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 : 0000-00-00

Article Type : Research Paper

Abstract :We study the moduli space MG insert ignore into journalissuearticles values(A); of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, MG insert ignore into journalissuearticles values(A); is the insert ignore into journalissuearticles values(coarse); moduli space of s-equivalence classes of holomorphic semi-stable G\cnums -bundles with trivial Chern classes. MG insert ignore into journalissuearticles values(A); has the structure of a hyperk\`ahler orbifold. We show that when G is Spinsert ignore into journalissuearticles values(n); or SU insert ignore into journalissuearticles values(n);, MG insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(n); or SU insert ignore into journalissuearticles values(n);, the moduli space MG insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(when it exists);. We directly compute the stringy Hodge numbers of MSU insert ignore into journalissuearticles values(n); insert ignore into journalissuearticles values(A); and MSp insert ignore into journalissuearticles values(n); insert ignore into journalissuearticles values(A);, thus deriving formulas insert ignore into journalissuearticles values(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 Turk J Math, vol 25, iss 1