Branched affine and projective structures on compact Riemann surfaces

Abstract

It is first established that there exist linear manifolds of branched affine structures having certain nonpolar branch divisors and simple polar divisors on an arbitrary compact Riemann surface M of genus g≤1. When ≥2, it is shown that these linear manifolds form a complex analytic vector bundle over the manifold of simple polar divisors on M. When g=1, elliptic functions are used to construct certain projective structures on M. A partial determination is made as to which of these projective structures are affine and which are not.