Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products ${\text{C}}^n{⋉}_{\varphi }N$ of Lie groups with lattices $\mathrm{\Gamma }$ 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 $G\mathrm{/}\mathrm{\Gamma }$ by using the Dolbeaut cohomology of the Lie algebras of the direct product ${\text{C}}^n\times N$. As a corollary of this computation, we can compute the Dolbeault cohomology $H^{p,q}(G\mathrm{/}\mathrm{\Gamma })$ of $G\mathrm{/}\mathrm{\Gamma }$ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices $\mathrm{\Gamma }$.