Categorical Torelli theorem for hypersurfaces

TL;DR

This paper proves that for certain smooth Fano hypersurfaces, the Kuznetsov component of their derived category, along with a specific autoequivalence, uniquely determines the hypersurface, extending previous results.

Contribution

It generalizes the categorical Torelli theorem for hypersurfaces by removing the divisibility condition on the degree.

Findings

The Kuznetsov component and rotation functor determine the hypersurface uniquely for d > 3.

The result extends to cases where d = 3 and n > 3, without divisibility assumptions.

It builds on and generalizes prior work by Huybrechts and Rennemo.

Abstract

Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $\mathcal{A}_X$. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines $X$ uniquely if $d > 3$ or if $d = 3$ and $n > 3$. This generalizes a result by D. Huybrechts and J. Rennemo, who proved the same statement under the additional assumption that $d$ divides $n+2$.