论文标题
邀请Kähler-Einstein指标和随机点过程
An invitation to Kähler-Einstein metrics and random point processes
论文作者
论文摘要
这是对概率方法的邀请，该方法是在复杂的投影代数代数歧管上构建Kähler-Einstein指标。所讨论的指标从x上对x进行x采样n点进行采样的方式出现，即从x上的随机点过程中进行x，从x上定义为algebro-decor-decor-decor-decor-decor-n-limit。解释了与负RICCI曲率的Kähler-Einstein指标的收敛证明。在阳性RICCI曲率的情况下，引入了一种变异方法来证明猜想的收敛，可以将其视为Yau-Tian-Donaldson猜想的概率的构建类似物。变化方法特别表明，收敛性存在于没有相变的假设下，从代数几何的角度来看，这相当于某个Archimedean Zeta函数的分析特性。
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.
