多复变Hartogs区域上的逆紧映射

This project mainly involves the study of the holomorphicity of proper harmonic mappings and rigidity of proper holomorphic mappings on Hartogs domains in several complex variables, which is a frontier topic in the fields of several complex variables, complex geometry and CR geometry. The applicant of the project has obtained some important results in Hartogs domains in several complex variables. In this project, we want to make further study about the holomorphicity of proper harmonic mappings on Hartogs domains in several complex variables and the rigidity of proper holomorphic mappings between Hartogs domains in several complex variables, especially the rigidity of proper holomorphic mappings in nonequidimensional case. We aim at the hot and difficult problems in several complex variables and complex geometry and expect to make great progress.

本项目重点研究多复变Hartogs区域上的不变度量下的逆紧调和映射的全纯性和逆紧全纯映射的刚性，属于多复变、复几何与CR几何的交叉前沿课题。本项目申请人在多复变Hartogs区域的研究领域已有充分的前期研究工作基础。我们希望在自己的工作基础上进一步研究多复变Hartogs区域上不变度量下的逆紧调和映射的全纯性以及多复变Hartogs区域上的逆紧全纯映射的刚性，特别是不等维数情形下的逆紧全纯映射的刚性。本项目采用的方法新颖、应用性强，瞄准多复变与复几何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。

本项目主要完成下面四项工作：（1）证明了从纤维的维数等于1的Cartan-Hartogs域到自身的逆紧全纯映射的刚性定理；（2）对不必覆盖单位球的拟椭球证明了其全纯自同构的局部化原理成立的充要条件；（3）证明了复射影空间中的全纯曲线交超曲面的第二基本定理和亏量关系，解决了P. Griffiths在1972年提出的一个猜想；（4）证明了两个复射影空间的乘积空间中的全纯曲线交超曲面的第二基本定理。