Vol. 289, No. 1, 2017

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On the asymptotic behavior of Bergman kernels for positive line bundles

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

Vol. 289 (2017), No. 1, 71–89
DOI: 10.2140/pjm.2017.289.71
Abstract

Let $L$ be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers ${L}^{p}$ of $L$ as $p$ tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on $L$. This situation appears naturally when we approximate a semipositive singular metric by smooth positively curved metrics.

Keywords
Bergman kernel, Dirac operator, Laplacian operator
Primary: 32U15
Secondary: 32L05
Milestones
Received: 8 March 2016
Accepted: 10 October 2016
Published: 12 May 2017
Authors
 Tien-Cuong Dinh Department of Mathematics National University of Singapore 10 Lower Kent Ridge Road Singapore 119076 Singapore Xiaonan Ma Institut de Mathématiques de Jussieu – Paris Rive Gauche UFR de Mathématiques Université Paris Diderot - Paris 7 Case 7012 75205 Paris Cedex 13 France Viêt-Anh Nguyên Laboratoire de Mathematiques Paul Painlevé Université de Lille 1 CNRS UMR 8524 59655 Villeneuve d’Asq CEDEX France