Derived equivalences of generalized Kummer varieties

In this article we study derived (auto)equivalences of generalized Kummer varieties ${\mathrm{K}\mathrm{u}\mathrm{m}}^n(A)$. We provide an answer to a question raised by Namikawa by showing that the generalized Kummer varieties ${\mathrm{K}\mathrm{u}\mathrm{m}}^n(A)$ and ${\mathrm{K}\mathrm{u}\mathrm{m}}^n(A^{\vee })$ 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.