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

Let Y be a cubic threefold with a non-Eckardt type involution. Our first main result is that the -equivariant category of the Kuznetsov component Kuₙ䃒 (Y) determines the isomorphism class of Y for general (Y, ). We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction in perry2020integral, 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, ). To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category Dᵇ (X) of a smooth projective variety X and on the Hochschild cohomology of a semi-orthogonal component of Dᵇ (X). Another key ingredient is a generalization of a result in macri2009infinitesimal 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.