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

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.