Inventiones mathematicae

, Volume 194, Issue 3, pp 515–534 | Cite as

Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics

Article

Abstract

This paper is intended to start a series of works aimed at proving that if in a (smooth) complex analytic family of compact complex manifolds all the fibres, except one, are supposed to be projective, then the remaining (limit) fibre must be Moishezon. A new method of attack, whose starting point originates in Demailly’s work, is introduced. While we hope to be able to address the general case in the near future, two important special cases are established here: the one where the Hodge numbers h0,1 of the fibres are supposed to be locally constant and the one where the limit fibre is assumed to be a strongly Gauduchon manifold. The latter is a rather weak metric assumption giving rise to a new, rather general, class of compact complex manifolds that we hereby introduce and whose relevance to this type of problems we underscore.

References

  1. 1.
    Barlet, D.: Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie. In: Fonctions de Plusieurs Variables Complexes, II (Sém. François Norguet, 1974–1975). Lecture Notes in Math., vol. 482, pp. 1–158. Springer, Berlin (1975) CrossRefGoogle Scholar
  2. 2.
    Boucksom, S.: On the volume of a line bundle. Int. J. Math. 13(10), 1043–1063 (2002) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier 49(1), 287–302 (1999) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Demailly, J.-P.: Champs magnétiques et inégalités de Morse pour la d″-cohomologie. Ann. Inst. Fourier (Grenoble) 35, 189–229 (1985) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Demailly, J.-P.: Singular hermitian metrics on positive line bundles. In: Hulek, K., Peternell, T., Schneider, M., Schreyer, F. (eds.) Proceedings of the Bayreuth Conference Complex Algebraic Varieties, April 2–6, 1990. Lecture Notes in Math., vol. 1507. Springer, Berlin (1992) Google Scholar
  6. 6.
    Demailly, J.-P., Paun, M.: Numerical charaterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gauduchon, P.: Le théorème de l’excentricité nulle. C. R. Acad. Sci. Paris, Sér. A 285, 387–390 (1977) MathSciNetMATHGoogle Scholar
  8. 8.
    Harvey, R., Lawson, H.B.: An intrinsic characterization of Kähler manifolds. Invent. Math. 74, 169–198 (1983) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hironaka, H.: An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures. Ann. Math. (2) 75(1), 190–208 (1962) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ji, S., Shiffman, B.: Properties of compact complex manifolds carrying closed positive currents. J. Geom. Anal. 3(1), 37–61 (1993) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kodaira, K.: Complex Manifolds and Deformations of Complex Structures. Grundlehren der Math. Wiss., vol. 283. Springer, Berlin (1986) CrossRefGoogle Scholar
  12. 12.
    Lamari, A.: Courants kählériens et surfaces compactes. Ann. Inst. Fourier 49(1), 263–285 (1999) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Moishezon, B.G.: On n-dimensional compact varieties with n algebraically independent meromorphic functions. Transl. Am. Math. Soc. 63, 51–177 (1967) MATHGoogle Scholar
  15. 15.
    Popovici, D.: Regularisation of currents with mass control and singular Morse inequalities. J. Differ. Geom. 80, 281–326 (2008) MathSciNetMATHGoogle Scholar
  16. 16.
    Siu, Y.-T.: Every K3 surface is Kähler. Invent. Math. 73, 139–150 (1983) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Siu, Y.-T.: An effective Matsusaka big theorem. Ann. Inst. Fourier 43(5), 1387–1405 (1993) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yau, S.T.: On the Ricci curvature of a complex Kähler manifold and the complex Monge-Ampère equation I. Commun. Pure Appl. Math. 31, 339–411 (1978) CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance

Personalised recommendations