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arXiv · 2501.18150

Stability thresholds for big classes

Chenzi Jin·Yanir A. Rubinstein·Gang Tian
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Abstract

In 1987, the $α$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the $δ$-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles $τ\in[0,1]$. The special degenerate (collapsing) case $τ=0$ on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.

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Chenzi Jin, Yanir A. Rubinstein, Gang Tian. 2025-01-30. Stability thresholds for big classes. https://doi.org/10.1515/crelle-2026-0050

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