Relative stability theory and properness of K-moduli spaces
Harold Blum, Yuchen Liu, Chenyang Xu, Ziquan Zhuang
TL;DR
The paper proves a birational, valuation-based proof of the properness of the K-moduli space for K-polystable log Fano pairs by introducing a relative stability threshold δ(X,Δ;L) and showing the existence of a divisorial minimizer computing this invariant. It develops the relative stability theory (valuations, basis-type divisors, and S/T invariants), and uses MMP-based extensions guided by minimizers to replace unstable special fibers with more stable ones, ultimately yielding a direct, birational proof of properness. A key technical achievement is finite generation of the associated graded ring gr_v R via special complements and cone constructions, enabling a divisorial minimizer and a controlled degeneration. The framework further yields a systematic method to pass from a family with unstable central fiber to a family with a stable or improved central fiber, with potential extensions to weighted K-stability and related moduli problems, thereby broadening the reach of K-moduli theory in birational geometry.
Abstract
We define the relative stability threshold of a family of Fano varieties over a DVR and show that it is computed by a divisorial valuation. In the case when the special fiber is K-unstable, but the generic fiber is K-semistable, we use the divisorial valuation computing the threshold to replace the special fiber by a new one with a strictly larger stability threshold. Iterating this process yields a new and more direct proof of the properness of the K-moduli space that uses only birational geometry arguments.