Special hermitian metrics in complex geometry

这一建议涉及几何微分方程（类似于广义相对论中的爱因斯坦场方程）解的存在性、模量和特殊性质，这些方程在复变的研究中自然产生。其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，该问题的解决引出了著名的紧致复曲线均匀化定理。在高维中，自然推广是Calabi在20世纪80年代提出的在给定的紧化Kaehler变的上同类中寻找正则Kaehler度量的问题，称为极值。提案中还考虑了一些其他相关的问题，这些问题自然会出现在复杂几何中，一个反复出现的主题是代数几何、微分几何和全局分析的强烈交织，每一个都提供了对解空间结构的互补见解。

This proposal deals with the existence, moduli, and special properties of solutions of geometric differential equations (analogous to the Einstein 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 theorem for compact complex curves. In higher dimensions, the natural extension is the problem proposed by Calabi in the 1980's of finding canonical Kaehler metrics, called extremal, in a given cohomology class of a compact Kaehler variety. There are some other related problems considered in the proposal, which naturally arise in complex geometry, and one recurrent theme is a strong intertwining of algebraic geometry, differential geometry, and global analysis, with each providing complementary insight into the structure of the spaces of solutions.