Abstract: A  CP^1-structure is a geometric structure on a Riemann surface, and it corresponds to a holomorphic  quadratic differential on the Riemann surface.  In addition,  the holonomy of a CP^1-structure is a homomorphism from the fundamental group of the base surface into PSL(2, C).

The set of CP^1-structures on a compact Riemann surface property embeds into the PSL(2,C)-character variety, so that its image is a half-dimensional complex analytic submanifold (Poincare holonomy variety).  In the first lecture, we first go over some basics of CP^1-structures, including a cut-and-paste operation, called grafting.   Then we discuss some further properties of  those half-dimensional submanifolds.