A. Schmitt (Essen)
Title: Decorated Principal Bundles
In this talk (which is a continuation of my VBAC talk in Bad Honnef in the
year 2000), I will present a solution to the following moduli problem:
Fix a compact Riemann surface X, a complex reductive group G, a representation
\rho\colon G->GL(V), and a line bundle L on X and classify pairs (P,\sigma)
which consist of a principal G-bundle P and a section \sigma\colon X->
P_\rho\otimes L, P_\rho being the vector bundle with fiber V that is
associated to P by means of \rho.
The most prominent example is the case when \rho is the adjoint representation
and L is the canonical bundle of X, i.e., the case of Higgs bundles. Higgs
bundles are important in studying representation spaces of the fundamental
group \pi_1(X) in G. We will discuss a new example of a concrete application
of Higgs bundles to the determination of Betti numbers of representation
spaces.
Other moduli problems of this kind arise from representations of \pi_1(X) in
non-compact real forms of G. Finally, some related moduli spaces have also
been used in Arithmetic Geometry.