Special metrics in complex geometry and applications

PI建议研究复杂流形上的特殊度量及其应用。这些度量是由非线性偏微分方程控制的，包括并推广了爱因斯坦的引力方程。本提案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、几个复变量和数学物理。这种跨学科的性质将导致PI与其他领域的专家之间进一步的互动。此外，该提案的主题将引起石溪大学许多在密切相关领域工作的研究生的兴趣，并将有助于扩大本科生和研究生的培训。S.-T之后。尤在卡拉比猜想上的著名工作，在过去的几十年里，对卡勒几何中的特殊度量进行了广泛的研究。指导性猜想是几何不变量理论中存在的极值Kahler度量与某些稳定性概念的对应关系。最近PI与X.X. Chen和Simon Donaldson爵士的联合工作证明了法诺流形上Kahler-Einstein度量的猜想。本研究将以这一结果为出发点，进一步推动理论的发展。首先，PI想要研究Kahler-Einstein度量、代数几何中的模和物理学中的AdS/CFT对应之间的相互作用；其次，他想研究一般极化Kahler流形上的几何演化方程，即Calabi流，并研究其与代数几何和Kahler度量空间几何中的退化的联系。

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.