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 $(X,\, K^{1/2}_X,\, F)$, where X is a compact connected Riemann surface of genus g, with $g\, \ge \, 2$, $K^{1/2}_X$ is a theta characteristic on X, and F is a stable vector bundle on X of rank r and degree zero. We construct a $T^*{\mathcal B}_g(r)$-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 algebraic 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 $T^*{{\mathcal {B}}}_g(r)$-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 $T^*{{\mathcal {B}}}_g(r)$-torsor given by the sheaf of holomorphic connections on the theta line bundle over ${{\mathcal {B}}}_g(r)$.