Foliation Theory in Birational Geometry

最小模型程序的目标是概括的主要成果的Castelnuovo-恩里克-塞维里分类的代数曲面，更高的维度投影品种。该计划由Mori在20世纪80年代开始，为此他赢得了着名的菲尔兹奖。这是成功地进行了投影三倍由川又，Kollár，宫冈，森，里德，Shokurov和许多其他人。最近，其中一些结果也被推广到任何dimension.The建议的研究的目的是开发新的技术，usingFoliation理论，以解决目前的一些障碍，在MinimalModel程序。这将通过将来自不同数学领域的工具结合在一起来实现，例如数论，Kähler几何和分析。特别是，我计划使用的最小模型程序中的新成果，研究的分类叶理的投影品种。

The goal of the Minimal Model Program is to generalise the main results of the Castelnuovo-Enrique-Severi classification of algebraic surfaces, to higher dimensional projective varieties. The programme was started by Mori in the 1980s, for which he won the prestigious Fields Medal. It was successfully carried out for projective threefolds by Kawamata, Kollár, Miyaoka, Mori, Reid, Shokurov and many others. More recently, some of these results were also generalised to any dimension.The purpose of the proposed research is to develop new techniques, usingFoliation Theory, to solve some of the current obstacles in the MinimalModel Program. This will be achieved by combining together tools from different areas of mathematics, such as Number Theory, Kähler Geometry and Analysis. In particular, I plan to use the new results in the Minimal Model Program to study the classification of foliations over a projective variety.