Equidistribution speed for Fekete points associated with an ample line bundle

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.

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 $\le p$. 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 $\le p$. 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.

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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