## Jordan property for Cremona groups.(English)Zbl 1343.14010

A group is Jordan if there exists a constant $$I$$ such that for any finite subgroup $$G$$ there is a normal abelian subgroup $$A\subseteq G$$ of index at most $$I$$. The group $$\mathrm{GL}_n$$ is Jordan and the group $$\mathrm{Bir}(\mathbb{P}^2)$$ of birational selfmaps of the projective plane is Jordan by [J.-P. Serre, Mosc. Math. J. 9, No. 1, 183–198 (2009; Zbl 1203.14017)]. The paper under review deals with the $$n$$-dimensional version of this last result. The question was asked by Serre in the aforementioned paper and the authors establish a relation with the BAB conjecture [A. Borisov, J. Algebr. Geom. 5, No. 1, 119–133 (1996; Zbl 0858.14022)]:
Conjecture: For a given $$n$$, Fano varieties of dimension $$n$$ with terminal singularities are bounded, that is, are contained in a finite number of algebraic families.
Their main result is the following:
Theorem: Assume the BAB conjecture in dimension $$n$$. Then there is a constant $$J=J(n)$$ such that for any rationally connected variety $$X$$ of dimension $$n$$ defined over a field of characteristic 0 and for any finite subgroup $$G\subseteq\mathrm{Bir}(X)$$ there is a normal abelian subgroup $$A\subseteq G$$ of index at most $$J$$.
The proof uses the $$G$$-MMP for rationally connected varieties and the Jordan property for $$\mathrm{GL}_n$$. The key point is to prove that, for every finite subgroup $$G\subseteq\mathrm{Bir}(X)$$, there is a subgroup $$F\subseteq G$$ of bounded degree that acts on a birational model $$W$$ of $$X$$ with a fixed point $$p$$. Thus $$F$$ acts on $$T_p W$$ and one concludes by using the Jordan property for $$\mathrm{GL}_n$$.

### MSC:

 14E07 Birational automorphisms, Cremona group and generalizations 14J45 Fano varieties

### Keywords:

birational selfmaps; jordan property; BAB conjecture

### Citations:

Zbl 1203.14017; Zbl 0858.14022
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