Fr\"olicher spectral sequence and Hodge structures on the cohomology of complex parallelisable manifolds

TL;DR

This paper investigates the behavior of the Fr"olicher spectral sequence and Hodge structures on the cohomology of complex parallelisable manifolds, revealing degeneracy at the second page and conditions for purity in different Lie group cases.

Contribution

It establishes the degeneracy of the Fr"olicher spectral sequence at the second page for these manifolds and characterizes the purity of the Hodge structure in solvable and semisimple cases.

Findings

Spectral sequence degenerates at second page for these manifolds.

De-Rham cohomology has a pure Hodge structure in the solvable case.

A pure Hodge structure exists as a direct summand in the semisimple case, independent of the lattice.

Abstract

For complex parallelisable manifolds $\Gamma\backslash G$, with $G$ a solvable or semisimple complex Lie group, the Fr\"olicher spectral sequence degenerates at the second page. In the solvable case, the de-Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.