极化赝凹域的Bergman核与平衡测度
Bergman kernels and equilibrium measures for polarized pseudoconcave domains
摘要：
设X是闭极化复流形（Y，L）中的严格伪凹区域，其中L是Y上的（半）正线丛.任何给定的L上的埃尔米特度量，连同一个体积形式，通过对X的限制，在Y上的整体全纯截面空间上导出一个希尔伯特空间结构，其值为L的k次张量幂幂。本文根据L的曲率和X的边界（在一定的相容性假设下），得到了相应的Bergman核和度量的领先大k渐近性. Bergman度量的收敛性是在一个非常一般的设置，其中X被替换为一个紧凑的子集。作为应用，我们显式地计算了极化伪凹域X的（广义）平衡测度.随机全纯截面的零点分布和质量分布以及Toeplitz算子的本征值分布的应用将在其他地方出现。
英文摘要：
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.
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