复流形及其全纯向量丛的若干问题研究

本项目拟研究复流形及其全纯向量丛的若干问题：复流形的线性不变量理论，紧Kähler流形的分类与direct image层的挠元问题。复流形的线性不变量理论是多复变与复几何的一个新的热点问题。紧Kähler流形的分类与direct image层的挠元的研究与Kähler几何和代数几何的重要基础性问题的研究有深刻的联系，一直是国际研究的热点问题。在前期工作中，我们得到了特殊的复流形的线性不变量理论、多次调和函数的新刻画、全纯向量丛的Nakano正性的新刻画以及direct image层的正性研究的新方法。本项目将结合这些新方法与Hormander L^2-方法，Ohsawa-Takegoshi型延拓定理，Gromov-Hausdorff极限，foliation理论，多重位势理论等前沿方法和技术对上述三个问题进行研究，进一步推动多复变与复几何相关问题的研究。

The main purpose of this project is to study some problems on complex manifolds and holomorphic vector bundles: linear invariant theory of complex manifolds, classification of compact Kähler manifolds, and torsion of first direct image sheaves. The theory of linear invariants of complex manifolds is a new hot topic in several complex variables and complex geometry. The classification of compact Kähler manifolds and the torsion of direct image sheaves have been a hot issue for a long time, and has attracted a lot of outstanding mathematicians. In our previous work, we obtained the linear invariant of special complex manifolds, new characterizations of plurisubharmonic functions, new characterizations of Nakano positivity of holomorphic vector bundles, and new methods to study the positivity of direct image sheaves. This project will combine these new techniques with Hormander's L^2-estimate, Ohsawa-Takegoshi type extension theorems, Gromov-Hausdorff limit, foliation theory and pluripotential theory to study above three problems, to push the research of related topics in the field of several complex variables and complex geometry.