Lifting derived equivalences of abelian surfaces to generalized Kummer varieties

Abstract

In this article, we study the $G$-autoequivalences of the derived category ${\mathbf{D}}_G^b(A)$ of $G$-equivariant objects for an abelian variety $A$ with $G$ being a finite subgroup of ${\mathrm{Pic}}^0(A)$. We provide a result analogue to Orlov's short exact sequence for derived equivalences of abelian varieties. It can be generalized to the derived equivalences of abelian varieties for a same $G$ in general. Furthermore, we find derived equivalences of generalized Kummer varieties by lifting derived equivalences of abelian surfaces using the $G$-equivariant version of Orlov's short exact sequence and some ``splitting" propositions.

arXiv commentBased on the mistake on previous Lemma 1.4 (It is deleted now.), the contents of Proposition 1.26, Proposition 1.29 and Lemma 1.30 are slightly different. Also, Corollary 0.4 and subsection 2.3 are added to emphasize the finite index property of the lifting results