Vector bundles and connections on Riemann surfaces with projective structure

Abstract

Let ${\mathcal{B}}_g(r)$ be the moduli space of triples of the form $(X,K_X^{1/2},F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g\ge 2$, $K_X^{1/2}$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^{\ast }{\mathcal{B}}_g(r)$--torsor ${\mathcal{H}}_g(r)$ over ${\mathcal{B}}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundles of rank $r$ on $Y$, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that ${\mathcal{H}}_g(r)$ has a holomorphic symplectic structure compatible with the $T^{\ast }{\mathcal{B}}_g(r)$--torsor structure. We also describe ${\mathcal{H}}_g(r)$ in terms of the second order matrix valued differential operators. It is shown that ${\mathcal{H}}_g(r)$ is identified with the $T^{\ast }{\mathcal{B}}_g(r)$--torsor given by the sheaf of holomorphic connections on the theta line bundle over ${\mathcal{B}}_g(r)$.