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Super-rigidity of certain skeleta using relative symplectic cohomology
TL;DR: In this paper, the authors used relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds.
Abstract: This article uses relative symplectic cohomology, recently studied by the second author, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi-Yau symplectic manifold $M$ whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of $M$ exhibits strong rigidity properties akin to super-heavy subsets of Entov-Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behaviour of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.
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14 Nov 2022
TL;DR: In this paper , the relative symplectic cohomology sheaf is computed on the bases of nodal Lagrangian torus fibrations on four dimensional symplectic cluster manifolds.
Abstract: We compute the relative symplectic cohomology sheaf in degree 0 on the bases of nodal Lagrangian torus fibrations on four dimensional symplectic cluster manifolds. We show that it is the pushforward of the structure sheaf of a certain rigid analytic manifold under a non-archimedean torus fibration. The rigid analytic manifold is constructed in a canonical way from the relative SH sheaf and is referred as the closed string mirror. The construction relies on computing relative SH for local models by applying general axiomatic properties rather than ad hoc analysis of holomorphic curves. These axiomatic properties include previously established ones such as the Mayer-Vietoris property and locality for complete embeddings; and new ones such as the Hartogs property and the holomorphic volume form preservation property of wall crossing in relative SH.
4 citations
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TL;DR: In this article, it was shown that the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the complement of a simple normal crossing divisor.
Abstract: We prove that under certain conditions, the quantum cohomology of a positively monotone compact symplectic manifold is a deformation of the symplectic cohomology of the complement of a simple normal crossings divisor. We also prove rigidity results for the skeleton of the divisor complement.
3 citations
31 May 2022
TL;DR: In this paper , the authors used relative symplectic cohomology to detect heavy sets, with the help of index bounded contact forms, by using the Viterbo restriction map, which establishes a relation between SH-heaviness and heaviness.
Abstract: We use relative symplectic cohomology to detect heavy sets, with the help of index bounded contact forms. The main result is a criterion for heaviness by using the Viterbo restriction map, which establishes a relation between SH-heaviness and heaviness. This partly answers a conjecture of Dickstein-Ganor-Polterovich-Zapolsky in the current setting. Along the proof we also construct spectral invariants of the relative symplectic cohomology for index bounded Liouville domains.
1 citations
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TL;DR: In this article, the authors adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections.
Abstract: We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.
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TL;DR: In this paper, the displacement energy of Lagrangian 3-spheres in a symplectic 6-manifold is estimated by estimating the displacement of a one-parameter family $L_{\lambda}$ of tori near the sphere, which is motivated by the change of open Gromov Witten invariants under the conifold transition.
Abstract: We estimate the displacement energy of Lagrangian 3-spheres in a symplectic 6-manifold $X$, by estimating the displacement energy of a one-parameter family $L_{\lambda}$ of Lagrangian tori near the sphere. The proof establishes a new version of Lagrangian Floer theory with cylinder corrections, which is motivated by the change of open Gromov-Witten invariants under the conifold transition. We also make observations and computations on the classical Floer theory by using symplectic sum formula and Welschinger invariants.
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Book•
30 Sep 2009
TL;DR: The Floer cohomology as mentioned in this paper is an algebra associated to a Lagrangian submanifold and is a homotopy equivalence of $A_\infty$ algebras.
Abstract: Part I Introduction Review: Floer cohomology The $A_\infty$ algebra associated to a Lagrangian submanifold Homotopy equivalence of $A_\infty$ algebras Homotopy equivalence of $A_\infty$ bimodules Spectral sequences Part II Transversality Orientation Appendices Bibliography Index.
992 citations
"Super-rigidity of certain skeleta u..." refers background in this paper
...[7], presents a powerful machine for detecting non-displaceability....
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...2 can be upgraded to Lagrangian submanifolds admitting bounding cochains (in the sense of [7]) with nonzero self-Floer cohomology....
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Book•
01 Jan 2002
TL;DR: The homotopy principle in symplectic geometry: Symplectic and contact structures on open manifold and contact structure on closed manifolds Embeddings into symplectic or contact manifolds Microflexibility and holonomic $\mathcal{R}$-approximation First applications of microflexibility Microflexible $mathfrak{U})-invariant differential relations Further applications to symplectic geometrical geometry Convex integration: One-dimensional convex integration Homotopy principles for ample differential relations Directed immersions and embeddings First order linear differential operators
Abstract: Intrigue Holonomic approximation: Jets and holonomy Thom transversality theorem Holonomic approximation Applications Differential relations and Gromov's $h$-principle: Differential relations Homotopy principle Open Diff $V$-invariant differential relations Applications to closed manifolds The homotopy principle in symplectic geometry: Symplectic and contact basics Symplectic and contact structures on open manifolds Symplectic and contact structures on closed manifolds Embeddings into symplectic and contact manifolds Microflexibility and holonomic $\mathcal{R}$-approximation First applications of microflexibility Microflexible $\mathfrak{U}$-invariant differential relations Further applications to symplectic geometry Convex integration: One-dimensional convex integration Homotopy principle for ample differential relations Directed immersions and embeddings First order linear differential operators Nash-Kuiper theorem Bibliography Index.
397 citations
Additional excerpts
...17 (Eliashberg-Mishachev [4], McLean [12])....
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TL;DR: In this article, the authors constructed an A∞-structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A ∞-homomorphism realizing the restriction to a Liouville subdomain.
Abstract: Liouville domains are a special type of symplectic manifolds with boundary (they have an everywhere defined Liouville flow, pointing outwards along the boundary). Symplectic cohomology for Liouville domains was introduced by Cieliebak–Floer–Hofer–Wysocki and Viterbo. The latter constructed a restriction (or transfer) map associated to an embedding of one Liouville domain into another. In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called “wrapped Floer cohomology”. We construct an A∞–structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A∞–homomorphism realizing the restriction to a Liouville subdomain. The construction of the A∞–structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations.
338 citations
"Super-rigidity of certain skeleta u..." refers background or methods in this paper
...which is a chain complex over Λ≥0 generated by the 1-chords of H, and the differential counts Floer solutions u : R× [0, 1]→M with boundary mapping to L with weights...
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...Another ingredient is the integrated maximum principle of [1]....
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...For a monotone homotopy H : [0, 1] × [0, 1] × M → R with H|0 = H0 and H|1 = H1, and a generic [0, 1]t × [0, 1]s-dependent almost complex structure J :...
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...We also fix a Morse function on [0, 1] with critical values at the end points once and for all, which turns a [0, 1]-family of Hamiltonians to a (−∞,∞)-family which is then used to write down the Floer equations....
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...For a Hamiltonian H : [0, 1]×M → R such that φH(L) is transverse to L, and a generic [0, 1]-dependent almost complex structure J := {Jt}t∈[0,1] with Jt(x) = JL(x) for every x ∈ L and t ∈ [0, 1], we obtain CF ∗(L, , JL, H, J,Λ≥0) := ( ⊕...
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TL;DR: In this paper, it was shown that I-adic completion is exact on a much larger class of modules than might be expected from the key role played by the Artin-Rees lemma and that deviations from exactness can be computed in terms of torsion products.
215 citations
"Super-rigidity of certain skeleta u..." refers background in this paper
...1 from [13] the reader might question why we had to introduce E instead of using lim −→ (C)⊗ lim −→ (C ′)....
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...1 of [13] would have helped us by removing this restriction....
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...8During the revision, we noticed the existence of the relevant paper [13]....
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TL;DR: In this paper, a link between the theory of quasi-state and topological measures has been established, which can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced by Yong-Geun Oh.
Abstract: We establish a link between symplectic topology and a recently emerg\-ed branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures) In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds This work is a part of a joint project with Paul Biran
175 citations