Pub Date : 2023-01-01DOI: 10.4310/jsg.2023.v21.n2.a4
Ilaria Di Dedda
We prove that the Fukaya-Seidel categories of a certain family of Lefschetz fibrations on $mathbb{C}^2$ are equivalent to the perfect derived categories of Auslander algebras of Dynkin type $mathbb{A}$. We give an explicit equivalence between these categories and the partially wrapped Fukaya categories considered by Dyckerhoff-Jasso-Lekili. We provide a complete description of the Milnor fibre of such fibrations.
{"title":"Realising perfect derived categories of Auslander algebras of type $mathbb{A}$ as Fukaya–Seidel categories","authors":"Ilaria Di Dedda","doi":"10.4310/jsg.2023.v21.n2.a4","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n2.a4","url":null,"abstract":"We prove that the Fukaya-Seidel categories of a certain family of Lefschetz fibrations on $mathbb{C}^2$ are equivalent to the perfect derived categories of Auslander algebras of Dynkin type $mathbb{A}$. We give an explicit equivalence between these categories and the partially wrapped Fukaya categories considered by Dyckerhoff-Jasso-Lekili. We provide a complete description of the Milnor fibre of such fibrations.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.4310/jsg.2021.v19.n6.a1
Benjamin Delarue, Pablo Ramacher
We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden–Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.
{"title":"Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions","authors":"Benjamin Delarue, Pablo Ramacher","doi":"10.4310/jsg.2021.v19.n6.a1","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n6.a1","url":null,"abstract":"We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden–Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138506810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/jsg.2022.v20.n4.a3
D. Joksimović, F. Ziltener
{"title":"Generating systems and representability for symplectic capacities","authors":"D. Joksimović, F. Ziltener","doi":"10.4310/jsg.2022.v20.n4.a3","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n4.a3","url":null,"abstract":"","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85749566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/jsg.2022.v20.n4.a2
Chi Hong Chow, N. Leung
{"title":"Twisted cyclic group actions on Fukaya categories and mirror symmetry","authors":"Chi Hong Chow, N. Leung","doi":"10.4310/jsg.2022.v20.n4.a2","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n4.a2","url":null,"abstract":"","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76642891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-18DOI: 10.4310/jsg.2023.v21.n1.a4
Ipsita Datta
In this paper, we present new obstructions to the existence of Lagrangian cobordisms in $mathbb{R}^4$ that depend only on the enriched knot diagrams of the boundary knots or links, using holomorphic curve techniques. We define enriched knot diagrams for generic smooth links. The existence of Lagrangian cobordisms gives a well-defined transitive relation on equivalence classes of enriched knot diagrams that is a strict partial order when restricted to exact enriched knot diagrams To establish obstructions we study $1$-dimensional moduli spaces of holomorphic disks with corners that have boundary on Lagrangian tangles - an appropriate immersed Lagrangian closely related to embedded Lagrangian cobordisms. We adapt existing techniques to prove compactness and transversality, and compute dimensions of these moduli spaces. We produce obstructions as a consequence of characterizing all boundary points of such moduli spaces. We use these obstructions to recover and extend results about ``growing"and ``shrinking"Lagrangian slices. We hope that this investigation will open up new directions in studying Lagrangian surfaces in $mathbb{R}^4$.
{"title":"Lagrangian cobordisms between enriched knot diagrams","authors":"Ipsita Datta","doi":"10.4310/jsg.2023.v21.n1.a4","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n1.a4","url":null,"abstract":"In this paper, we present new obstructions to the existence of Lagrangian cobordisms in $mathbb{R}^4$ that depend only on the enriched knot diagrams of the boundary knots or links, using holomorphic curve techniques. We define enriched knot diagrams for generic smooth links. The existence of Lagrangian cobordisms gives a well-defined transitive relation on equivalence classes of enriched knot diagrams that is a strict partial order when restricted to exact enriched knot diagrams To establish obstructions we study $1$-dimensional moduli spaces of holomorphic disks with corners that have boundary on Lagrangian tangles - an appropriate immersed Lagrangian closely related to embedded Lagrangian cobordisms. We adapt existing techniques to prove compactness and transversality, and compute dimensions of these moduli spaces. We produce obstructions as a consequence of characterizing all boundary points of such moduli spaces. We use these obstructions to recover and extend results about ``growing\"and ``shrinking\"Lagrangian slices. We hope that this investigation will open up new directions in studying Lagrangian surfaces in $mathbb{R}^4$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86990306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-08DOI: 10.4310/jsg.2021.v19.n4.a2
Russell Avdek
The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a contact manifold $(M,xi)$ gives an exact—and in many cases, Weinstein—cobordism whose concave boundary is $(M,xi)$ and whose convex boundary is the surgered manifold. These cobordisms are used to establish the existence of “fillability” and “non-vanishing contact homology” monoids in symplectomorphism groups of Liouville domains, study the symplectic fillability of a family of contact manifolds which fiber over the circle, associate cobordisms to certain branched coverings of contact manifolds, and construct exact symplectic cobordisms that do not admit Weinstein structures. The Liouville connect sum generalizes the Weinstein handle attachment and is used to extend the definition of contact $(1/k)$-surgery along Legendrian knots in contact $3$-manifolds to contact $(1/k)$-surgery along Legendrian spheres in contact manifolds of arbitrary dimension. We use contact surgery to construct exotic contact structures on $5$- and $13$-dimensional spheres after establishing that $S^2$ and $S^6$ are the only spheres along which generalized Dehn twists smoothly square to the identity mapping. The exoticity of these contact structures implies that Dehn twists along $S^2$ and $S^6$ do not symplectically square to the identity, generalizing a theorem of Seidel. A similar argument shows that the $(2n + 1)$-dimensional contact manifold determined by an open book whose page is $(T^ast S^n , -lambda_{can})$ and whose monodromy is any negative power of a symplectic Dehn twist is not exactly fillable.
{"title":"Liouville hypersurfaces and connect sum cobordisms","authors":"Russell Avdek","doi":"10.4310/jsg.2021.v19.n4.a2","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n4.a2","url":null,"abstract":"The purpose of this paper is to introduce <i>Liouville hypersurfaces</i> in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the <i>Liouville connect sum</i>. Performing this operation on a contact manifold $(M,xi)$ gives an exact—and in many cases, Weinstein—cobordism whose concave boundary is $(M,xi)$ and whose convex boundary is the surgered manifold. These cobordisms are used to establish the existence of “fillability” and “non-vanishing contact homology” monoids in symplectomorphism groups of Liouville domains, study the symplectic fillability of a family of contact manifolds which fiber over the circle, associate cobordisms to certain branched coverings of contact manifolds, and construct exact symplectic cobordisms that do not admit Weinstein structures. The Liouville connect sum generalizes the Weinstein handle attachment and is used to extend the definition of contact $(1/k)$-surgery along Legendrian knots in contact $3$-manifolds to contact $(1/k)$-surgery along Legendrian spheres in contact manifolds of arbitrary dimension. We use contact surgery to construct exotic contact structures on $5$- and $13$-dimensional spheres after establishing that $S^2$ and $S^6$ are the only spheres along which generalized Dehn twists smoothly square to the identity mapping. The exoticity of these contact structures implies that Dehn twists along $S^2$ and $S^6$ do not symplectically square to the identity, generalizing a theorem of Seidel. A similar argument shows that the $(2n + 1)$-dimensional contact manifold determined by an open book whose page is $(T^ast S^n , -lambda_{can})$ and whose monodromy is any negative power of a symplectic Dehn twist is not exactly fillable.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-28DOI: 10.4310/JSG.2023.v21.n1.a3
Yannis Bahni
Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces.
{"title":"First steps in twisted Rabinowitz–Floer homology","authors":"Yannis Bahni","doi":"10.4310/JSG.2023.v21.n1.a3","DOIUrl":"https://doi.org/10.4310/JSG.2023.v21.n1.a3","url":null,"abstract":"Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81239680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-12DOI: 10.4310/jsg.2022.v20.n5.a1
Daniel Álvarez-Gavela, David Darrow
For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $pi_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.
{"title":"Caustics of Lagrangian homotopy spheres with stably trivial Gauss map","authors":"Daniel Álvarez-Gavela, David Darrow","doi":"10.4310/jsg.2022.v20.n5.a1","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n5.a1","url":null,"abstract":"For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $pi_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89264581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-20DOI: 10.4310/jsg.2022.v20.n5.a3
Yin Li
Let $Msubsetmathbb{C}^{n+1}$ be a smooth affine hypersurface defined by the equation $xy+p(z_1,cdots,z_{n-1})=1$, where $p$ is a Brieskorn-Pham polynomial and $ngeq2$. We prove that if $Lsubset M$ is an orientable exact Lagrangian submanifold, then $L$ does not admit a Riemannian metric with non-positive sectional curvature. The key point of the proof is to establish a version of homological mirror symmetry for the wrapped Fukaya category of $M$, from which the finite-dimensionality of the symplectic cohomology group $mathit{SH}^0(M)$ follows by a Hochschild cohomology computation.
{"title":"Nonexistence of exact Lagrangian tori in affine conic bundles over $mathbb{C}^n$","authors":"Yin Li","doi":"10.4310/jsg.2022.v20.n5.a3","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n5.a3","url":null,"abstract":"Let $Msubsetmathbb{C}^{n+1}$ be a smooth affine hypersurface defined by the equation $xy+p(z_1,cdots,z_{n-1})=1$, where $p$ is a Brieskorn-Pham polynomial and $ngeq2$. We prove that if $Lsubset M$ is an orientable exact Lagrangian submanifold, then $L$ does not admit a Riemannian metric with non-positive sectional curvature. The key point of the proof is to establish a version of homological mirror symmetry for the wrapped Fukaya category of $M$, from which the finite-dimensionality of the symplectic cohomology group $mathit{SH}^0(M)$ follows by a Hochschild cohomology computation.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79840806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-12DOI: 10.4310/JSG.2022.v20.n6.a3
M. Cengiz, Ferit Ozturk
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, which is called a real structure. A real contact 3-manifold is a real 3-manifold with a contact distribution that is antisymmetric with respect to the real structure. We show that every real 3-manifold can be obtained via surgery along invariant knots starting from the standard real $S^3$ and that this operation can be performed in the contact setting too. Using this result we prove that any real 3-manifold admits a real contact structure. As a corollary we show that any oriented overtwisted contact structure on an integer homology real 3-sphere can be isotoped to be real. Finally we give construction examples on $S^1times S^2$ and lens spaces. For instance on every lens space there exists a unique real structure that acts on each Heegaard torus as hyperellipic involution. We show that any tight contact structure on any lens space is real with respect to that real structure.
{"title":"Every real $3$-manifold is real contact","authors":"M. Cengiz, Ferit Ozturk","doi":"10.4310/JSG.2022.v20.n6.a3","DOIUrl":"https://doi.org/10.4310/JSG.2022.v20.n6.a3","url":null,"abstract":"A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, which is called a real structure. A real contact 3-manifold is a real 3-manifold with a contact distribution that is antisymmetric with respect to the real structure. We show that every real 3-manifold can be obtained via surgery along invariant knots starting from the standard real $S^3$ and that this operation can be performed in the contact setting too. Using this result we prove that any real 3-manifold admits a real contact structure. As a corollary we show that any oriented overtwisted contact structure on an integer homology real 3-sphere can be isotoped to be real. Finally we give construction examples on $S^1times S^2$ and lens spaces. For instance on every lens space there exists a unique real structure that acts on each Heegaard torus as hyperellipic involution. We show that any tight contact structure on any lens space is real with respect to that real structure.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72465567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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