Koszul duality via suspending Lefschetz fibrations

TL;DR

This paper demonstrates a Koszul duality relationship between the compact and wrapped Fukaya categories of certain Liouville 6-manifolds derived from suspending Lefschetz fibrations, leading to new insights into their structure and generation.

Contribution

It establishes an $A_$-Koszul duality between Fukaya categories of specific Liouville manifolds, linking them to cyclic and Calabi-Yau completions of quiver algebras, and introduces new examples with quasi-dilations.

Findings

Fukaya categories are related via $A_$-Koszul duality.

Split-generation of the compact Fukaya category by vanishing cycles.

New examples of Liouville manifolds with quasi-dilations.

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.