拟凸域上全纯不变量的研究

Pseudoconvex domains and quasi-homogeneous manifolds are both of the important research fields in several complex variables and complex geometry, which are mainstreams of current international mathematical researches. Some related problems and results give fresh impetus for the developments of several complex variables and also promote the intersection and infiltration between different branches in the mathematics. One of the fundamental problems in several complex variables is to classify the bounded domains in C^n, in which the study of holomorphic invariants plays an important role. The Hua domains are a class of non homogeneous pseudoconvex domains which have abundant analytic and geometric properties. The study of Hua domains, related non homogeneous pseudoconvex domain and generalized quasi-homogeneous manifold will be of great significance. What we will study in this research project includes: the holomorhic invariants on non homogeneous pseudoconvex domains and generalized quasi-homogeneous manifolds, the comparison theorem between the invariant metrics, Lu Qikeng's conjecture, the K?hler immersion submanifolds of infinite complex projective space, Schwarz-Pick type estimates et al. All these researches which we have traditional advantages and special feature in China are inheritances and developments of Hua's theory on classical domains and classical manifolds. Our previous work on this research field has already produced certain international influence and raised some research interests in the world.

拟凸域、拟齐性流形是多复变函数论、复几何等国际数学主流方向的重要研究领域之一。与之相关的一些问题及成果，已成为推动多复变函数论发展的重要因素，也促进了分支学科间的交叉和渗透。域的分类问题是多复变的一个基本问题，全纯不变量的研究对分类问题起到重要作用。华罗庚域是一类非齐性的拟凸域，具有丰富的分析和几何性质，对华罗庚域以及类似的非齐性拟凸域、广义拟齐性流形的研究是很有意义的。本项目将对非齐性拟凸域和广义拟齐性流形上的不变量，比较定理，陆启铿猜想，无穷维复投影空间的K?hler浸入子流形以及Schwarz-Pick 型估计等内容进行研究。这些是在我国有传统优势和特色的研究方向，也是对华罗庚的典型域理论的继承与发展，而且在国际上产生一定的影响并具有一定的规模。

拟凸域、拟齐性流形是多复变函数论、复几何等国际数学主流方向的重要研究领域之一。与之相关的一些问题及研究成果，促进了分支学科间的发展以及各方向间的交叉和渗透。域的分类问题是多复变中的一个基本问题，而全纯不变量的研究对分类问题起到重要作用。华罗庚域作为一类非齐性的拟凸域，具有丰富的分析和几何性质。对华罗庚域的研究，可以为类似的非齐性拟凸域、广义拟齐性流形的研究提供更多的方法和工具。本项目主要研究非齐性拟凸域和广义拟齐性流形上的不变量，比较定理，陆启铿猜想以及无穷维复投影空间的Kahler浸入子流形等内容。主要研究结果如下：给出了一类Reinhardt域的Bergman核函数零点的边界性质；给出了一类复Monge-Ampere方程的精确解；给出了Bergman-Hartogs型域的最大全纯自同构群及其上Kahler-Einstein度量，并给出了其上Bergman度量，Kahler-Einstein度量，Caratheodary度量及Kobayashi度量的比较定理。给出了广义Bergman-Hartogs域的Bergman核函数；给出了非齐性Kahler-Einstein流形到无穷维复投影空间的全纯浸入；给出了具有非紧全纯自同构群的有界强拟凸域上的不变度量。