SYZ for index 1 Fano hypersurfaces in projective space

TL;DR

This paper investigates homological mirror symmetry for a specific singular Fano hypersurface using SYZ methods, producing an LG-model that recovers line bundles and confirming a conjecture about Fukaya categories.

Contribution

It introduces an SYZ-based approach to construct an LG-model for the hypersurface, linking Fukaya-Seidel categories to line bundles and resolving a conjecture on Fukaya category generation.

Findings

Constructed an LG-model via SYZ for the hypersurface

Fukaya-Seidel category recovers line bundles on the hypersurface

Confirmed Sheridan's conjecture on Fukaya category generation

Abstract

We study homological mirror symmetry of the singular hypersurface $X_0=V(t^{n+1}-x_0\dotsi x_n)\subseteq\mathbb{P}^{n+1}$. Following an SYZ type approach, we produce an LG-model, whose Fukaya-Seidel category recovers line bundles on $X_0$. As a byproduct of our approach, we answer a conjecture of N.Sheridan about generating the small component of the Fukaya category of the smooth index 1 Fano hypersurface in $\mathbb{P}^{n+1}$, without bounding co-chains.