Kodaira Dimension of SU(m)-structures

We study the Kodaira dimension of almost complex manifolds admitting an SU(m)-structure. We introduce the notion of almost complex structure of splitting type and of associated SU(m)-structure. When the latter is pseudoholomorphic, we provide two constructions that allow to obtain non-invariant almost complex structures with Kodaira dimension 0, resp. with Kodaira dimension −∞. Our results apply, in particular, to complex structures of splitting type and to several almost complex manifolds already well-studied in the literature.