基除子与平衡度量
Basis divisors and balanced metrics
AI总结：
本文为Fujita-Odaka代数不变量$δ_m$提供了解析解释，证明其为Bergman空间上量化Ding泛函的强制阈值并刻画平衡度量存在性，同时推广至Kähler-Ricci孤立子及逼近结果。
摘要：
利用对数典范阈值和基除子，Fujita--Odaka引入了纯代数几何不变量$δ_m$，其在$m$中的极限现已知可刻画Fano簇上的一致K-稳定性。正如Blum-Jonsson所示，这可推广至一般极化，结合Berman、Boucksom和Jonsson的工作，现已知这些$δ_m$-不变量的极限可刻画一致Ding稳定性。自Fujita-Odaka的工作以来，一个基本问题是为这些不变量寻找解析解释。我们证明每个$δ_m$是第$m$个Bergman空间上量化Ding泛函的强制阈值，从而刻画了平衡度量的存在性。该方法有诸多应用。最基本的是提供了一种计算这些不变量的替代方法，这对$\mathbb{P}^n$而言也是全新的。其次，它允许我们引入代数定义的不变量，以刻画Kähler-Ricci孤立子（及更一般的Berman-Witt Nyström的$g$-孤立子）及其耦合版本的存在性。第三，它导出了在有自同构存在时涉及平衡度量的逼近结果，推广了Donaldson的某些结论。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
