邀请Kähler-Einstein度量和随机点过程
An invitation to Kähler-Einstein metrics and random point processes
摘要：
这是对在复射影代数流形X上构造Kähler-Einstein度量的概率方法的邀请。所讨论的度量出现在大的N-极限中，来自X上的N个点的典型采样方式，即来自X上的随机点过程，根据代数几何数据定义。给出了收敛于具有负里奇曲率的Kähler-Einstein度量的证明。在正里奇曲率的情形下，引入变分方法证明了该方法的渐近收敛性，这可看作是Yau-Tian-Donaldson猜想的概率构造性模拟.变分方法揭示，特别是，收敛的假设下，没有相变，这-从代数几何的角度来看-相当于一个分析性质的某一阿基米德zeta函数。
英文摘要：
This is an invitation to the probabilistic approach for constructing Kähler-Einstein metrics on complex projective algebraic manifolds X. The metrics in question emerge in the large N-limit from a canonical way of sampling N points on X, i.e. from random point processes on X, defined in terms of algebro-geometric data. The proof of the convergence towards Kähler-Einstein metrics with negative Ricci curvature is explained. In the case of positive Ricci curvature a variational approach is introduced to prove the conjectural convergence, which can be viewed as a probabilistic constructive analog of the Yau-Tian-Donaldson conjecture. The variational approach reveals, in particular, that the convergence holds under the hypothesis that there is no phase transition, which - from the algebro-geometric point of view - amounts to an analytic property of a certain Archimedean zeta function.
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