Finite generation for valuations computing stability thresholds and applications to K-stability
Yuchen Liu, Chenyang Xu, Ziquan Zhuang
Abstract
We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.
Topics & Concepts
Fano planeMathematicsConjectureDimension (graph theory)Moduli spaceStability (learning theory)Equivalence (formal languages)CombinatoricsModuliPure mathematicsDiscrete mathematicsMathematical analysisQuantum mechanicsComputer sciencePhysicsMachine learningGeometry and complex manifoldsGeometric Analysis and Curvature FlowsAlgebraic Geometry and Number Theory