Acta Mathematica

, Volume 207, Issue 1, pp 1–27

Fekete points and convergence towards equilibrium measures on complex manifolds

  • Robert Berman
  • Sébastien Boucksom
  • David Witt Nyström
Article

Abstract

Building on [BB1] we prove a general criterion for convergence of (possibly singular) Bergman measures towards pluripotential-theoretic equilibrium measures on complex manifolds. The criterion may be formulated in terms of the growth properties of the unit-balls of certain norms on holomorphic sections, or equivalently as an asymptotic minimization property for generalized Donaldson L-functionals. Our result settles in particular a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points and it gives the convergence of Bergman measures towards the equilibrium measure for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.

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References

  1. 1.
    [BT] Bedford, E. & Taylor, B. A., Fine topology, Šilov boundary, and (dd c)n. J. Funct. Anal., 72 (1987), 225–251.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    [B1] Berman, R. J., Bergman kernels for weighted polynomials and weighted equilibrium measures of \( {\mathbb{C}^n} \). Indiana Univ. Math. J., 58 (2009), 1921–1946.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    [B2] — Bergman kernels and equilibrium measures for line bundles over projective manifolds. Amer. J. Math., 131 (2009), 1485–1524.Google Scholar
  4. 4.
    [BB1] Berman, R. J. & Boucksom, S., Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math., 181 (2010), 337–394.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    [BB2] — Equidistribution of Fekete points on complex manifolds. Preprint, 2008. arXiv:0807.0035 [math.CV].Google Scholar
  6. 6.
    [BBGZ] Berman, R. J., Boucksom, S., Guedj, V. & Zeriahi, A., A variational approach to complex Monge–Ampère equations. Preprint, 2009. arXiv:0907.4490 [math.CV].Google Scholar
  7. 7.
    [BW] Berman, R. J. & Witt Nyström, D., Convergence of Bergman measures for high powers of a line bundle. Preprint, 2008. arXiv:0805.2846 [math.CV].Google Scholar
  8. 8.
    [BBLW] Bloom, T., Bos, L., Levenberg, N. &Waldron, S., On the convergence of optimal measures. Constr. Approx., 32 (2010), 159–179.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    [BL1] Bloom, T. & Levenberg, N., Distribution of nodes on algebraic curves in \( {\mathbb{C}^N} \). Ann. Inst. Fourier (Grenoble), 53 (2003), 1365–1385.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    [BL2] — Asymptotics for Christoffel functions of planar measures. J. Anal. Math., 106 (2008), 353–371.Google Scholar
  11. 11.
    [BL3] — Transfinite diameter notions in \( {\mathbb{C}^N} \) and integrals of Vandermonde determinants. Ark. Mat., 48 (2010), 17–40.Google Scholar
  12. 12.
    [Bos] Bos, L., Some remarks on the Fejér problem for Lagrange interpolation in several variables. J. Approx. Theory, 60 (1990), 133–140.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    [Bou] Bouche, T., Convergence de la métrique de Fubini–Study d’un fibré linéaire positif. Ann. Inst. Fourier (Grenoble), 40:1 (1990), 117–130.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    [BEGZ] Boucksom, S., Eyssidieux, P., Guedj, V. & Zeriahi, A., Monge–Ampère equations in big cohomology classes. Acta Math., 205 (2010), 199–262.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    [C] Catlin, D., The Bergman kernel and a theorem of Tian, in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., pp. 1–23. Birkhäuser, Boston, MA, 1999.Google Scholar
  16. 16.
    [Dei] Deift, P. A., Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York, 1999.Google Scholar
  17. 17.
    [Dem] Demailly, J. P., Potential Theory in Several Complex Variables. Manuscript available at www-fourier.ujf-grenoble.fr/~demailly/.
  18. 18.
    [D1] Donaldson, S. K., Scalar curvature and projective embeddings. II. Q. J. Math., 56 (2005), 345–356.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    [D2] — Some numerical results in complex differential geometry. Pure Appl. Math. Q., 5 (2009), 571–618.Google Scholar
  20. 20.
    [GMS] Götz, M., Maymeskul, V. V. & Saff, E.B., Asymptotic distribution of nodes for near-optimal polynomial interpolation on certain curves in \( {\mathbb{R}^2} \). Constr. Approx., 18 (2002), 255–283.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    [GZ] Guedj, V. & Zeriahi, A., Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal., 15 (2005), 607–639.CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    [HKPV] Hough, J. B., Krishnapur, M., Peres, Y. & Virág, B., Determinantal processes and independence. Probab. Surv., 3 (2006), 206–229.CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    [KW] Kiefer, J. & Wolfowitz, J., The equivalence of two extremum problems. Canad. J. Math., 12 (1960), 363–366.CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    [K] Klimek, M., Pluripotential Theory. London Mathematical Society Monographs, 6. Oxford University Press, New York, 1991.Google Scholar
  25. 25.
    [L] Levenberg, N., Approximation in \( {\mathbb{C}^N} \). Surv. Approx. Theory, 2 (2006), 92–140.MATHMathSciNetGoogle Scholar
  26. 26.
    [M] Marzo, J., Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics. J. Funct. Anal., 250 (2007), 559–587. fekete points 27CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    [MO] Marzo, J. & Ortega-Cerdà, J., Equidistribution of Fekete points on the sphere. Constr. Approx., 32 (2010), 513–521.CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    [N] Nguyen, Q. D., Regularity of certain sets in \( {\mathbb{C}^N} \). Ann. Polon. Math., 82 (2003), 219–232.CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    [NZ] Nguyen, T. V. & Zériahi, A., Familles de polynômes presque partout bornées. Bull. Sci. Math., 107 (1983), 81–91.Google Scholar
  30. 30.
    [ST] Saff, E. B. & Totik, V., Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, 316. Springer, Berlin–Heidelberg, 1997.Google Scholar
  31. 31.
    [S] Siciak, J., Families of polynomials and determining measures. Ann. Fac. Sci. Toulouse Math., 9 (1988), 193–211.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    [SW] Sloan, I. H. & Womersley, R. S., Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math., 21 (2004), 107–125.Google Scholar
  33. 33.
    [SUZ] Szpiro, L., Ullmo, E. & Zhang, S., Équirépartition des petits points. Invent. Math., 127 (1997), 337–347.CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    [T] Tian, G., On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom., 32 (1990), 99–130.MATHMathSciNetGoogle Scholar
  35. 35.
    [Y] Yuan, X., Big line bundles over arithmetic varieties. Invent. Math., 173 (2008), 603–649.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    [ZZ] Zeitouni, O. & Zelditch, S., Large deviations of empirical measures of zeros of random polynomials. Int. Math. Res. Not., 20 (2010), 3935–3992.MathSciNetGoogle Scholar
  37. 37.
    [Z] Zelditch, S., Szegő kernels and a theorem of Tian. Int. Math. Res. Not., 6 (1998), 317–331.CrossRefMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2011

Authors and Affiliations

  • Robert Berman
    • 1
  • Sébastien Boucksom
    • 2
  • David Witt Nyström
    • 1
  1. 1.Department of MathematicsChalmers University of Technology and University of GöteborgGöteborgSweden
  2. 2.Institut de MathématiquesCNRS-Université Pierre et Marie CurieParis Cedex 05France

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