Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below

Abstract

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

11 pages, comments are welcome; v2: typos corrected, examples and references added; v3: 12 pages, introduction modified; final version, to appear in Annales scientifiques de l'ENS

References in corpus (4)

• K-stability of constant scalar curvature polarization
• A stronger concept of K-stability
• K-stability of constant scalar curvature Kähler manifolds
• Geometry of Kähler Metrics and Foliations by Holomorphic Discs

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• K-stability of Fano varieties: an algebro-geometric approach
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• Wall crossing for K-moduli spaces of plane curves
• The existence of the Kähler-Ricci soliton degeneration
• Uniqueness of the minimizer of the normalized volume function
• On toric geometry and K-stability of Fano varieties
• On properness of K-moduli spaces and optimal degenerations of Fano varieties
• On K-stability of some del Pezzo surfaces of Fano index 2
• Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below
• Boundedness results for singular Fano varieties and applications to Cremona groups
• Equivariant K-stability under finite group action
• A Minimizing Valuation is Quasi-monomial
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• On the sharpness of Tian's criterion for K-stability
• On boundedness of singularities and minimal log discrepancies of Kollár components, II
• K-moduli of Fano 3-folds can have embedded points
• Symmetries of Fano varieties
• On wall crossing for K-stability
• Log K-stability of GIT-stable divisors on Fano varieties
• On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves
• On local delta invariant of del Pezzo surfaces
• Bounded Complements for Weak Fano Pairs with Alpha-invariants and Volumes Bounded Below