Skip to main content
SpringerLink
Log in

A criterion for the properness of the \(K\)-energy in a general Kähler class

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper, we give a criterion for the properness of the \(K\)-energy in a general Kähler class of a compact Kähler manifold by using Song–Weinkove’s result in (Commun Pure Appl Math 61(2):210–229, 2008). As applications, we give some Kähler classes on \(\mathbb {C}\mathbb {P}^2\#3\overline{\mathbb {C}\mathbb {P}^2}\) and \(\mathbb {C}\mathbb {P}^2\#8\overline{\mathbb {C}\mathbb {P}^2}\) in which the \(K\)-energy is proper. Finally, we prove Song-Weinkove’s result on the existence of critical points of \(\hat{J}\) functional by the continuity method.

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. In Fulton’s terminology, it is called “convex”. We choose this name according to the usual notation.

  2. Just compute the intersection number, or use the fact that \(\underline{R}={Vol(\partial P_L)\over Vol(P_L)}\), where \(Vol(\partial P_L)\) is computed using Donaldson’s special boundary measure, see [8] and [33].

  3. We can also prove this by observing that \((DL(f),\eta )_{L^2}=-\int _X \eta <\frac{\sqrt{-1}}{2} \partial \bar{\partial }f, \omega _t>_{\chi _{t_0}} {\chi _{t_0}^n\over n!}= -\int _X \eta \frac{\sqrt{-1}}{2} \partial \bar{\partial }f\wedge \star \omega _t\). Here \(\star \) is the Hodge star operator associated with \(\chi _{t_0}\). Since \(\partial \) and \(\bar{\partial }\) commute with \(\star \) It is obvious that this equals \(-\int _X f \frac{\sqrt{-1}}{2} \partial \bar{\partial }\eta \wedge \star \omega _t= (f,DL(\eta ))_{L^2}\). A good reference for Hodge star operator on Kähler manifold is [32].

  4. The adaptation of the classical Evans–Krylov theorem to the complex case has been carried out by Siu [22] P100–107.

References

  1. Cheltsov, I., Shramov, K.: Log canonical thresholds of smooth Fano threefolds. Russ. Math. Surv. 63, 859–958 (2008)

    Article  MathSciNet  Google Scholar 

  2. Chen, X.X.: On the lower bound of the Mabuchi energy and its application. Internat. Math. Res. Not. 12, 607–623 (2000)

    Article  Google Scholar 

  3. Chen, X.X., He, W.: The Calabi flow on Kähler Surfaces with bounded Sobolev constant (I). Math. Ann. 354, 227–261 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, X.X., Tian, G.: Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes études Sci. 107, 1–107 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Demailly, J.-P.: On Tian’s invariant and log canonical thresholds. Appendix of Cheltsov and Shramov’s paper Log canonical thresholds of smooth Fano threefolds. Russ. Math. Surv. 63, 945–950 (2008)

    MathSciNet  Google Scholar 

  6. Dervan, R.: Alpha invariants and K-stability for general polarisations of Fano varieties. arXiv:1307.6527

  7. Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3(1), 1–15 (1999)

    MATH  MathSciNet  Google Scholar 

  8. Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geometry 62(2), 289–349 (2002)

    MATH  MathSciNet  Google Scholar 

  9. Fang, H., Lai, X., Ma, X.: On a class of fully nonlinear flows in Kähler geometry. J. Reine Angew. Math. 653, 189–220 (2011)

    MATH  MathSciNet  Google Scholar 

  10. Fang, H., Lai, X., Song, J., Weinkove, B.: The J-flow on Kähler surfaces: a boundary case. arXiv:1204.4068

  11. Fulton, W.: Introduction to Toric Varieties. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  12. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edn. Classics in Mathematics. Springer-Verlag, Berlin (2001)

    Google Scholar 

  13. Guan, B., Li, Q.: A Monge-Ampére type fully nonlinear equation on Hermitian manifolds. Discrete Contin. Dyn. Syst. Ser. B 17(6), 1991–1999 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Guan, B., Li, Q.: The Dirichlet problem for a complex Monge-Ampére type equation on Hermitian manifolds. Adv. Math. 246, 351–367 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Guan, B., Sun, W.: On a class of fully nonlinear elliptic equations on Hermitian manifolds. arXiv:1301.5863

  16. Lejmi, X., Székelyhidi, G.: The J-flow and stability. arXiv:1309.2821

  17. Paul, S.: Hyperdiscriminant polytopes, chow polytopes, and mabuchi energy asymptotics. Ann. Math. 175(1), 255–296 (2012)

    Article  MATH  Google Scholar 

  18. Paul, S.: A numerical criterion for K-energy maps of algebraic manifolds. arXiv:1210.0924

  19. Paul, S.: Stable pairs and coercive estimates for the Mabuchi functional. arXiv:1308.4377

  20. Ross, J.: Unstable products of smooth curves. Invent. Math. 165(1), 153–162 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ross, J.: Priviate communication

  22. Siu, Y.-T.: Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics. DMV Seminar, Bd. 8, Birkhäuser Verlag Basel (1987)

  23. Song, J.: The \(\alpha \)-invariant on toric Fano manifolds. Am. J. Math. 127(6), 1247–1259 (2005)

    Article  MATH  Google Scholar 

  24. Song, J., Weinkove, B.: On the convergence and singularities of the J-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61(2), 210–229 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Song, J., Weinkove, B.: The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type. arXiv:1309.3810

  26. Sun, W.: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, arXiv: 1310.0363

  27. Gang, T.: Canonical Metrics in Kähler Geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2000). (Notes taken by Meike Akveld)

    Google Scholar 

  28. Wang, L.: On the regularity theory of fully nonlinear parabolic equations. I. Commun. Pure Appl. Math. 45(1), 27–76 (1992)

    Article  MATH  Google Scholar 

  29. Wang, L.: On the regularity theory of fully nonlinear parabolic equations. II. Commun. Pure Appl. Math. 45(2), 141–178 (1992)

    Article  MATH  Google Scholar 

  30. Weinkove, B.: Convergence of the J-flow on Kähler surfaces. Commun. Anal. Geom. 12(4), 949–965 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Weinkove, B.: On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differ. Geom. 73(2), 351–358 (2006)

    MATH  MathSciNet  Google Scholar 

  32. Wells, R.O.: Differential Analysis on Complex Manifolds, GTM 65. Springer, Berlin (1980)

    Book  Google Scholar 

  33. Zhou, B., Zhu, X.: Relative K-stability and modified K-energy on toric manifolds. Adv. Math. 219, 1327–1362 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Professor Julius Ross for bringing our attention to this problem and many helpful discussions. We are also grateful to the anonymous referee for valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Haozhao Li

  2. Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, 230026, Anhui, China

    Haozhao Li

  3. Department of Mathematics and Institute of Mathematical Science, Nanjing University, Nanjing, 210093, Jiangsu, China

    Yalong Shi & Yi Yao

Authors
  1. Haozhao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yalong Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yalong Shi.

Additional information

H. Li’s research partially supported by NSFC Grant No. 11001080 and No. 11131007.

Y. Shi’s research partially supported by NSFC Grants No. 11101206, No. 11171143, No. 11171144, No. 11331001 and by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Shi, Y. & Yao, Y. A criterion for the properness of the \(K\)-energy in a general Kähler class. Math. Ann. 361, 135–156 (2015). https://doi.org/10.1007/s00208-014-1073-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-014-1073-z

Search

Navigation