Bergman kernels and equilibrium measures for polarized pseudoconcave domains

摘要

Let X be a strictly pseudoconcave domain in a closed polarized complexmanifold (Y,L) where L is a (semi-)positive line bundle over Y. Any givenHermitian metric on L, together with a volume form, induces by restriction to Xa Hilbert space structure on the space of global holomorphic sections on Y withvalues in the k:th tensor power of L. In this paper the leading large kasymptotics for the corresponding Bergman kernels and metrics are obtained interms of the curvature of L and of the boundary of X (undere a certaincompatibility assumption). The convergence of the Bergman metrics is obtainedin a very general setting where X is replaced by a compact subset. As anapplication the (generalized) equilibrium measure of the polarizedpseudoconcave domain X is computed explicitely. Applications to the zero andmass distribution of random holomorphic sections and the eigenvaluedistributionof Toeplitz operators will appear elsewhere.