A categorical invariant for cubic threefolds

TL;DR

This paper establishes a categorical Torelli theorem for cubic threefolds, demonstrating that a specific component of their derived category uniquely determines their isomorphism class.

Contribution

It introduces a categorical invariant via semi-orthogonal decomposition that characterizes cubic threefolds up to isomorphism, extending classical Torelli results.

Findings

Categorical invariant uniquely determines cubic threefolds.

Semi-orthogonal decomposition encodes the isomorphism class.

New perspective on cubic threefold classification through derived categories.

Abstract

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.