  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 $d$.
