Vector bundles and connections on Riemann surfaces with projective structure

TL;DR

This paper constructs and analyzes a moduli space torsor of vector bundles with connections on Riemann surfaces with projective structures, revealing its symplectic geometry and relation to differential operators.

Contribution

It introduces a new torsor over the moduli space of vector bundles with projective structures, generalizing known torsors and describing its symplectic and differential operator structures.

Findings

Constructed a $T^*{oldsymbol{eta}}_g(r)$-torsor over the moduli space.

Proved the torsor has a compatible holomorphic symplectic structure.

Identified the torsor with sheaves of holomorphic connections on theta line bundles.

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 $g\, \geq\, 2$, $K^{1/2}_X$ 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 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 $T^*{\mathcal B}_g(r)$--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 ${\mathcal B}_g(r)$.