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Jordan property for groups of birational selfmaps. (English) Zbl 1314.14022

The Borisov-Alexeev-Borisov conjecture (BAB conjecture, for short) states that if \(k\) is an algebraically closed field, fixed \(n\in\mathbb{N}\), Fano varieties of dimension \(n\) with terminal singularities defined over \(k\) are bounded, i.e. they are contained in a finite number of algebraic families. In the paper under review the authors prove the following result:
Let \(k\) be a finitely generated field over \(\mathbb{Q}\) and \(X\) a variety of dimension \(n\). Let \(\mathrm{Bir}(X)\) be the group of birational automorphisms of \(X\) over \(k\). Suppose that the BAB conjecture holds in dimension \(n\). Then there exists a constant \(B=B(\mathrm{Bir}(X))\) such that for any finite subgroup \(G\subseteq \mathrm{Bir}(X)\) the cardinality of \(G\) is bounded by \(B\).
Moreover, the same conclusion is proved without assuming the BAB conjecture, under the additional hypothesis that \(X\) is not uniruled and the irregularity of the variety is zero, \(q(X)=0\).
The Jordan property is also studied: a group \(\Gamma\) is Jordan if there exists \(J\in\mathbb{N}\) such that for any finite subgroup \(G\subseteq \Gamma\) there exists a normal abelian subgroup \(A\subseteq G\) of index at most \(J\). In the paper under review it is proved that, if \(q(X)=0\) and the BAB conjecture holds in dimension \(n\) or \(X\) is non-uniruled, then \(\mathrm{Bir}(X)\) is Jordan.
The proofs rely on a previous work by the authors [“Jordan property for Cremona groups”, arXiv:1211.3563, to appear in Am. J. Math. (2015)] where it is proved that, modulo the BAB conjecture, the family of groups \(\{\mathrm{Bir}(X)\}\) where \(X\) is rationally connected of dimension \(n\), is uniformly Jordan (i.e. \(\mathrm{Bir}(X)\) is Jordan for every \(X\) and there is a uniform constant \(J\)). The existence of a \(G\)-equivariant version of the minimal model program and the study of the maximal rationally connected fibration are important tools as well.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
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References:

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