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Probability Measures Associated to Geodesics in the Space of Kähler Metrics

  • Bo BerndtssonEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)

Abstract

We associate certain probability measures on \({\mathbb R}\) to geodesics in the space \({\mathcal H}_L\) of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on \(H^0(X, kL)\). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in \({\mathcal H}_L\) as k goes to infinity. The convergence of second order moments implies a recent result of Chen and Sun on geodesic distances in the respective spaces, while the convergence of first order moments gives convergence of Donaldson’s Z-functional to the Aubin–Yau energy. We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm.

References

  1. 1.
    Arezzo, C., Tian, G.: Infinite geodesic rays in the space of Kähler potentials. Ann. Sc. Norm. Super. Pisa. Cl. Sci. 2(5), 617–630 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Berndtsson, B.: Positivity of direct image bundles and convexity on the space of Kähler metrics. J. Differ. Geom. 3, 81 (2009)zbMATHGoogle Scholar
  3. 3.
    Bouche, T.: Convergence de la métrique de Fubini-Study d’un fibré linéaire positif. Ann. Inst. Fourier (Grenoble) 40(1), 117–130 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boutet de Monvel, L., Guillemin, V.: The spectral Theory of Toeplitz Operators. Annals of Mathematics Studies, vol. 99. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1981)Google Scholar
  5. 5.
    Błocki, Z.: On geodesics in the space of Kähler metrics. Conference in “Geometry" dedicated to Shing-Tung Yau (Warsaw, April 2009). In: Janeczko, S., Li, J., Phong, D. (eds.) Advances in Geometric Analysis. Advanced Lectures in Mathematics, vol. 21, pp. 3–20. International Press, Vienna (2012)Google Scholar
  6. 6.
    Chen, X.X.: The space of Kähler metrics. J. Differ. Geom. 56(2), 189–234 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, X.X., Sun, S.: Space of Kähler Metrics (V)– Kähler Quantization arXiv:0902.4149
  8. 8.
    Darvas, T.: Weak geodesic rays in the space of Kähler metrics and the class \(E(X,\omega _0)\). arXiv:1307.7318
  9. 9.
    Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1, 361–409 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Donaldson, S.K.: Scalar curvature and projective embeddings. I. J. Differ. Geom. 59(3), 479–522 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Donaldson, S.K.: Scalar curvature and projective embeddings. II. Q. J. Math. 56(3), 345–356 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lempert, L., Vivas, L.: Geodesics in the space of Kähler metrics. Duke Math. J. 162, 1369–1381 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mabuchi, T.: \(K\)-energy maps integrating Futaki invariants. Tohoku Math. J. (2) 38(4), 575–593 (1986)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Phong, D.H., Sturm, J.: The Monge-Ampere operator and geodesics in the space of Kähler potentials. Invent. Math. 166(1), 125–149 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rochberg, R.: Interpolation of Banach spaces and negatively curved vector bundles. Pac. J. Math. 110(2), 355–376 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rubinstein, Y.: Geometric Quantization and Dynamical Constructions on the Space of Kähler Metrics. MIT Thesis (2008)Google Scholar
  17. 17.
    Rubinstein, Y., Zelditch, S.: Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties. J. Symplectic Geom. 8, 239–265 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Semmes, S.: Complex Monge-Ampäre and symplectic manifolds. Am. J. Math. 114(3), 495–550 (1992)CrossRefGoogle Scholar
  19. 19.
    Semmes, S.: Interpolation of Banach spaces, differential geometry and differential equations. Rev. Mat. Iberoamericana 4(1), 155–176 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Song, J., Zelditch, S.: Bergman metrics and geodesics in the space of Kähler metrics on toric varieties Anal. PDE 3, 295–358 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32(1), 99–130 (1990)CrossRefGoogle Scholar
  22. 22.
    Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsChalmers University of TechnologyGöteborgSweden

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