Title: Arcs, stability of pairs and the Mabuchi functional Show Chinese title

Title: 弧，对的稳定性以及Mabuchi泛函

Abstract: 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. Show Chinese abstract

Abstract: 我们在K稳定性理论中证明了涉及弧的一些结果，这些结果推广了测试配置。 我们的主要结果通过弧的均匀K-半稳定性的角度表征了Fubini-Study度量空间上Mabuchi泛函的强制性，从而证明了Tian的一个猜想的一个版本。 主要的新工具是Paul关于对稳定性理论的数值准则的弧版本，我们也提供了适用于具有非平凡稳定子的对的适当推广。 我们给出了两个应用。 首先，我们沿着Tian最初设想的路线，给出了Fano流形的一个Yau-Tian-Donaldson猜想版本的新证明，使我们能够将一般的Yau-Tian-Donaldson猜想简化为部分C^0估计的类似物。 其次，对于一个(可能奇异的)极化代数簇，如果它相对于弧是均匀K-半稳定的，我们证明了其自同构群的相关Cartan子群是约化的。 特别是，相对于弧的均匀K稳定性意味着自同构群的有限性。 这推广了Blum-Xu在Fano流形上的工作。