The Geometry of Moduli Spaces of Vector Bundles over Algebraic Surfaces

Abstract

The study of moduli problems is one of the central topics in algebraic geometry. After the development of GIT theory, the moduli of vector bundles over curves were constructed and in the 1970's, Gieseker constructed the moduli space of vector bundles over algebraic surfaces. Since then, many mathematicians have studied the geometry of this moduli space. For projective plane, Horrocks discovered the very powerful monad constructions of vector bundles over CP 2 . The proof that the moduli space of bundles over CP 2 is either rational or unirational and is irreducible, and the recent development in understanding its cohomology ring rest on this construction. Brosius [Br] gave a simple description of vector bundles over ruled surfaces. In [Mu], Mukai studied the geometry of moduli of vector bundles over K3 surfaces. In particular, he constructed nondegenerate symplectic forms on these moduli spaces. Recently, Friedman has provided us with a description of the global structure of the moduli of bundles over regular elliptic surfaces [Fri]. As to the geometry of moduli of vector bundles over general surfaces, the picture has emerged only recently. To begin with, Bogomolov's inequality says that the Chern numbers of any stable sheaf E obey 2r -c 2 (E) -ci(E) 2 > 0, r = rank E. r — 1 On the other hand, works of [Gi2], [Ta2] show that when r — 2, stable E with ci(E) = 0 does exist if c 2 (E) > 2(p g (X) + 1). The major breakthrough in this area comes from Donaldson's generic smoothness theorem, which point out that deformation of general vector bundles of sufficiently large second Chern classes is unobstructed. This discovery contrasts sharply to the counter examples of [Gi2] for small 02(E). Speaking of moduli of vector bundles over algebraic surfaces, we have to discuss the influence of gauge theory. In short, based on works of [Do], the under-standing of the moduli of vector bundles will provide us valuable information on the topology of the underlining algebraic surfaces. In this lecture I will report on the recent progress in understanding the ge-ometry of these moduli spaces. I will concentrate on topics that arise from both algebraic geometry and gauge theory and are solved (mainly) by using algebraic geometry means.