We classify generic germs of contracting holomorphic mappings which factorize through blowing-ups, under the relation of conjugation by invertible germs of mappings. As for Hopf surfaces, this is the key to the study of compact complex surfaces with and which contain a global spherical shell. We study automorphisms and deformations and we show that these generic surfaces are endowed with a holomorphic foliation which is unique and stable under any deformation.
@article{bwmeta1.element.bwnjournal-article-apmv70z1p49bwm,
author = {Georges Dloussky and Franz Kohler},
title = {Classification of singular germs of mappings and deformations of compact surfaces of class VII0},
journal = {Annales Polonici Mathematici},
volume = {69},
year = {1998},
pages = {49-83},
zbl = {0930.32013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv70z1p49bwm}
}
Georges Dloussky; Franz Kohler. Classification of singular germs of mappings and deformations of compact surfaces of class VII₀. Annales Polonici Mathematici, Tome 69 (1998) pp. 49-83. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv70z1p49bwm/
[000] [B] F. A. Bogomolov, Surfaces of class VII-0 and affine geometry, Math. USSR-Izv. 21 (1983), 31-73. | Zbl 0527.14029
[001] [BR] M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, prépublication 86 (1996), Université de Bourgogne, Dijon.
[002] [DA] K. Dąbrowski, Kuranishi families for Hopf surfaces, Ann. Polon. Math. 45 (1985), 61-84. | Zbl 0578.32053
[003] [D1] G. Dloussky, Structure des surfaces de Kato, Mém. Soc. Math. France 112 (1984), no. 14. | Zbl 0543.32012
[004] [D2] G. Dloussky, Sur la classification des germes d'applications holmorphes contractantes, Math. Ann. 280 (1988), 649-661. | Zbl 0677.32004
[005] [D3] G. Dloussky, Une construction élémentaire des surfaces d'Inoue-Hirzebruch, ibid. 280 (1988), 663-682. | Zbl 0617.14025
[006] [E1] I. Enoki, Surfaces of class VII₀ with curves, Tôhoku Math. J. 33 (1981), 453-492.
[007] [E2] I. Enoki, Deformations of surfaces containing global spherical shells, in: Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progr. Math. 39, Birkhäuser, 1983, 45-64.
[008] [H] J. H. Hubbard and R. W. Oberste-Vorth, Hénon mappings in the complex domain I: The global topology of dynamical space, Publ. Math. IHES 79 (1994), 5-46. | Zbl 0839.54029
[009] [I] M. Inoue, New surfaces with no meromorphic functions II, in: Complex Analysis and Algebraic Geometry, W. L. Baily and T. Shioda (ed.), Cambridge Univ. Press and Iwanami Shoten Publ., 1977, 91-106.
[010] [IKO] M. Inoue, S. Kobayashi and T. Ochiai, Holomorphic affine connections on compact complex surfaces, J. Fac. Sci. Univ. Tokyo Sci. IA 27 (1980), 247-264. | Zbl 0467.32014
[011] [KA] M. Kato, Compact complex manifolds containing 'global spherical shells', in: Proc. Internat. Sympos. Algebraic Geometry (Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, 45-84.
[012] [KW] Y. Kawamata, On deformations of compactifiable complex manifolds, Math. Ann. 235 (1978), 247-265. | Zbl 0363.32015
[013] [KO] K. Kodaira, On the structure of compact complex analytic surface, I, Amer. J. Math. 86 (1964), 751-798; II, ibid. 88 (1966), 682-721. | Zbl 0137.17501
[014] [KH1] F. Kohler, Feuilletages holomorphes singuliers et déformations sur les surfaces contenant une coquille sphérique globale, thèse, Université d'Aix-Marseille I, 1994.
[015] [KH2] F. Kohler, Feuilletages holomorphes singuliers sur les surfaces contenant une coquille sphérique globale, Ann. Inst. Fourier 45 (1995), 161-182; erratum, ibid., 46 (1996), 589. | Zbl 0814.57022
[016] [KH3] F. Kohler, Feuilletages holomorphes singuliers sur les surfaces contenant une coquille sphérique globale, in: Estado actual y persectivas en singularidades de ecuaciones diferenciales y foliaciones holomorfas (Medina, 1995), J. Mozo (ed.), Serie Ciencias 15, Secretariado de Publicacioes e intercambio cientifico, 1997, 143-159.
[017] [LYZ] J. Li, S. T. Yau and F. A. Zheng, A simple proof of Bogomolov's theorem on class VII₀ surfaces with b₂ = 0, Illinois J. Math. 34 (1990), 217-220.
[018] [N1] I. Nakamura, On surfaces of class VII₀ with curves, Invent. Math. 78 (1984), 393-443. | Zbl 0575.14033
[019] [N2] I. Nakamura, On surfaces of class VII₀ with curves II, Tôhoku Math. J. 42 (1990), 475-516. | Zbl 0732.14019
[020] [NA] M. Namba, Automorphism groups of Hopf surfaces, ibid. J. 26 (1974), 133-152.
[021] [T] A.-D. Teleman, Projectively flat surfaces and Bogomolov's theorem on VII₀ surfaces, Internat. J. Math. 5 (1994), 253-264. | Zbl 0803.53038
[022] [W] J. Wavrik, Obstructions to the existence of a space of moduli, in: Global Analysis, Princeton Univ. Press, Princeton, 1969, 403-413. | Zbl 0191.38003