Meromorphic Projective Structures, Opers and Monodromy

2023/09/05 by Sérandour, Titouan
#30Fxx #34M40 #34M45 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.02203

Abstract

The complex projective structures considered is this article are compact curves locally modeled on \mathbbCP¹. To such a geometric object, modulo marked isomorphism, the monodromy map associates an algebraic one: a representation of its fundamental group into PGL(2,ℂ), modulo conjugacy. This correspondence is neither surjective nor injective. Nonetheless, it is a local diffeomorphism [Hejhal, 1975]. We generalize this theorem to projective structures admitting poles (without apparent singularity and with fixed residues): the corresponding monodromy map (including Stokes data) is a local biholomorphism.

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