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Complex Manifolds

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From non-Kählerian surfaces to Cremona group of P2(C)

Georges Dloussky
Published Online: 2014-04-23 | DOI: https://doi.org/10.2478/coma-2014-0001

Abstract

For any minimal compact complex surface S with n = b2(S) > 0 containing global spherical shells (GSS) we study the effectiveness of the 2n parameters given by the n blown up points. There exists a family of surfaces S → B with GSS which contains as fibers S, some Inoue-Hirzebruch surface and non minimal surfaces, such that blown up points are generically effective parameters. These families are versal outside a non empty hypersurface T ⊂ B. We deduce that, for any configuration of rational curves, there is a non empty open set in the Oeljeklaus-Toma moduli space such that the corresponding surfaces are defined by a contracting germ in Cremona group, in particular admit a birational structure.

References

  • [1] Baum B., Bott R., Singularities of holomorphic foliations. J. Diff. Geom. 7 (1972) 279-342Google Scholar

  • [2] Banica C., Stanasila O., Algebraic methods in the global theory of complex spaces, John Wiley & Sons, 1976.Google Scholar

  • [3] Bruasse L., Thèse: Stabilité et filtration de Harder-Narasimhan. Université d’Aix-Marseille 1 (2001).Google Scholar

  • [4] Brunella M., Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Scient. Ec. Norm. Sup., 30, p569-594 (1997).Google Scholar

  • [5] Dloussky G., Structure des surfaces de Kato, Mémoire de la S.M.F 112.n°14 (1984).Google Scholar

  • [6] Dloussky G., Une construction élémentaire des surfaces d’Inoue-Hirzebruch. Math. Ann. 280, (1988), 663-682.Google Scholar

  • [7] Dloussky G., Quadratic forms and singularities of genus one or two. Annales de la faculté des sciences de Toulouse vol 20 (2011), p15-69.Google Scholar

  • [8] Dloussky G., From non-Ḱ’ahlerian surfaces to Cremona group of P2(C), arXiv:1206.2518 (2012).Google Scholar

  • [9] Dloussky G., Kohler F., Classification of singular germs of mappings and deformations of compact surfaces of class VII0, Ann. Polonici Mathematici LXX, (1998), 49-83Google Scholar

  • [10] Dloussky G., Oeljeklaus K., Vector fields and foliations on surfaces of class VII0, Ann. Inst. Fourier 49, (1999), 1503-1545Google Scholar

  • [11] Dloussky G., Oeljeklaus K., Surfaces de la classe VII0 et automorphismes de Hénon. C.R.A.S. 328, série I, p.609-612, (1999)Google Scholar

  • [12] Dloussky G., Oeljeklaus K., Toma M., Class VII0 surfaces with b2 curves. Tohoku Math. J. 55, 283-309 (2003).Google Scholar

  • [13] Dloussky G., Teleman A., Infinite bubbling phenomenon in non Kȁhler geometry. Math. Ann. 353, 1283-1314 (2012)Web of ScienceGoogle Scholar

  • [14] Favre Ch., Classification of 2-dimensional contracting rigid germs, Jour. Math. Pures Appl. 79, (2000), 475-514Google Scholar

  • [15] Gauduchon P., Le théorème de l’excentricité nulle. C.R. Acad. Sci. Paris 285, 387-390 (1977).Google Scholar

  • [16] Griffiths P., Harris J., Principles of algebraic geometry, Pure and applied mathematics, John Wiley (1978)Google Scholar

  • [17] Hubbard John H., Oberste-Vorth Ralph W., Hénon mappings in the complex domain. I. Publ. IHES, (79):5-46, 1994.CrossrefGoogle Scholar

  • [18] Inoue M., Kobayashi S., Ochiai T., Holomorphic affine connections on compact complex surfaces. J. Fac. Sci. Univ. Tokyo 27 (1980), 247-264.Google Scholar

  • [19] Klingler B., Structures affines et projectives sur les surfaces complexes. Ann. Inst. Fourier 48, 2 (1998), 441-477.CrossrefGoogle Scholar

  • [20] Kohler F., Feuilletages holomorphes singuliers sur les surfaces contenant une coquille sphérique globale. Ann. Inst. Fourier, 45 (1995), 161-182. Erratum Ann. Inst. Fourier 46 (1996), 589.CrossrefGoogle Scholar

  • [21] Kobayashi S., Differential geometry of complex vector bundles Publ. of the Math. Soc. of Japan 15, Iwanami Shoten and Princeton univ. press (1987).Google Scholar

  • [22] Kobayashi S., Ochiai T., Holomorphic Projective Structures on Compact Complex Surfaces Math. Ann. 249. p75-94 (1980).Google Scholar

  • [23] Lübke, Teleman A., The Kobayashi-Hitchin correspondence. World Scientific 1995.Google Scholar

  • [24] Nakamura I., On surfaces of class VII0 with curves. Invent. Math. 78,(1984), 393-443.Google Scholar

  • [25] Nakamura I., On surfaces of class VII0 with curves II. Tohoku Math. J. 42 (1990), 475-516.Google Scholar

  • [26] Oeljeklaus K., Toma M., Logarithmic moduli spaces for surfaces of class VII, Math. Ann. 341 (2008), 323-345Web of ScienceGoogle Scholar

  • [27] Potters J., On Almost Homogeneous Compact Complex Surfaces. Invent. Math. 8, 244-266 (1969).Google Scholar

  • [28] Teleman A., Donaldson theory on non-Kählerian surfaces and class VII surfaces with b2 = 1, Invent. math. 162, 493-521 (2005)Google Scholar

  • [29] Teleman A., Instantons and curves on class VII surfaces, Annals of Math., 172-3 (2010), 1749-1804.Google Scholar

  • [30] Teleman A., On the torsion of the first direct image of a locally free sheaf. arxiv:1309.0342 (2013) Google Scholar

About the article

Received: 2013-11-03

Accepted: 2014-02-11

Published Online: 2014-04-23

Published in Print: 2014-01-01


Citation Information: Complex Manifolds, Volume 1, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.2478/coma-2014-0001.

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© 2014 Georges Dloussky. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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