Junior Geometry Seminar – Carmine Imbriani (Milan)

Abstract: Complex projective structures provide a compelling example of (G,X)-structures on surfaces, serving as a bridge between classical uniformization theory and modern representation theory. This seminar explores the geometric and algebraic properties of these structures, with a primary focus on the holonomy map hol from the space of projective structures P(S) to the PSL(2, C)-character variety of the fundamental group \pi_1(S).

While the holonomy map is a general construction that is always open, its behavior in the context of projective structures is particularly rich: it is known to be a local homeomorphism but, notably, it fails to be globally injective due to phenomena such as grafting. In the final part of the talk, we will discuss a landmark result by R.C. Gunning, which provides a criterion for the injectivity of the holonomy map. By restricting the domain to a special class of projective structures, we will show how it is possible to recover a unique correspondence between projective structures and their associated representations.