$p$-K\"ahler and balanced structures on nilmanifolds with nilpotent complex structures

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