Branched projective structures on a Riemann surface and logarithmic connections

2018/08/14 by Biswas, Indranil, Dumitrescu, Sorin, Gupta, Subhojoy
#30F30 #32G15 #51N15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.04555

Abstract

We study the set \mathcal P_S consisting of all branched holomorphic projective structures on a compact Riemann surface X of genus g ≥ 1 and with a fixed branching divisor S:= ∑_i=1^d n_i⋅ x_i, where x_i ∈ X. Under the hypothesis that n_i=1, for all i, with d a positive even integer such that d ≠ 2g-2, we show that \mathcal P_S coincides with a subset of the set of all logarithmic connections with singular locus S, satisfying certain geometric conditions, on the rank two holomorphic jet bundle J¹(Q), where Q is a fixed holomorphic line bundle on X such that Q^{⊗ 2}= TX⊗ \mathcal O_X(S). The space of all logarithmic connections of the above type is an affine space over the vector space H⁰(X, K^{⊗ 2}_X ⊗ \mathcal O_X(S)) of dimension 3g-3+d. We conclude that \mathcal P_S is a subset of this affine space that has codimenison d at a generic point.

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