Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components

TL;DR

This paper establishes a categorical Torelli theorem for cubic threefolds with non-Eckardt involutions using three different approaches, and proves an equivariant infinitesimal version, advancing the understanding of the relationship between derived categories and geometric structures.

Contribution

It introduces three novel approaches to prove a categorical Torelli theorem for cubic threefolds with involutions and extends results to an infinitesimal setting with equivariance considerations.

Findings

The $ au$-equivariant Kuznetsov component determines the isomorphism class of the threefold.

Three different methods (Hodge theory, Bridgeland stability, Chow theory) successfully prove the categorical Torelli theorem.

An equivariant infinitesimal Torelli theorem is established for non-Eckardt cubic threefolds.

Abstract

Let $Y$ be a cubic threefold with a non-Eckardt type involution $\tau$. Our first main result is that the $\tau$-equivariant category of the Kuznetsov component $\mathcal{K}u_{\mathbb{Z}_2}(Y)$ determines the isomorphism class of $Y$ for general $(Y,\tau)$. We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction due to Alexander Perry), a Bridgeland moduli theoretical one (using equivariant stability conditions), and a Chow theoretical one (using some techniques in [kuznetsovnonclodedfield2021]).The remaining part of the paper is devoted to proving an equivariant infinitesimal categorical Torelli for non-Eckardt cubic threefolds $(Y,\tau)$. To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category $D^b(X)$ of a smooth projective variety $X$ and on the Hochschild cohomology of a semi-orthogonal component of $D^b(X)$. Another key ingredient is a generalization of a result in [macri2009infinitesima] which shows that the twisted Hochschild-Kostant-Rosenberg isomorphism is compatible with the actions on the Hochschild cohomology and on the singular cohomology induced by an automorphism of $X$. In appendix, we prove an equivariant categorical Torelli theorem for arbitrary cubic threefold with a geometric involution under a natural assumption.