On the Fukaya category of a Fano hypersurface in projective space

TL;DR

This paper studies the Fukaya category of Fano hypersurfaces in projective space, establishing foundational structures, computing the category explicitly, and proving homological mirror symmetry, linking symplectic geometry with mirror symmetry and Gromov-Witten invariants.

Contribution

It provides a detailed construction and computation of the Fukaya category for Fano hypersurfaces, confirming homological mirror symmetry and connecting to Gromov-Witten invariants.

Findings

Fukaya category structures are established for monotone Fano hypersurfaces.

Explicit computation of the Fukaya category matches the mirror superpotential.

Proof of Kontsevich's homological mirror symmetry conjecture for these hypersurfaces.

Abstract

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. 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.