On properness of K-moduli spaces and optimal degenerations of Fano varieties

Harold Blum, Daniel Halpern-Leistner, Yuchen Liu, Chenyang Xu

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15 Scopus citations


We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Θ -stratification on the moduli stack of Fano varieties.

Original languageEnglish (US)
Article number73
JournalSelecta Mathematica, New Series
Issue number4
StatePublished - Sep 2021

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Physics and Astronomy


  • Fano varieties
  • K-stability
  • Moduli


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