弧、配对稳定性与Mabuchi泛函
Arcs, stability of pairs and the Mabuchi functional
AI总结：
本文在K-稳定性理论中引入弧的概念，刻画了Mabuchi泛函的强制性，证明Tian猜想的一个版本，并给出Yau-Tian-Donaldson猜想的新证明及自同构群有限性的推广。
摘要：
我们在K-稳定性理论中证明了涉及弧（它是测试配置的推广）的多个结果。我们的主要结果刻画了Fubini-Study度量空间上Mabuchi泛函的强制性，其条件是关于弧的一致K-多稳定性，从而证明了Tian猜想的一个版本。主要的新工具是Paul的配对稳定性理论中数值判据的弧版本，我们还为该判据提供了适用于具有非平凡稳定化子的配对的一个适当推广。我们给出两个应用。首先，我们给出了Fano流形的Yau-Tian-Donaldson猜想一个版本的新证明，其思路最初由Tian设想——这使我们能够将一般的Yau-Tian-Donaldson猜想归结为部分C^0估计的类似物。其次，对于（可能奇异的）极化簇，若其关于弧是一致K-多稳定的，我们证明其自同构群的相应Cartan子群是约化的。特别地，关于弧的一致K-稳定性蕴含自同构群的有限性。这推广了Blum-Xu关于Fano簇的工作。
英文摘要：
We prove various results involving arcs - which generalise test configurations - within the theory of K-stability. Our main result characterises coercivity of the Mabuchi functional on spaces of Fubini-Study metrics in terms of uniform K-polystability with respect to arcs, thereby proving a version of a conjecture of Tian. The main new tool is an arc version of a numerical criterion for Paul's theory of stability of pairs, for which we also provide a suitable generalisation applicable to pairs with nontrivial stabiliser. We give two applications. Firstly, we give a new proof of a version of the Yau-Tian-Donaldson conjecture for Fano manifolds, along the lines originally envisaged by Tian - allowing us to reduce the general Yau-Tian-Donaldson conjecture to an analogue of the partial C^0-estimate. Secondly, for a (possibly singular) polarised variety which is uniformly K-polystable with respect to arcs, we show that the associated Cartan subgroup of its automorphism group is reductive. In particular, uniform K-stability with respect to arcs implies finiteness of the automorphism group. This generalises work of Blum-Xu for Fano varieties.
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