  Assuming Borisov--Alexeev--Borisov conjecture, we prove that there is a
constant $J=J(n)$ such that for any rationally connected variety $X$ of
dimension $n$ and any finite subgroup $G\subset Bir(X)$ there exists a normal
abelian subgroup $A\subset G$ of index at most $J$. In particular, we obtain
that the Cremona group $Cr_3=Bir(P^3)$ enjoys the Jordan property.

Algebraic Geometry