Projective structures with discrete holonomy representations

Abstract

Let $K(X)$ denote the set of projective structures on a compact Riemann surface $X$ whose holonomy representations are discrete. We will show that each component of the interior of $K(X)$ is holomorphically equivalent to a complex submanifold of the product of Teichm\"uller spaces and the holonomy representation of every projective structure in the interior of $K(X)$ is a quasifuchsian group.