Finite subgroups of the birational automorphism group are "almost nilpotent"

Investigating finite subgroups of the birational automorphism group of varieties was initiated by J.-P. Serre and V. L. Popov in the beginning of 2010’s. It turned out to be a fruitful field of research.
The Jordan property lies in the center of attention. A group $G$ is called Jordan, if there exists a constant $J$ such that all finite subgroups of $G$ has an Abelian subgroup of index at most $J$. In analogy we can introduce the notations of solvably and nilpotently Jordan properties by requiring a solvable or a nilpotent subgroup in the finite subgroups of $G$ with small enough index.
Investigating surfaces Yu. G. Zarhin found that the birational automorphism group of a product of an elliptic curve and the projective line does not enjoy the Jordan property. (It turned out to be the only counterexample amongst surfaces.) On the contrary C. Shramov and Yu. Prokhorov showed that in many important cases the birational automorphism group is Jordan, moreover it is solvably Jordan for all varieties.
Therefore we know that the birational automorphism group is solvably Jordan, however it is not necessarily Jordan. Hence it is natural to ask what holds between the two properties. In my talk I will show that the birational automorphism group is nilpotently Jordan and give a bound for the nilpotency class in terms of the dimension of the variety.