Kahler geometry and canonical metrics

项目编号：DMS 1306298，项目负责人：Gabor szekelyhidi该项目主要研究代数变量上的正则Kahler度量，特别是Calabi在80年代引入的极值度量。该领域的主要猜想将这类度量的存在性与几何不变理论意义上的变化的稳定性联系起来。本文提出的研究重点是不存在极值度量的情况。在代数几何方面，目标是构造变种的规范退化，PI的建议是使用齐次坐标环的滤波，类似于不稳定向量束的Harder-Narasimhan滤波。在微分几何方面，我们需要理解极值度量族可能的极限行为。一般来说，这比更彻底理解的卡勒-爱因斯坦度规的情况要复杂得多，PI建议首先将注意力限制在曲率有限的度规上，并将极限行为与过滤联系起来。在建议中，特别强调了极值度量的新例子的构建，以及这些思想在卡勒几何中其他问题的应用。几何偏微分方程支配着物理世界的大部分。例如，爱因斯坦方程的解与我们对宇宙的理解密切相关。提出的研究研究与爱因斯坦方程相关的微分方程，关键问题是空间的全局结构如何影响局部解析性质，例如此类方程解的奇点。理解这一现象将在物理学和一般科学中得到应用。

AbstractAward: DMS 1306298, Principal Investigator: Gabor SzekelyhidiThe PI's proposed research is concerned with the study of canonical Kahler metrics on algebraic varieties, in particular the extremal metrics introduced by Calabi in the 80's. The main conjecture in the field relates the existence of such metrics to the stability of the variety in the sense of geometric invariant theory. The proposed research focuses on situations when no extremal metric exists. On the algebro-geometric side the goal is to construct canonical degenerations of the variety and the PI's proposal is to use filtrations of the homogeneous coordinate ring, in analogy with Harder-Narasimhan filtrations of unstable vector bundles. On the differential geometric side one needs to understand the possible limiting behavior of families of extremal metrics. In general this is much more intricate than the much more thoroughly understood case of Kahler-Einstein metrics, and the PI proposes to first restrict attention to metrics with bounded curvature, and to relate the limiting behavior to filtrations. In the proposal a special emphasis is placed on the construction of new examples of extremal metrics, and the applications of these ideas to other problems in Kahler geometry.Geometric partial differential equations govern much of the physical world. For example solutions of Einstein's equations are intimately related to our understanding of the universe. The proposed research studies differential equations related to Einstein's equations and the key question is how the global structure of a space influences the local, analytic properties, such as singularities of the solutions of such equations. Understanding this phenomenon will have applications in physics and the sciences in general.