Kodaira dimension of $\mathrm{SU}(m)$-structures

TL;DR

This paper investigates the Kodaira dimension of almost complex manifolds with $ ext{SU}(m)$-structures, introducing new concepts and constructions to analyze their complex geometric properties.

Contribution

It introduces the notions of splitting type and associated $ ext{SU}(m)$-structures, and provides methods to construct non-invariant almost complex structures with specific Kodaira dimensions.

Findings

Constructed non-invariant almost complex structures with Kodaira dimension 0

Constructed non-invariant almost complex structures with Kodaira dimension -∞

Applicable to complex structures of splitting type and well-studied almost complex manifolds

Abstract

We study the Kodaira dimension of almost complex manifolds admitting an $\mathrm{SU} (m)$-structure. We introduce the notion of almost complex structure of splitting type and of associated $\mathrm{SU} (m)$-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 $-\infty$. Our results apply, in particular, to complex structures of splitting type and to several almost complex manifolds already well-studied in the literature