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**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 \(\mathrm{GL}_n\).

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 \(\mathrm{GL}_n\).

Reviewer: Enrica Floris (Basel)