Moduli spaces of marked branched projective structures on surfaces

Gustave Billon

doi:10.5802/jep.279

Moduli spaces of marked branched projective structures on surfaces
[Espaces de modules de structures projectives branchées sur les surfaces]

Gustave Billon ¹

¹ IRMA, CNRS, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1373-1410.

Résumé
Abstract

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 $g \geq 2$ , on montre que $𝒫_{g} (n)$ est de dimension $6 g - 6 + n$ 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 $n < 2 g - 2$ , l’espace $𝒫_{g} (n)$ est un fibré affine sur $𝒜_{g} (n)$ , tandis que si $n > 4 g - 4$ , $𝒫_{g} (n)$ est un sous-espace analytique de $𝒜_{g} (n)$ .

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 $g \geq 2$ , we show that $𝒫_{g} (n)$ is of dimension $6 g - 6 + n$ 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 $n < 2 g - 2$ the space $𝒫_{g} (n)$ is an affine bundle over $𝒜_{g} (n)$ , while if $n > 4 g - 4$ , $𝒫_{g} (n)$ is an analytic subspace of $𝒜_{g} (n)$ .

Reçu le : 2023-10-31
Accepté le : 2024-10-17
Publié le : 2024-10-31

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

Affiliations des auteurs :

Gustave Billon ¹

¹ IRMA, CNRS, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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, Tome 11 (2024), pp. 1373-1410. doi : 10.5802/jep.279. https://jep.centre-mersenne.org/articles/10.5802/jep.279/

Bibliographie
Cité par

[AB20] D. G. L. Allegretti & T. Bridgeland - “The monodromy of meromorphic projective structures”, Trans. Amer. Math. Soc. 373 (2020) no. 9, p. 6321-6367 | DOI | Zbl

[BD05] A. Beilinson & V. Drinfeld - “Opers”, 2005 | arXiv

[BDG19] I. Biswas, S. Dumitrescu & S. Gupta - “Branched projective structures on a Riemann surface and logarithmic connections”, Doc. Math. 24 (2019), p. 2299-2337 | DOI | Zbl

[BDH22] I. Biswas, S. Dumitrescu & S. Heller - “Branched $SL (r, ℂ)$ -opers”, Internat. Math. Res. Notices 10 (2022), p. 8311-8355 | DOI | Zbl

[Blu79] R. A. Blumenthal - “Transversely homogeneous foliations”, Ann. Inst. Fourier (Grenoble) 29 (1979) no. 4, p. 143-158 | DOI | Numdam | Zbl

[BM14] D. Barlet & J. Magnússon - Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Cours Spécialisés, vol. 22, Société Mathématique de France, Paris, 2014

[CDF14] G. Calsamiglia, B. Deroin & S. Francaviglia - “Branched projective structures with Fuchsian holonomy”, Geom. Topol. 18 (2014) no. 1, p. 379-446 | DOI | Zbl

[DG23] B. Deroin & A. Guillot - “Foliated affine and projective structures”, Compositio Math. 159 (2023) no. 6, p. 1153-1187 | DOI | Zbl

[dSG16] H. P. de Saint-Gervais - Uniformization of Riemann surfaces, Heritage of European Math., European Mathematical Society, Zürich, 2016 | DOI

[Dum09] D. Dumas - “Complex projective structures”, in Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, European Mathematical Society, Zürich, 2009, p. 455-508 | DOI | Zbl

[EE69] C. J. Earle & J. Eells - “A fibre bundle description of Teichmüller theory”, J. Differential Geometry 3 (1969), p. 19-43 | Zbl

[FG10] E. Frenkel & D. Gaitsgory - “Weyl modules and opers without monodromy”, in Arithmetic and geometry around quantization, Progress in Math., vol. 279, Birkhäuser Boston, Boston, MA, 2010, p. 101-121 | DOI | Zbl

[FR21] S. Francaviglia & L. Ruffoni - “Local deformations of branched projective structures: Schiffer variations and the Teichmüller map”, Geom. Dedicata 214 (2021), p. 21-48 | DOI | Zbl

[Fre07] E. Frenkel - “Lectures on the Langlands program and conformal field theory”, in Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, p. 387-533 | DOI | Zbl

[GKM00] D. Gallo, M. Kapovich & A. Marden - “The monodromy groups of Schwarzian equations on closed Riemann surfaces”, Ann. of Math. (2) 151 (2000) no. 2, p. 625-704 | DOI | Zbl

[GM20] S. Gupta & M. Mj - “Monodromy representations of meromorphic projective structures”, Proc. Amer. Math. Soc. 148 (2020) no. 5, p. 2069-2078 | DOI | Zbl

[GM21] S. Gupta & M. Mj - “Meromorphic projective structures, grafting and the monodromy map”, Adv. Math. 383 (2021), article ID 107673, 49 pages | DOI | Zbl

[God91] C. Godbillon - Feuilletages, Progress in Math., vol. 98, Birkhäuser Verlag, Basel, 1991

[Gun66] R. C. Gunning - Lectures on Riemann surfaces, Princeton Math. Notes, Princeton University Press, Princeton, NJ, 1966

[Hej75] D. A. Hejhal - “Monodromy groups and linearly polymorphic functions”, Acta Math. 135 (1975) no. 1, p. 1-55 | DOI | Zbl

[Hub81] J. H. Hubbard - “The monodromy of projective structures”, in Riemann surfaces and related topics (Stony Brook, NY, 1978), Annals of Math. Studies, vol. 97, Princeton University Press, Princeton, NJ, 1981, p. 257-275 | DOI | Zbl

[LF23] T. Le Fils - “Holonomy of complex projective structures on surfaces with prescribed branch data”, J. Topology 16 (2023) no. 1, p. 430-487 | DOI | Zbl

[LMP09] F. Loray & D. Marín Pérez - “Projective structures and projective bundles over compact Riemann surfaces”, in Équations différentielles et singularités, Astérisque, vol. 323, Société Mathématique de France, Paris, 2009, p. 223-252 | Numdam | Zbl

[Man72] R. Mandelbaum - “Branched structures on Riemann surfaces”, Trans. Amer. Math. Soc. 163 (1972), p. 261-275 | DOI | Zbl

[Man73] R. Mandelbaum - “Branched structures and affine and projective bundles on Riemann surfaces”, Trans. Amer. Math. Soc. 183 (1973), p. 37-58 | DOI | Zbl

[Man75] R. Mandelbaum - “Unstable bundles and branched structures on Riemann surfaces”, Math. Ann. 214 (1975), p. 49-59 | DOI | Zbl

[Scá97] B. A. Scárdua - “Transversely affine and transversely projective holomorphic foliations”, Ann. Sci. École Norm. Sup. (4) 30 (1997) no. 2, p. 169-204 | DOI | Numdam | Zbl

[Ste43] N. E. Steenrod - “Homology with local coefficients”, Ann. of Math. (2) 44 (1943), p. 610-627 | DOI | Zbl

[Sun17] F. Sun - “An elementary proof for Poincaré duality with local coefficients”, 2017 | arXiv

[Sér22] T. Sérandour - Meromorphic projective structures, opers and monodromy, Ph. D. Thesis, Université de Rennes, 2022

[Voi07] C. Voisin - Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Math., vol. 76, Cambridge University Press, Cambridge, 2007

[Whi78] G. W. Whitehead - Elements of homotopy theory, Graduate Texts in Math., vol. 61, Springer-Verlag, New York-Berlin, 1978 | DOI

Cité par Sources :