Derived equivalences of generalized Kummer varieties

TL;DR

This paper investigates derived autoequivalences of generalized Kummer varieties, establishing conditions for their derived equivalence, describing autoequivalence groups, and providing examples of non-birational but derived equivalent fourfolds.

Contribution

It proves derived equivalence between certain generalized Kummer varieties and describes their autoequivalence groups, extending known results to new cases.

Findings

Kum^n(A) and Kum^n(A^∨) are derived equivalent for even n under specific polarization conditions.

Exact sequences for autoequivalence groups analogous to Orlov's sequence are obtained.

Examples of non-birational but derived equivalent fourfolds are provided.

Abstract

In this article we study derived (auto)equivalences of generalized Kummer varieties $\mathrm{Kum}^n(A)$. We provide an answer to a question raised by Namikawa by showing that the generalized Kummer varieties $\mathrm{Kum}^n(A)$ and $\mathrm{Kum}^n(A^\vee)$ are derived equivalent as long as $n$ is even and the abelian surface $A$ admits a polarization whose exponent is coprime to $n+1$. Furthermore we obtain exact sequences involving groups of autoequivalences in the style of Orlov's short exact sequence for autoequivalences of abelian varieties. Finally, we exhibit generalized Kummer fourfolds which are not birationally equivalent but still derived equivalent.