Skip to main content
Log in

The Monge-Ampère operator and geodesics in the space of Kähler potentials

  • Published:
Inventiones mathematicae Aims and scope

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37, 1–44 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blocki, Z.: The complex Monge-Ampère operator and pluripotential theory. Lecture notes available from the author’s website

  4. Catlin, D.: The Bergman kernel and a theorem of Tian. Analysis and geometry in several complex variables (Katata 1997), pp. 1–23, Trends Math. Boston, MA: Birkhäuser 1999

  5. Cegrell, U.: Discontinuité de l’opérateur de Monge-Ampère complexe. C. R. Acad. Sci., Paris, Sér. I 296, 869–871 (1983)

    MATH  MathSciNet  Google Scholar 

  6. Chen, X.X.: The space of Kähler metrics. J. Differ. Geom. 56, 189–234 (2000)

    MATH  Google Scholar 

  7. Chen, X.X., Tian, G.: Geometry of Kähler metrics and foliations by discs. arXiv: math.DG/0409433

  8. Chern, S.S., Levine, H., Nirenberg, L.: Intrinsic norms on a complex manifold. Global Analysis, Papers in honor of K. Kodaira, pp. 119–139. University of Tokyo Press 1969

  9. Demailly, J.P.: Complex analytic and differential geometry. Book available from the author’s website

  10. Donaldson, S.K.: Remarks on gauge theory, complex geometry and 4-manifold topology. Fields Medallists’ lectures. World Sci. Ser. 20th Century Math., vol. 5, pp. 384–403. River Edge, NJ: World Sci. Publishing 1997

  11. Donaldson, S.K.: Symmetric spaces, Kähler geometry, and Hamiltonian dynamics. Am. Math. Soc. Transl. 196, 13–33 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Donaldson, S.K.: Scalar curvature and projective imbeddings I. J. Differ. Geom. 59, 479–522 (2001)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  14. Donaldson, S.K.: Scalar curvature and projective imbeddings II. Q. J. Math. 56, 345–356 (2005)

    Article  MathSciNet  Google Scholar 

  15. Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds. arXiv: math.CV/0401302

  16. Klimek, M.: Pluripotential theory. London Mathematical Society Monographs, New Series, vol. 6. New York: Oxford University Press 1991

  17. Lelong, P.: Fonctions plurisousharmoniques et formes différentielles positives. Paris, London, New York: Gordon & Breach 1968

  18. Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  19. Lu, Z.: On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Am. J. Math. 122, 235–273 (2000)

    MATH  Google Scholar 

  20. Mabuchi, T.: Some symplectic geometry on compact Kähler manifolds. Osaka J. Math. 24, 227–252 (1987)

    MATH  MathSciNet  Google Scholar 

  21. Paul, S.: Geometric analysis of Chow Mumford stability. Adv. Math. 182, 333–356 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Phong, D.H., Sturm, J.: Stability, energy functionals, and Kähler-Einstein metrics. Commun. Anal. Geom. 11, 563–597 (2003). arXiv: math.DG/0203254

    MathSciNet  Google Scholar 

  23. Phong, D.H., Sturm, J.: Scalar curvature, moment maps, and the Deligne pairing. Am. J. Math. 126, 693–712 (2004). arXiv: math.DG/0209098

    MATH  MathSciNet  Google Scholar 

  24. Phong, D.H., Sturm, J.: The Futaki invariant and the Mabuchi energy of a complete intersection. Commun. Anal. Geom. 12, 321–343 (2004). arXiv: math.DG/0312529

    MATH  MathSciNet  Google Scholar 

  25. Semmes, S.: Complex Monge-Ampère and symplectic manifolds. Am. J. Math. 114, 495–550 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990)

    MATH  Google Scholar 

  27. Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. 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)

    MATH  Google Scholar 

  29. Yau, S.T.: Nonlinear analysis in geometry. Enseign. Math., II. Sér. 33, 109–158 (1987)

    MATH  Google Scholar 

  30. Yau, S.T.: Open problems in geometry. Proc. Symp. Pure Math., vol. 54, pp. 1–28. Providence, RI: AMS 1993

  31. Zelditch, S.: The Szegö kernel and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhang, S.: Heights and reductions of semi-stable varieties. Compos. Math. 104, 77–105 (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Phong, D., Sturm, J. The Monge-Ampère operator and geodesics in the space of Kähler potentials. Invent. math. 166, 125–149 (2006). https://doi.org/10.1007/s00222-006-0512-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-006-0512-1

Keywords

Navigation