Koszul duality patterns in Floer theory

Etgü, Tolga; Lekili, Yankı

doi:10.2140/gt.2017.21.3313

Cited by 37 publications

(65 citation statements)

References 64 publications

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“…Together with Seidel–Solomon's inductive argument based on Lefschetz fibration techniques , this implies that the total space

M_{p, q, r}

also admits a quasi‐dilation. However, this argument does not apply when

K = Z / 2

, since in this case Koszul duality does not hold for

D_{4}

Milnor fibers (see [, Theorem 14] for details).…”

Section: Split‐generation and Quasi‐dilationsmentioning

confidence: 99%

“…Note that

F_{M}

and

W_{M}

are equipped with augmentations

ε_{scriptF} : {scriptF}_{M} \to k

and

ε_{scriptW} : {scriptW}_{M} \to k

, where

ε_{F}

is defined by projecting to

k ≅ {scriptF}_{M}^{0}

, while

ε_{W}

is induced from the exact Lagrangian filling

false(⋃_{v \in T_{0}} Q_{v} false) \cap D^{2 n}

of the Legendrian submanifold

false(⋃_{v \in T_{0}} Q_{v} false) \cap \partial D^{2 n} \subset false(S^{2 n - 1}, ξ_{std} false)

. In it is proved that there are quasi‐isomorphisms

R {Hom}_{W_{M}} false(double-struckk, double-struckk false) ≅ {scriptF}_{M}, R {Hom}_{F_{M}} false(double-struckk, double-struckk false) ≅ {scriptW}_{M}

when

{prefixdim}_{double-struckR} false(M false) = 4

and

M

is a plumbing of

T^{*} S^{2}

with

T = A_{n}

D_{n}

and

char (K) \neq 2

;

prefixdim<...>

…”

Section: Introductionmentioning

confidence: 99%

“…namely C * (Q; K) and C − * (Ω q Q; K) are Koszul dual as dg algebras. Generalizations of the above Koszul duality in the context of symplectic topology have been obtained by Etgü-Lekili [24] and Ekholm-Lekili [22]. More specifically, they considered symplectic manifolds M which are plumbings of cotangent bundles T * Q v of simply connected manifolds Q v according a tree T = (T 0 , T 1 ), where T 0 is the set of vertices, and T 1 is the set of edges.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Together with Seidel–Solomon's inductive argument based on Lefschetz fibration techniques , this implies that the total space

M_{p, q, r}

also admits a quasi‐dilation. However, this argument does not apply when

K = Z / 2

, since in this case Koszul duality does not hold for

D_{4}

Milnor fibers (see [, Theorem 14] for details).…”

Section: Split‐generation and Quasi‐dilationsmentioning

confidence: 99%

“…Note that

F_{M}

and

W_{M}

are equipped with augmentations

ε_{scriptF} : {scriptF}_{M} \to k

and

ε_{scriptW} : {scriptW}_{M} \to k

, where

ε_{F}

is defined by projecting to

k ≅ {scriptF}_{M}^{0}

, while

ε_{W}

is induced from the exact Lagrangian filling

false(⋃_{v \in T_{0}} Q_{v} false) \cap D^{2 n}

of the Legendrian submanifold

false(⋃_{v \in T_{0}} Q_{v} false) \cap \partial D^{2 n} \subset false(S^{2 n - 1}, ξ_{std} false)

. In it is proved that there are quasi‐isomorphisms

R {Hom}_{W_{M}} false(double-struckk, double-struckk false) ≅ {scriptF}_{M}, R {Hom}_{F_{M}} false(double-struckk, double-struckk false) ≅ {scriptW}_{M}

when

{prefixdim}_{double-struckR} false(M false) = 4

and

M

is a plumbing of

T^{*} S^{2}

with

T = A_{n}

D_{n}

and

char (K) \neq 2

;

prefixdim<...>

…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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“…We improve on the results of [14] in two aspects. Firstly, no restriction is imposed on Γ which was assumed to be a tree in [14] and only plumbings of T * S 2 's were considered.…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

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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Arithmetic mirror symmetry for genus 1 curves with n marked points

Lekili

Polishchuk

2016

Sel. Math. New Ser.

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Abstract. We establish a Z[[t 1 , . . . , t n ]]-linear derived equivalence between the relative Fukaya category of the 2-torus with n distinct marked points and the derived category of perfect complexes on the n-Tate curve. Specialising to t 1 = . . . = t n = 0 gives a Z-linear derived equivalence between the Fukaya category of the n-punctured torus and the derived category of perfect complexes on the standard (Néron) n-gon. We prove that this equivalence extends to a Z-linear derived equivalence between the wrapped Fukaya category of the n-punctured torus and the derived category of coherent sheaves on the standard n-gon. The corresponding results for n = 1 were established in [31]. IntroductionOver the past few decades, Kontsevich's Homological Mirror Symmetry (HMS) conjecture ([28]), which predicts a remarkable equivalence between an A-model category associated to a symplectic manifold and a B -model category associated to a complex manifold, has been studied extensively in many instances: abelian varieties, toric varieties, hypersurfaces in projective spaces, etc. From the beginning, the case of the symplectic 2-torus played a special role as this is the simplest instance of a Calabi-Yau manifold where concrete computations in the Fukaya category can be made, and the non-trivial nature of the HMS conjecture becomes manifest. Calculations provided in [38]

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Koszul duality patterns in Floer theory

Cited by 37 publications

References 64 publications

Koszul duality via suspending Lefschetz fibrations

Koszul duality via suspending Lefschetz fibrations

Fukaya categories of plumbings and multiplicative preprojective algebras

Arithmetic mirror symmetry for genus 1 curves with n marked points

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