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Log canonical thresholds of smooth Fano threefolds. (English. Russian original) Zbl 1167.14024
Russ. Math. Surv. 63, No. 5, 859-958 (2008); translation from Usp. Mat. Nauk 63, No. 5, 73-180 (2008).
The log canonical threshold of an effective \({\mathbb{Q}}\)-divisor is a subtle invariant to associate to a pair \((X,D)\), where \(X\) is a variety and \(D\) a \({\mathbb{Q}}\)-divisor. It is both related to the complex singularity exponent of holomorphic functions and to other invariants like Tian’s \(\alpha\)-invariant. Actually in the appendix Demailly proves the equality between the global log canonical threshold (that is the log canonical threshold of the anticanonical class, glct for short) of a Fano variety and its \(\alpha\)-invariant.
In this monumental paper the authors study glct of smooth Fano 3-folds. There are 105 deformation families of those. The authors are able to determine the glct for 64 families and give partial result for other 34 families. The techniques used are mainly of birational nature. The idea is to study the glct on a birational model that one can control more easily and then translate it via subtle techniques of birational geometry.

14J45 Fano varieties
14J30 \(3\)-folds
14J17 Singularities of surfaces or higher-dimensional varieties
32Q20 Kähler-Einstein manifolds
32S05 Local complex singularities
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