2006/08/09 by Robert J. Berman, Berman, Robert · 1 citation
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.math/0608226
openalex publication_date 2006/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a strictly pseudoconcave domain in a closed polarized complex manifold (Y,L) where L is a (semi-)positive line bundle over Y. Any given Hermitian metric on L, together with a volume form, induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k:th tensor power of L. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in terms of the curvature of L and of the boundary of X (undere a certain compatibility assumption). The convergence of the Bergman metrics is obtained in a very general setting where X is replaced by a compact subset. As an application the (generalized) equilibrium measure of the polarized pseudoconcave domain X is computed explicitely. Applications to the zero and mass distribution of random holomorphic sections and the eigenvaluedistribution of Toeplitz operators will appear elsewhere.