Skip to main content
SpringerLink
Log in

Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Let M be a compact Kähler manifold and N be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler–Einstein metrics on \(M-N\). As an application, let E be an exceptional divisor of M. Then \(M-E\) cannot admit any complete Kähler–Einstein metric if blow-down of E is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.

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

References

  1. Berman, R.J., Guenancia, H.: Kähler–Einstein metrics on stable varieties and log canonical pairs. Geom. Funct. Anal. 24(6), 1683–1730 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge–Ampére equations in big cohomology classes. Acta Math. 205(2), 199–262 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, S.-Y., Yau, S.T.: On the existence of a complete Kähler metric in non-compact complex manifolds and the regularity of Fefferman’ss equation. Commun. Pure Appl. Math. 33, 507–544 (1980)

    Article  MATH  Google Scholar 

  4. Fu, J., Yau, S.-T., Zhou, W.: Complete cscK metrics on the local models of the conifold transition. Commun. Math. Phys. 335, 1215–1233 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fu, J., Yau, S.-T., Zhou, W.: On complete cscK metrics with Poincaré-Mok-Yau asymptotic property. Commum. Anal. Geom. 24(3), 521–557 (2016)

    Article  MATH  Google Scholar 

  6. Greb, D., Kebekus, S., Kovacs, S., Peternell, T.: Differential Forms on Log Canonical Spaces. Publ. Math. Inst. Hautes Études Sci. No. 114, 87–169 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guenancia, H.: Kähler–Einstein metrics with cone singularities on klt pairs. Int. J. Math. 24(05), 1350035 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Igusa, J.: On the desingularization of Satake compactifications. In: Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics. American Mathematical Society, pp. 301–305 (1966)

  9. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  10. Kollár, J.: Singularities of pairs. arXiv:alg-geom/9601026 (preprint)

  11. Mok, N., Yau, S.-T.: Completeness of the Kähler–Einstein metric on bound domains and the Characterization of domain of holomorphy by curvature condition. In: Proceedings of Symposia in Pure mathematics, vol. 39, Part 1, pp. 41–59 (1983)

  12. Philippe, E., Guedj, V., Zeriahi, A.: Singular Käler–Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009)

    Article  MATH  Google Scholar 

  13. Reid, M.: Young person’s guide to canonical singularities. Proc. Symp. Pure Math. 46, 343–416 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Reid, M.: Canonical 3-folds, Journées de Géométrie Algébrique d’Angers. In: Beauville, A. (ed.) Sijthoff and Noordhoff, Alphen aan den Rijn, pp. 273–310 (1980)

  15. Tian, G., Yau, S.T.: Complete Kähler manifolds with zero Ricci curvature. I. J. Am. Math. Soc. 3, 579–609 (1990)

    MATH  Google Scholar 

  16. Tian, G., Yau, S.T.: Complete Kähler manifolds with zero Ricci curvature. II. Invent. Math. 106, 27–60 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tian, G., Yau, S.T.: Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry. In: Advances in Theoretical and Mathematical Physics. Mathematical aspects of string theory (San Diego, Calif, 1986), pp. 574–628 (1987)

  18. Wu, D.: Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds. Commun. Anal. Geom. 16(2), 395–435 (2008)

    Article  MATH  Google Scholar 

  19. Yau, S.-T.: Some function theoretic properties of complete Riemannian manifolds and their application to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)

    Article  MATH  Google Scholar 

  21. Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure. Appl. Math. 31, 339–411 (1978)

    Article  MATH  Google Scholar 

  22. Yau, S.-T.: The role of partial differential equations in differential geometry. In: Proceedings of the International Congress of Mathematicians, Helsinki (1978)

Download references

Acknowledgments

We thank Chenglong Yu for discussions on material in Sect. 2, where he made a critical observation for Lemma 2.1. P. Gao and S.-T. Yau are supported by NSF Grants DMS-1308244 and PHY-0937443, and W. Zhou is partially supported by China Postdoctoral Science Foundation Grant No. 2015M571479.

Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Peng Gao & Shing-Tung Yau

  2. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Wubin Zhou

Authors
  1. Peng Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shing-Tung Yau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wubin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wubin Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, P., Yau, ST. & Zhou, W. Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds. Math. Ann. 369, 1271–1282 (2017). https://doi.org/10.1007/s00208-016-1486-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-016-1486-y

Keywords

Search

Navigation