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We show that any
-dimensional Fano manifold
is K-stable, where
is the alpha invariant of
introduced by Tian. In particular, any such
admits Kähler–Einstein metrics and the holomorphic automorphism group
The notion of Berman–Gibbs stability was originally introduced by Berman for
. We show that the pair
is K-stable (respectively K-semistable) provided that
is Berman–Gibbs stable (respectively semistable).
A projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.
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