Moduli spaces of marked branched projective structures on surfaces

Billon, Gustave

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

We show that the moduli space ${𝒫}_g\left(n\right)$ of marked branched projective structures of genus $g$ and branching degree $n$ is a complex analytic space. In the case $g\ge 2$, we show that ${𝒫}_g\left(n\right)$ 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 ${𝒜}_g\left(n\right)$ of marked branching classes of genus $g$ and branching degree $n$ is a complex manifold. We show that if $n<2g-2$ the space ${𝒫}_g\left(n\right)$ is an affine bundle over ${𝒜}_g\left(n\right)$, while if $n>4g-4$, ${𝒫}_g\left(n\right)$ is an analytic subspace of ${𝒜}_g\left(n\right)$.

On montre que l’espace de modules ${𝒫}_g\left(n\right)$ des structures projectives branchées de genre $g$ et de degré de branchement $n$ est un espace analytique complexe. Dans le cas où $g\ge 2$, on montre que ${𝒫}_g\left(n\right)$ 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 ${𝒜}_g\left(n\right)$ 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 ${𝒫}_g\left(n\right)$ est un fibré affine sur ${𝒜}_g\left(n\right)$, tandis que si $n>4g-4$, ${𝒫}_g\left(n\right)$ est un sous-espace analytique de ${𝒜}_g\left(n\right)$.

Classification : 57M50, 14H15, 32G15, 14H30
Keywords: Complex projective structures, moduli spaces, families of Riemann surfaces
Mots-clés : Structures projectives complexes, espaces de modules, familles de surfaces de Riemann

[AB20] Allegretti, Dylan G. L.; Bridgeland, Tom The monodromy of meromorphic projective structures, Trans. Amer. Math. Soc., Volume 373 (2020) no. 9, pp. 6321-6367 | DOI | Zbl | MR

[BD05] Beilinson, Alexander; Drinfeld, Vladimir Opers, 2005 | arXiv

[BDG19] Biswas, Indranil; Dumitrescu, Sorin; Gupta, Subhojoy Branched projective structures on a Riemann surface and logarithmic connections, Doc. Math., Volume 24 (2019), pp. 2299-2337 https://ems.press/journals/dm/articles/8965657 | DOI | Zbl | MR

[BDH22] Biswas, Indranil; Dumitrescu, Sorin; Heller, Sebastian Branched $\mathrm{SL}(r,ℂ)$-opers, Internat. Math. Res. Notices, Volume 10 (2022), pp. 8311-8355 | DOI | Zbl | MR

[Blu79] Blumenthal, Robert A. Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble), Volume 29 (1979) no. 4, pp. 143-158 | DOI | Numdam | Zbl | MR

[BM14] Barlet, Daniel; Magnússon, Jón Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Cours Spécialisés, 22, Société Mathématique de France, Paris, 2014, 525 pages | MR

[CDF14] Calsamiglia, Gabriel; Deroin, Bertrand; Francaviglia, Stefano Branched projective structures with Fuchsian holonomy, Geom. Topol., Volume 18 (2014) no. 1, pp. 379-446 | DOI | Zbl | MR

[DG23] Deroin, Bertrand; Guillot, Adolfo Foliated affine and projective structures, Compositio Math., Volume 159 (2023) no. 6, pp. 1153-1187 | DOI | Zbl | MR

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

[Dum09] Dumas, David Complex projective structures, Handbook of Teichmüller theory. Vol. II (IRMA Lect. Math. Theor. Phys.), Volume 13, European Mathematical Society, Zürich, 2009, pp. 455-508 | DOI | Zbl

[EE69] Earle, Clifford J.; Eells, James A fibre bundle description of Teichmüller theory, J. Differential Geometry, Volume 3 (1969), pp. 19-43 http://projecteuclid.org/euclid.jdg/1214428816 | Zbl | MR

[FG10] Frenkel, Edward; Gaitsgory, Dennis Weyl modules and opers without monodromy, Arithmetic and geometry around quantization (Progress in Math.), Volume 279, Birkhäuser Boston, Boston, MA, 2010, pp. 101-121 | DOI | Zbl

[FR21] Francaviglia, Stefano; Ruffoni, Lorenzo Local deformations of branched projective structures: Schiffer variations and the Teichmüller map, Geom. Dedicata, Volume 214 (2021), pp. 21-48 | DOI | Zbl | MR

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

[GKM00] Gallo, Daniel; Kapovich, Michael; Marden, Albert The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2), Volume 151 (2000) no. 2, pp. 625-704 | DOI | Zbl | MR

[GM20] Gupta, Subhojoy; Mj, Mahan Monodromy representations of meromorphic projective structures, Proc. Amer. Math. Soc., Volume 148 (2020) no. 5, pp. 2069-2078 | DOI | Zbl | MR

[GM21] Gupta, Subhojoy; Mj, Mahan Meromorphic projective structures, grafting and the monodromy map, Adv. Math., Volume 383 (2021), 107673, 49 pages | DOI | Zbl | MR

[God91] Godbillon, Claude Feuilletages, Progress in Math., 98, Birkhäuser Verlag, Basel, 1991 | MR

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

[Hej75] Hejhal, Dennis A. Monodromy groups and linearly polymorphic functions, Acta Math., Volume 135 (1975) no. 1, pp. 1-55 | DOI | Zbl | MR

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

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

[LMP09] Loray, Frank; Marín Pérez, David Projective structures and projective bundles over compact Riemann surfaces, Équations différentielles et singularités (Astérisque), Volume 323, Société Mathématique de France, Paris, 2009, pp. 223-252 | Numdam | Zbl

[Man72] Mandelbaum, Richard Branched structures on Riemann surfaces, Trans. Amer. Math. Soc., Volume 163 (1972), pp. 261-275 | DOI | Zbl | MR

[Man73] Mandelbaum, Richard Branched structures and affine and projective bundles on Riemann surfaces, Trans. Amer. Math. Soc., Volume 183 (1973), pp. 37-58 | DOI | Zbl | MR

[Man75] Mandelbaum, Richard Unstable bundles and branched structures on Riemann surfaces, Math. Ann., Volume 214 (1975), pp. 49-59 | DOI | Zbl | MR

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

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

[Sun17] Sun, Fang An elementary proof for Poincaré duality with local coefficients, 2017 | arXiv

[Sér22] Sérandour, Titouan Meromorphic projective structures, opers and monodromy, Ph. D. Thesis, Université de Rennes (2022)

[Voi07] Voisin, Claire Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Math., 76, Cambridge University Press, Cambridge, 2007 | MR

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