We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{ N})$ if there exists a constant $J\in\mathbb{ Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index at most $J$. We show that the birational automorphism group of a $d$ dimensional variety over a field of characteristic zero is nilpotently Jordan of class at most $d$.
Categories: math.AG