标题： 实曲线上的实射影结构 显示英文标题

标题： Real projective structures on a real curve

摘要： 给定一个带有反全纯对合 $\tau$的紧致连通黎曼曲面 $X$，我们考虑满足相对于 $\tau$的相容条件的 $X$上的射影结构。 对于射影结构$P$在$X$上，存在使用$P$构造的在$X$上的全纯连接和全纯微分算子。 当射影结构$P$与$\tau$相容时，研究了$\tau$与与$P$相关的全纯联络或微分算子之间的关系。 紧致定向$C^\infty$曲面在亏格$g\, \geq\, 2$的射影结构的模空间具有自然的全纯辛结构。 已知这个全纯辛流形与由 Teichmüller 空间${\mathcal T}_g$的全纯余切丛配备 Liouville 辛形式所定义的全纯辛流形同构。 我们证明这两个全纯辛流形之间存在一个同构，该同构与$\tau$相容。 显示英文摘要

摘要： 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 $\tau$.