Special metrics in classical and generalized Kaehler geometry

经典和广义凯勒几何中的特殊度量

• 批准号: 238790-2011
• 负责人: Apostolov, Vestislav
• 金额: $ 1.89万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570492
• 关键词: Special; metrics; classical; generalized; Kaehler

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.

这一提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、模和特殊几何性质，这些解是在复杂多样性的研究中自然产生的。 原型问题是紧致复曲线上常高斯曲率度量的存在唯一性问题，其解决导致了著名的紧致复曲线的一致化。在20世纪80年代，卡拉比提出了一个更一般的问题，即在紧致凯勒簇的给定上同调类中找到规范的凯勒度量，称为极值。从全局分析的角度出发，给出了极值Kaehler度量的解的非线性四阶偏微分方程，目前还没有通用的方法。特殊情况包括常数标量曲率的凯勒度量和凯勒-爱因斯坦度量。 本提案中的项目试图解决以下问题： (a)通过将相应的方程简化为更简单的偏微分方程，找到特殊簇上的极值度量。将这些特殊情况下解的存在性（不存在性）与相应变种的稳定性（不稳定性）的各种代数几何概念联系起来，从而为一般理论的主要理论提供了肥沃的试验场。 (b)将极值Kaehler度量的概念推广到广义Kahler几何，并找到非平凡的例子。 （内部或外部）对称性的研究将在实现这些目标方面发挥重要作用。