Vector Bundles and Connections on Riemann Surfaces with Projective Structure

Let 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 ≥ 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^*ℬ_g(r) -torsor ℋ_g(r) over 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 ℋ_g(r) has a holomorphic symplectic structure compatible with the T^*ℬ_g(r) -torsor structure. We also describe ℋ_g(r) in terms of the second order matrix valued differential operators. It is shown that ℋ_g(r) is identified with the T^*ℬ_g(r) -torsor given by the sheaf of holomorphic connections on the theta line bundle over ℬ_g(r) .