Geometry of Moduli Spaces of Flat Bundles on Punctured Surfaces

Abstract

We consider the moduli spaces of flat $SL(n, C)$-bundles on Riemann surfaces with one puncture when we fix the conjugacy class ${\cal C}$ of the monodromy transformation around the puncture. We show that under a certain condition on the class ${\cal C}$ (namely the product of $k<n$ eigenvalues is not equal to $1$) that we call property P the moduli space in question is smooth and its natural closure is a normal algebraic variety with rational singularities. The set of conjugacy classes having property P constitutes a Zariski open subset of $SL(n, C)$ and it is also possible to define property P for the groups $SO(n, C)}$ and $Sp(2n, C)$ to prove similar results. There are a few other applications of our techniques, one of which is that if $G$ is a classical reductive algebraic group and $A_1, A_2, ..., A_p\in G$, $p>1$ and $A_1A_2\cdots A_pA_1^{-1}A_2^{-1}\cdots A_p^{-1}$ belongs to a class which satisfies property P then the $p$ -tuple $(A_1, ..., A_p)$ algebraically generates the whole group $G$.