Kähler锥中一致K-稳定性的开性
Openness of uniform K-stability in the Kähler cone
摘要：
我们证明了与一致K-稳定性相关的新的稳定性阈值的连续性结果，并推导出一致K-稳定性是任何紧致Kähler流形的Kähler锥中的开条件，从而建立了LeBrun-Simanca关于常数数量曲率（cscK）度量的经典结果的代数几何对应.这解决了该领域的一个民间传说的猜想，特别是意味着开放的均匀K-稳定光滑极化品种。此外，它加强了支持任意极化的Yau-Tian-Donaldson猜想的统一版本的证据，包括无理极化和非投射Kähler流形的情况。作为一个关键的工具，我们引入了一个新的规范测试配置，并建立估计非阿基米德能量泛函在此规范。这导致了新的一致K-稳定性的特征，限制测试配置，满足一定的统一的界限。作为一个副产品，我们得到了连续性的结果，有关的非阿基米德熵的稳定性阈值，并推导出开放的一致J-稳定性，以及开放的J-稳定性的投影的情况下。
英文摘要：
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.
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