The Geometry of Moduli Spaces of Vector Bundles over Algebraic Surfaces

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 CP2. The proof that the moduli space of bundles over CP2 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 [Fr1].