Cyclic covers: Hodge theory and categorical Torelli theorems

Author:

Hannah Dell, Augustinas Jacovskis, Franco Rota

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)

journal:

date:

2023-10-19 16:00:00

Abstract

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 a categorical Torelli theorem for the prime Fano threefolds of index 1 and genus 2.

PDF: Cyclic covers: Hodge theory and categorical Torelli theorems.pdf