On K-stability of Fano weighted hypersurfaces

Abstract

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, with a 0 ≤ . . . ≤ a n .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.