Degenerations of bundle moduli

Abstract

Over a family X of complete curves of genus g, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of holomorphic SL(n,C) bundles over the generic smooth curve Xt in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by (C⁎)s of a moduli space on the desingularization. Taking a “maximal” degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a (C⁎)(3g−3)(n−1)-quotient of a (2g−2)-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli space of SL(n,C) bundles on the smooth curve is a space of representations of the fundamental group into SU(n) (the “symplectic picture”). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a (S1)(3g−3)(n−1) symplectic quotient of a (2g−2)-th power of a space associated to the three-pointed sphere.