Derived categories of cyclic covers and their branch divisors

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 ${\mathrm{D}}^{\mathrm{b}}(X)$ and ${\mathrm{D}}^{\mathrm{b}}(Z)$ 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.

27 pages, minor changes

References in corpus (3)

• Hodge theory and derived categories of cubic fourfolds
• Categorical resolutions of irrational singularities
• Descent theory for semiorthogonal decompositions

Cited by in corpus (9)

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• Categorical Torelli theorems for Gushel-Mukai threefolds
• Perverse sheaves of categories and some applications
• Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces
• Derived categories of quartic double fivefolds
• On two families of Enriques categories over K3 surfaces