$\widehat{\mathrm K}$-稳定性和 $\mathrm K^\beta$-稳定性的代数比较
An algebraic comparison of $\widehat{\mathrm K}$-stability and $\mathrm K^β$-stability
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过交集理论建立非阿基米德Mabuchi泛函与量子化泛函的定量比较，证明一致$\widehat{\mathrm K}$-多重稳定性与一致$\mathrm K^\beta$-稳定性等价，并刻画K-半稳定性及阈值收敛。
AI中文摘要：
我们建立了非阿基米德Mabuchi泛函与Darvas--Zhang量子化Mabuchi泛函之间的统一定量比较。对于具有有限自同构群的光滑极化簇，我们提供了一个代数证明，表明对于每个充分大的有理数 $\beta>1$，一致 $\widehat{\mathrm K}$-多重稳定性等价于一致 $\mathrm K^\beta$-稳定性，而无需通过cscK度量的存在性。主要输入是通过交集理论证明的非阿基米德能量的方向导数的平方根估计。我们还研究了当极化自同构群的单位连通分支是约化的且Futaki特征标消失时的约化 $\mathrm{K}^\beta$-稳定性条件。最后，我们通过渐近 $\mathrm K^\beta$-半稳定性来刻画K-半稳定性，并证明了相应稳定性阈值的定量收敛性。
英文摘要：
We establish a uniform quantitative comparison between the non-Archimedean Mabuchi functional and the Darvas--Zhang quantised Mabuchi functional. For a smooth polarised variety with finite automorphism group, we provide an algebraic proof that uniform $\widehat{\mathrm K}$-polystability is equivalent to uniform $\mathrm K^β$-stability for every sufficiently large rational $β>1$, without passing through the existence of a cscK metric. The main input is a square-root estimate for directional derivatives of non-Archimedean energies, proved by intersection theory. We also study a reduced $\mathrm{K}^β$-stability condition when the identity component of the polarised automorphism group is reductive and the Futaki character vanishes. We conclude by characterising K-semistability in terms of asymptotic $\mathrm K^β$-semistability and prove the quantitative convergence of the corresponding stability thresholds.
