Derived categories of cyclic covers and their branch divisors

Alexander Kuznetsov, Alexander Perry

Published 2014-11-07Version 1

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 $\mathcal{A}_Z$.