G-bundles on Abelian surfaces, hyperkahler manifolds, and stringy Hodge numbers

Abstract

We study the moduli space $M_G$ (A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, $M_G$ (A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable Gcnums -bundles with trivial Chern classes. $M_G$ (A) has the structure of a hyperkahler orbifold. We show that when G is Sp(n) or SU (n), $M_G$ (A) has a natural hyperkahler desingularization which we exhibit as a moduli space of Gcnums -bundles with an altered stability condition. In this way, we obtain the two known families of hyperkahler manifolds, the Hilbert scheme of points on a K3 surface and the generalized Kummer varieties. We show that for G not Sp (n) or SU (n), the moduli space $M_G$ (A) does emph{not} admit a hyperkahler resolution. sloppy{Inspired by the physicists Vafa and Zaslow, Batyrev and Dais define stringy Hodge numbers'' for certain orbifolds. These numbers have been proven to agree with the Hodge numbers of a crepant resolution (when it exists). We directly compute the stringy Hodge numbers of $M_{SU_(n)} (A)$ and $M_{Sp (n)} (A)$, thus deriving formulas (originally due to Gottsche and Gottsche-Soergel) for the Hodge numbers of the Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.}