The Fr\"olicher spectral sequence of certain solvmanifolds

TL;DR

This paper investigates the degeneracy properties of the Fr"olicher spectral sequence on certain complex solvmanifolds, establishing conditions under which degeneracy at a specific stage is preserved in more complex structures.

Contribution

It proves that the Fr"olicher spectral sequence degenerates at E2 for complex parallelizable solvmanifolds and extends this degeneracy to certain semi-direct product solvmanifolds under specific conditions.

Findings

Degeneracy at E2 for complex parallelizable solvmanifolds.

Degeneracy at E_r for nilmanifolds implies degeneracy at E_r for related solvmanifolds.

Conditions for degeneracy preservation in semi-direct product solvmanifolds.

Abstract

We show that the Fr\"olicher spectral sequence of a complex parallelizable solvmanifold is degenerate at $E_{2}$-term. For a semi-direct product $G=\C^{n}\ltimes_{\phi}N$ of Lie-groups with lattice $\Gamma=\Gamma^{\prime}\ltimes \Gamma^{\prime\prime}$ such that $N$ is a nilpotent Lie-group with a left-invariant complex structure and $\phi$ is a semi-simple action, we also show that, if the Fr\"olicher spectral sequence of the nilmanifold $N/\Gamma^{\prime\prime}$ is degenerate at $E_{r}$-term for $r\ge 2$, then the Fr\"olicher spectral sequence of the solvmanifold $G/\Gamma$ is also degenerate at $E_{r}$-term.