Fukaya categories and deformations

TL;DR

This paper explores the structure and deformations of Fukaya categories associated with symplectic manifolds derived from Calabi-Yau varieties, focusing on Hochschild cohomology and geometric implications.

Contribution

It offers an informal, conjectural discussion on the deformation theory of Fukaya categories and their relation to Hochschild cohomology in specific symplectic contexts.

Findings

Hochschild cohomology's potential geometric significance

Connections between Fukaya categories of exact and closed Calabi-Yau manifolds

Role of Lefschetz pencils in deformation analysis

Abstract

This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.