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

TL;DR

The paper analyzes the Kodaira dimension $\kappa_J$ for compact almost complex manifolds with an $\mathrm{SU}(m)$-structure, introducing almost complex structures of splitting type and their standard SU$(m)$-structures. It shows that if the standard structure is pseudoholomorphic, then invariant structures on compact quotients have $\kappa_J=0$, and it provides two constructions yielding non-invariant structures with $\kappa_J=0$ or $\kappa_J=-\infty$ by exploiting a pseudoholomorphic fibration over a complex torus. A key theme is relating the vanishing or triviality of tensor powers of the canonical bundle $K_J$ to the Kodaira dimension, with explicit arguments on invariant vs non-invariant trivializations on compact quotients of Lie groups. The results unify and extend previous computations of $\kappa_J$ on Lie-group–based manifolds and supply new non-invariant examples across tori, Iwasawa/Nakamura-type manifolds, and higher-dimensional solvmanifolds, highlighting the utility of splitting-type SU$(m)$-structures for controlling $\kappa_J$ in non-integrable settings.

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

Theorems & Definitions (21)

• Remark 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Remark 2.6
• Lemma 2.7
• proof