Special riemannian metrics and kaehler geometry

这一建议的目的是从Kahler几何的角度研究特殊的黎曼度量。这样的度量出现在广义相对论和弦理论中的物理模型的数学对象中，但也出现在射影变量的稳定性的代数几何概念中。我计划致力于以下具体问题：1.射影丛上极值Kahler度量的存在性。这个项目将为将极值Kahler度量与射影变差的稳定性联系起来的猜想提供一个肥沃的试验场。2.将哈密顿2-形式的概念推广到Sasakian几何的顶点。这将为弦理论家发现的佐佐木-爱因斯坦度规的一些新构造提供一种系统的方法。3.广义Kahler度量。Hitchin引入了广义复结构的概念，为弦理论中当前感兴趣的一些概念提供了一个自然的微分几何框架。我在这里的项目旨在对某种类型的广义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.