BRANCHED PROJECTIVE STRUCTURES ON A RIEMANN SURFACE AND LOGARITHMIC CONNECTIONS

Indranil Biswas (1) , Sorin Dumitrescu (2) , Subhojoy Gupta (3)

1 TIFR - Tata Institute for Fundamental Research
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 Department of Mathematics, Indian Institute of Science

We study the set P(S) of all branched holomorphic projective structures on a compact Riemann surface X of genus g ≥ 1 and with a fixed branching divisor S. We show that P(S) coincides with a subset of the set of logarithmic connections with singular locus S, satisfying certain geometric conditions, on the rank two holomorphic jet bundle J^1(Q), where Q is a fixed holomorphic line bundle on X such that Q^⊗2 = T X ⊗ O X (S). The space of all logarithmic connections of the above type is an affine space over the vector space of meromorphic quadratic differentials with at most simple poles at S. We conclude that P(S) is a subset of this affine space of dimension 3g-3+deg(S), that has codimension deg(S) at a generic point.

Domaines: Mathématiques [math]

Indranil Biswas, Sorin Dumitrescu, Subhojoy Gupta. BRANCHED PROJECTIVE STRUCTURES ON A RIEMANN SURFACE AND LOGARITHMIC CONNECTIONS. Documenta Mathematica, 2019, 24, pp.2159-2197. ⟨hal-01863576⟩