On the Fukaya category of a Fano hypersurface in projective space

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Abstract

This paper is about the Fukaya category of a Fano hypersurface XCPn. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface X: we construct a configuration of monotone Lagrangian spheres in X, and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of X (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for X. We also explain how to extract non-trivial information about Gromov–Witten invariants of X from its Fukaya category.
Original languageEnglish
Pages (from-to)165-317
Number of pages153
JournalPublications mathématiques de l'IHÉS
Volume124
Issue number1
Early online date15 Feb 2016
DOIs
Publication statusPublished - 30 Nov 2016

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