This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR singularities, a research direction originated from a fundamental and classical paper of E. Bishop. Earlier works along these lines include those by many prominent mathematicians working both on complex analysis and geometry. We prove that a compact smooth (or, real analytic) real codimension two submanifold $M$, that is contained in the boundary of a smoothly bounded strongly pseudoconvex domain, with a natural and necessary condition called CR non-minimal condition at CR points and with two elliptic CR singular points bounds a smooth-up-to-boundary (real analytic-up-to-boundary, respectively) Levi-flat hypersurface $widehat{M}$. This answers a well-known question left open from the work of Dolbeault-Tomassini-Zaitsev, or a generalized version of a problem already asked by Bishop in 1965. Our study here reveals an intricate interaction of several complex analysis with other fields such as symplectic geometry and foliation theory.
{"title":"Bounding smooth Levi-flat hypersurfaces in a Stein manifold","authors":"Hanlong Fang, Xiaojun Huang, Wanke Yin, Zhengyi Zhou","doi":"arxiv-2409.08470","DOIUrl":"https://doi.org/arxiv-2409.08470","url":null,"abstract":"This paper is concerned with the problem of constructing a smooth Levi-flat\u0000hypersurface locally or globally attached to a real codimension two submanifold\u0000in $mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR\u0000singularities, a research direction originated from a fundamental and classical\u0000paper of E. Bishop. Earlier works along these lines include those by many\u0000prominent mathematicians working both on complex analysis and geometry. We\u0000prove that a compact smooth (or, real analytic) real codimension two\u0000submanifold $M$, that is contained in the boundary of a smoothly bounded\u0000strongly pseudoconvex domain, with a natural and necessary condition called CR\u0000non-minimal condition at CR points and with two elliptic CR singular points\u0000bounds a smooth-up-to-boundary (real analytic-up-to-boundary, respectively)\u0000Levi-flat hypersurface $widehat{M}$. This answers a well-known question left\u0000open from the work of Dolbeault-Tomassini-Zaitsev, or a generalized version of\u0000a problem already asked by Bishop in 1965. Our study here reveals an intricate\u0000interaction of several complex analysis with other fields such as symplectic\u0000geometry and foliation theory.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256771","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 there exist contact isotopies of the standard contact sphere whose time-1 maps do not have any translated points which are optimally close to the identity in the Shelukhin-Hofer distance. This proves the sharpness of a theorem of Shelukhin on the existence of translated points for contact isotopies of Liouville fillable contact manifolds with small enough Shelukhin-Hofer norm.
{"title":"On the rigidity of translated points","authors":"Dylan Cant, Jakob Hedicke","doi":"arxiv-2409.08962","DOIUrl":"https://doi.org/arxiv-2409.08962","url":null,"abstract":"We show that there exist contact isotopies of the standard contact sphere\u0000whose time-1 maps do not have any translated points which are optimally close\u0000to the identity in the Shelukhin-Hofer distance. This proves the sharpness of a\u0000theorem of Shelukhin on the existence of translated points for contact\u0000isotopies of Liouville fillable contact manifolds with small enough\u0000Shelukhin-Hofer norm.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142256773","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}
A submanifold of the standard symplectic space determines a partially defined, multi-valued symplectic map, the outer symplectic billiard correspondence. Two points are in this correspondence if the midpoint of the segment connecting them is on the submanifold, and this segment is symplectically orthogonal to the tangent space of the submanifold at its midpoint. This is a far-reaching generalization of the outer billiard map in the plane; the particular cases, when the submanifold is a closed convex hypersurface or a Lagrangian submanifold, were considered earlier. Using a variational approach, we establish the existence of odd-periodic orbits of the outer symplectic billiard correspondence. On the other hand, we give examples of curves in 4-space which do not admit 4-periodic orbits at all. If the submanifold satisfies 49 pages, certain conditions (which are always satisfied if its dimension is at least half of the ambient dimension) we prove the existence of two $n$-reflection orbits connecting two transverse affine Lagrangian subspaces for every $ngeq1$. In addition, for every immersed closed submanifold, the number of single outer symplectic billiard ``shots" from one affine Lagrangian subspace to another is no less than the number of critical points of a smooth function on this submanifold. We study, in detail, the behavior of this correspondence when the submanifold is a curve or a Lagrangian submanifold. For Lagrangian submanifolds in 4-dimensional space we present a criterion for the outer symplectic billiard correspondence to be an actual map. We show, in every dimension, that if a Lagrangian submanifold has a cubic generating function, then the outer symplectic billiard correspondence is completely integrable in the Liouville sense.
{"title":"Outer symplectic billiards","authors":"Peter Albers, Ana Chavez Caliz, Serge Tabachnikov","doi":"arxiv-2409.07990","DOIUrl":"https://doi.org/arxiv-2409.07990","url":null,"abstract":"A submanifold of the standard symplectic space determines a partially\u0000defined, multi-valued symplectic map, the outer symplectic billiard\u0000correspondence. Two points are in this correspondence if the midpoint of the\u0000segment connecting them is on the submanifold, and this segment is\u0000symplectically orthogonal to the tangent space of the submanifold at its\u0000midpoint. This is a far-reaching generalization of the outer billiard map in\u0000the plane; the particular cases, when the submanifold is a closed convex\u0000hypersurface or a Lagrangian submanifold, were considered earlier. Using a variational approach, we establish the existence of odd-periodic\u0000orbits of the outer symplectic billiard correspondence. On the other hand, we\u0000give examples of curves in 4-space which do not admit 4-periodic orbits at all.\u0000If the submanifold satisfies 49 pages, certain conditions (which are always\u0000satisfied if its dimension is at least half of the ambient dimension) we prove\u0000the existence of two $n$-reflection orbits connecting two transverse affine\u0000Lagrangian subspaces for every $ngeq1$. In addition, for every immersed closed\u0000submanifold, the number of single outer symplectic billiard ``shots\" from one\u0000affine Lagrangian subspace to another is no less than the number of critical\u0000points of a smooth function on this submanifold. We study, in detail, the behavior of this correspondence when the submanifold\u0000is a curve or a Lagrangian submanifold. For Lagrangian submanifolds in\u00004-dimensional space we present a criterion for the outer symplectic billiard\u0000correspondence to be an actual map. We show, in every dimension, that if a\u0000Lagrangian submanifold has a cubic generating function, then the outer\u0000symplectic billiard correspondence is completely integrable in the Liouville\u0000sense.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202194","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}
Let $L$ be a monotone Lagrangian torus inside a compact symplectic manifold $X$, with superpotential $W_L$. We show that a geometrically-defined closed-open map induces a decomposition of the quantum cohomology $operatorname{QH}^*(X)$ into a product, where one factor is the localisation of the Jacobian ring $operatorname{Jac} W_L$ at the set of isolated critical points of $W_L$. The proof involves describing the summands of the Fukaya category corresponding to this factor -- verifying the expectations of mirror symmetry -- and establishing an automatic generation criterion in the style of Ganatra and Sanda, which may be of independent interest. We apply our results to understanding the structure of quantum cohomology and to constraining the possible superpotentials of monotone tori
{"title":"Quantum cohomology and Fukaya summands from monotone Lagrangian tori","authors":"Jack Smith","doi":"arxiv-2409.07922","DOIUrl":"https://doi.org/arxiv-2409.07922","url":null,"abstract":"Let $L$ be a monotone Lagrangian torus inside a compact symplectic manifold\u0000$X$, with superpotential $W_L$. We show that a geometrically-defined\u0000closed-open map induces a decomposition of the quantum cohomology\u0000$operatorname{QH}^*(X)$ into a product, where one factor is the localisation\u0000of the Jacobian ring $operatorname{Jac} W_L$ at the set of isolated critical\u0000points of $W_L$. The proof involves describing the summands of the Fukaya\u0000category corresponding to this factor -- verifying the expectations of mirror\u0000symmetry -- and establishing an automatic generation criterion in the style of\u0000Ganatra and Sanda, which may be of independent interest. We apply our results\u0000to understanding the structure of quantum cohomology and to constraining the\u0000possible superpotentials of monotone tori","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202195","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 central fiber of a Gross-Siebert type toric degeneration is known to satisfy homological mirror symmetry: its category of coherent sheaves is equivalent to the wrapped Fukaya category of a certain exact symplectic manifold. Here we show that, in the Calabi-Yau case, the images of line bundles are represented by Lagrangian spheres.
{"title":"Toric mirror monodromies and Lagrangian spheres","authors":"Vivek Shende","doi":"arxiv-2409.08261","DOIUrl":"https://doi.org/arxiv-2409.08261","url":null,"abstract":"The central fiber of a Gross-Siebert type toric degeneration is known to\u0000satisfy homological mirror symmetry: its category of coherent sheaves is\u0000equivalent to the wrapped Fukaya category of a certain exact symplectic\u0000manifold. Here we show that, in the Calabi-Yau case, the images of line bundles\u0000are represented by Lagrangian spheres.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202193","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}
Given a mirror pair of a symplectic manifold $X$ and a Landau-Ginzburg potential $W$, we are interested in the problem whether the quantum cohomology of $X$ and the Jacobian algebra of $W$ are isomorphic. Since those can be equipped with Frobenius algebra structures, we might ask whether they are isomorphic as Frobenius algebras. We show that the Kodaira-Spencer map gives a Frobenius algebra isomorphism for elliptic orbispheres, under the Floer theoretic modification of the residue pairing.
{"title":"Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras","authors":"Sangwook Lee","doi":"arxiv-2409.07814","DOIUrl":"https://doi.org/arxiv-2409.07814","url":null,"abstract":"Given a mirror pair of a symplectic manifold $X$ and a Landau-Ginzburg\u0000potential $W$, we are interested in the problem whether the quantum cohomology\u0000of $X$ and the Jacobian algebra of $W$ are isomorphic. Since those can be\u0000equipped with Frobenius algebra structures, we might ask whether they are\u0000isomorphic as Frobenius algebras. We show that the Kodaira-Spencer map gives a\u0000Frobenius algebra isomorphism for elliptic orbispheres, under the Floer\u0000theoretic modification of the residue pairing.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227776","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 establish a geometric criterion for local microlocal holonomies to be globally regular on the moduli space of Lagrangian fillings. This local-to-global regularity result holds for arbitrary Legendrian links and it is a key input for the study of cluster structures on such moduli spaces. Specifically, we construct regular functions on derived moduli stacks of sheaves with Legendrian microsupport by studying the Hochschild homology of the associated dg-categories via relative Lagrangian skeleta. In this construction, a key geometric result is that local microlocal merodromies along positive relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for these dg-categories.
{"title":"Positive microlocal holonomies are globally regular","authors":"Roger Casals, Wenyuan Li","doi":"arxiv-2409.07435","DOIUrl":"https://doi.org/arxiv-2409.07435","url":null,"abstract":"We establish a geometric criterion for local microlocal holonomies to be\u0000globally regular on the moduli space of Lagrangian fillings. This\u0000local-to-global regularity result holds for arbitrary Legendrian links and it\u0000is a key input for the study of cluster structures on such moduli spaces.\u0000Specifically, we construct regular functions on derived moduli stacks of\u0000sheaves with Legendrian microsupport by studying the Hochschild homology of the\u0000associated dg-categories via relative Lagrangian skeleta. In this construction,\u0000a key geometric result is that local microlocal merodromies along positive\u0000relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for\u0000these dg-categories.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202196","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 study multiplicity-free representations of nilpotent Lie groups over a quasi-symmetric Siegel domain, with a focus on certain two-step nilpotent Lie groups. We provide necessary and sufficient conditions for the multiplicity-freeness property. Specifically, we establish the equivalence between the disjointness of irreducible unitary representations realized over the domain, the multiplicity-freeness of the unitary representation on the space of $L^2$ holomorphic functions, and the coisotropicity of the group action.
{"title":"Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains","authors":"Koichi Arashi","doi":"arxiv-2409.05507","DOIUrl":"https://doi.org/arxiv-2409.05507","url":null,"abstract":"We study multiplicity-free representations of nilpotent Lie groups over a\u0000quasi-symmetric Siegel domain, with a focus on certain two-step nilpotent Lie\u0000groups. We provide necessary and sufficient conditions for the\u0000multiplicity-freeness property. Specifically, we establish the equivalence\u0000between the disjointness of irreducible unitary representations realized over\u0000the domain, the multiplicity-freeness of the unitary representation on the\u0000space of $L^2$ holomorphic functions, and the coisotropicity of the group\u0000action.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202199","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 define an SFT-type invariant for Legendrian knots in the standard contact $mathbb{R}^3$. The invariant is a deformation of the Chekanov-Eliashberg differential graded algebra. The differential consists of a part that counts index zero $J$-holomorphic disks with up to two positive punctures, annuli with one positive puncture, and a string topological part. We describe the invariant and demonstrate its invariance combinatorially from the Lagrangian knot projection, and compute some simple examples where the deformation is non-vanishing.
{"title":"Extension of Chekanov-Eliashberg algebra using annuli","authors":"Milica Dukic","doi":"arxiv-2409.05856","DOIUrl":"https://doi.org/arxiv-2409.05856","url":null,"abstract":"We define an SFT-type invariant for Legendrian knots in the standard contact\u0000$mathbb{R}^3$. The invariant is a deformation of the Chekanov-Eliashberg\u0000differential graded algebra. The differential consists of a part that counts\u0000index zero $J$-holomorphic disks with up to two positive punctures, annuli with\u0000one positive puncture, and a string topological part. We describe the invariant\u0000and demonstrate its invariance combinatorially from the Lagrangian knot\u0000projection, and compute some simple examples where the deformation is\u0000non-vanishing.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"401 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202197","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 paper develops a theory of symplectic reduction in the infinite-dimensional setting, covering both the regular and singular case. Extending the classical work of Marsden, Weinstein, Sjamaar and Lerman, we address challenges unique to infinite dimensions, such as the failure of the Darboux theorem and the absence of the Marle-Guillemin-Sternberg normal form. Our novel approach centers on a normal form of only the momentum map, for which we utilize new local normal form theorems for smooth equivariant maps in the infinite-dimensional setting. This normal form is then used to formulate the theory of singular symplectic reduction in infinite dimensions. We apply our results to important examples like the Yang-Mills equation and the Teichm"uller space over a Riemann surface.
{"title":"Symplectic Reduction in Infinite Dimensions","authors":"Tobias Diez, Gerd Rudolph","doi":"arxiv-2409.05829","DOIUrl":"https://doi.org/arxiv-2409.05829","url":null,"abstract":"This paper develops a theory of symplectic reduction in the\u0000infinite-dimensional setting, covering both the regular and singular case.\u0000Extending the classical work of Marsden, Weinstein, Sjamaar and Lerman, we\u0000address challenges unique to infinite dimensions, such as the failure of the\u0000Darboux theorem and the absence of the Marle-Guillemin-Sternberg normal form.\u0000Our novel approach centers on a normal form of only the momentum map, for which\u0000we utilize new local normal form theorems for smooth equivariant maps in the\u0000infinite-dimensional setting. This normal form is then used to formulate the\u0000theory of singular symplectic reduction in infinite dimensions. We apply our\u0000results to important examples like the Yang-Mills equation and the\u0000Teichm\"uller space over a Riemann surface.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202198","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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