The Fr\"olicher spectral sequence can be arbitrarily non degenerate

TL;DR

This paper constructs examples of compact complex manifolds where the Frölicher spectral sequence does not degenerate at a finite stage, showing the sequence can be arbitrarily non-degenerate.

Contribution

It provides explicit nilmanifold examples with left-invariant complex structures where the spectral sequence's differentials do not vanish at the nth stage, correcting previous inaccuracies.

Findings

Constructed nilmanifolds with non-vanishing d_n differentials

Demonstrated the spectral sequence can be arbitrarily non-degenerate

Corrected earlier incorrect example

Abstract

The Fr\"olicher spectral sequence of a compact complex manifold $X$ measures the difference between Dolbeault cohomology and de Rham cohomology. We construct for $n\geq 2$ nilmanifolds with left-invariant complex structure $X_n$ such that the $n$-th differential $d_n$ does not vanish. This replaces an earlier incorrect example by the second author.