Symplectic cohomology rings of affine varieties in the topological limit

Ganatra, Sheel; Pomerleano, Daniel

doi:10.1007/s00039-020-00529-1

Cited by 26 publications

(52 citation statements)

References 52 publications

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“…It is expected by several methods that c 0,3 = 6, while c 1,3 , c 2,3 vanish for minimal Chern number reasons. It follows that W L,3 = (W L ) 3 − 6; compare with [36,Remark 5.39].…”

Section: 2mentioning

confidence: 86%

“…Related work. The recent works of Ganatra and Pomerleano [35,36], while they do not discuss Lagrangian enumerative geometry, use related methods and ideas. In order to get control on the broken curves arising in the proof of Theorem 1.1, we use a collection of existing tools surrounding the theory of symplectic cohomology.…”

Section: 2mentioning

confidence: 99%

“…The structure constants c i,k need not be trivial, and are of big interest as they are expected to encode certain closed log Gromov-Witten invariants of X, see e.g. [37], [36,Remark 5.39].…”

Section: 2mentioning

confidence: 99%

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From symplectic cohomology to Lagrangian enumerative geometry

Tonkonog

2019

Advances in Mathematics

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We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials.We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains. arXiv:1711.03292v3 [math.SG] 27 Jun 2019As a general prediction, rings of regular functions are mirror to degree zero symplectic cohomology. (This was proven by Ganatra and Pomerleano [35] for the complement to the anticanonical divisor itself, see Section 2.) So one expects:These rings can be complicated (much bigger than C[r]), but they carry a distinguished element, the restriction of W :It is natural to ask what is the symplectic counterpart, or the mirror, of this element. This is answered by Theorem 1.1, which can be summarised in the language of mirror symmetry as follows:The Borman-Sheridan class BS ∈ SH 0 (M ) is mirror to W |M .

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“…It is expected by several methods that c 0,3 = 6, while c 1,3 , c 2,3 vanish for minimal Chern number reasons. It follows that W L,3 = (W L ) 3 − 6; compare with [36,Remark 5.39].…”

Section: 2mentioning

confidence: 86%

Section: 2mentioning

confidence: 99%

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From symplectic cohomology to Lagrangian enumerative geometry

Tonkonog

2019

Advances in Mathematics

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“…Allowing for

normalΣ

to be a normal crossings divisor would be a significant improvement in generality, but is currently beyond our reach. Important progress has been made in that direction, including .…”

Section: Introductionmentioning

confidence: 99%

Symplectic homology of complements of smooth divisors

Diogo

Lisi

2019

Journal of Topology

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If (X, ω) is a closed symplectic manifold, and Σ is a smooth symplectic submanifold Poincaré dual to a positive multiple of ω, then X \ Σ can be completed to a Liouville manifold (W, dλ).Under monotonicity assumptions on X and on Σ, we construct a chain complex whose homology computes the symplectic homology of W . We show that the differential is given in terms of Morse contributions, Gromov-Witten invariants of X relative to Σ and Gromov-Witten invariants of Σ.We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to Σ. Contents

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“…For example, the works of Gross–Hacking–Keel and Abouzaid–Auroux–Katzarkov suggest that the

A_{\infty}

‐algebras

F_{M}

and

W_{M}

are not Koszul dual when

M

is a log Calabi–Yau surface or an affine conic bundle over

{false(C^{*} false)}^{n - 1}

. In the case when

M

is the complement of an anticanonical divisor in some smooth projective variety, see for computations of

{italicSH}^{0} false(M false)

using Morse–Bott spectral sequences. In general, the

A_{\infty}

‐algebras

F_{M}

and

W_{M}

will mean the endomorphism algebras of a set of split‐generators in the compact and wrapped Fukaya categories, respectively.…”

Section: Introductionmentioning

confidence: 99%

Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Let M be a Liouville 6‐manifold which is the smooth fiber of a Lefschetz fibration on C4 constructed by suspending a Lefschetz fibration on C3. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category F(M) and the wrapped Fukaya category W(M) are related through A∞‐Koszul duality, by identifying them with cyclic and Calabi–Yau completions of the same quiver algebra. This implies the split‐generation of the compact Fukaya category F(M) by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi‐dilations in the sense of Seidel–Solomon are obtained.

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Symplectic cohomology rings of affine varieties in the topological limit

Cited by 26 publications

References 52 publications

From symplectic cohomology to Lagrangian enumerative geometry

From symplectic cohomology to Lagrangian enumerative geometry

Symplectic homology of complements of smooth divisors

Koszul duality via suspending Lefschetz fibrations

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