论文标题
Kähler锥体中均匀K稳定性的开放性
Openness of uniform K-stability in the Kähler cone
论文作者
论文摘要
我们证明了与均匀K稳定性有关的新稳定性阈值的连续性结果，并推断出均匀的K稳定性是任何紧凑的Kähler歧管的Kähler锥中的开放条件，从而确立了代数几何的对应物，以与Lebrun-Simanca的经典结果相对，以实现恒定标量级曲率（CSCK）的恒定标度曲率。这会在田野中解决民间传说的猜想，特别是意味着对于平滑极化品种的均匀K稳定性的开放性。此外，它加强了支持Yau-tian-Donaldson统一版本的构想的证据，包括任意极化，包括非理性极化和非侵入性Kähler歧管的情况。作为一个关键工具，我们在测试配置上引入了新的规范，并根据该规范建立了非架构能量功能的估计。这通过限制满足某些统一界限的测试配置，从而导致均匀K稳定性的新特征。作为副产品，我们获得了与非架构熵相关的稳定性阈值的连续性结果，并推断出均匀J稳定性的开放性，以及在投射情况下J稳定性的开放性。
We prove continuity results for new stability thresholds related to uniform K-stability and deduce that uniform K-stability is an open condition in the Kähler cone of any compact Kähler manifold, thus establishing an algebro-geometric counterpart to a classical result of LeBrun-Simanca for constant scalar curvature (cscK) metrics. This settles a folklore conjecture in the field, and in particular implies openness of uniform K-stability for smooth polarized varieties. Moreover, it strengthens evidence supporting the uniform version of the Yau-Tian-Donaldson conjecture for arbitrary polarizations, including the case of irrational polarizations and non-projective Kähler manifolds. As a key tool we introduce a new norm on test configurations and establish estimates for non-archimedean energy functionals in terms of this norm. This leads to new characterizations of uniform K-stability by restricting to test configurations that satisfy certain uniform bounds. As a byproduct we obtain continuity results for a stability threshold related to non-archimedean entropy and deduce openness of uniform J-stability, as well as openness of J-stability in the projective case.
