On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry

We study complex solvmanifolds $\mathrm{\Gamma }\mathrm{\backslash }G$ 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 $\psi$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ 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 $\psi$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of $(M,J_{\alpha })$ is trivial only for $\alpha =1$, so that $M$ is not an $\mathrm{SL}(n,\mathbb{H})$-manifold.