Complex Projective Structures

TL;DR

This survey reviews the theory of complex projective structures on compact surfaces, focusing on parameterizations via Schwarzian derivatives and grafting, and exploring their holonomy representations and interrelations.

Contribution

It provides a comprehensive overview of the main parameterizations of CP^1 structures and compares their effectiveness and relationships.

Findings

Schwarzian derivative effectively parameterizes the moduli space

Grafting offers an alternative parameterization of the moduli space

Comparison reveals connections between different parameterizations and holonomy

Abstract

This is a survey of the theory of complex projective (CP^1) 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 CP^1 structures We also discuss some results comparing the two parameterizations of the space of projective structures and relating these parameterizations to the holonomy map.