- Turkish Journal of Mathematics
- Vol: 25 Issue: 1
- G-bundles on Abelian surfaces, hyperk"aler manifolds, and stringy Hodge numbers
G-bundles on Abelian surfaces, hyperk"aler manifolds, and stringy Hodge numbers
Authors : Jim Bryan, Ron Donagi, Naichung Conan Leung
Pages : 195-236
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Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :We study the moduli space MG (A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, MG (A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G\cnums -bundles with trivial Chern classes. MG (A) has the structure of a hyperk\"ahler orbifold. We show that when G is Sp(n) or SU (n), MG (A) has a natural hyperk\"ahler desingularization which we exhibit as a moduli space of G\cnums -bundles with an altered stability condition. In this way, we obtain the two known families of hyperk\"ahler 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 MG (A) does \emph{not} admit a hyperk\"ahler 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 MSU (n) (A) and MSp (n) (A), thus deriving formulas (originally due to G\"ottsche and G\"ottsche-Soergel) for the Hodge numbers of the Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.}Keywords : Turk. J. Math., 25, (2001), 195-236. Full text: pdf Other articles published in the same issue: Turk. J. Math., vol.25, iss.1.