Skip to main content
SpringerLink
Log in

Uniformisation in dimension four: towards a conjecture of Iitaka

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let \(X\) be a compact Kähler manifold whose universal cover is \(\mathbb{C }^n\). A conjecture of Iitaka claims that some finite étale cover of \(X\) is a torus. We prove this conjecture in various cases in dimension four. We also show that in the projective case Iitaka’s conjecture is a consequence of the non-vanishing conjecture.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The induced MMP might extract or flip a divisor on \(S\), cf. [23, Ch.4].

  2. We do not suppose that \(k=n\), this additional flexibility is useful for the induction argument in the proofs below.

References

  1. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boucksom, S., Demailly, J.-P., Păun, M., Peternell, M.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J.A.G. (2004, in press). Preprint, arxiv:0405285

  3. Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37(1), 45–76 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Brunella, M.: A positivity property for foliations on compact Kähler manifolds. Int. J. Math. 17(1), 35–43 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bando, S., Siu, Y.-T.: Stable sheaves and Einstein–Hermitian metrics. In: Geometry and Analysis on Complex Manifolds, pp. 39–50. World Scientific Publishing, River Edge (1994)

  6. Campana, F.: Coréduction algébrique d’un espace analytique faiblement kählérien compact. Invent. Math. 63(2), 187–223 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Campana, F.: Réduction d’Albanèse d’un morphisme propre et faiblement kählérien II. Groupes d’automorphismes relatifs. Compos. Math. 54(3), 399–416 (1985)

    MathSciNet  Google Scholar 

  8. Campana, F.: Fundamental group and positivity of cotangent bundles of compact Kähler manifolds. J. Algebraic Geom. 4(3), 487–502 (1995)

    Google Scholar 

  9. Campana, F.: Connexité abélienne des variétés kählériennes compactes. Bull. Soc. Math. France 126(4), 483–506 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Campana, F.: Orbifolds, special varieties and classification theory: an appendix. Ann. Inst. Fourier (Grenoble) 54(3), 631–665 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Claudon, B., Höring, A.: Compact Kähler manifolds with compactifiable universal cover. Preprint, arXiv:1205.1415 (2012) (With an appendix by Frédéric Campana)

  12. Claudon, B.: Invariance de la \(\Gamma \)-dimension pour certaines familles kählériennes de dimension 3. Math. Z. 266(2), 265–284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Campana, F., Oguiso, K., Peternell, T.: Non-algebraic hyperkähler manifolds. J. Differ. Geom. 85(3), 397–424 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Campana, F., Peternell, T.: Geometric stability of the cotangent bundle and the universal cover of a projective manifold. Bulletin de la S.M.F. 139(1), 41–74 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Campana, F., Zhang, Q.: Compact Kähler threefolds of \(\pi _1\)-general type. In: Recent progress in arithmetic and algebraic geometry. Contemp. Math., vol. 386 , pp. 1–12. American Mathematical Society, Providence (2005)

  16. Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1(3), 361–409 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Demailly, J.-P., Hacon, C.D., Pǎun, M.: Extension theorems, non-vanishing and the existence of good minimal models. arxiv:1012.0493 (2010)

  18. Demailly, J.-P., Peternell, T.: A Kawamata–Viehweg vanishing theorem on compact Kähler manifolds. J. Differ. Geom. 63(2), 231–277 (2003)

    Google Scholar 

  19. Demailly, J.-P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12(6), 689–741 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fischer, G., Forster, O.: Ein Endlichkeitssatz für Hyperflächen auf kompakten komplexen Räumen. J. Reine Angew. Math. 306, 88–93 (1979)

    Google Scholar 

  21. Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Jpn., 779–794 (1978)

  22. Fujiki, A.: On the structure of compact complex manifolds in c. In: Algebraic varieties and analytic varieties (Tokyo, 1981). Adv. Stud. Pure Math., vol. 1, pp. 231–302. North-Holland, Amsterdam (1983)

  23. Fujino, O.: Special termination and reduction to pl flips. In: Flips for 3-folds and 4-folds. Oxford Lecture Ser. Math. Appl., vol. 35, pp. 63–75. Oxford University Press, Oxford (2007)

  24. Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

  25. Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985)

    MathSciNet  MATH  Google Scholar 

  26. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, vol 134. Cambridge University Press, Cambridge (1998)

  27. Kobayashi, S., Ochiai, T.: Meromorphic mappings onto compact complex spaces of general type. Invent. Math. 31(1), 7–16 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kollár, J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113(1), 177–215 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kollár, J. (with 14 coauthors): Flips and abundance for algebraic threefolds. Société Mathématique de France, Paris, 1992. Astérisque No. 211 (1992)

  30. Lai, C.-J.: Varieties fibered by good minimal models. Math. Ann. 350, 533–547 (2011)

    Google Scholar 

  31. Miyaoka, Y.: Abundance conjecture for \(3\)-folds: case \(\nu =1\). Compos. Math. 68(2), 203–220 (1988)

    MathSciNet  MATH  Google Scholar 

  32. Mok, N.: Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups. Ann. Inst. Fourier (Grenoble) 50(2), 633–675 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nakayama, N.: Compact Kähler manifolds whose universal covering spaces are biholomorphic to \({ C}^n\). RIMS preprint, 1230 (1999)

  34. Nakayama, N.: Projective algebraic varieties whose universal covering spaces are biholomorphic to \({ C}^n\). J. Math. Soc. Jpn. 51(3), 643–654 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ran, Z.: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1(2), 279–291 (1992)

    MathSciNet  MATH  Google Scholar 

  36. Reid, M.: Nonnormal del Pezzo surfaces. Publ. Res. Inst. Math. Sci. 30(5), 695–727 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ueno, K.: On compact analytic threefolds with nontrivial Albanese tori. Math. Ann. 278(1–4), 41–70 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank the Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds” of the Deutsche Forschungsgemeinschaft for financial support. A. Höring was partially supported by the A.N.R. project “CLASS”, he acknowledges the support of the Albert-Ludwigs-Universität Freiburg where the main part of this work was done.

Author information

Authors and Affiliations

  1. Université Pierre et Marie Curie, 4 Place Jussieu, Case 247, 75252 , Paris Cedex 05, France

    Andreas Höring

  2. Mathematisches Institut, Universität Bayreuth, 75252 , Bayreuth, Germany

    Thomas Peternell

  3. Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 , Tübingen, Germany

    Ivo Radloff

Authors
  1. Andreas Höring
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Peternell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivo Radloff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivo Radloff.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Höring, A., Peternell, T. & Radloff, I. Uniformisation in dimension four: towards a conjecture of Iitaka. Math. Z. 274, 483–497 (2013). https://doi.org/10.1007/s00209-012-1081-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-012-1081-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation