Techniques of computations of Dolbeault cohomology of solvmanifolds

Abstract

We consider semi-direct products $\C^{n}\ltimes_ϕN$ 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 $\C^{n}\times N$. As a corollary of this computation, we can compute the Dolbeault cohomology $H^{p,q}(G/Γ)$ of $G/Γ$ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices $Γ$.