Hodge theory and categorical Torelli for double covers

The work of Kuznetsov and Perry describes the derived category of a cyclic cover X of weighted projective space. The description is in terms of the categories of the base and of the branch locus Z and works under the assumption that both Z and X are Fano. We relax this assumption on Z and obtain an analogous result for canonically polarized branch loci.  This extended framework unlocks information about two deformation families of low degree Fano threefolds, which arise as double covers. We then use categorical K-theory to show that a specific component of the derived category of the threefolds determines them up to isomorphism (solving a so-called categorical Torelli problem). This is joint work with A. Jacovskis and H. Dell.