Kahler geometry and canonical metrics

AbstractAward：DMS 1306298，首席研究员：Gabor Szekelyhidi PI的研究计划是关于代数簇上的规范Kahler度量的研究，特别是Calabi在80年代引入的极值度量。该领域的主要猜想涉及到几何不变理论意义上的稳定性的各种存在这样的度量。所提出的研究重点的情况下，不存在极值度量。在代数几何方面的目标是构建规范退化的品种和PI的建议是使用过滤的齐次坐标环，在类比哈德-Narasimhan过滤的不稳定向量丛。在微分几何方面，人们需要理解极值度量族的可能极限行为。一般来说，这比更透彻理解的卡勒-爱因斯坦度量的情况要复杂得多，PI建议首先将注意力限制在有界曲率的度量上，并将限制行为与过滤联系起来。在建议中特别强调的是建设新的例子极值度量，以及应用这些想法的其他问题在Kahler geometrics.Geometric偏微分方程治理大部分的物理世界。例如，爱因斯坦方程的解与我们对宇宙的理解密切相关。 拟议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分析性质，例如此类方程解的奇异性。理解这种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.