Techniques of computations of Dolbeault cohomology of solvmanifolds

Abstract: We consider semi-direct products $\C^{n}\ltimes_{\phi}N$ of Lie groups with lattices $\Gamma$ 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 $G/\Gamma$ 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 $H^{p,q}(G/\Gamma)$ of $G/\Gamma$ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices $\Gamma$.