The Kähler–Ricci flow on surfaces of positive Kodaira dimension
The Kähler–Ricci flow on surfaces of positive Kodaira dimension
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DOI：
10.1007/s00222-007-0076-8
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发表时间：
2006-02
影响因子：
3.1
通讯作者：
Jian Song;G. Tian
中科院分区：
文献类型：
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作者：
Jian Song;G. Tian
紧化K\ ahler流形上K\ ahler- einstein度量的存在性在过去几十年里一直是继Yau对Calabi猜想的解之后深入研究的主题。理查德·汉密尔顿提出的里奇流已经成为几何分析中最强大的工具之一。我们研究了Kodaira维1的最小曲面上的K\ ahler-Ricci流，证明了该流在其正则模型上坍缩并收敛到一个唯一的正则度量。这样的正则就是广义的K\ ahler-Einstein度规。结合Cao， Tsuji， Tian和Zhang的结果，我们给出了K\ \ ahler-Ricci流对具有数值有效规范线束的K\ \ ahher曲面的度量分类。在一般情况下，我们提出了在正Kodaira维的射影变体的正则模型上寻找正则度量的程序。
The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized K\"ahler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for K\"aher surfaces with a numerical effective canonical line bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding canonical metrics on canonical models of projective varieties of positive Kodaira dimension.