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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
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