Anticanonically balanced metrics and the Hilbert–Mumford criterion for the \(\delta _m\)-invariant of Fujita–Odaka

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 \(\delta _m\)-invariant of Fujita–Odaka satisfies \(\delta _m >1\) if and only if the Fano manifold is stable in the sense of Saito–Takahashi, establishing a Hilbert–Mumford-type criterion for \(\delta _m >1\). 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.