A smooth but non-symplectic moduli of sheaves on a hyperkähler variety

Abstract

For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $K_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles of degree $0$ is isomorphic to the blowup of the dual abelian surface in one point. We believe that this is the first explicit example of a component which is smooth with a non-trivial canonical bundle.