Real projective structures on a real curve

Given a compact connected Riemann surface XX equipped with an antiholomorphic involution ττ, we consider the projective structures on XX satisfying a compatibility condition with respect to ττ. For a projective structure PP on XX, there are holomorphic connections and holomorphic differential operators on XX that are constructed using PP. When the projective structure PP is compatible with ττ, the relationships between ττ and the holomorphic connections, or the differential operators, associated to PP are investigated. The moduli space of projective structures on a compact oriented C∞C∞ surface of genus g≥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 Teichmüller space TgTg equipped with the Liouville symplectic form. We show that there is an isomorphism between these two holomorphic symplectic manifolds that is compatible with ττ.