On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

Abstract: The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact $4$-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: $\mathfrak{r}\mathfrak{r}{3, -1}$, $\mathfrak{nil}^4$ and $\mathfrak{r}{4, \lambda, -(1 + \lambda)}$ with $-1 < \lambda < -\frac{1}{2}$. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover we prove Ricci flatness of the canonical connection for the almost K\"ahler structure. Regarding the other two manifolds we compute the Kodaira dimension for certain almost complex structures. Finally we construct a natural hypercomplex structure providing a twistorial description.