Uniqueness of K-polystable degenerations of Fano varieties

Abstract

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.

arXiv commentv2: minor revisions, to appear in Ann. of Math

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Cited by in corpus (9)

• Optimal destabilization of K-unstable Fano varieties via stability thresholds
• On toric geometry and K-stability of Fano varieties
• Applications of the moduli continuity method to log K-stable pairs
• Moduli of polarised manifolds via canonical Kähler metrics
• Hybrid convergence of Kähler-Einstein measures
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• On the uniform K-stability for some asymptotically log del Pezzo surfaces
• Minimizing CM degree and slope stability of projective varieties
• On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line