Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

Abstract:

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 $G$.