Geodesic
Branched Projective Structures on a Riemann Surface and Logarithmic Connections
Documenta mathematica, Tome 24 (2019), pp. 2299-2337
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We study the set PS​ 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=1d​ni​⋅xi​, where xi​∈X. Under the hypothesis that ni​,=1, for all i, with d a positive even integer such that d=2g−2, we show that PS​ 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⊗OX​(S). The space of all logarithmic connections of the above type is an affine space over the vector space H0(X,KX⊗2​⊗OX​(S)) of dimension 3g−3+d. We conclude that PS​ is a subset of this affine space that has codimenison d at a generic point.
DOI : 10.4171/dm/726
Classification : 30F10, 30F30
Mots-clés : Riemann surface, quadratic differential, branched projective structure, logarithmic connection 
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     author = {Sorin Dumitrescu and Indranil Biswas and Subhojoy Gupta},
     title = {Branched {Projective} {Structures} on a {Riemann} {Surface} and {Logarithmic} {Connections}},
     journal = {Documenta mathematica},
     pages = {2299--2337},
     year = {2019},
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     doi = {10.4171/dm/726},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/726/}
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Sorin Dumitrescu; Indranil Biswas; Subhojoy Gupta. Branched Projective Structures on a Riemann Surface and Logarithmic Connections. Documenta mathematica, Tome 24 (2019), pp. 2299-2337. doi: 10.4171/dm/726

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