p-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

Let (X, J) be a nilmanifold with an invariant nilpotent complex structure. We study the existence of p-Kähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no p-Kähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.