Ricci熵、RCD结构与Kahler空间

中文摘要 AI 辅助

本文是我们关于复Monge-Ampere方程几何理论系列的最后一篇。我们研究具有klt奇点的紧致正规Kahler空间上的奇异Kahler度量，其体积密度属于某个$p>1$的$L^p$空间。我们证明，在多重位势理论意义下Ricci电流的下界等价于RCD理论意义下的合成Ricci下界。作为推论，具有klt奇点的紧致正规Kahler空间上的每个Kahler类都 admits 一个RCD结构，这使得微分几何分析可用于奇异复空间。作为应用，我们证明具有klt奇点的紧致Kahler Calabi-Yau空间的基本群几乎是阿贝尔的。我们进一步建立了奇异Kahler-Einstein空间的紧致性结果。

英文摘要

This paper is the final installment in our series on the geometric theory of complex Monge-Ampere equations. We study singular Kahler metrics on compact normal Kahler spaces with klt singularities whose volume densities lie in $L^p$ for some $p>1$. We show that a lower bound for the Ricci current in the sense of pluripotential theory is equivalent to a synthetic Ricci lower bound in the sense of RCD theory. As a consequence, every Kahler class on a compact normal Kahler space with klt singularities admits an RCD structure, which makes differential geometric analysis available on singular complex spaces. As an application, we show that the fundamental group of a compact Kahler Calabi-Yau space with klt singularities is almost Abelian. We further establish compactness results for singular Kahler-Einstein spaces.

发表机构

• Rutgers University（罗格斯大学）

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