Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Seidel, P.

doi:10.1007/978-3-319-59939-7_8

Cited by 29 publications

(81 citation statements)

References 91 publications

(216 reference statements)

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“…This is because the Fukaya category

A (π)

is defined to be the

Z / 2

‐invariant subcategory of the Fukaya category

F (\tilde{E})

, where

\overset{E}{true}

is a double cover of

E

branched along the smooth fiber

M_{*}

. However, this assumption can actually be removed since the Fukaya category

A (π)

can also be defined directly on

E

using a particular class of Hamiltonian perturbations specified in , without passing to the double branched cover

\overset{E}{true}

. Alternatively,

A (π)

can be defined as a version of partially wrapped Fukaya category .…”

Section: Formality Of A∞‐structuresmentioning

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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“…This is because the Fukaya category

A (π)

is defined to be the

Z / 2

‐invariant subcategory of the Fukaya category

F (\tilde{E})

, where

\overset{E}{true}

is a double cover of

E

branched along the smooth fiber

M_{*}

. However, this assumption can actually be removed since the Fukaya category

A (π)

can also be defined directly on

E

using a particular class of Hamiltonian perturbations specified in , without passing to the double branched cover

\overset{E}{true}

. Alternatively,

A (π)

can be defined as a version of partially wrapped Fukaya category .…”

Section: Formality Of A∞‐structuresmentioning

confidence: 99%

“…Lemma 4.2]. Assume that(58) holds, and as an A-bimodule, B is quasiisomorphic to A ⊕ (B/A)[−1], then the A ∞ -category B σ is quasi-isomorphic to the trivial extension constructed from A and the A-bimodule (B/A)[−1].Geometrically, the construction above can be applied to the pairing (A, B) = (A(π), V(M )),…”

mentioning

confidence: 99%

Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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“…Iterating this argument infinitely many times (which we can do as each quasiisomorphism increases the length), we obtain the following lemma (cf. [62,Lemma A.5]). …”

Section: Let Us Write (Lcamentioning

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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“…One candidate is the theory of Seidel-Fukaya category [23][24][25], which studies J-holomorphic sections of exact Morse fibrations. Seidel also studied a related theory called Fukaya category of Lefschetz fibration [26,27]. Haydys [11] studied the perturbed Cauchy-Riemann equation on C, and considered its Floer theory.…”

Section: Introductionmentioning

confidence: 98%