Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 3, p. 495-525

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of 0 in p , for some p>0) or differentiable (parametrized by an open neighborhood of 0 in p , for some p>0) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point t of the parameter space, the fiber over t of the first family is biholomorphic to the fiber over t of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.

Considérons le problème d’uniformisation suivant. Prenons deux familles de déformation holomorphes (paramétrées par un ensemble analytique défini dans un voisinage de 0 dans p pour p>0) ou différentiables (paramétrées par un voisinage de 0 dans p pour p>0) de variétés compactes complexes. Supposons-les ponctuellement isomorphes, c’est-à-dire que, pour tout point t de l’espace des paramètres, la fibre en t de la première famille est biholomorphe à la fibre en t de la deuxième famille. Sous quelle(s) condition(s) les deux familles sont-elles localement isomorphes en 0 ? Dans cet article, nous donnons une condition suffisante dans le cas de familles holomorphes. Nous montrons ensuite que, de façon surprenante, la condition n’est pas suffisante dans le cas des familles différentiables. Nous décrivons également plusieurs types de contre-exemples et donnons quelques éléments de classifications de ces contre-exemples. Ces résultats reposent sur une étude géométrique de l’espace de Kuranishi d’une variété compacte complexe.

DOI : https://doi.org/10.24033/asens.2148
Classification:  32G07,  57R30
Keywords: deformations of complex manifolds, foliations, uniformization
@article{ASENS_2011_4_44_3_495_0,
     author = {Meersseman, Laurent},
     title = {Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 44},
     number = {3},
     year = {2011},
     pages = {495-525},
     doi = {10.24033/asens.2148},
     zbl = {1239.32012},
     mrnumber = {2839457},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2011_4_44_3_495_0}
}
Meersseman, Laurent. Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 44 (2011) no. 3, pp. 495-525. doi : 10.24033/asens.2148. http://www.numdam.org/item/ASENS_2011_4_44_3_495_0/

[1] F. Catanese, Moduli of algebraic surfaces, in Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math. 1337, Springer, 1988, 1-83. | Zbl 0658.14017

[2] W. Fischer & H. Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1965 (1965), 89-94. | MR 184258 | Zbl 0135.12601

[3] O. Forster, Lectures on Riemann surfaces, Graduate Texts in Math. 81, Springer, 1981. | MR 648106 | Zbl 0475.30002

[4] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Math. I.H.É.S. 5 (1960). | Numdam | MR 121814 | Zbl 0100.08001

[5] H. Grauert & H. Kerner, Deformationen von Singularitäten komplexer Räume, Math. Ann. 153 (1964), 236-260. | MR 170354 | Zbl 0118.30401

[6] P. A. Griffiths, The extension problem for compact submanifolds of complex manifolds. I. The case of a trivial normal bundle, in Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, 1965, 113-142. | MR 190952 | Zbl 0141.27403

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

[8] K. Kodaira & D. C. Spencer, On deformations of complex analytic structures. II, Ann. of Math. 67 (1958), 403-466. | MR 112154 | Zbl 0128.16901

[9] M. Kuranishi, New proof for the existence of locally complete families of complex structures, in Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, 1965, 142-154. | MR 176496 | Zbl 0144.21102

[10] M. Kuranishi, A note on families of complex structures, in Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, 1969, 309-313. | MR 254883 | Zbl 0211.10301

[11] D. Mumford, Further pathologies in algebraic geometry, Amer. J. Math. 84 (1962), 642-648. | MR 148670 | Zbl 0114.13106

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

[13] J. J. Wavrik, Obstructions to the existence of a space of moduli, in Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, 1969, 403-414. | MR 254882 | Zbl 0191.38003

[14] J. J. Wavrik, Deforming cohomology classes, Trans. Amer. Math. Soc. 181 (1973), 341-350. | MR 326002 | Zbl 0238.32011

[15] J. Wehler, Isomorphie von Familien kompakter komplexer Mannigfaltigkeiten, Math. Ann. 231 (1977/78), 77-90. | MR 499327 | Zbl 0363.32016