Anticanonically balanced metrics and the Hilbert-Mumford criterion for the ${\delta }_m$-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 ${\delta }_m$-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.

v2: 36 pages, details added

References in corpus (7)

• Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow
• Kähler metrics with constant weighted scalar curvature and weighted K-stability
• Coupled Kähler-Ricci solitons on toric Fano manifolds
• Quantisation of extremal Kähler metrics
• Coupled Sasaki-Ricci solitons
• Asymptotic stability for Kähler-Ricci solitons
• Ricci iteration for coupled Kähler-Einstein metrics