Derived categories of cyclic covers and their branch divisors

TL;DR

This paper studies the derived categories of cyclic covers of varieties and their branch divisors, constructing semiorthogonal decompositions that relate the categories of the cover and the branch divisor, with applications to specific classes of varieties.

Contribution

It introduces a method to decompose the derived category of a cyclic cover and relate it to the branch divisor's category, extending the understanding of derived categories in branched covers.

Findings

Constructed semiorthogonal decompositions for derived categories of cyclic covers.

Established relations between categories of covers and branch divisors via equivariant categories.

Applied results to examples like quartic double solids and Gushel-Mukai varieties.

Abstract

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, we consider a degree $n$ cyclic cover $X \to Y$ ramified over a divisor $Z \subset Y$. We construct semiorthogonal decompositions of $\mathrm{D^b}(X)$ and $\mathrm{D^b}(Z)$ with distinguished components $\mathcal{A}_X$ and $\mathcal{A}_Z$, and prove the equivariant category of $\mathcal{A}_X$ (with respect to an action of the $n$-th roots of unity) admits a semiorthogonal decomposition into $n-1$ copies of $\mathcal{A}_Z$. As examples we consider quartic double solids, Gushel-Mukai varieties, and cyclic cubic hypersurfaces.