Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

Source abstract

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