Special metrics in complex geometry and applications

PI提出研究复流形上的特殊度量及其应用。这些度量由非线性偏微分方程控制，包括和推广爱因斯坦的引力方程。在这个建议中提出的问题围绕着许多活跃的数学领域，包括微分几何，代数几何，偏微分方程，几个复杂的变量，和数学物理。这种跨学科的性质将导致PI和其他领域的专家之间的进一步互动。此外，这一建议的主题将是感兴趣的许多研究生在斯托尼布鲁克在密切相关的领域工作，并将促进扩大培训的本科生和研究生。在S之后- T.丘的著名工作卡拉比猜想，有广泛的研究特殊度量在卡勒几何在过去几十年中。指导猜想是极值Kahler度量的存在性和几何不变理论所产生的某些稳定性概念之间的对应关系。PI与X. X最近的联合工作。Chen和Simon唐纳森爵士在Fano流形上的Kahler-Einstein度量的情况下证明了这个猜想。这项研究将以这一结果为出发点，旨在进一步推动这一理论。首先，PI想调查之间的相互作用卡勒爱因斯坦度量，模代数几何，和AdS/CFT对应物理;其次，他想研究几何演化方程，即卡拉比流，一般极化卡勒流形，并调查其连接退化代数几何和几何空间的卡勒度量。

The PI proposes to study special metrics on complex manifolds and applications. These metrics are governed by non-linear partial differential equations, including and generalizing Einstein's equation for gravity. The problems raised in this proposal center around many active areas of mathematics, including differential geometry, algebraic geometry, partial differential equations, several complex variables, and mathematical physics. This interdisciplinary nature will lead to further interactions between the PI and experts in other areas. Also the topic of this proposal will be of interest to many graduate students in Stony Brook working in closely-related fields, and will facilitate the broadening training of both undergraduate and graduate students. After S.-T. Yau's renowned work on the Calabi conjecture, there has been extensive study of special metrics in Kahler geometry during the past few decades. The guiding conjecture is a correspondence between the existence of extremal Kahler metrics and certain stability notion arising from geometric invariant theory. Recent joint work of the PI with X.X. Chen and Sir Simon Donaldson proved the conjecture in the case of Kahler-Einstein metrics on Fano manifolds. The proposed research will take this result as a starting point and is intended to push the theory further. Firstly the PI would like to investigate the interplay between Kahler-Einstein metrics, moduli in algebraic geometry, and the AdS/CFT correspondence in physics; secondly he would like to study a geometric evolution equation, namely the Calabi flow, on a general polarized Kahler manifold, and investigate its connection with degenerations in algebraic geometry and geometry of the space of Kahler metrics.