On Kodaira dimension and scalar curvature in almost Hermitian geometry

In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $\kappa(M, J)=-\infty$; or $\kappa(M, J)=0$, in which case $(M,J,g)$ is a K\"{a}hler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero.