Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if Γ ∖ G Γ G is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology H ∂ ¯ ∗ , ∗ ( Γ ∖ G ) H,∂ ̄(Γ G) is canonically isomorphic to the ∂ ¯ ∂ ̄ –cohomology H ∂ ¯ ∗ , ∗ ( g C ) H,∂ ̄(gC) of the bigraded complex ( Λ ∗ , ∗ ( g C ) ∗ , <mml:mi mathvari