e_0 &=-s_{i_0} \bigwedge _{i=1}^{i_0-1} e_i \wedge \sum _{i=1}^d -e_i \wedge \bigwedge _{i=i_0+1}^{d} e_i \\
&=s_{i_0} \bigwedge _{i=1}^{d} e_i.In either case, \pi (\tau _d)=\sigma (\rho _0)... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
"Yuto Yamamoto"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.004641605541110039,
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0.039756860584020615,
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0.022822817787528038,
-0.006907105911523104,
-0.0033029010519385338,
0.01583561860024929,
... | |
6c7eda7d7481b02872315ea73016b695c78a6aad | subsection | 25 | 48 | Proof of th:1 | This coincides with the affine length between n_{\mu _0} and n_{\mu _1}.
On the other hand, we also haven_{\mu _1}-n_{\mu _0}= - \sum _{i=1}^{d+1} e_i^{\prime } +\sum _{i=0}^{d} e_i =2 -\sum _{i=1}^d s_i e_0.From (REF ) and (REF ), we also get
m_{\rho _{d+1}}= -m_{\rho _0}+\sum _{i=1}^d s_i m_{\rho _i}.
Since the point... | {
"cite_spans": []
} | 1806.04239 | Periods of tropical Calabi--Yau hypersurfaces | [
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1ec1d1f2206c4e3e26767c10a0d4b10c5fbabebc | subsection | 26 | 48 | Proof of th:1 | Hence, we haveY \cdot D_{\rho _1} \cdots D_{\rho _{i_0}} \cdots D_{\rho _d}
&= - \sum _{\rho \notin \rho _0, \cdots \rho _{d} } a_\rho Y \cdot D_{\rho _1} \cdots D_{\rho } \cdots D_{\rho _d} \\
&= - \sum _{\rho \notin \rho _0, \cdots \rho _{d} } a_\rho \psi (x_{\rho _1}) \wedge \cdots \wedge \psi (x_{\rho }) \wedge \c... | {
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b43a643915263b1cfa560122cf92cf50e1ab8b89 | subsection | 27 | 48 | Proof of th:1 | By the Poincare duality for X_, we obtain x \cup Y =0, i.e., x \in \mathrm {Ann}(Y ).
Hence, we get \psi ^{-1} T \subset \mathrm {Ann}(Y ).Next, we check \psi ^{-1} T \supset \mathrm {Ann}(Y ).
We will show\psi ^{-1} T \cap H^{2i} X_, =\mathrm {Ann}(Y ) \cap H^{2i} X_,for any 0 \le i \le d.
When 0 \le i \le d/2, we kno... | {
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7cd67577870ae8464dd82f94b5d945d43203e928 | subsection | 28 | 48 | Proof of th:1 | Therefore, we obtain (REF ) also for d/2 < i \le d.Therefore, the map \psi (REF ) defines the injective graded ring homomorphism\psi \colon H^\bullet _\mathrm {amb} Y, \hookrightarrow \bigoplus _{i=0}^d H^i B, \iota _\ast \bigwedge ^i _/ T \hookrightarrow \bigoplus _{i=0}^d H^i B, \iota _\ast \bigwedge ^i _of th:1.1.In... | {
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99aa952636ec2297ada45f790e6be5f146cdef50 | subsection | 29 | 48 | Proof of th:1 | Therefore, we get\sum _{\rho \in (1)} \check{h}(m_\rho ) \psi (D_\rho ) (U_{\tau _0}, U_{\tau _1}) &=
\sum _{\rho \in (1)} \check{h}(m_\rho ) n \xi _1 \pi (\tau _1) , \rho - n \xi _1 \pi (\tau _{0}) , \rho \\
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b0a619047328707838e91760742bd87421f02e38 | subsection | 30 | 48 | PLH on the standard log point | We recall the definition of polarized logarithmic Hodge structures (PLH) on the standard log point.
We refer the reader to for the definition of PLH on general fs logarithmic analytic spaces.The standard log point is the point 0 equipped with the logarithmic structure given byM_{0 }:= ^\times \oplus \rightarrow _{0}=, ... | {
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c3b710d0e7ea5dbfea0797eaa07ef6b5f7daf870 | subsection | 31 | 48 | PLH on the standard log point | (Positivity) Let y \in 0 ^\mathrm {log} and s \in _{-\mathrm {alg}}_{0 ,y}^\mathrm {log}, .
Let further F(s)=F^p(s) _p be the decreasing filtration on the -vector space H_{, y}:=\otimes _H_{,y} defined by
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7d6d63e0eeaf64ff802b2e3eef8622b5e3a7b1a1 | subsection | 32 | 48 | Extension of variations of polarized Hodge structure | We briefly recall how variations of polarized Hodge structure on a punctured disk extend to logarithmic variations of polarized Hodge structure on the disk.
This subsection is based on .Consider a variation of polarized Hodge structure (H_, Q, ) on the small punctured disk D_\varepsilon \setminus 0 , where D_\varepsilo... | {
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99a5046922ad052af8c640b4117a287813e95d74 | subsection | 33 | 48 | Extension of variations of polarized Hodge structure | By Schmid's nilpotent orbit theorem , we know that this extends to a holomorphic map \Psi \colon D_\varepsilon \rightarrow \check{D}.The logarithmic structure on D_\varepsilon associated with the divisor 0 \subset D_\varepsilon is given by\alpha \colon M_{D_\varepsilon } :=\bigcup _{n \ge 0} _{D_\varepsilon }^\times \c... | {
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cd8fa427a566cb220594a1630df38de91ef58107 | subsection | 34 | 48 | Extension of variations of polarized Hodge structure | \xi ^{-1} 1 \otimes H_0 \right|_{q_0}, \quad v \mapsto \xi ^{-1} (1 \otimes v)of the stalks at q_0 preserves the actions of \pi _1(D_\varepsilon ^\mathrm {log}) \cong \pi _1(D_\varepsilon \setminus 0 ) (See the proof of ).
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cd135da30543cc6c501f13ea6e36ef87e7a4905b | subsection | 35 | 48 | Mirror symmetry for Calabi–Yau hypersurfaces | We refer the reader to for the details of the context of this chapter.
There is also a review in the case of K3 hypersurfaces in .
For A-model Hodge structure, see also .Let M be a free -module of rank d+1 and N:=(M, ) be the dual lattice.
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73f6129f10271fdb3538135bfbbce2d1e8f73622 | subsection | 36 | 48 | Mirror symmetry for Calabi–Yau hypersurfaces | Consider the residue part of H^d(\check{Y}_\alpha , ) defined byH^d_\mathrm {res}(\check{Y}_\alpha , ) := \mathrm {Im} \mathrm {Res} \colon H^0 X_\Sigma , \Omega ^{d+1}_{X_\Sigma } (\ast \check{Y}_\alpha ) \rightarrow H^d(\check{Y}_\alpha , ) ,where H^0 X_\Sigma , \Omega ^{d+1}_{X_\Sigma } (\ast \check{Y}_\alpha ) is t... | {
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0d721fa276f41b5e774a410b5c1869f5d0c23c6a | subsection | 37 | 48 | Mirror symmetry for Calabi–Yau hypersurfaces | The vanishing cycle integral structure H_{B, }^\mathrm {vc} \subset H_{B, } on the residual B-model VHS is the image of H_{d+1} ()^{d+1}, _\alpha ^\circ ; by the map \mathrm {VC}.
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58dbe3bb5bfa66f59f06aa6e2270acb3436d7707 | subsection | 38 | 48 | Mirror symmetry for Calabi–Yau hypersurfaces | The ambient \widehat{\Gamma }-integral structure on the ambient A-model VHS is the local subsystem H_{A, }^\mathrm {amb} \subset H_{A, }:=\nabla ^A defined byH^\mathrm {amb}_{A, }:=
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0341d4c953f859fce8840e5d40d20589e0d96d12 | subsection | 39 | 48 | Tropical periods and logarithmic Hodge theory | Let M be a free -module of rank d+1 and N:=(M, ) be the dual lattice.
Let further \Sigma \subset N_, \subset M_ be unimodular fans whose fan polytopes \check{\Delta } \subset N_, \Delta \subset M_ are polar dual to each other.
Let K := t be the convergent Laurent series field, equipped with the standard non-archimedean... | {
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"Yuto Yamamoto"
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872a5d9f10653b21571c8ba7103b85e6013a1a29 | subsection | 40 | 48 | Tropical periods and logarithmic Hodge theory | We write it also as (_B, \nabla ^B, H_{B,}^\mathrm {vc}, _B^\bullet , Q_B) in the following.We fix a point q_0 \in D_\varepsilon \setminus 0 and set\check{}&:=F^p _{p=1}^d F^p \in \mathrm {Gr}(r_p, H_{B, , q_0}^\mathrm {vc} ), F^1 \supset \cdots \supset F^d , \\
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4adbe71bd9c88fc47e91048d9be0df608cac9eed | subsection | 41 | 48 | Tropical periods and logarithmic Hodge theory | Here we replace the real number \varepsilon with a smaller one again if necessary.
We will show the theorem by using this isomorphism.First, we compute the monodromy of H_{B,}^\mathrm {vc} \cong H_{A, }^\mathrm {amb}.
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4301bacba52864da0799918f0ed95aca28788207 | subsection | 42 | 48 | Tropical periods and logarithmic Hodge theory | (See also .)
Hence, it turns out that the monodromy of flat sections of _{A} is given by the cup product of\prod _{i=1}^r \exp -2 \pi \sqrt{-1} p_i \sum _{m \in A} b_{m,i} \mathrm {val}(k_m) &= \exp -2 \pi \sqrt{-1} \sum _{i=1}^r p_i \sum _{m \in A} b_{m,i} \mathrm {val}(k_m) \\
&=\exp -2 \pi \sqrt{-1} \sum _{m \in A} ... | {
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16097ed6db565c93b24408bebb6de1d35603d2c0 | subsection | 43 | 48 | Tropical periods and logarithmic Hodge theory | For a section s \in H^0 _R, \pi ^\ast H_A , we define a holomorphic section \varphi (s) \in H^0 _R, \pi ^\ast _{A} by\varphi (s)(z):=\exp (-z N) \cdot s(z).This section \varphi (s) is invariant under z \mapsto z+1, and descends to the section \tilde{\varphi }(s) \in H^0 D_\varepsilon \setminus 0, _{A} defined by\tilde{... | {
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9b68dbb96232bc93e564555a84f53af359f65652 | subsection | 44 | 48 | Tropical periods and logarithmic Hodge theory | The map \tilde{\varphi } induces an identification\pi ^\ast H_{A, }^\mathrm {amb} \cong \widetilde{}_{A, }(0) = H^\mathrm {amb}_{A, , 0}.This identification preservers the pairing Q_A.Now we haveH_{B, , q_0}^\mathrm {vc} \cong \pi ^\ast H_{B, }^\mathrm {vc} \cong \pi ^\ast H_{A, }^\mathrm {amb} \cong H_{A, , 0}^\mathrm... | {
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80961c12261c3c00491d0a75f4ca4e7c8fec3f76 | subsection | 45 | 48 | Tropical periods and logarithmic Hodge theory | By the isomorphism \psi \otimes _\colon H^\bullet _\mathrm {amb} Y, \rightarrow H^\bullet _\psi B, \iota _\ast \bigwedge ^\bullet _, the _{0 }^\mathrm {log}-modules of (REF ) are isomorphic to_{0 }^\mathrm {log} \otimes _H_^\xrightarrow{} _{0}^\mathrm {log} \otimes _H^\bullet _{\psi } B, \iota _\ast \bigwedge ^\bullet ... | {
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"raw": ", Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010), no. 4, 679–780. MR 2669728",
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bb3614b57f63cd92de976a3964151fd7d3d67751 | subsection | 46 | 48 | Tropical periods and logarithmic Hodge theory | The family V_q _q which we consider in this paper is a one-parameter subfamily of this family.For the canonical family over Aq , the period integral of the holomorphic volume form over a d-cycle \beta constructed from a tropical 1-cycle \beta _\in H_1\check{B}, \iota _\ast _ is computed in .
Here H_1\check{B}, \iota _\... | {
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05fe400d01e5eb0eb46b965a0a12c890306675a7 | subsection | 47 | 48 | Tropical periods and logarithmic Hodge theory | The positivity condition (REF ) of PLH is equivalent to eq:HR2.When p=d, q=0 or p=0, q=d, the left hand side of (REF ) is equal to&\sqrt{-1} ^{d} Q_\exp -2 \pi \sqrt{-1} z \cdot c_B , \exp -2 \pi \sqrt{-1} \overline{z} \cdot c_B \\
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eca9a357f6a58565adc0ed3e26b87faf1c5e29eb | abstract | 0 | 39 | Abstract | The explosion of data in recent years has generated an increasing need for
new analysis techniques in order to extract knowledge from massive datasets.
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automatized methods have recently gathered great popularity, even though those
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"Antonio D'Isanto",
"Stefano Cavuoti",
"Fabian Gieseke",
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800098bdd5ef58d594c0d4ff4b118889ed0d825f | subsection | 1 | 39 | Introduction | In recent years, astronomy has experienced a true explosion in the amount and complexity of the available data.
The new generation of digital surveys is opening a new era for astronomical research, characterized by the necessity to analyse data-sets that fall into the Tera-scale and Peta-scale regime.
This is leading t... | {
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