The Teichmüller and Riemann moduli stacks
[Les champs de modules de Teichmüller et de Riemann]
Journal de l’École polytechnique — Mathématiques, Tome 6 (2019) , pp. 879-945.

Le but de cet article est d’étudier la structure des espaces de modules de Teichmüller et de Riemann de variétés de dimension plus grande que 1, considérés comme des champs sur la catégorie des espaces analytiques complexes. Nous montrons que ces deux champs sont analytiques, c’est-à-dire isomorphes à la champification d’un groupoïde analytique lisse. Nous donnons ensuite une construction explicite d’atlas comme groupoïde d’holonomie généralisé. Ces résultats sont valables dès que la dimension du groupe d’automorphismes de chaque structure est bornée par un entier fixé. On peut voir ce travail comme une réponse à la question 1.8 de [48].

The aim of this paper is to study the structure of the Teichmüller and Riemann moduli spaces, viewed as stacks over the category of complex analytic spaces, for higher-dimensional manifolds. We show that both stacks are analytic in the sense that they are isomorphic to the stackification of a smooth analytic groupoid. We then show how to construct explicitly such an atlas as a sort of generalized holonomy groupoid. This is achieved under the sole condition that the dimension of the automorphism group of each structure is bounded by a fixed integer. All this can be seen as an answer to Question 1.8 of [48].

Reçu le : 2016-11-02
Accepté le : 2019-10-03
Publié le : 2019-10-11
DOI : https://doi.org/10.5802/jep.108
Classification : 32G05,  58H05,  14D23
Mots clés : Espace de Teichmüller, déformations de structures complexes, groupoïdes analytiques, champs et problèmes de modules
@article{JEP_2019__6__879_0,
     author = {Laurent Meersseman},
     title = {The Teichm\"uller and Riemann moduli stacks},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     pages = {879--945},
     publisher = {\'Ecole polytechnique},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.108},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2019__6__879_0/}
}
Laurent Meersseman. The Teichmüller and Riemann moduli stacks. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019) , pp. 879-945. doi : 10.5802/jep.108. https://jep.centre-mersenne.org/item/JEP_2019__6__879_0/

[1] K. D. An - “A counter-example to the equivariance structure on semi-universal deformation”, 2019 | arXiv:1906.00082

[2] W. P. Barth, K. Hulek, C. A. M. Peters & A. Van de Ven - Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), vol. 4, Springer-Verlag, Berlin, 2004 | Article | MR 2030225 | Zbl 1036.14016

[3] K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. Göttsche & A. Kresch - “Algebraic stacks”, 2014

[4] E. Brieskorn & A. van de Ven - “Some complex structures on products of homotopy spheres”, Topology 7 (1968), p. 389-393 | Article | MR 233360 | Zbl 0174.54901

[5] M. Brunella - “Uniformisation de feuilletages et feuilles entières”, in Complex manifolds, foliations and uniformization, Panoramas & Synthèses, vol. 34-35, Société Mathématique de France, Paris, 2011, p. 1-52 | MR 3088901 | Zbl 1288.32032

[6] C. Camacho & A. Lins Neto - Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985 | Article | Zbl 0568.57002

[7] F. Catanese - “Moduli of algebraic surfaces”, in Theory of moduli (Montecatini Terme, 1985), Lect. Notes in Math., vol. 1337, Springer, Berlin, 1988, p. 1-83 | Article | MR 963062 | Zbl 0658.14017

[8] F. Catanese - “A superficial working guide to deformations and moduli”, in Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, Int. Press, Somerville, MA, 2013, p. 161-215 | MR 3184164 | Zbl 1322.14002

[9] F. Catanese - “Topological methods in moduli theory”, Bull. Math. Sci. 5 (2015) no. 3, p. 287-449 | Article | MR 3404712 | Zbl 1375.14129

[10] F. Catanese - “Moduli spaces of surfaces and real structures”, Ann. of Math. (2) 158 (2003) no. 2, p. 577-592 | Article | MR 2018929 | Zbl 1042.14011

[11] K. Dąbrowski - “Moduli spaces for Hopf surfaces”, Math. Ann. 259 (1982) no. 2, p. 201-225 | Article | MR 656662 | Zbl 0497.32017

[12] A. Douady - “Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné”, Ann. Inst. Fourier (Grenoble) 16 (1966) no. 1, p. 1-95 | Article | Zbl 0146.31103

[13] A. Douady - “Le problème des modules pour les variétés analytiques complexes (d’après Masatake Kuranishi)”, in Séminaire Bourbaki, Vol. 9, Société Mathématique de France, Paris, 1995, p. 7-13, Exp. No. 277 | MR 1608786 | Zbl 0191.38002

[14] R. Friedman & J. W. Morgan - “Complex versus differentiable classification of algebraic surfaces”, Topology Appl. 32 (1989) no. 2, p. 135-139 | Article | MR 1007985 | Zbl 0694.14013

[15] R. Friedman & Z. Qin - “On complex surfaces diffeomorphic to rational surfaces”, Invent. Math. 120 (1995) no. 1, p. 81-117 | Article | MR 1323983 | Zbl 0823.14022

[16] C. Fromenteau - Sur le champ de Teichmüller des surfaces de Hopf, Ph. D. Thesis, Univ. Angers, 2017

[17] A. González, E. Lupercio, C. Segovia & B. Uribe - “Orbifold topological quantum field theories in dimension 2”, 2013

[18] H. Grauert - “Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen”, Publ. Math. Inst. Hautes Études Sci. 5 (1960), p. 5-64 | Article

[19] A. Haefliger - “Groupoids and foliations”, in Groupoids in analysis, geometry, and physics (Boulder, CO, 1999), Contemp. Math., vol. 282, American Mathematical Society, Providence, RI, 2001, p. 83-100 | Article | MR 1855244 | Zbl 0994.57025

[20] R. S. Hamilton - “The inverse function theorem of Nash and Moser”, Bull. Amer. Math. Soc. (N.S.) 7 (1982) no. 1, p. 65-222 | Article | MR 656198 | Zbl 0499.58003

[21] D. Knutson - Algebraic spaces, Lect. Notes in Math., vol. 203, Springer-Verlag, Berlin-New York, 1971 | MR 302647 | Zbl 0221.14001

[22] K. Kodaira - “Complex structures on S 1 ×S 3 ”, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), p. 240-243 | Article | Zbl 0141.27402

[23] K. Kodaira - Complex manifolds and deformation of complex structures, Grundlehren Math. Wiss., vol. 283, Springer-Verlag, New York, 1986 | Article | MR 815922 | Zbl 0581.32012

[24] K. Kodaira & D. C. Spencer - “On deformations of complex analytic structures. I”, Ann. of Math. (2) 67 (1958), p. 328-402 | Article | MR 112154 | Zbl 0128.16901

[25] M. Kreck & Y. Su - “Finiteness and infiniteness results for Torelli groups of (hyper) Kähler manifolds”, 2019 | arXiv:1907.05693

[26] M. Kuranishi - “On the locally complete families of complex analytic structures”, Ann. of Math. (2) 75 (1962), p. 536-577 | Article | MR 141139 | Zbl 0106.15303

[27] M. Kuranishi - “New proof for the existence of locally complete families of complex structures”, in Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, 1965, p. 142-154 | Article | Zbl 0144.21102

[28] M. Kuranishi - “A note on families of complex structures”, in Global Analysis (Papers in honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, p. 309-313

[29] M. Kuranishi - Deformations of compact complex manifolds, Les Presses de l’Université de Montréal, Montreal, Que., 1971, Séminaire de Mathématiques Supérieures, No. 39 (Été 1969) | MR 355111 | Zbl 0382.32014

[30] S. Lang - Fundamentals of differential geometry, Graduate Texts in Math., vol. 191, Springer-Verlag, New York, 1999 | Article | MR 1666820 | Zbl 0932.53001

[31] G. Laumon & L. Moret-Bailly - Champs algébriques, Ergeb. Math. Grenzgeb. (3), vol. 39, Springer-Verlag, Berlin, 2000 | Zbl 0945.14005

[32] C. LeBrun - “Topology versus Chern numbers for complex 3-folds”, Pacific J. Math. 191 (1999) no. 1, p. 123-131 | Article | MR 1725466 | Zbl 0951.57010

[33] J. A. Leslie - “On a differential structure for the group of diffeomorphisms”, Topology 6 (1967), p. 263-271 | Article | MR 210147 | Zbl 0147.23601

[34] L. Meersseman - “Feuilletages par variétés complexes et problèmes d’uniformisation”, in Complex manifolds, foliations and uniformization, Panoramas & Synthèses, vol. 34-35, Société Mathématique de France, Paris, 2011, p. 205-257 | MR 3088905 | Zbl 1287.32013

[35] L. Meersseman - “Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds”, Ann. Sci. École Norm. Sup. (4) 44 (2011) no. 3, p. 495-525 | Article | MR 2839457 | Zbl 1239.32012

[36] L. Meersseman - “A note on the automorphism group of a compact complex manifold”, Enseign. Math. 63 (2017) no. 3-4, p. 263-272 | Article | MR 3852172 | Zbl 06980526

[37] L. Meersseman - “The Teichmüller stack”, in Complex and symplectic geometry, Springer INdAM Ser., vol. 21, Springer, Cham, 2017, p. 123-136 | Article | MR 3645311 | Zbl 1390.32009

[38] L. Meersseman - “Kuranishi-type moduli spaces for proper CR-submersions fibering over the circle”, J. reine angew. Math. 749 (2019), p. 87-132 | Article | MR 3935900 | Zbl 07050842

[39] L. Meersseman, M. Nicolau & J. Ribón - “On the automorphism group of foliations with geometric transverse structure”, 2018 | arXiv:1810.07244

[40] I. Moerdijk & J. Mrčun - Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Math., vol. 91, Cambridge University Press, Cambridge, 2003 | Article | MR 2012261 | Zbl 1029.58012

[41] S. Morita - “A topological classification of complex structures on S 1 ×S 2n-1 ”, Topology 14 (1975), p. 13-22 | Article | MR 405444 | Zbl 0301.57010

[42] J. Morrow & K. Kodaira - Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971 | Zbl 0325.32001

[43] M. Namba - “On deformations of automorphism groups of compact complex manifolds”, Tôhoku Math. J. (2) 26 (1974), p. 237-283 | Article | MR 377115 | Zbl 0288.32019

[44] A. Newlander & L. Nirenberg - “Complex analytic coordinates in almost complex manifolds”, Ann. of Math. (2) 65 (1957), p. 391-404 | Article | MR 88770 | Zbl 0079.16102

[45] D. Ruberman - “A polynomial invariant of diffeomorphisms of 4-manifolds”, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, p. 473-488 | Article | MR 1734421 | Zbl 0952.57007

[46] - “Stacks project”, http://stacks.math.columbia.edu

[47] M. Verbitsky - “Mapping class group and a global Torelli theorem for hyperkähler manifolds”, Duke Math. J. 162 (2013) no. 15, p. 2929-2986, Appendix A by Eyal Markman | Article | Zbl 1295.53042

[48] M. Verbitsky - “Teichmüller spaces, ergodic theory and global Torelli theorem”, in Proceedings of the ICM (Seoul 2014) Vol. II, Kyung Moon Sa, Seoul, 2014, p. 793-811 | MR 3728638 | Zbl 1373.32011

[49] M. Verbitsky - “Ergodic complex structures on hyperkähler manifolds”, Acta Math. 215 (2015) no. 1, p. 161-182 | Article | Zbl 1332.53092

[50] M. Verbitsky - “Mapping class group and a global Torelli theorem for hyperkähler manifolds: an erratum”, 2019 | arXiv:1908.11772

[51] J. Wehler - “Versal deformation of Hopf surfaces”, J. reine angew. Math. 328 (1981), p. 22-32 | Article | MR 636192 | Zbl 0459.32009