Classification of singular germs of mappings and deformations of compact surfaces of class VII₀

Georges Dloussky; Franz Kohler

Annales Polonici Mathematici (1998)

  • Volume: 70, Issue: 1, page 49-83
  • ISSN: 0066-2216

Abstract

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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 b 1 = 1 and b > 0 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.

How to cite

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Georges Dloussky, and Franz Kohler. "Classification of singular germs of mappings and deformations of compact surfaces of class VII₀." Annales Polonici Mathematici 70.1 (1998): 49-83. <http://eudml.org/doc/262872>.

@article{GeorgesDloussky1998,
abstract = {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 $b_1=1$ and $b₂ >0$ 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.},
author = {Georges Dloussky, Franz Kohler},
journal = {Annales Polonici Mathematici},
keywords = {surfaces of class VII₀; germs of mappings; surfaces of class VII},
language = {eng},
number = {1},
pages = {49-83},
title = {Classification of singular germs of mappings and deformations of compact surfaces of class VII₀},
url = {http://eudml.org/doc/262872},
volume = {70},
year = {1998},
}

TY - JOUR
AU - Georges Dloussky
AU - Franz Kohler
TI - Classification of singular germs of mappings and deformations of compact surfaces of class VII₀
JO - Annales Polonici Mathematici
PY - 1998
VL - 70
IS - 1
SP - 49
EP - 83
AB - 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 $b_1=1$ and $b₂ >0$ 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.
LA - eng
KW - surfaces of class VII₀; germs of mappings; surfaces of class VII
UR - http://eudml.org/doc/262872
ER -

References

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  7. [E1] I. Enoki, Surfaces of class VII₀ with curves, Tôhoku Math. J. 33 (1981), 453-492. 
  8. [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. 
  9. [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. Zbl0839.54029
  10. [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. 
  11. [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. Zbl0467.32014
  12. [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. 
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  14. [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. Zbl0137.17501
  15. [KH1] F. Kohler, Feuilletages holomorphes singuliers et déformations sur les surfaces contenant une coquille sphérique globale, thèse, Université d'Aix-Marseille I, 1994. 
  16. [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. Zbl0814.57022
  17. [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. 
  18. [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. 
  19. [N1] I. Nakamura, On surfaces of class VII₀ with curves, Invent. Math. 78 (1984), 393-443. Zbl0575.14033
  20. [N2] I. Nakamura, On surfaces of class VII₀ with curves II, Tôhoku Math. J. 42 (1990), 475-516. Zbl0732.14019
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