G-Bundles on Abelian surfaces, Hyperkahler manifolds and Stringy Hodge numbers,

by J. Bryan, R. Donagi and N.C. Leung.

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 G_C-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 G_C-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 *not* admit a hyperkahler resolution.

	Inspired by the physicists Vafa and Zaslow, Batyrev and Dais define
'stringy Hodge numbers' for certain orbifolds. These numbers are
conjectured to agree with the Hodge numbers of a crepant resolution (when
it exists). We compute the stringy Hodge numbers of M_{SU(n)}(A) and
M_{Sp(n)}(A) and verify the conjecture in these cases.