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

TL;DR

This paper investigates the Kodaira dimension of certain 4-dimensional solvmanifolds with almost complex structures, revealing non-invariance under deformation and establishing Ricci flatness in specific cases.

Contribution

It introduces new almost complex structures on solvmanifolds, computes their Kodaira dimensions, and constructs a hypercomplex structure with a twistorial description.

Findings

Kodaira dimension is not a deformation invariant for these manifolds

Ricci flatness of the canonical connection is established for almost Kähler structures

A natural hypercomplex structure with a twistorial description is constructed

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.