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 $\Gamma \backslash G$ is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}(\Gamma \backslash G)$ is canonically isomorphic to the ${\bar{\partial}}$-cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}({\mathfrak g}^{{\mathbb C}})$of the bigraded complex $(\Lambda ^{\ast ,\ast } ({\mathfrak g}^{{\mathbb C}})^{\ast }, {\bar{\partial}})$of complex valued left invariant differential forms on the nilpotent Lie group $G$.