Kahler geometry and canonical metrics

摘要奖：DMS 1306298，首席研究员：Gabor Szekelyhidi PI建议的研究涉及代数簇上的规范Kahler度量的研究，特别是由Calabi在80年代的S引入的极值度量的研究。该领域的主要猜想将此类度量的存在与代数簇在几何不变量理论意义下的稳定性联系起来。建议的研究集中在不存在极值度量的情况下。在代数几何方面，我们的目标是构造簇的典范退化，PI的建议是使用齐次坐标环的滤子，类似于不稳定向量丛的Hard-Narasimhan滤子。在微分几何方面，我们需要了解极值度量族可能的极限行为。一般而言，这比更彻底地理解卡勒-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。在该提案中，特别强调了极值度量的新实例的构造，以及这些思想在Kahler几何中的其他问题上的应用。几何偏微分方程组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。这项拟议的研究研究与爱因斯坦方程有关的微分方程，关键问题是空间的整体结构如何影响局部的解析性质，如此类方程解的奇异性。对这一现象的理解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.