Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties

Abstract

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the delta-invariant) is a recently introduced invariant that is known to detect the K-semistability and uniform K-stability of a Q-Fano variety. We show that the stability threshold is lower semicontinuous in families and provide an interpretation of the invariant in terms of the K-stability of log pairs.

arXiv commentminor revisions, to appear in Ann. Sci. Éc. Norm. Supér

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• Uniqueness of K-polystable degenerations of Fano varieties
• Wall crossing for K-moduli spaces of plane curves
• The existence of the Kähler-Ricci soliton degeneration
• On properness of K-moduli spaces and optimal degenerations of Fano varieties
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Equivariant K-stability under finite group action
• K-stability of cubic fourfolds
• Birational superrigidity is not a locally closed property
• A Minimizing Valuation is Quasi-monomial
• On the sharpness of Tian's criterion for K-stability
• On the optimal volume upper bound for Kähler manifolds with positive Ricci curvature (with an appendix by Yuchen Liu)
• Openness of uniform K-stability in the Kähler cone
• Valuative invariants with higher moments
• On K-stability of Fano weighted hypersurfaces