Special hermitian metrics in complex geometry

这一建议涉及几何微分方程解的存在性、模和特殊性质(类似于广义相对论中的爱因斯坦场方程)，这是在复变数研究中自然产生的。其原型是紧复曲线上常高斯曲率度量的存在唯一性问题，它的解决导致了紧复曲线的著名的一致化定理。在高维中，自然扩张是Calabi在1980年代S提出的问题，即在紧的Kaehler簇的给定上同调类中寻找规范的Kaehler度量，称为极值。提案中还考虑了一些在复杂几何中自然出现的其他相关问题，一个反复出现的主题是代数几何、微分几何和整体分析的紧密交织，每一个都提供了对解空间结构的补充洞察。*本提案中描述的具体目标试图解决以下方向：*(A)通过将相应的方程化为更简单的偏微分方程组，找到特殊极化簇上的极值Kaehler度量，例如Delzant多面体为长方体的Toric簇或特殊簇上的投影丛。将这些特殊情况下解的(非)存在性与相应极化簇的稳定性的各种代数几何概念联系起来。*(B)将该理论推广到当前感兴趣的其他几何方程，例如对应于共形-Kaehler，Einstein-Maxwell度规的几何方程。*(C)寻找非Kahler紧致复杂曲面的极值Kaehler度量的自然替代品。*研究(内部或外部)对称性和一些相当微妙的分离变量技术将在实现这些目标方面发挥重要作用。

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.******The specific objectives described in this proposal attempt to address the following directions:****** (a) Find extremal Kaehler metrics on special polarized varieties, such as toric varieties whose Delzant polytopes are cuboids, or projective bundles over special varieties, by reducing the corresponding equations to simpler partial differential equations. Link the (non)existence of solutions in these special cases with various algebro-geometric notions of stability for the corresponding polarized varieties.****** (b) Extend the theory to other geometric equations of current interest, such as the one corresponding to conformally-Kaehler, Einstein-Maxwell metrics. ****** (c) Find natural substitutes for extremal Kaehler metrics for non-Kahler compact complex surfaces.******The study of (internal or external) symmetries and some rather subtle separation of variables techniques will play an important role in achieving these objectives.**