DOI: 10.1515/crelle-2026-0050 ISSN: 0075-4102
Stability thresholds for big classes
Chenzi Jin, Yanir A. Rubinstein, Gang Tian Abstract
In 1987, the α-invariant theorem gave a fundamental
criterion for existence of Kähler–Einstein metrics on smooth Fano manifolds.
In 2012, Odaka–Sano extended the framework to
ℚ
{{\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
∈
[
0
,
1
]
{\tau\in[0,1]}
.
The special degenerate (collapsing) case
τ
=
0
{\tau=0}
on ample classes
recovers Odaka–Sano’s theorem.
This can be used to obtain the existence of many new twisted Kähler–Einstein metrics on big classes without computing the δ-invariant.
Of independent interest,
the proof is centered around a generalization to
sub-barycenters
of the classical Neumann–Hammer Theorem from convex geometry.