On the Canonical Bundle of Complex Solvmanifolds and Applications to Hypercomplex Geometry

We study complex solvmanifolds \(\Gamma \backslash G\) $\mathrm{\Gamma }\backslash G$ with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of G. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form \(\psi \) $\psi$ canonically associated to \((\mathfrak {g},J)\) $(\mathfrak{g},J)$ , where \(\mathfrak {g}\) $\mathfrak{g}$ is the Lie algebra of G, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of \(\psi \) $\psi$ , for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold \((M^{4n},\{J_1,J_2,J_3\})\) $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of \((M,J_{\alpha })\) $(M,J_{\alpha })$ is trivial only for \(\alpha =1\) $\alpha =1$ , so that M is not an \({\text {SL}}(n,\mathbb {H})\) $\text{SL}(n,\mathbb{H})$ -manifold.