L2 方法、弱莱夫谢茨定理及全纯叶状结构

本项目将使用L2 方法研究复流形的拓扑，特别地，我们将用L2方法改进弱莱夫谢茨定理以及结合得到的相关结果来研究流形上的全纯叶状结构。另外，作为一个应用，我们还将讨论如何运用弱莱夫谢茨定理及全纯叶状结构来研究凯勒群。. 首先，我们将通过对相关d-bar方程进行更细致的L2估计给出弱莱夫谢茨定理中两个空间基本群间指标关系的更精细的结果，进而我们用得到的结果用于全纯叶状结构的研究，特别是尝试解决Ohsawa的一个猜想。最后使用L2方法，并结合前两项研究得到的结果研究紧致凯勒流形的基本群，特别是Shafarevich猜想。

In this project, we will apply the L2 method to study the topology of complex manifolds, especially, we will use the L2 method to give an improvement of the weak Lefschetz theorems and then, combined with the obtained results, to study holomorphic foliations. On the other hand, as an application, we will also discuss how to apply the weak Lefschetz theorems and holomorphic foliations with the study of Kähler groups.. Firstly, we will give a more precise result about the index relationship between the fundamental groups of two spaces in the weak Lefschetz theorems by a more careful L2 estimate about the related d-bar equations, and then we will apply the results thus obtained in the study of holomorphic foliation, especially, we will try to solve a conjecture given by Ohsawa. Finally, we will use the L2 method, together with the first two research results, to study the fundamental groups of compact Kähler manifolds, especially the Shafarevich conjecture.