Techniques of computations of Dolbeault cohomology of solvmanifolds

TL;DR

This paper develops methods to compute the Dolbeault cohomology of solvmanifolds formed by semi-direct products of complex Lie groups, using Lie algebra cohomology and finite-dimensional complexes, revealing lattice-dependent variations.

Contribution

It introduces a new computational approach for Dolbeault cohomology of solvmanifolds via Lie algebra techniques and finite cochain complexes, accounting for lattice choices.

Findings

Dolbeault cohomology can be computed using Lie algebra cohomology.

The cohomology varies with different lattice choices.

Finite-dimensional complexes suffice for computation.

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 $\Gamma$.