Projective structures and projective bundles over compact riemann surfaces

A projective structure on a compact Riemann surface C of genus g is given by an atlas with transition functions in PGL(2, C). Equivalently, a projective structure is given by a P1-bundle over C equipped with a section σ and a foliation F which is both transversal to the ℙ1- fibers and the section σ. From this latter geometric bundle picture, we survey on classical problems and results on projective structures. By the way, we will recall some basic facts about ℙ1-bundles. We will give a complete description of projective (actually affine) structures on the torus with an explicit versal family of foliated bundle picture. © Astérisque 323, SMF 2009.