Kodaira dimension of almost K\"ahler manifolds and curvature of the canonical connection

TL;DR

This paper extends the concept of Kodaira dimension to almost K"ahler manifolds, explicitly computes it for certain threefolds, and explores its relation to the curvature of the canonical connection, revealing cases with vanishing Ricci curvature.

Contribution

It provides explicit computations of Kodaira dimension for a family of almost K"ahler threefolds and investigates the connection between Kodaira dimension and canonical connection curvature.

Findings

Kodaira dimension computed for specific almost K"ahler threefolds

Ricci curvature vanishes in the studied examples

Links between Kodaira dimension and curvature are established

Abstract

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost K\"ahler manifolds, providing an explicit computation for a family of almost K\"ahler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost K\"ahler manifold, and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost K\"ahler metric vanishes for all the members of the family.