Heijal's theorem for parabolic projective structures on non compact surfaces

TL;DR

This paper proves that the holonomy map for parabolic projective structures on non-compact surfaces is a local biholomorphism, establishing a key local equivalence between geometric structures and their holonomy representations.

Contribution

It demonstrates that the holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces, extending understanding of their deformation theory.

Findings

Holonomy map is a local biholomorphism.

Establishes a correspondence between projective structures and representations.

Advances the theory of parabolic structures on non-compact surfaces.

Abstract

In this note, we prove that the holonomy map from the set of equivalence classes of projective structures of parabolic type on non compact surfaces to the set of equivalence classes of parabolic representations of the fundamental group of the surface to P SL 2 (C) is a local biholomorphism.