Dolbeault cohomology via left-invariant forms for all complex nilmanifolds

Establish that for every complex nilmanifold X = (Γ\G, J) with G a real nilpotent Lie group, Γ a lattice, and J a left-invariant complex structure with associated Lie algebra (g, J), the natural inclusion H^{p,q}(g, J) → H^{p,q}_{∂̄}(X) is an isomorphism for all bidegrees (p, q), thereby showing that Dolbeault cohomology can always be computed by left-invariant forms.

Background

For a complex nilmanifold X = (Γ\G, J) obtained from a real nilpotent Lie group G with lattice Γ and a left-invariant complex structure J, there is a natural map from the Lie algebra Dolbeault cohomology H^{p,q}(g, J) to the Dolbeault cohomology H^{p,q}_{∂̄}(X). It is known that this map is always injective and, in many cases, an isomorphism, in which case one says that Dolbeault cohomology is computed by left-invariant forms.

The paper proves this isomorphism for the class of almost abelian complex nilmanifolds and surveys prior results. However, the general assertion that the map is always an isomorphism for arbitrary complex nilmanifolds remains conjectural and is highlighted as an open problem, with references to overviews and recent progress.