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On the Cohomology of Pseudoeffective Line Bundles

  • Jean-Pierre DemaillyEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 10)

Abstract

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact Kähler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general Kähler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1, 1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact Kähler threefolds without nontrivial subvarieties, following a joint work with F. Campana and M. Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.

Keywords

Line Bundle Cohomology Class Bergman Kernel Lelong Number Analytic Subvariety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bayer, D.A.: The division algorithm and the Hilbert scheme. ProQuest LLC, Ann Arbor (1982). Thesis (Ph.D.)–Harvard UniversityGoogle Scholar
  2. 2.
    Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge-Ampère equations in big cohomology classes. Acta Math. 205(2), 199–262 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Berndtsson, B.: The openness conjecture for plurisubharmonic functions (2013). ArXiv e-prints, math.CV/1305.5781Google Scholar
  4. 4.
    Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44(2), 449–494 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bierstone, E., Milman, P.D.: Relations among analytic functions. I. Ann. Inst. Fourier (Grenoble) 37(1), 187–239 (1987)Google Scholar
  6. 6.
    Bierstone, E., Milman, P.D.: Uniformization of analytic spaces. J. Am. Math. Soc. 2(4), 801–836 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Boucksom, S.: Cônes positifs des variétés complexes compactes. Ph.D. thesis, Grenoble (2002). https://tel.archives-ouvertes.fr/tel-00002268/
  8. 8.
    Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37(1), 45–76 (2004)Google Scholar
  9. 9.
    Brunella, M.: Uniformisation of foliations by curves. In: Holomorphic Dynamical Systems. Volume 1998 of Lecture Notes in Mathematics, pp. 105–163. Springer, Berlin (2010). Also available from arXiv e-prints, math.CV/0802.4432Google Scholar
  10. 10.
    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37(1), 1–44 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Campana, F.: Algébricité et compacité dans l’espace des cycles d’un espace analytique complexe. Math. Ann. 251(1), 7–18 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Campana, F.: Réduction d’Albanese d’un morphisme propre et faiblement kählérien. II. Groupes d’automorphismes relatifs. Compos. Math. 54(3), 399–416 (1985)MathSciNetGoogle Scholar
  13. 13.
    Cao, J.: Théorèmes d’annulation et théorèmes de structure sur les variétés kähleriennes compactes. Ph.D. thesis, Grenoble (2013)Google Scholar
  14. 14.
    Cao, J.: Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact Kähler manifolds. Compos. Math. (2014). (to appear, cf. arXiv e-prints: math.AG/1210.5692)Google Scholar
  15. 15.
    Cao, J.: Additivity of the approximation functional of currents induced by Bergman kernels. C. R. Math. Acad. Sci. Paris (2015). (to appear, cf. doi:10.1016/j.crma.2014.11.004, arXiv e-prints: math.AG/1410.8288)Google Scholar
  16. 16.
    Campana, F., Demailly, J.-P., Verbitsky, M.: Compact Kähler 3-manifolds without non-trivial subvarieties (2013). ArXiv e-prints, math.CV/1305.5781Google Scholar
  17. 17.
    Campana, F., Höring, A., Peternell, T.: Abundance for Kähler threefolds (2014). ArXiv e-prints, math.AG/1403.3175Google Scholar
  18. 18.
    Demailly, J.-P., Ein, L., Lazarsfeld, R.: A subadditivity property of multiplier ideals. Michigan Math. J. 48, 137–156 (2000). Dedicated to William Fulton on the occasion of his 60th birthday.Google Scholar
  19. 19.
    Demailly, J.-P.: Estimations L 2 pour l’opérateur \(\bar{\partial }\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Sup. (4), 15(3), 457–511 (1982)Google Scholar
  20. 20.
    Demailly, J.-P.: Champs magnétiques et inégalités de Morse pour la d ′ ′-cohomologie. Ann. Inst. Fourier (Grenoble) 35(4), 189–229 (1985)Google Scholar
  21. 21.
    Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebra. Geom. 1(3), 361–409 (1992)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Demailly, J.-P.: A numerical criterion for very ample line bundles. J. Differ. Geom. 37(2), 323–374 (1993)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Demailly, J.-P.: Multiplier ideal sheaves and analytic methods in algebraic geometry. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000). Volume 6 of ICTP Lecture Notes, pp 1–148. Abdus Salam International Centre for Theoretical Physics, Trieste (2001)Google Scholar
  24. 24.
    Demailly, J.-P.: Complex analytic and differential geometry, online book at: http://www-fourier.ujf-grenoble.fr/~demailly/documents.html. Institut Fourier (2012)
  25. 25.
    de Fernex, T., Ein, L., Mustaţă, M.: Bounds for log canonical thresholds with applications to birational rigidity. Math. Res. Lett. 10(2–3), 219–236 (2003)zbMATHMathSciNetGoogle Scholar
  26. 26.
    de Fernex, T., Ein, L., Mustaţă, M.: Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties. Duke Math. J. 152(1), 93–114 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Demailly, J.-P., Kollár, J.: Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4) 34(4), 525–556 (2001)Google Scholar
  28. 28.
    Demailly, J.-P., Peternell, T.: A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds. In: Surveys in Differential Geometry (Boston, 2002). Surveys in Differential Geometry, vol. VIII, pp. 139–169. International Press, Somerville (2003)Google Scholar
  29. 29.
    Demailly, J.-P., Phạm, H.H.: A sharp lower bound for the log canonical threshold. Acta Math. 212(1), 1–9 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Demailly, J.-P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebra. Geom. 3(2), 295–345 (1994)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Demailly, J.-P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12(6), 689–741 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Volume 150 of Graduate Texts in Mathematics. Springer, New York (1995)Google Scholar
  33. 33.
    Enoki, I.: Strong-Lefschetz-type theorem for semi-positive line bundles over compact Kähler manifolds. In: Geometry and Global Analysis (Sendai, 1993), pp. 211–212. Tohoku University, Sendai (1993)Google Scholar
  34. 34.
    Favre, C., Jonsson, M.: Valuations and multiplier ideals. J. Am. Math. Soc. 18(3), 655–684 (electronic) (2005)Google Scholar
  35. 35.
    Galligo, A.: Théorème de division et stabilité en géométrie analytique locale. Ann. Inst. Fourier (Grenoble) 29(2), vii, 107–184 (1979)Google Scholar
  36. 36.
    Guan, Q., Zhou, X.: Strong openness conjecture for plurisubharmonic functions (2013). ArXiv e-printsGoogle Scholar
  37. 37.
    Guan, Q., Zhou, X.: Strong openness conjecture and related problems for plurisubharmonic functions. ArXiv e-prints, math.CV/1401.7158Google Scholar
  38. 38.
    Guan, Q., Zhou, X.: Effectiveness of demailly’s strong openness conjecture and related problems (2014). ArXiv e-prints, math.CV/1403.7247Google Scholar
  39. 39.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203; ibid. (2), 79, 205–326 (1964)Google Scholar
  40. 40.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Volume 7 of North-Holland Mathematical Library, 3rd edn. North-Holland, Amsterdam (1990); First edition 1966Google Scholar
  41. 41.
    Höring, A., Peternell, T., Radloff, I.: Uniformisation in dimension four: towards a conjecture of iitaka (2011). arXiv e-prints, math.AG/1103.5392, to appear in Math. ZeitschriftGoogle Scholar
  42. 42.
    Jonsson, M., Mustaţă, M.: Valuations and asymptotic invariants for sequences of ideals. Ann. Inst. Fourier (Grenoble) 62(6):2145–2209 (2012)Google Scholar
  43. 43.
    Jonsson, M., Mustaţă, M.: An algebraic approach to the openness conjecture of Demailly and Kollár. J. Inst. Math. Jussieu 13(1), 119–144 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Kim, D.: A remark on the approximation of plurisubharmonic functions. C. R. Math. Acad. Sci. Paris 352(5), 387–389 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Kiselman, C.O.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math. 60(2), 173–197 (1994)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Lazarsfeld, R.: Positivity in Algebraic Geometry. I, II. Volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (2004)Google Scholar
  47. 47.
    Lempert, L.: Modules of square integrable holomorphic germs (2014). ArXiv e-prints, math.CV/1404.0407Google Scholar
  48. 48.
    Moĭšezon, B.G.: On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. I, II, III. Izv. Akad. Nauk SSSR Ser. Mat. 30, 133–174, 345–386, 621–656 (1966)Google Scholar
  49. 49.
    Mourougane, C.: Versions kählériennes du théorème d’annulation de Bogomolov-Sommese. C. R. Acad. Sci. Paris Sér. I Math. 321(11), 1459–1462 (1995)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Nadel, A.M.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. (2), 132(3), 549–596 (1990)Google Scholar
  51. 51.
    Ohsawa, T., Takegoshi, K.: On the extension of L 2 holomorphic functions. Math. Z. 195(2), 197–204 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Peternell, T.: Algebraicity criteria for compact complex manifolds. Math. Ann. 275(4), 653–672 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Peternell, T.: Moishezon manifolds and rigidity theorems. Bayreuth. Math. Schr. 54, 1–108 (1998)zbMATHMathSciNetGoogle Scholar
  54. 54.
    Phạm, H.H.: The weighted log canonical threshold. C. R. Math. Acad. Sci. Paris 352(4), 283–288 (2014)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Phong, D.H., Sturm, J.: On a conjecture of Demailly and Kollár. Asian J. Math. 4(1), 221–226 (2000). Kodaira’s issueGoogle Scholar
  56. 56.
    Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 257–286 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Siu, Y.T.: A vanishing theorem for semipositive line bundles over non-Kähler manifolds. J. Differ. Geom. 19(2), 431–452 (1984)zbMATHMathSciNetGoogle Scholar
  59. 59.
    Siu, Y.T.: Some recent results in complex manifold theory related to vanishing theorems for the semipositive case. In: Workshop Bonn 1984 (Bonn, 1984). Volume 1111 of Lecture Notes in Mathematics, pp. 169–192. Springer, Berlin (1985)Google Scholar
  60. 60.
    Skoda, H.: Application des techniques L 2 à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids. Ann. Sci. École Norm. Sup. (4), 5, 545–579 (1972)Google Scholar
  61. 61.
    Skoda, H.: Sous-ensembles analytiques d’ordre fini ou infini dans \(\mathbb{C}^{n}\). Bull. Soc. Math. Fr. 100, 353–408 (1972)zbMATHMathSciNetGoogle Scholar
  62. 62.
    Takegoshi, K.: On cohomology groups of nef line bundles tensorized with multiplier ideal sheaves on compact Kähler manifolds. Osaka J. Math. 34(4), 783–802 (1997)zbMATHMathSciNetGoogle Scholar
  63. 63.
    Tsuji, H.: Extension of log pluricanonical forms from subvarieties (2007). ArXiv e-printsGoogle Scholar
  64. 64.
    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut Fourier, Université de Grenoble IGrenobleFrance

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