Forms and currents defining generalized p-Kähler structures

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Abstract

This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kähler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant role than “closed” forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.

Keywords

Kähler manifold Balanced manifold SKT manifold SG manifold p-Kähler manifold Positive forms and currents 

Mathematics Subject Classification

Primary 53C55 Secondary 53C56 32J27 

References

  1. 1.
    Alessandrini, L.: Classes of compact non-Kähler manifolds. Comptes Rendus Acad. Sci. Paris Ser. I 349, 1089–1092 (2011)CrossRefMATHGoogle Scholar
  2. 2.
    Alessandrini, L.: Proper modifications of generalized \(p\)-Kähler manifolds. J. Geom. Anal. 27, 947–967 (2017).  https://doi.org/10.1007/s12220-016-9705-z MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alessandrini, L.: Product of generalized \(p\)-Kähler manifolds. Ann. Mat. 196, 1835–1853 (2017).  https://doi.org/10.1007/s10231-017-0640-9 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alessandrini, L., Andreatta, M.: Closed transverse \((p, p)\)-forms on compact complex manifolds. Composit. Math. 61, 181–200 (1987); erratum ibid. 63 (1987), 143Google Scholar
  5. 5.
    Alessandrini, L., Bassanelli, G.: Compact \(p\)-Kähler manifolds. Geom. Dedicata 38, 199–210 (1991).  https://doi.org/10.1007/BF00181219 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Alessandrini, L., Bassanelli, G.: Positive \({\partial }\overline{\partial }\)-closed currents and non-Kähler geometry. J. Geom. Anal. 2, 291–316 (1992).  https://doi.org/10.1007/BF02934583 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Alessandrini, L., Bassanelli, G.: On the embedding of 1-convex manifolds with 1-dimensional exceptional set. Ann. Inst. Fourier 51, 99–108 (2001).  https://doi.org/10.5802/aif1818 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Alessandrini, L., Bassanelli, G., Leoni, M.: 1-convex manifolds are p-Kähler. Abh. Math. Sem. Univ. Hambg. 72, 255–268 (2002).  https://doi.org/10.1007/BF02941676 CrossRefMATHGoogle Scholar
  9. 9.
    Andreotti, A., Grauert, H.: Theoremes de finitude pour la cohomologie des espaces complexes. Bull. SMF 90, 153–259 (1962)MathSciNetMATHGoogle Scholar
  10. 10.
    Barlet, D.: Convexité de l’espace des cycles. Bull. SMF 106, 373–397 (1978)MathSciNetMATHGoogle Scholar
  11. 11.
    Bigolin, B.: Gruppi di Aeppli. Ann. SNS Pisa 23, 259–287 (1969)MathSciNetMATHGoogle Scholar
  12. 12.
    Bigolin, B.: Osservazioni sulla coomologia del \({\partial }\overline{\partial }\). Ann. SNS Pisa 24, 571–583 (1970)MathSciNetGoogle Scholar
  13. 13.
    Cassa, A.: Coomologia separata sulle varietá analitiche complesse. Ann. SNS Pisa 25, 291–323 (1971)MathSciNetGoogle Scholar
  14. 14.
    Cassa, A.: New spectral sequences for duality theorems on complex analytic spaces. Boll. UMI 18–B, 429–453 (1981)MathSciNetMATHGoogle Scholar
  15. 15.
    Coltoiu, M.: On the embedding of 1-convex manifolds with 1-dimensional exceptional set. Comment. Math. Helv. 60, 458–465 (1985)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Coltoiu, M., Mihalache, N.: Strongly plurisubharmonic exaustion functions on 1-convex spaces. Math. Ann. 270, 63–68 (1985)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Demailly, J. P.: Complex Analytic and Algebraic Geometry, free accessible book http://www.fourier.ujf-grenoble.fr/demailly/books.html
  18. 18.
    Egidi, N.: Special metrics on compact complex manifolds. Differ. Geom. Appl. 14, 217–234 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fino, A., Tomassini, A.: Blow-ups and resolutions of strong Kähler with torsion metrics. Adv. Math. 221, 914–935 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Fino, A., Tomassini, A.: A survey on strong KT structures. Bull. Math. Soc. Sci. Math. Roum. 52(100), 99–116 (2009)MathSciNetMATHGoogle Scholar
  21. 21.
    Gauduchon, P.: Le théoreme de l’excentricité nulle. Comptes Rendus Acad. Sci. Paris Ser. A 285, 387–390 (1977)MATHGoogle Scholar
  22. 22.
    Grauert, H., Remmert, G.: Über kompakte homogene komplexe Mannigfaltigkeiten. Arch. Math. 13, 498–507 (1962)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Griffiths, Ph.A., Harris, J.: Principles of Algebraic Geometry. Reprint of the 1978 original, Wiley Classics Library. Wiley, New York (1994)Google Scholar
  24. 24.
    Harvey, R., Knapp, A.W.: Positive \((p,p)\)-forms, Wirtinger’s inequality and currents. In: Proceedings of Tulane University 1972–73, pp. 43–62. Dekker, New York (1974)Google Scholar
  25. 25.
    Harvey, R., Lawson, H.B.: An intrinsic characterization of Kähler manifolds. Invent. Math. 74, 169–198 (1983)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Michelsohn, M.L.: On the existence of special metrics in complex geometry. Acta Math. 149, 261–295 (1982)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Norguet, F., Siu, Y.T.: Holomorphic convexity of spaces of analytic cycles. Bull. SMF 105, 191–223 (1977)MathSciNetMATHGoogle Scholar
  28. 28.
    Popovici, D.: Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics. Invent. Math. 194, 515–534 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Popovici, D.: Stability of strongly Gauduchon manifolds under modifications. J. Geom. Anal. 23, 653–659 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Popovici, D.: Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds. Bull. SMF 143, 763–800 (2015)MathSciNetMATHGoogle Scholar
  31. 31.
    Popovici, D., Ugarte, L.: The sGG Class of Compact Complex Manifolds. arXiv e-print arXiv:1407.5070v1 [math.DG]
  32. 32.
    Schäfer, H.H.: Topological Vector Spaces. Graduate Texts in Mathematics, vol. 3. Springer, New York (1970)Google Scholar
  33. 33.
    Serre, J.P.: Un théoreme de dualité. Comment. Math. Helv. 29, 9–26 (1955)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 16, 3101–3133 (2010)MathSciNetMATHGoogle Scholar
  35. 35.
    Sullivan, D.: Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–255 (1976)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Xiao, J.: On strongly Gauduchon metrics of compact complex manifolds. J. Geom. Anal. 25, 2011–2027 (2015)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Yachou, A.: Une classe de variétés semi-kählériennes. Comptes Rendus Acad. Sci. Paris Ser. I 321, 763–765 (1995)MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche Fisiche e InformaticheUniversità di ParmaParmaItaly

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