Betti numbers of the moduli space of rank 4 Higgs bundles on a curve.
Talk by Prof. Dr. Alexander Schmitt
Speaker invited by: Prof. Dr. Christian Okonek
Date: 07.11.11 Time: 13.15 - 14.45 Room: Y27H25
The completely reducible $n$-dimensional representations of a finitely generated group $\pi$ form in a natural way a complex affine variety $X_n(\pi)$, called character variety. It is an interesting and intensely studied problem to investigate the topology of these character varieties. Now, let $\pi$ be the canonical central extension of the fundamental group of a compact Riemann surface $X$ by the integers $\Z$. For $d\in\Z$, we look at the subvariety $X_{n,d}(\pi)$ of representations, such that $k\in\Z$ maps to $\exp(2\pi i k d/n)*Id$. By the non-abelian Hodge theorem, $X_{n,d}(\pi)$ is homeomorphic to the moduli space $H(n,d)$ of Higgs bundles of rank $n$ and degree $d$ on $X$. The latter space carries a richer structure, so that it can be studied by methods of Algebraic Geometry. In this way, Hitchin and Gothen could compute the Betti numbers of $H(n,1)$ for $n=2$ and $n=3$, respectively. There is a general conjecture by Hausel on the Betti numbers for arbitrary $n$. In the talk, I will survey the afore-mentioned topics and report on recent work with Garcia-Prada and Heinloth on the computation of the Betti numbers of $H(4,1)$.