Meromorphic Projective Structures, Opers and Monodromy

Understand

The complex projective structures considered is this article are compact curves locally modeled on $\mathbb{CP}^1$.

• To such a geometric object, modulo marked isomorphism, the monodromy map associates an algebraic one: a representation of its fundamental group into $\operatorname{PGL}(2,\mathbb{C})$, modulo conjugacy.
• This correspondence is neither surjective nor injective.
• Nonetheless, it is a local diffeomorphism [Hejhal, 1975].