Complex projective structures

Abstract

This is a survey of the theory of complex projective ℂℙ1-structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics:

1. Using the Schwarzian derivative to parameterize the moduli space
2. Thurston’s parameterization of the moduli space using grafting
3. Holonomy representations of ℂℙ1-structures

We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map.