We construct new families of non-Kähler compact complex threefolds belonging to Kato's Class L. The construction uses certain polynomial automorphisms of C3. We also study basic properties of our manifolds.

We study compact complex manifolds covered by a domain in $n$-dimensional projective space whose complement $E$ is non-empty with $(2n-2)$-dimensional Hausdorff measure zero. Such manifolds only exist for $n\ge 3$. They do not belong to the class $𝒞$, so they are neither Kähler nor Moishezon, their Kodaira dimension is $-\infty$, their fundamental groups are generalized Kleinian groups, and they are rationally chain connected. We also consider the two main classes of known 3-dimensional examples: Blanchard manifolds, for which...