## 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:

 1.4e+08 Birational automorphisms, Cremona group and generalizations
Full Text:

### References:

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