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Extremal metrics on two Fano varieties. (English. Russian original) Zbl 1205.14052
Sb. Math. 200, No. 1, 95-132 (2009); translation from Mat. Sb. 200, No. 1, 97-136 (2009).
In the paper under review, the author studies the problem of existence of an orbifold Kähler–Einstein metric on three-dimensional Fano varieties $$X$$ with quotient singularities. As the main result (see Theorems 1.43 and 1.44), it is proved that such a metric exists on a general hypersurface of degree $$6$$ in $$\mathbb{P}(1^{3}, 2, 2)$$ and a general hypersurface of degree $$7$$ in $$\mathbb{P}(1^{3}, 2, 3)$$. From this and classification of well-formed quasismooth hypersurfaces in weighted projective spaces $$\mathbb{P}(1, a_{1}, \ldots, a_{4})$$, which have degree $$\sum_{i = 1}^{4} a_i$$ and only terminal singularities, the author deduces that every such a hypersurface admits an orbifold Kähler–Einstein metric.
The proof of the main result, on the one hand, employs a sufficient condition, introduced by G. Tian [Invent. Math. 89, 225–246 (1987; Zbl 0599.53046)] and J.-P. Demailly and J. Kollár [Ann. Sci. Ec. Norm. Super. 34(4), 525–556 (2001; Zbl 0994.32021)], for existence of an orbifold Kähler–Einstein metric on $$X$$. Namely, such a metric exists if $$\alpha(X) > \dim X / (\dim X + 1)$$ for the alpha-invariant $$\alpha(X)$$ of $$X$$. On the other hand, the author employs the fact that $$\alpha(X)$$ coincides the global log canonical threshold $$\mathrm{lct}(X)$$ of $$X$$. Here $$\mathrm{lct}(X)$$ ($$=\alpha(X)$$) is a numerical invariant of singularities of $$X$$ which can be defined in purely algebro-geometric terms [see I. A. Cheltsov and K. A. Shramov, Russ. Math. Surv. 63, No. 5, 859–958 (2008; Zbl 1167.14024)]. The latter allows one calculate $$\mathrm{lct}(X)$$ explicitly in the above two cases of $$X$$ by exploring the explicit geometry of $$X$$ and utilizing the theory of singularities of pairs.

##### MSC:
 14J45 Fano varieties 32Q20 Kähler-Einstein manifolds 14J17 Singularities of surfaces or higher-dimensional varieties
##### Citations:
Zbl 0599.53046; Zbl 0994.32021; Zbl 1167.14024
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