Techniques of computations of Dolbeault cohomology of solvmanifolds

TLDR: In this article, the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ is computed for semi-direct products of Lie groups with lattices Γ such that N are nilpotent Lie groups.

Abstract: We consider semi-direct products \({\mathbb{C}^{n}\ltimes_{\phi}N}\) of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product \({\mathbb{C}^{n}\times N}\) . As a corollary of this computation, we can compute the Dolbeault cohomology Hp,q(G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.