Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold

Ekholm, Tobias; Ng, Lenhard

doi:10.4310/jdg/1433975484

Cited by 33 publications

(82 citation statements)

References 34 publications

(99 reference statements)

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“…While this strategy has been carried out in interesting examples (see e.g. [BEE2,BEE1,EN,EL]) and remains a remarkably effective tool for identifying instances when SH * (X) vanishes, general computations can run into two difficult issues: computing the relevant open-string invariant (a process that in some cases can be combinatorialized), and computing its Hochschild invariants (which is known to be a difficult algebraic problem except in special circumstances). Also, realizing an affine variety as an explicit handlebody or calculating a skeleton may be challenging in practice, though there is some work to systematize the process [CM1].…”

Section: Comparison With Other Methods For Computing Symplectic Cohommentioning

confidence: 99%

Symplectic cohomology rings of affine varieties in the topological limit

Ganatra

Pomerleano

2020

Geom. Funct. Anal.

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We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification (M, D) of X. We exhibit a broad class of pairs (M, D) (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be described topologically. Sample applications include: (a) a complete topological description of the symplectic cohomology ring of the complement, in any projective M , of the union of sufficiently many generic ample divisors whose homology classes span a rank one subspace, (b) complete additive and partial multiplicative computations of degree zero symplectic cohomology rings of many log Calabi-Yau varieties, and (c) a proof in many cases that symplectic cohomology is finitely generated as a ring. A key technical ingredient in our results is a logarithmic version of the PSS morphism, introduced in our earlier work [GP].

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Section: Comparison With Other Methods For Computing Symplectic Cohommentioning

confidence: 99%

Symplectic cohomology rings of affine varieties in the topological limit

Ganatra

Pomerleano

2020

Geom. Funct. Anal.

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“…In order to give a combinatorial description of the Chekanov-Eliashberg's Legendrian DG-algebra ( [7,13]) associated with a Legendrian link L ⊂ # k (S 1 × S 2 ) given in Gompf's standard form, Ekholm and Ng [12] described a procedure called resolution (analogous to the resolution of a Legendrian in S 3 in [20]). We give an example in Figure 1 and refer the readers to the original references [15,12] for precise definitions. We next recall the description of the Legendrian DG-algebra, here denoted CE * (L), that Ekholm and Ng provide coming from a resolution diagram.…”

Section: Introductionmentioning

confidence: 99%

“…We next recall the description of the Legendrian DG-algebra, here denoted CE * (L), that Ekholm and Ng provide coming from a resolution diagram. All our complexes are cohomological, thus we reverse the gradings from [12].…”

Section: Introductionmentioning

confidence: 99%

Fukaya categories of plumbings and multiplicative preprojective algebras

Etgü

Lekili

2019

Quantum Topol.

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Given an arbitrary graph Γ and non-negative integers g v for each vertex v of Γ, let X Γ be the Weinstein 4-manifold obtained by plumbing copies of T * Σ v according to this graph, where Σ v is a surface of genus g v . We compute the wrapped Fukaya category of X Γ (with bulk parameters) using Legendrian surgery extending our previous work [14] where it was assumed that g v = 0 for all v and Γ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw [8]. Along the way, we find a smaller model for the internal DG-algebra of Ekholm-Ng [12] associated to 1-handles in the Legendrian surgery presentation of Weinstein 4-manifolds which might be of independent interest. Legendrian DG-algebra on a subcritical Weinstein 4-manifoldRecall that any Weinstein 4-manifold X has a Weinstein handlebody decomposition consisting of 0-, 1-and 2-handles. Assuming X is connected, one can find a handlebody decomposition with a unique 0-handle. Furthermore, Weinstein 1-handles can be attached

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“…However, [14] and [15] recently outlined a proof of a surgery formula for symplectic (co)homology. Combining this with the very recent [26], one obtains a purely combinatorial description of symplectic cohomology of any 4-dimensional Weinstein manifold. (In the absence of 1-handles and when the coefficient field is Z 2 , one had [18] as a precursor to [26]).…”

Section: Now Eqn (1) Becomes An Eilenberg-moore Equivalence (Of Dgamentioning

confidence: 99%

“…Combining this with the very recent [26], one obtains a purely combinatorial description of symplectic cohomology of any 4-dimensional Weinstein manifold. (In the absence of 1-handles and when the coefficient field is Z 2 , one had [18] as a precursor to [26]). This combinatorial description is in general still highly complicated.…”

Section: Now Eqn (1) Becomes An Eilenberg-moore Equivalence (Of Dgamentioning

confidence: 99%

Koszul duality patterns in Floer theory

Etgü

Lekili

2017

Geom. Topol.

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We study symplectic invariants of the open symplectic manifolds X Γ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree Γ. For any tree Γ, we calculate (DG-)algebra models of the Fukaya category F(X Γ ) of closed exact Lagrangians in X Γ and the wrapped Fukaya category W(X Γ ). When Γ is a Dynkin tree of type A n or D n (and conjecturally also for E 6 , E 7 , E 8 ), we prove that these models for the Fukaya category F(X Γ ) and W(X Γ ) are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of X Γ for Γ = A n , D n , based on the Legendrian surgery formula of [14]. In the case that Γ is non-Dynkin, we obtain a spectral sequence that converges to symplectic cohomology whose E 2 -page is given by the Hochschild cohomology of the preprojective algebra associated to the corresponding Γ. It is conjectured that this spectral sequence is degenerate if the ground field has characteristic zero.

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Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold

Cited by 33 publications

References 34 publications

Symplectic cohomology rings of affine varieties in the topological limit

Symplectic cohomology rings of affine varieties in the topological limit

Fukaya categories of plumbings and multiplicative preprojective algebras

Koszul duality patterns in Floer theory

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