Lie群紧化空间上的Kähler-Ricci流

设G是连通的约化复Lie群，M是K-不稳定的Fano G-紧化空间。本项目将研究M上Kähler-Ricci流的极限行为，并得到Kähler-Ricci流的极限。根据极限的代数唯一性，我们将首先分类M的所有G×G-等变R-试验配置。对每一等变R-试验配置计算相关代数不变量，从代数几何上确定Kähler-Ricci流的极限空间。之后，我们将研究极限空间的几何性质，分析其奇点集的具体结构。特别地，将给出极限空间上Kähler-Ricci孤立子的存在性判据。最后，我们将把M上的Kähler-Ricci流约化为实发展方程，进一步从分析上描述M的复结构沿Kähler-Ricci流收敛或突变的过程。由此刻画Kähler-Ricci流极限度量的奇性。这些将对研究一般Fano流形，特别是K-不稳定Fano流形上Kähler-Ricci流的极限行为提供有意义的模型。

Let G be a connected, complex reductive Lie group and M be a K-unstable Fano G-compactification. In this project we will study the limiting behavior of the Kähler-Ricci flow on M and find its limit. In view of the algebraic uniqueness of the limiting space, we will first classify the G×G-equivariant R-test configurations of M and determine the limiting space of Kähler-Ricci flow via computing corresponding algebraic invariants. Then we study the geometric structure of the limiting space and get a existence criterion of Kähler-Ricci solitons. Finally we will reduce the Kähler-Ricci flow on M into a real evolution equation and give an analytical description of the convergence/jumping progress of the complex structure. Accordingly we will describe the singularity of the limiting metric of the Kähler-Ricci flow. This will provide examples and models for the study of the limiting behavior of Kähler-Ricci flow on a general Fano manifold, especially in the K-unstable cases.