多复变Bergman核函数渐近展开及相关问题研究

本项目的课题属于多复变与复几何,是现代数学的主流方向之一。Bergman核函数及其渐进展开在Kähler流形的量子化,紧Kähler流形上Kähler-Einstein度量和常曲率Kähler度量的存在性研究中起着重要作用。本项目将研究Bergman核函数渐近展开以及相关方面，主要围绕下面三个问题：用矩映射(the moment mapping)的方法研究紧和非紧环流形上的不变平衡度量存在性和它的正则量子化、结合Calabi方法与矩映射研究全纯向量丛上的平衡度量存在性和它的正则量子化、在给定Kähler上同调类上研究Bergman核函数渐近展开系数为常数的Kähler度量的存在性。本项目瞄准多复变的热点和难点问题进行探索，有望获得显著进展并产生丰硕的研究成果。

The subject of this project belongs to several complex variables and complex geometry, which is one of the active topics in the modern mathematics. The Bergman functions and its asymptotic expansions play important roles in the quantizations of Kähler manifolds, and the existence of the Kähler-Einstein metrics and the constant curvature Kähler metrics on compact Kähler manifolds. This project will study the asymptotic expansions of Bergman functions and related problems, especially such problems as the following: using the moment mapping to study the existence of invariant balanced metrics on compact and non-compact toric manifolds and their regular quantizations; combining the Calabi's method and the moment mapping to study the existence of balanced metrics on some holomorphic vector bundles and their regular quantizations; for a given Kähler cohomology class, study the existence of Kähler metrics which Bergman function expansion coefficients are constants. This project aims to explore the hot and difficult problems in several complex variables and complex geometry, and is expected to achieve remarkable progress and produce fruitful research results.