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arXiv · 2510.06197

Relative stability theory and properness of K-moduli spaces

Harold Blum·Yuchen Liu·Chenyang Xu·Ziquan Zhuang
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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.

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Harold Blum, Yuchen Liu, Chenyang Xu, Ziquan Zhuang. 2025-10-07. Relative stability theory and properness of K-moduli spaces. https://arxiv.org/abs/2510.06197

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