Geodesic
Stable degeneration of families of klt singularities with constant local volume
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e158

Voir la notice de l'article provenant de la source Cambridge University Press

For a klt singularity, C. Xu and Z. Zhuang [33] proved the associated graded algebra of a minimizing valuation of the normalized volume function is finitely generated, finishing the proof of the stable degeneration conjecture proposed by C. Li and C. Xu. We prove a family version of the stable degeneration: for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form families of ideals with flat cosupport, which induce a degeneration to a locally stable family of K-semistable log Fano cone singularities. In the proof, we give a method to construct families of Kollár models, which are a crucial tool introduced by Xu–Zhuang to prove finite generation for valuations of higher rational rank. 
Chen, Zhiyuan. Stable degeneration of families of klt singularities with constant local volume. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e158. doi: 10.1017/fms.2025.10111
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