Cyclic covers: Hodge theory and categorical Torelli theorems

Let $Y$ admit a rectangular Lefschetz decomposition of its derived category, and consider a cyclic cover $X\to Y$ ramified over a divisor $Z$. In a setting not considered by Kuznetsov and Perry, we define a subcategory ${\mathcal{A}}_Z$ of the equivariant derived category of $X$ which contains, rather than is contained in, ${\mathrm{D}}^{\mathrm{b}}(Z)$. We then show that the equivariant category of the Kuznetsov component of $X$ is decomposed into copies of ${\mathcal{A}}_Z$. As an application, we relate ${\mathcal{A}}_Z$ with the cohomology of $Z$ under some numerical assumptions. In particular, we obtain categorical Torelli theorems for the lowest degree prime Fano threefolds of index 1 and 2.