Equidistribution results for singular metrics on line bundles

Coman, Dan; Marinescu, George

doi:10.24033/asens.2250

Equidistribution results for singular metrics on line bundles
[Résultats d'équidistribution pour métriques singulières sur des fibrés en droites]

Coman, Dan ; Marinescu, George

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 497-536.

Résumé
Abstract

Considérons un fibré holomorphe en droites $L$ muni d'une métrique singulière $h$ au-dessus d'une variété complexe $X$ . Soit $γ_{p}$ le courant de Fubini-Study associé naturellement à l'espace des sections holomorphes de carré intégrable de $L^{\otimes p}$ . En supposant que le lieu singulier de la métrique $h$ est contenu dans un ensemble analytique compact $Σ \subset X$ tel que $codim Σ \geq k$ et que le logarithme du noyau de Bergman associé à $L^{\otimes p} ↾_{X ∖ Σ}$ a l'ordre de croissance $o (p)$ , $p \to \infty$ , nous prouvons que :

1) Les courants $γ_{p}^{k}$ convergent faiblement sur $X$ vers $c_{1} {(L, h)}^{k}$ , où $c_{1} (L, h)$ est le courant de courbure de $h$ .

2) Les moyennes des zéros communs d'un $k$ -vecteur aléatoire de sections holomphes $L^{2}$ -intégrables convergent faiblement dans le sens des courants vers $c_{1} {(L, h)}^{k}$ .

L'hypothèse de croissance du noyau de Bergman est la conséquence de son développement asymptotique dans le cas d'une métrique lisse $h$ . Nous la démontrons ici sous des conditions assez générales. Nous montrons ensuite que nos résultats s'appliquent à nombre de situations géométriques (métriques singulières sur un fibré gros, métriques de Kähler-Einstein sur des ouverts de Zariski, quotients arithmétiques...).

Let $(L, h)$ be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold $X$ . One can define naturally the sequence of Fubini-Study currents $γ_{p}$ associated to the space of $L^{2}$ -holomorphic sections of $L^{\otimes p}$ . Assuming that the singular set of the metric is contained in a compact analytic subset $Σ$ of $X$ and that the logarithm of the Bergman density function of $L^{\otimes p} ↾_{X ∖ Σ}$ grows like $o (p)$ as $p \to \infty$ , we prove the following:

1) the currents $γ_{p}^{k}$ converge weakly on the whole $X$ to $c_{1} {(L, h)}^{k}$ , where $c_{1} (L, h)$ is the curvature current of $h$ .

2) the expectations of the common zeros of a random $k$ -tuple of $L^{2}$ -holomorphic sections converge weakly in the sense of currents to $c_{1} {(L, h)}^{k}$ .

Here $k$ is so that $codim Σ \geq k$ . Our weak asymptotic condition on the Bergman density function is known to hold in many cases, as it is a consequence of its asymptotic expansion. We also prove it here in a quite general setting. We then show that many important geometric situations (singular metrics on big line bundles, Kähler-Einstein metrics on Zariski-open sets, arithmetic quotients) fit into our framework.

Publié le : 2015-09-24

DOI : 10.24033/asens.2250

Classification : 32L10; 32U40, 32W20, 53C55
Keywords: Bergman density function, Fubini-Study currents, singular Hermitian metric, equidistribution of zeros, random holomorphic sections.
Mot clés : Noyau de Bergman, courants de Fubini-Study, métrique hermitienne singulière, équidistribution des zéros, sections holomorphes aléatoires.

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     author = {Coman, Dan and Marinescu, George},
     title = {Equidistribution results for singular metrics on line bundles},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {497--536},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {3},
     year = {2015},
     doi = {10.24033/asens.2250},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2250/}
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Coman, Dan; Marinescu, George. Equidistribution results for singular metrics on line bundles. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 497-536. doi : 10.24033/asens.2250. https://www.numdam.org/articles/10.24033/asens.2250/

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