具有复双曲尖点的Kähler流形上的峰截面与Bergman核

Bergman bundles and applications to the geometry of compact complex manifolds

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample Hilbert bundle whose fibers are isomorphic to the standard L${}^2$ Hardy space on the complex unit ball; however the bundle is locally trivial only in the real analytic category, and its complex structure is strongly twisted. We compute the Chern curvature of the Bergman bundle, and show that it is strictly positive. As a potential application, we investigate a l…

Boundary asymptotics of the relative Bergman kernel metric for curves

We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical…

二维Picard模群的一个显式基本区

An explicit fundamental domain for the Picard modular group in two complex dimensions

本文的主要目的是构造作用在复双曲空间${\bf CH}^{2}$上的Picard模群的第一显式基本区.复双曲空间是一个Hermite对称空间，它的有界实现是${\bf C}^2$中的单位球，它具有Bergman度量. Picard模群是$SU(2,1)$的具有高斯整数元的不连续全纯自同构子群.这个基本区具有有限体积、一个尖点、明确给定的边界表面和有趣的对称性。