Fukaya \(A_\infty\)-structures associated to Lefschetz fibrations. I
type: Item
instance of: scholarly article
title: Fukaya \(A_\infty\)-structures associated to Lefschetz fibrations. I
zbMATH Open document ID: 1267.53094
author: Paul Seidel
published in: The Journal of Symplectic Geometry
publication date: 2012-12-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.3932
full work available at URL: https://projecteuclid.org/euclid.jsg/1350392490
review text: Algebraic structures such as \(A_\infty\)-algebras, bimodules and morphisms between bimodules obtained from the fibre and total space with vanishing cycles and Lefschetz thimbles of an exact symplectic Lefschetz fibration \(\pi: E\to\mathbb{C}\) are compared (Theorem 7.1). The main result says that the associated directed \(A_\infty\)-algebra \({\mathcal A}\) has two maps from \({\mathcal A}^\vee\), the dual diagonal bimodule of \({\mathcal A}\) by working in terms of the fibre and total space of the fibration, that agree up to homotopy and composition with a quasi-morphism from \({\mathcal A}^\vee\) to itself (\S1. Theorem cf. Corollary 7.1). The author says that the theorem of \S1 corresponds to a weak version of conjecture 4 of \textit{P. Seidel} [Proc. Symp. Pure Math. 80, Pt. 1, 415--434 (2009; Zbl 1179.53085)] and is intended to be the start of a wider project concerning Fukaya categories of Lefschetz fibration [\textit{M. Maydanskiy},``Exotic symplectic manifolds from Lefschetz fibration'', (2009), \url{arXiv:0906.2224}; \textit{M. Maydanskiy} and \textit{P. Seidel}, J. Topol. 3, No. 1, 157--180 (2010; Zbl 1235.53088)]. The author also remarks that this paper has a significant overlap in the eventual goals, even though not necessary in intermediate steps, with [\textit{F. Bourgeois} et al., Geom. Topol. 16, No. 1, 301--389 (2012; Zbl 1322.53080)]. An exact symplectic manifold \(M\) has an exact symplectic form \(\omega= d\theta\) and a \(\omega\)-compatible almost complex structure \(I\) which is weakly convex (Definition 2.1). A Lefschetz fibration \(\pi: E\to\mathbb{C}\) is said to be an exact symplectic Lefschetz fibration if \(E\) is an exact symplectic manifold and \(\partial E\) is weakly convex (Definition 5.1). Moreover, the fibration is assumed to be trivial near infinity (as for detailed meaning of ``trivial near infinity'' cf. \S4.1). Vanishing cycles in the fibre \(M\) of \(\pi: E\to\mathbb{C}\), denoted by \((V_1,\dots, V_m)\), are assumed to be …
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 14D05
zbMATH DE Number: 6117817
zbMATH Keywords: Lefschetz fibration
zbMATH Keywords: exact symplectic manifold
zbMATH Keywords: vanishing cycle
zbMATH Keywords: Lefschetz thimble
zbMATH Keywords: \(A_\infty\)-structure
zbMATH Keywords: Floer cochain
zbMATH Keywords: Hochschild cochain
MaRDI profile type: Publication
arXiv ID: 0912.3932
OpenAlex ID: W3016247203
DOI: 10.4310/JSG.2012.V10.N3.A1
description: scientific article