Topology of moduli of parabolic connections with fixed determinant

Abstract

Let $X$ be a compact Riemann surface of genus $g \geq 2$ and $D\subset X$ be a fixed finite subset. Let $ξ$ be a line bundle of degree $d$ over $X$. Let $\mathcal{M}(α, r, ξ)$ (respectively, $\mathcal{M}_{\mathrm{conn}}(α, r, ξ)$) denote the moduli space of stable parabolic bundles (respectively, parabolic connections) of rank $r$ $(\geq 2)$, determinant $ξ$ and full flag generic rational parabolic weight type $α$. We show that $ π_k(\mathcal{M}_{\mathrm{conn}}(α, r, ξ)) \cong π_k(\mathcal{M}(α, r, ξ)) $ for $k \leq2(r-1)(g-1)-1$. As a consequence, we deduce that the moduli space $\mathcal{M}_{\mathrm{conn}}(α, r, ξ)$ is simply connected. We also show that the Hodge structures on the torsion-free parts of both the cohomologies $H^k(\mathcal{M}_{\mathrm{conn}}(α, r, ξ),\mathbb{Z})$ and $H^k(\mathcal{M}(α, r, ξ),\mathbb{Z})$ are isomorphic for all $k\leq 2(r-1)(g-1)+1$.