Equidistribution speed for Fekete points  associated with an ample line bundle

Dinh, Tien-Cuong; Ma, Xiaonan; Nguyên, Viêt-Anh

doi:10.24033/asens.2327

Equidistribution speed for Fekete points associated with an ample line bundle
[Vitesse d'équidistribution pour les points de Fekete associés à un fibré en droites ample]

Dinh, Tien-Cuong ; Ma, Xiaonan ; Nguyên, Viêt-Anh

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 545-578.

Résumé
Abstract

Soit $K$ l'adhérence d'un ouvert borné à bord lisse dans $ℂ^{n}$ . Une configuration de Fekete d'ordre $p$ pour $K$ est un sous-ensemble fini de $K$ qui maximise le déterminant de Vandermonde associé aux polynômes de degré $\leq p$ . Un théorème récent de Berman, Boucksom et Witt Nyström implique que les configurations de Fekete sont asymptotiquement équiréparties par rapport à une mesure d'équilibre canonique quand $p \to \infty$ . Nous donnons ici une estimation précise de la vitesse de convergence. Le résultat est aussi valable dans un cadre général des points de Fekete associés à un fibré en droites ample au-dessus d'une variété projective. Notre approche nécessite une estimation nouvelle sur les noyaux de Bergman pour les fibrés en droites et des résultats quantitatifs de la théorie du pluripotentiel qui sont d'intérêt indépendant.

Let $K$ be the closure of a bounded open set with smooth boundary in $ℂ^{n}$ . A Fekete configuration of order $p$ for $K$ is a finite subset of $K$ maximizing the Vandermonde determinant associated with polynomials of degree $\leq p$ . A recent theorem by Berman, Boucksom and Witt Nyström implies that Fekete configurations for $K$ are asymptotically equidistributed with respect to a canonical equilibrium measure, as $p \to \infty$ . We give here an explicit estimate for the speed of convergence. The result also holds in a general setting of Fekete points associated with an ample line bundle over a projective manifold. Our approach requires a new estimate on Bergman kernels for line bundles and quantitative results in pluripotential theory which are of independent interest.

Publié le : 2017-11-12

MR Zbl

DOI : 10.24033/asens.2327

Classification : 32U15 ; 32L05.
Keywords: Fekete points, equilibrium measure, equidistribution, Bergman kernel, Monge-Ampère operator, Bernstein-Markov property.
Mot clés : Points de Fekete, mesure d'équilibre, équidistribution, noyau de Bergman, opérateur de Monge-Ampère, propriété de Bernstein-Markov.

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     author = {Dinh, Tien-Cuong and Ma, Xiaonan and Nguy\^en, Vi\^et-Anh},
     title = {Equidistribution speed for {Fekete} points  associated with an ample line bundle},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {545--578},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
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     year = {2017},
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     mrnumber = {3665550},
     zbl = {1379.32016},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2327/}
}

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Dinh, Tien-Cuong; Ma, Xiaonan; Nguyên, Viêt-Anh. Equidistribution speed for Fekete points  associated with an ample line bundle. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 545-578. doi : 10.24033/asens.2327. http://archive.numdam.org/articles/10.24033/asens.2327/

Bibliographie
Cité par

Ameur, Y.; Ortega-Cerdà, J. Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates, J. Funct. Anal., Volume 263 (2012), pp. 1825-1861 (ISSN: 0022-1236) | DOI | MR | Zbl

Berman, R.; Boucksom, S. Growth of balls of holomorphic sections and energy at equilibrium, Invent. math., Volume 181 (2010), pp. 337-394 (ISSN: 0020-9910) | DOI | MR | Zbl

Berman, R.; Boucksom, S.; Witt Nyström, D. Fekete points and convergence towards equilibrium measures on complex manifolds, Acta Math., Volume 207 (2011), pp. 1-27 (ISSN: 0001-5962) | DOI | MR | Zbl

Berman, R.; Demailly, J.-P., Perspectives in analysis, geometry, and topology (Progr. Math.), Volume 296, Birkhäuser, 2012, pp. 39-66 | DOI | MR | Zbl

Berndtsson, B., Explorations in complex and Riemannian geometry (Contemp. Math.), Volume 332, Amer. Math. Soc., Providence, RI, 2003, pp. 1-17 | DOI | MR | Zbl

Berndtsson, B. Positivity of direct image bundles and convexity on the space of Kähler metrics, J. Differential Geom., Volume 81 (2009), pp. 457-482 http://projecteuclid.org/euclid.jdg/1236604342 (ISSN: 0022-040X) | MR | Zbl

Błocki, Z.; Kołodziej, S. On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 2089-2093 (ISSN: 0002-9939) | DOI | MR | Zbl

Catlin, D., Analysis and geometry in several complex variables (Katata, 1997) (Trends Math.), Birkhäuser, 1999, pp. 1-23 | MR | Zbl

Coman, D.; Ma, X.; Marinescu, G. Equidistribution for sequences of line bundles on normal Kähler spaces (preprint arXiv:1412.8184, to appear in Geometry & Topology ) | MR

Demailly, J.-P. Complex Analytic and Differential Geometry (2012) (preprint http://www-fourier.ujf-grenoble.fr/~demailly/books.html )

Demailly, J.-P., Contributions to complex analysis and analytic geometry (Aspects Math., E26), Friedr. Vieweg, Braunschweig, 1994, pp. 105-126 | MR | Zbl

Dinh, T.-C.; Ma, X.; Marinescu, G. Equidistribution and convergence speed for zeros of holomorphic sections of singular Hermitian line bundles, J. Funct. Anal., Volume 271 (2016), pp. 3082-3110 (ISSN: 0022-1236) | DOI | MR | Zbl

Dinh, T.-C.; Nguyên, V.-A. Characterization of Monge-Ampère measures with Hölder continuous potentials, J. Funct. Anal., Volume 266 (2014), pp. 67-84 (ISSN: 0022-1236) | DOI | MR | Zbl

Donaldson, S. K. Scalar curvature and projective embeddings. II, Q. J. Math., Volume 56 (2005), pp. 345-356 (ISSN: 0033-5606) | DOI | MR | Zbl

Dinh, T.-C.; Sibony, N., Holomorphic dynamical systems (Lecture Notes in Math.), Volume 1998, Springer, Berlin, 2010, pp. 165-294 | DOI | MR | Zbl

Hörmander, L., Progress in Math., 127, Birkhäuser, 1994, 414 pages (ISBN: 0-8176-3799-0) | MR | Zbl

Kiselman, C. O. The partial Legendre transformation for plurisubharmonic functions, Invent. math., Volume 49 (1978), pp. 137-148 (ISSN: 0020-9910) | DOI | MR | Zbl

Kiselman, C. O. Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math., Volume 60 (1994), pp. 173-197 (ISSN: 0066-2216) | DOI | MR | Zbl

Leja, F. Propriétés des points extrémaux des ensembles plans et leur application à la représentation conforme, Ann. Polon. Math., Volume 3 (1957), pp. 319-342 (ISSN: 0066-2216) | DOI | MR | Zbl

Leja, F. Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme, Ann. Polon. Math., Volume 4 (1957), pp. 8-13 (ISSN: 0066-2216) | DOI | MR | Zbl

Levenberg, N. Weighted Pluripotential Theory Results of Berman-Boucksom (preprint arXiv:1010.4035 )

Levenberg, N. Approximation in $ℂ^{N}$ , Surv. Approx. Theory, Volume 2 (2006), pp. 92-140 (ISSN: 1555-578X) | MR | Zbl

Lindholm, N. Sampling in weighted $L^{p}$ spaces of entire functions in $ℂ^{n}$ and estimates of the Bergman kernel, J. Funct. Anal., Volume 182 (2001), pp. 390-426 (ISSN: 0022-1236) | DOI | MR | Zbl

Lev, N.; Ortega-Cerdà, J. Equidistribution estimates for Fekete points on complex manifolds, J. Eur. Math. Soc. (JEMS), Volume 18 (2016), pp. 425-464 (ISSN: 1435-9855) | DOI | MR | Zbl

Ma, X.; Marinescu, G., Progress in Math., 254, Birkhäuser, 2007, 422 pages (ISBN: 978-3-7643-8096-0) | MR | Zbl

Ma, X.; Marinescu, G. Exponential estimate for the asymptotics of Bergman kernels, Math. Ann., Volume 362 (2015), pp. 1327-1347 (ISSN: 0025-5831) | DOI | MR | Zbl

Pawłucki, W.; Pleśniak, W. Markov's inequality and $C^{\infty}$ functions on sets with polynomial cusps, Math. Ann., Volume 275 (1986), pp. 467-480 (ISSN: 0025-5831) | DOI | MR | Zbl

Siciak, J. Extremal plurisubharmonic functions in $ℂ^{n}$ , Ann. Polon. Math., Volume 39 (1981), pp. 175-211 (ISSN: 0066-2216) | DOI | MR | Zbl

Saff, E. B.; Totik, V., Grundl. math. Wiss., 316, Springer, Berlin, 1997, 505 pages (ISBN: 3-540-57078-0) | DOI | MR | Zbl

Tian, G. On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., Volume 32 (1990), pp. 99-130 http://projecteuclid.org/euclid.jdg/1214445039 (ISSN: 0022-040X) | MR | Zbl

Triebel, H., Johann Ambrosius Barth, Heidelberg, 1995, 532 pages (ISBN: 3-335-00420-5) | MR | Zbl

Zaharjuta, V. P. Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernšteĭn-Walsh theorem for functions of several complex variables, Proceedings of the Sixth Conference on Analytic Functions (Krakow, 1974), Volume 33 (1976/77), pp. 137-148 (ISSN: 0066-2216) | MR | Zbl

Zelditch, S. Szegő kernels and a theorem of Tian, Int. Math. Res. Not., Volume 1998 (1998), pp. 317-331 (ISSN: 1073-7928) | DOI | MR | Zbl

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