Abstract
Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover \(X \rightarrow Y\) 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 nth 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.
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Acknowledgments
A.K. is grateful to Alexey Elagin for his clarifications concerning equivariant categories. A.P. thanks Joe Harris and Johan de Jong for useful conversations related to this work.
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A.K. was partially supported by the Russian Academic Excellence Project “5-100”, by RFBR grants 14-01-00416, 15-01-02164, 15-51-50045, and by the Simons foundation. A.P. was partially supported by NSF GRFP Grant DGE1144152, and thanks the Laboratory of Algebraic Geometry SU-HSE for its hospitality in November 2013, when this work was initiated.
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Kuznetsov, A., Perry, A. Derived categories of cyclic covers and their branch divisors. Sel. Math. New Ser. 23, 389–423 (2017). https://doi.org/10.1007/s00029-016-0243-0
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DOI: https://doi.org/10.1007/s00029-016-0243-0