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 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.
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Kasuya, H. Techniques of computations of Dolbeault cohomology of solvmanifolds. Math. Z. 273, 437–447 (2013). https://doi.org/10.1007/s00209-012-1013-0
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DOI: https://doi.org/10.1007/s00209-012-1013-0