Special metrics in classical and generalized Kaehler geometry

这个建议处理在复变研究中自然产生的几何微分方程（类似于广义相对论中的爱因斯坦场方程）解的存在性、模量和特殊几何性质。原型是紧致复曲线上的常高斯曲率度规的存在唯一性问题，该问题的解决导致了著名的紧致复曲线的均匀化问题。在20世纪80年代，Calabi提出了一个更一般的问题，即在给定的紧化Kaehler变的上同调类中寻找正则Kaehler度量，称为极值。从全局分析的角度出发，用四阶非线性偏微分方程的解给出了极值Kaehler度量。特殊情况包括常数标量曲率的凯勒度规和凯勒-爱因斯坦度规。

This proposal deals with the existence, moduli, and special geometrical properties of solutions of geometric differential equations (analogous to Einstein's field equation in general relativity) naturally arising in the study of complex varieties. The prototype is the problem of existence and uniqueness of a metric of constant Gauss curvature on a compact complex curve, whose resolution leads to the famous uniformization of compact complex curves. In the 1980's, Calabi formulated a more general problem of finding canonical Kaehler metrics, called extremal, in a given cohomology class of a compact Kaehler variety. From the point of view of global analysis, extremal Kaehler metrics are given by solutions to a non-linear 4th order PDE for which no general methods are currently available. Particular cases include Kaehler metrics of constant scalar curvature and Kaehler-Einstein metrics. The projects in this proposal attempt to address the following problems: (a) Find extremal metrics on special varieties by reducing the corresponding equations to simpler PDE's. Link the (non)existence of solutions in these special cases with various algebro-geometric notions of (un)stability for the corresponding varieties, thus providing a fertile testing ground for the main conjectures of the general theory. (b) Extend the notion of extremal Kaehler metric to generalized Kahler geometry and find non-trivial examples. The study of (internal or external) symmetries will plain an important role in achieving these objectives.