On the singular Kähler geometry of certain moduli spaces

Moduli spaces of semistable holomorphic vector bundles on a Riemann surface and generalizations thereof to moduli spaces of semistable holomorphic principal bundles for a complex reductive Lie group may be obtained via extended moduli spaces as well as by a geometric invariant theory construction. Both constructions are finite dimensional; the former leads to the stratified symplectic structure while the latter yields the complex analytic one. We will explain a finite dimensional approach which involves suitable extended moduli spaces arising from spaces of holomorphic maps; this approach is aimed at providing the stratified symplectic and complex analytic structures at the same time, thereby establishing the fact that the two structure combine to a stratified Kähler structure. Our approach includes a construction of the familiar line bundle on such a moduli space.