Reductivity of the automorphism group of K-polystable Fano varieties

Abstract

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

arXiv comment32 pages. Final version. To appear in Inventiones Math

References in corpus (6)

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• K-semistability is equivariant volume minimization
• Optimal destabilizing centers and equivariant K-stability
• On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties
• The volume of singular Kähler-Einstein Fano varieties
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Cited by in corpus (10)

• Optimal destabilizing centers and equivariant K-stability
• 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
• On K-stability of some del Pezzo surfaces of Fano index 2
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• On K-stability of Calabi-Yau fibrations
• Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability
• On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves
• On local delta invariant of del Pezzo surfaces