We show that the moduli space of marked branched projective structures of genus and branching degree is a complex analytic space. In the case , we show that is of dimension 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 of marked branching classes of genus and branching degree is a complex manifold. We show that if the space is an affine bundle over , while if , is an analytic subspace of .
On montre que l’espace de modules des structures projectives branchées de genre et de degré de branchement est un espace analytique complexe. Dans le cas où , on montre que est de dimension 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 des classes de branchement marquées de genre et de degré de branchement est une variété différentielle complexe. On montre que si , l’espace est un fibré affine sur , tandis que si , est un sous-espace analytique de .
Accepted:
Published online:
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
@article{JEP_2024__11__1373_0, author = {Gustave Billon}, title = {Moduli spaces of marked branched projective structures on surfaces}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1373--1410}, publisher = {\'Ecole polytechnique}, volume = {11}, year = {2024}, doi = {10.5802/jep.279}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.279/} }
TY - JOUR AU - Gustave Billon TI - Moduli spaces of marked branched projective structures on surfaces JO - Journal de l’École polytechnique — Mathématiques PY - 2024 SP - 1373 EP - 1410 VL - 11 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.279/ DO - 10.5802/jep.279 LA - en ID - JEP_2024__11__1373_0 ER -
%0 Journal Article %A Gustave Billon %T Moduli spaces of marked branched projective structures on surfaces %J Journal de l’École polytechnique — Mathématiques %D 2024 %P 1373-1410 %V 11 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.279/ %R 10.5802/jep.279 %G en %F JEP_2024__11__1373_0
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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