Moduli spaces of marked branched projective structures on surfaces
Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 1373-1410.

We show that the moduli space 𝒫 g (n) of marked branched projective structures of genus g and branching degree n is a complex analytic space. In the case g2, we show that 𝒫 g (n) is of dimension 6g-6+n and we characterize its singular points in terms of their monodromy. We introduce a notion of branching class, that is an infinitesimal description of branched projective structures at the branched points. We show that the space 𝒜 g (n) of marked branching classes of genus g and branching degree n is a complex manifold. We show that if n<2g-2 the space 𝒫 g (n) is an affine bundle over 𝒜 g (n), while if n>4g-4, 𝒫 g (n) is an analytic subspace of 𝒜 g (n).

On montre que l’espace de modules 𝒫 g (n) des structures projectives branchées de genre g et de degré de branchement n est un espace analytique complexe. Dans le cas où g2, on montre que 𝒫 g (n) est de dimension 6g-6+n et on caractérise ses points singuliers en termes de leur monodromie. On introduit une notion de classe de branchement, qui est une description infinitésimale des structures projectives branchées aux points de branchement. On montre que l’espace 𝒜 g (n) des classes de branchement marquées de genre g et de degré de branchement n est une variété différentielle complexe. On montre que si n<2g-2, l’espace 𝒫 g (n) est un fibré affine sur 𝒜 g (n), tandis que si n>4g-4, 𝒫 g (n) est un sous-espace analytique de 𝒜 g (n).

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Accepted:
Published online:
DOI: 10.5802/jep.279
Classification: 57M50, 14H15, 32G15, 14H30
Keywords: Complex projective structures, moduli spaces, families of Riemann surfaces
Mot clés : Structures projectives complexes, espaces de modules, familles de surfaces de Riemann

Gustave Billon 1

1 IRMA, CNRS, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gustave Billon. Moduli spaces of marked branched projective structures on surfaces. Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 1373-1410. doi : 10.5802/jep.279. https://jep.centre-mersenne.org/articles/10.5802/jep.279/

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