Fr\"olicher spectral sequence of compact complex manifolds with special Hermitian metrics

TL;DR

This paper investigates how special Hermitian metrics influence the behavior of the Fr"olicher spectral sequence on compact complex manifolds, providing new examples and counterexamples related to spectral sequence degeneration.

Contribution

It introduces infinite families of balanced manifolds with non-degenerate spectral sequences and provides a counterexample to a conjecture on spectral sequence degeneration.

Findings

Balanced metrics can lead to non-degeneration of the spectral sequence at arbitrary pages.

Constructs a compact SKT manifold with non-degenerate spectral sequence, countering previous conjectures.

Extends results to generalized Gauduchon manifolds regarding spectral sequence behavior.

Abstract

In this paper we focus on the interplay between the behaviour of the Fr\"olicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Fr\"olicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Fr\"olicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Fr\"olicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.