Stability thresholds for big classes

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> {{\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>τ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {\tau\in[0,1]} . The special degenerate (collapsing) case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>τ</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {\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.