Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups

TL;DR

This paper proves that the Dolbeault cohomology of certain complex nilmanifolds can be computed using left-invariant forms, especially when the manifold is foliated in toroidal groups, confirming the conjecture in low dimensions.

Contribution

It establishes the conjecture for complex nilmanifolds foliated in toroidal groups and extends previous methods by removing the need for a holomorphic fibration.

Findings

Conjecture holds in real dimension up to six.

Dolbeault cohomology can be computed by left-invariant forms under specific foliations.

Generalizes previous approaches to include foliations in toroidal groups.

Abstract

It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.