Arcs, stability of pairs and the Mabuchi functional

TL;DR

This paper advances the understanding of K-stability by introducing arc-based techniques, characterizing the Mabuchi functional's coercivity, and providing new proofs and implications for Fano manifolds and automorphism groups.

Contribution

It develops an arc version of stability criteria, proves a conjecture of Tian relating Mabuchi functional coercivity to uniform K-polystability, and extends results on automorphism groups for K-stable varieties.

Findings

Characterizes coercivity of the Mabuchi functional via arcs and uniform K-polystability.

Provides a new proof of a version of the Yau-Tian-Donaldson conjecture for Fano manifolds.

Shows that K-polystability with respect to arcs implies the automorphism group is reductive.

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.