多复变Bergman几何中的若干问题研究

该申请项目主要研究多复变现代Bergman核理论中与复几何、复代数几何密切相关的几个前沿问题，主要内容包括：（1）复流形多典则丛（或者有界域）的L^p型（奇异）Bergman度量，研究度量在奇点附近的渐进性质，并利用渐进性质来构造新的不变完备Kähler度量；（2）有界拟凸域上Bergman度量的d-有界性质及其在超凸指标估计、Bergman距离边界估计中的应用；（3）研究Bergman度量下(p,q)阶L^2上同调群的消没和非消没定理与Bergman度量增长性条件的关系，并尝试给出光滑边界有界拟凸域上Bergman度量的最优增长估计；（4）讨论Bergman度量完备的Stein流形上有界多重次调和函数的存在性问题。

We shall study some recent problems in Bergman geometry, which are closely related to several known questions in Complex geometry and complex algebraic geometry. The main ingredients consist of the following. (1) The L^p type (singular) Bergman metric on the pluricanonical bundles of complex manifolds, especially the asymptotic behaviors of the metric near the singularities. We would like to construct some new invariant complete Kähler metric via the asymptotic behaviors. (2) The d-boundedness property of the Bergman metric, as well as its applications to the estimates of hyperconvex indices and the optimal estimates for the growth of the Bergman metric near the boundary. (3) The vanishing and non-vanishing theorems of the L^2 cohomology with respect to the Bergman metric, and its relations with the growth condition of the Bergman metric. The optimal growth speed of the Bergman metric near the smooth pseudoconvex boundary will also be explored. (4) The existence problem of bounded strictly plurisubharmonic functions on Bergman complete Stein manifolds.