This paper establishes closed-string mirror symmetry for all log Calabi-Yau surfaces with generic parameters, where the exceptional divisor are sufficiently small. We demonstrate that blowing down a $(-1)$-divisor removes a single geometric critical point, ensuring that the resulting potential remains a Morse function. Additionally, we show that the critical values are distinct, which implies that the quantum cohomology $QH^{ast}(X)$ is semi-simple.
{"title":"Closed-String Mirror Symmetry for Log Calabi-Yau Surfaces","authors":"Hyunbin Kim","doi":"arxiv-2408.02592","DOIUrl":"https://doi.org/arxiv-2408.02592","url":null,"abstract":"This paper establishes closed-string mirror symmetry for all log Calabi-Yau\u0000surfaces with generic parameters, where the exceptional divisor are\u0000sufficiently small. We demonstrate that blowing down a $(-1)$-divisor removes a\u0000single geometric critical point, ensuring that the resulting potential remains\u0000a Morse function. Additionally, we show that the critical values are distinct,\u0000which implies that the quantum cohomology $QH^{ast}(X)$ is semi-simple.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a stable homotopy type invariant for any Legendrian submanifold in a jet bundle equipped with a linear-at-infinity generating family. We show that this spectrum lifts the generating family homology groups. When the generating family extends to a generating family for an embedded Lagrangian filling, we lift the Seidel isomorphism to the spectrum level. As applications, we establish topological constraints on Lagrangian fillings arising from generating families, algebraic constraints on whether generating families admit fillings, and lower bounds on how many fiber dimensions are needed to construct a generating family for a Legendrian.
{"title":"A stable homotopy invariant for Legendrians with generating families","authors":"Hiro Lee Tanaka, Lisa Traynor","doi":"arxiv-2408.01587","DOIUrl":"https://doi.org/arxiv-2408.01587","url":null,"abstract":"We construct a stable homotopy type invariant for any Legendrian submanifold\u0000in a jet bundle equipped with a linear-at-infinity generating family. We show\u0000that this spectrum lifts the generating family homology groups. When the\u0000generating family extends to a generating family for an embedded Lagrangian\u0000filling, we lift the Seidel isomorphism to the spectrum level. As applications,\u0000we establish topological constraints on Lagrangian fillings arising from\u0000generating families, algebraic constraints on whether generating families admit\u0000fillings, and lower bounds on how many fiber dimensions are needed to construct\u0000a generating family for a Legendrian.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By slicing the Heegaard diagram for a given $3$-manifold in a particular way, it is possible to construct $mathcal{A}_{infty}$-bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. In this paper, we use the graphical calculus for $mathcal{A}_{infty}$-structures introduced in arXiv:2009.05222v3 to construct Koszul dual $mathcal{A}_{infty}$ algebras $mathcal{A}$ and $mathcal{B}$ for a particular star-shaped class of slice. Using $mathcal{A}_{infty}$-bimodules over $mathcal{A}$ and $mathcal{B}$, we then verify the Koszul duality relation.
{"title":"Koszul Duality for star-shaped partial Heegaard diagrams","authors":"Isabella Khan","doi":"arxiv-2408.01564","DOIUrl":"https://doi.org/arxiv-2408.01564","url":null,"abstract":"By slicing the Heegaard diagram for a given $3$-manifold in a particular way,\u0000it is possible to construct $mathcal{A}_{infty}$-bimodules, the tensor\u0000product of which retrieves the Heegaard Floer homology of the original\u00003-manifold. The first step in this is to construct algebras corresponding to\u0000the individual slices. In this paper, we use the graphical calculus for\u0000$mathcal{A}_{infty}$-structures introduced in arXiv:2009.05222v3 to construct\u0000Koszul dual $mathcal{A}_{infty}$ algebras $mathcal{A}$ and $mathcal{B}$ for\u0000a particular star-shaped class of slice. Using $mathcal{A}_{infty}$-bimodules\u0000over $mathcal{A}$ and $mathcal{B}$, we then verify the Koszul duality\u0000relation.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we prove that the space of Floer Hessians has infinitely many connected components.
在这篇文章中，我们证明了 Floer Hessians 空间具有无限多的连接成分。
{"title":"Growth of eigenvalues of Floer Hessians","authors":"Urs Frauenfelder, Joa Weber","doi":"arxiv-2408.00269","DOIUrl":"https://doi.org/arxiv-2408.00269","url":null,"abstract":"In this article we prove that the space of Floer Hessians has infinitely many\u0000connected components.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the category of linearly topologized vector spaces over discrete fields constitutes the correct framework for algebraic structures on Floer homologies with field coefficients. Our case in point is the Poincar'e duality theorem for Rabinowitz Floer homology. We prove that Rabinowitz Floer homology is a locally linearly compact vector space in the sense of Lefschetz, or, equivalently, a Tate vector space in the sense of Beilinson-Feigin-Mazur. Poincar'e duality and the graded Frobenius algebra structure on Rabinowitz Floer homology then hold in the topological sense. Along the way, we develop in a largely self-contained manner the theory of linearly topologized vector spaces, with special emphasis on duality and completed tensor products, complementing results of Beilinson-Drinfeld, Beilinson, Rojas, Positselski, and Esposito-Penkov.
{"title":"Rabinowitz Floer homology as a Tate vector space","authors":"Kai Cieliebak, Alexandru Oancea","doi":"arxiv-2407.21741","DOIUrl":"https://doi.org/arxiv-2407.21741","url":null,"abstract":"We show that the category of linearly topologized vector spaces over discrete\u0000fields constitutes the correct framework for algebraic structures on Floer\u0000homologies with field coefficients. Our case in point is the Poincar'e duality\u0000theorem for Rabinowitz Floer homology. We prove that Rabinowitz Floer homology\u0000is a locally linearly compact vector space in the sense of Lefschetz, or,\u0000equivalently, a Tate vector space in the sense of Beilinson-Feigin-Mazur.\u0000Poincar'e duality and the graded Frobenius algebra structure on Rabinowitz\u0000Floer homology then hold in the topological sense. Along the way, we develop in\u0000a largely self-contained manner the theory of linearly topologized vector\u0000spaces, with special emphasis on duality and completed tensor products,\u0000complementing results of Beilinson-Drinfeld, Beilinson, Rojas, Positselski, and\u0000Esposito-Penkov.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We resolve the periodic square peg problem using a simple Lagrangian Floer homology argument. Inscribed squares are interpreted as intersections between two non-displaceable Lagrangian sub-manifolds of a symplectic 4-torus.
{"title":"A Solution to the Periodic Square Peg Problem","authors":"Cole Hugelmeyer","doi":"arxiv-2407.20412","DOIUrl":"https://doi.org/arxiv-2407.20412","url":null,"abstract":"We resolve the periodic square peg problem using a simple Lagrangian Floer\u0000homology argument. Inscribed squares are interpreted as intersections between\u0000two non-displaceable Lagrangian sub-manifolds of a symplectic 4-torus.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"124 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures that have been introduced in mathematical shape analysis. We also derive Hamiltonian vector fields for several classical Hamiltonian functions with respect to these new symplectic structures.
{"title":"Symplectic structures on the space of space curves","authors":"Martin Bauer, Sadashige Ishida, Peter W. Michor","doi":"arxiv-2407.19908","DOIUrl":"https://doi.org/arxiv-2407.19908","url":null,"abstract":"We present symplectic structures on the shape space of unparameterized space\u0000curves that generalize the classical Marsden-Weinstein structure. Our method\u0000integrates the Liouville 1-form of the Marsden-Weinstein structure with\u0000Riemannian structures that have been introduced in mathematical shape analysis.\u0000We also derive Hamiltonian vector fields for several classical Hamiltonian\u0000functions with respect to these new symplectic structures.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the recent paper arXiv:2405.13258, the first author of this note proved that if a billiard in a convex domain in $mathbb{R}^n$ is simultaneously projective and Minkowski, then it is the standard Euclidean billiard in an appropriate Euclidean structure. The proof was quite complicated and required high smoothness. Here we present a direct simple proof of this result which works in $C^1$-smoothness. In addition we prove the semi-local and local versions of the result
{"title":"If a Minkowski billiard is projective, it is the standard billiard","authors":"Alexey Glutsyuk, Vladimir S. Matveev","doi":"arxiv-2407.20159","DOIUrl":"https://doi.org/arxiv-2407.20159","url":null,"abstract":"In the recent paper arXiv:2405.13258, the first author of this note proved\u0000that if a billiard in a convex domain in $mathbb{R}^n$ is simultaneously\u0000projective and Minkowski, then it is the standard Euclidean billiard in an\u0000appropriate Euclidean structure. The proof was quite complicated and required\u0000high smoothness. Here we present a direct simple proof of this result which\u0000works in $C^1$-smoothness. In addition we prove the semi-local and local\u0000versions of the result","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau 3-folds of Picard number one, up to a linear term and finitely many degree, without assuming the conjecture of Bayer-Macr`i-Toda. Furthermore, we prove an effective vanishing theorem for surface-counting invariants of Calabi-Yau 4-folds of Picard number one. We also apply our techniques to study low-degree curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is a combination of classical techniques and the wall-crossing of weak stability conditions on derived categories, and works for any projective 3-fold with at worst isolated singularities over any algebraically closed field.
{"title":"Castelnuovo bound for curves in projective 3-folds","authors":"Zhiyu Liu","doi":"arxiv-2407.20161","DOIUrl":"https://doi.org/arxiv-2407.20161","url":null,"abstract":"The Castelnuovo bound conjecture, which is proposed by physicists, predicts\u0000an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau\u00003-folds of Picard number one. Previously, it is only known for a few cases and\u0000all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau\u00003-folds of Picard number one, up to a linear term and finitely many degree,\u0000without assuming the conjecture of Bayer-Macr`i-Toda. Furthermore, we prove an\u0000effective vanishing theorem for surface-counting invariants of Calabi-Yau\u00004-folds of Picard number one. We also apply our techniques to study low-degree\u0000curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds\u0000for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is\u0000a combination of classical techniques and the wall-crossing of weak stability\u0000conditions on derived categories, and works for any projective 3-fold with at\u0000worst isolated singularities over any algebraically closed field.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study introduces a new districting approach using the US Postal Service network to measure community connectivity. We combine Topological Data Analysis with Markov Chain Monte Carlo methods to assess district boundaries' impact on community integrity. Using Iowa as a case study, we generate and refine districting plans using KMeans clustering and stochastic rebalancing. Our method produces plans with fewer cut edges and more compact shapes than the official Iowa plan under relaxed conditions. The low likelihood of finding plans as disruptive as the official one suggests potential inefficiencies in existing boundaries. Gaussian Mixture Model analysis reveals three distinct distributions in the districting landscape. This framework offers a more accurate reflection of community interactions for fairer political representation.
{"title":"The Traveling Mailman: Topological Optimization Methods for User-Centric Redistricting","authors":"Nelson A. Colón Vargas","doi":"arxiv-2407.19535","DOIUrl":"https://doi.org/arxiv-2407.19535","url":null,"abstract":"This study introduces a new districting approach using the US Postal Service\u0000network to measure community connectivity. We combine Topological Data Analysis\u0000with Markov Chain Monte Carlo methods to assess district boundaries' impact on\u0000community integrity. Using Iowa as a case study, we generate and refine\u0000districting plans using KMeans clustering and stochastic rebalancing. Our\u0000method produces plans with fewer cut edges and more compact shapes than the\u0000official Iowa plan under relaxed conditions. The low likelihood of finding\u0000plans as disruptive as the official one suggests potential inefficiencies in\u0000existing boundaries. Gaussian Mixture Model analysis reveals three distinct\u0000distributions in the districting landscape. This framework offers a more\u0000accurate reflection of community interactions for fairer political\u0000representation.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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