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

Abstract

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 Theta-stratification on the moduli stack of Fano varieties.

arXiv commentv2: to appear in Selecta Math

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Cited by in corpus (4)

• Optimal destabilizing centers and equivariant K-stability
• K-stability of Fano varieties: an algebro-geometric approach
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• K-moduli of Fano 3-folds can have embedded points