Let X ⊂ P(a 0 , . . ., a n ) be a quasi-smooth weighted Fano hypersurface of degree d and index I X such that a i | d for all i.If I X = 1, we show that, under a suitable condition, the α-invariant of X is greater than or equal to dim X/(dim X + 1) and X is K-stable.This can be applied in particular to any X as above such that dim X ⩽ 3.If X is general and I X < dim X, then we show that X is K-stable.We also give a sufficient condition for the finiteness of automorphism groups of quasi-smooth Fano weighted complete intersections.
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Sano et al. (2024) studied this question.
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