The Frölicher spectral sequence can be arbitrarily non-degenerate

The Frölicher spectral sequence of a compact complex manifold X measures the difference between Dolbeault cohomology and de Rham cohomology. If X is Kähler then the spectral sequence collapses at the E1term and no example with dn ≠  0 for n > 3 has been described in the literature.We construct for n ≥  2 nilmanifolds with left-invariant complex structure Xn such that the n-th differential dn does not vanish. This answers a question mentioned in the book of Griffiths and Harris.