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On exceptional quotient singularities. (English) Zbl 1232.14001

Let \(X\) be a smooth Fano variety of dimension \(n\). Let \(\bar{G}\subset \text{Aut}(X)\) be a compact subgroup. The \(\bar{G}\)-invariant \(\alpha\)-invariant of \(X\) can be defined by using a Kähler metric. A natural question is whether there exists a finite subgroup of the automorphism groups of \(\mathbb{P}^n\) whose \(\alpha\)-invariant is greater than \(1\). In the paper under review, the authors give an affirmative answer to the question for \(n\leq 4\) by studying exceptional quotient singularities.

MSC:

14B05 Singularities in algebraic geometry
14J45 Fano varieties
32Q20 Kähler-Einstein manifolds
14Exx Birational geometry
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