Louis Dailly

When given a connected compact complex manifold, a natural question is how to describe its universal cover: this leads to the study of uniformization problems. In the early 20th century, Poincaré and Koebe established such a description for Riemann surfaces, and this description depends on the genus of the surface.

On the other hand, when a group G acts freely and properly by biholomorphisms on a compact complex manifold X, the quotient map defines a covering, and the orbit space naturally inherits a manifold structure that makes this quotient map holomorphic. If the freeness assumption is dropped, the resulting quotient spaces may have singularities. The objects that appear particularly well-suited for these uniformization questions are orbifolds, topological spaces whose local models are quotients of Cn by a finite group acting linearly. Can the results of Poincaré and Koebe be adapted to the orbifold setting and to higher dimensions?

In this talk, I will present the various concepts involved and, time permitting, illustrate an orbifold version of the Poincaré–Koebe theorem.