Real projective structures on a real curve

TL;DR

This paper studies projective structures on real Riemann surfaces with antiholomorphic involutions, exploring their compatibility conditions, associated holomorphic connections, and the symplectic geometry of their moduli spaces, including real structure compatibility.

Contribution

It establishes an isomorphism between the moduli space of compatible projective structures and the cotangent bundle of Teichmüller space that respects the real structure.

Findings

Compatibility conditions between projective structures and antiholomorphic involutions.

Construction of holomorphic connections and differential operators from projective structures.

An isomorphism of holomorphic symplectic manifolds compatible with the real structure.

Abstract

Given a compact connected Riemann surface $X$ equipped with an antiholomorphic involution $\tau$, we consider the projective structures on $X$ satisfying a compatibility condition with respect to $\tau$. For a projective structure $P$ on $X$, there are holomorphic connections and holomorphic differential operators on $X$ that are constructed using $P$. When the projective structure $P$ is compatible with $\tau$, the relationships between $\tau$ and the holomorphic connections, or the differential operators, associated to $P$ are investigated. The moduli space of projective structures on a compact oriented $C^\infty$ surface of genus $g\, \geq\, 2$ has a natural holomorphic symplectic structure. It is known that this holomorphic symplectic manifold is isomorphic to the holomorphic symplectic manifold defined by the total space of the holomorphic cotangent bundle of the Teichm\"uller space ${\mathcal T}_g$ equipped with the Liouville symplectic form. We show that there is an isomorphism between these two holomorphic symplectic manifolds that is compatible with $\tau$.