论文标题
基础除数和平衡指标
Basis divisors and balanced metrics
论文作者
论文摘要
使用日志规范阈值和基础除数Fujita-Odaka引入了纯粹的代数几何不变式$Δ_M$，其$ m $中的极限现在已知是在Fano品种上表征均匀的K稳定性。如Blum-Jonsson所示，这将延续到一般的极化，并与Berman，Boucksom和Jonsson的作品一起，现在众所周知，这些$Δ_M$ $ invariants的极限表征了统一的滴定稳定性。一个基本问题以来，藤田 - 奥达卡（Fujita-Odaka）的工作是找到对这些不变的分析解释。我们表明，每个$Δ_M$都是量化的ding在$ m $ th伯格曼空间上起作用的强制性阈值，因此表征了平衡指标的存在。这种方法有许多应用程序。最基本的是，它提供了一种计算这些不变性的替代方法，即使对于$ \ Mathbb {p}^n $，这也是新的。其次，它使我们能够引入代数定义的不变性，这些代数定义为Kähler-ricci solitons的存在（以及更通用的$ g $ $ solitons of Berman-WittNyström），以及其耦合版本。第三，它导致近似结果，涉及在存在唐纳森（Donaldson）的一些结果的情况下，涉及平衡指标。
Using log canonical thresholds and basis divisors Fujita--Odaka introduced purely algebro-geometric invariants $δ_m$ whose limit in $m$ is now known to characterize uniform K-stability on a Fano variety. As shown by Blum-Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these $δ_m$-invariants characterizes uniform Ding stability. A basic question since Fujita-Odaka's work has been to find an analytic interpretation of these invariants. We show that each $δ_m$ is the coercivity threshold of a quantized Ding functional on the $m$-th Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for $\mathbb{P}^n$. Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler-Ricci solitons (and the more general $g$-solitons of Berman-Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.
