Koszul duality via suspending Lefschetz fibrations

Abstract

Let $M$ be a Liouville 6-manifold which is the smooth fiber of a Lefschetz fibration on ${\mathbb{C}}^4$ constructed by suspending a Lefschetz fibration on ${\mathbb{C}}^3$. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category $\mathcal{F}(M)$ and the wrapped Fukaya category $\mathcal{W}(M)$ are related through $A_{\infty }$-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 $\mathcal{F}(M)$ by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi-dilations in the sense of Seidel-Solomon are obtained.

arXiv comment68 pages, 23 figures; v6: final version, accepted by Journal of Topology

References in corpus (21)

• An open string analogue of Viterbo functoriality
• Symplectic cohomology and duality for the wrapped Fukaya category
• Duality between Lagrangian and Legendrian invariants
• Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors
• Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry
• Symplectic cohomology and q-intersection numbers
• Koszul duality patterns in Floer theory
• Morse flow trees and Legendrian contact homology in 1-jet spaces
• Legendrian Fronts for Affine Varieties
• On the symplectic cohomology of log Calabi-Yau surfaces
• Existence of Lefschetz fibrations on Stein and Weinstein domains
• Quiver algebras as Fukaya categories
• Picard-Lefschetz theory and dilating C^*-actions
• Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules
• A bordered Chekanov-Eliashberg algebra
• Disjoinable Lagrangian spheres and dilations
• Hochschild (co)homology and Koszul duality
• Orientations of Morse flow trees in Legendrian contact homology
• The interplay between 2-and-3-Calabi--Yau triangulated categories
• To compute orientations of Morse flow trees in Legendrian contact homology
• Curved String Topology and Tangential Fukaya Categories II

Cited by in corpus (3)

• Simplicial descent for Chekanov-Eliashberg dg-algebras
• Singular Legendrian unknot links and relative Ginzburg algebras
• Persistence of unknottedness of clean Lagrangian intersections