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

TL;DR

This paper establishes a link between stability conditions for Fano manifolds, anticanonically balanced metrics, and the $oldsymbol{ ext{delta}_m}$-invariant, providing a Hilbert-Mumford criterion and extending to Kähler-Ricci solitons and coupled Kähler-Einstein metrics.

Contribution

It proves the equivalence between Saito-Takahashi stability and the existence of anticanonically balanced metrics, and establishes a Hilbert-Mumford criterion for the $oldsymbol{ ext{delta}_m}$-invariant of Fujita-Odaka.

Findings

$oldsymbol{ ext{delta}_m} > 1$ if and only if Fano manifold is Saito-Takahashi stable

Extension of results to Kähler-Ricci $g$-solitons and coupled Kähler-Einstein metrics

Derived a formula for the asymptotic slope of the coupled Ding functional

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 $\delta_m$-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 K\"ahler-Ricci $g$-solitons and the coupled K\"ahler-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.