Fukaya A_∞ structures associated to Lefschetz fibrations. I

2009/12/19 by Paul Seidel, Seidel, Paul · 2 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.0912.3932

Abstract

This (partially expository) paper discusses Lagrangian Floer cohomology in the context of Lefschetz fibrations, with emphasis on the algebraic structures encountered there. In addition to the well-known directed A_infinity algebras which appear in this situation, one has additional information encoded in a certain bimodule homomorphism. There are two approaches to constructing this homomorphism: in terms of the (noncompact) Lefschetz thimbles in the total space, or else in terms of vanishing cycle in the fibre. We prove a comparison result, which shows that (up to a certain remaining ambiguity) the two approaches are equivalent.

Cited by

• Lagrangian Floer theory, from geometry to algebra and back again
• Semi-orthogonality in Fukaya-Seidel mirrors to blowups of abelian varieties

Related

• Fukaya A_∞-structures associated to Lefschetz fibrations. V
• Fukaya A_∞ structures associated to Lefschetz fibrations. II
• Fukaya A_∞-structures associated to Lefschetz fibrations. IV
• Fukaya A_∞-structures associated to Lefschetz fibrations. III
• Fukaya A_∞-structures associated to Lefschetz fibrations. IV 1/2