Lifting derived equivalences of abelian surfaces to generalized Kummer varieties

TL;DR

This paper investigates how derived equivalences of abelian surfaces can be extended to their associated generalized Kummer varieties, utilizing $G$-equivariant categories and Orlov's sequence.

Contribution

It introduces a method to lift derived equivalences from abelian surfaces to generalized Kummer varieties via $G$-equivariant categories and an adapted Orlov's sequence.

Findings

Derived equivalences of abelian surfaces can be extended to generalized Kummer varieties.

A $G$-equivariant version of Orlov's short exact sequence is established.

The approach generalizes to other cases with the same subgroup $G$.

Abstract

In this article, we study the $G$-autoequivalences of the derived category $\mathbf{D}^b_G(A)$ 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.