Special riemannian metrics and kaehler geometry

这个建议的目的是从卡勒几何的角度研究特殊的黎曼度量。这些度量在广义相对论和弦理论中作为物理模型的数学对象出现，但也在射影变量的代数几何稳定性概念中出现。我计划在以下几个具体问题上进行工作：射影束上极值Kahler度量的存在性。这个项目将提供一个肥沃的试验场，有关极端Kahler指标的推测，以稳定的投影品种。2. 将哈密顿2型的概念推广到sasaki几何。这将为弦理论家发现的Sasaki-Einstein度规的一些新构造提供一个系统的方法。3. 广义Kahler度量。广义复杂结构的概念是由Hitchin引入的，它为当前弦理论中一些感兴趣的概念提供了一个自然的微分几何框架。我的课题是对一类广义Kahler流形进行分类。

The aim of this proposal is the study of special Riemannian metrics from the point of view of  Kahler geometry. Such metrics appear as mathematical objects of physical models arising in General Relativity and String Theory, but also in the algebro-geometric notion of stability of projective varities. I plan to work on the following specific problems: 1. Existence of extremal Kahler metrics on projective bundles.  This project would provide a fertile testing ground for the conjectures relating extremal Kahler metrics to stability of projective varities. 2. Extension of the notion of  a Hamiltonian 2-form to the contex of Sasakian geometry. This would give a systematic approach to some new constructions of Sasaki-Einstein metrics found by string theorists. 3. Generalized Kahler metrics. The concept of a generalized complex structure has been introduced by Hitchin to provide a natural differential geometric framework for some notions of current interest in string theory. My project here aims at classifying a certain  type of generalized Kahler manifolds.