Projective structures and projective bundles over compact Riemann surfaces

TL;DR

This paper surveys classical problems and results on projective structures over compact Riemann surfaces, providing a detailed description of affine structures on the torus through a foliated bundle perspective.

Contribution

It offers a comprehensive survey of projective structures and introduces an explicit versal family for affine structures on the torus using foliated bundle models.

Findings

Complete description of projective structures on the torus

Explicit versal family of foliated bundle models

Survey of classical problems and results

Abstract

A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From this latter geometric bundle picture, we survey on classical problems and results on projective structures. We will give a complete description of projective (actually affine) structures on the torus with an explicit versal family of foliated bundle picture.