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

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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 h 0,1 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.

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References

  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)

    Chapter  Google Scholar 

  2. Boucksom, S.: On the volume of a line bundle. Int. J. Math. 13(10), 1043–1063 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier 49(1), 287–302 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Demailly, J.-P.: Champs magnétiques et inégalités de Morse pour la d″-cohomologie. Ann. Inst. Fourier (Grenoble) 35, 189–229 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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. Demailly, J.-P., Paun, M.: Numerical charaterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3), 1247–1274 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gauduchon, P.: Le théorème de l’excentricité nulle. C. R. Acad. Sci. Paris, Sér. A 285, 387–390 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Harvey, R., Lawson, H.B.: An intrinsic characterization of Kähler manifolds. Invent. Math. 74, 169–198 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hironaka, H.: An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures. Ann. Math. (2) 75(1), 190–208 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ji, S., Shiffman, B.: Properties of compact complex manifolds carrying closed positive currents. J. Geom. Anal. 3(1), 37–61 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kodaira, K.: Complex Manifolds and Deformations of Complex Structures. Grundlehren der Math. Wiss., vol. 283. Springer, Berlin (1986)

    Book  Google Scholar 

  12. Lamari, A.: Courants kählériens et surfaces compactes. Ann. Inst. Fourier 49(1), 263–285 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149(3–4), 261–295 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Moishezon, B.G.: On n-dimensional compact varieties with n algebraically independent meromorphic functions. Transl. Am. Math. Soc. 63, 51–177 (1967)

    MATH  Google Scholar 

  15. Popovici, D.: Regularisation of currents with mass control and singular Morse inequalities. J. Differ. Geom. 80, 281–326 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Siu, Y.-T.: Every K3 surface is Kähler. Invent. Math. 73, 139–150 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Siu, Y.-T.: An effective Matsusaka big theorem. Ann. Inst. Fourier 43(5), 1387–1405 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

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Popovici, D. Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics. Invent. math. 194, 515–534 (2013). https://doi.org/10.1007/s00222-013-0449-0

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  • DOI: https://doi.org/10.1007/s00222-013-0449-0

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