Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka

Abstract

We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the δm-invariant of Fujita-Odaka satisfies δm > 1 if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for δm > 1. We also extend this result to the Kähler-Ricci g-solitons and the coupled Kähler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations. m A.3. Coupled Kähler-Einstein version δ coupled m