Holonomy of complex projective structures on punctured surfaces

Let $S$ be a punctured surface of finite type. We determine all possible representations $\rho:\pi_1(S) \to PSL_2\mathbb{C}$ that appear as the holonomy of a complex projective structure on $S$. This answers a question of F. Luo, as well as a question of Gallo-Kapovich-Marden.  Our proof involves, in particular, explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. This is a joint work with S. Gupta.

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