Scalar curvature, Kodaira dimension and $\hat A$-genus

Let $(X,g)$ be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure $J$ compatible with $g$, then the canonical bundle $K_X$ is not pseudo-effective and the Kodaira dimension $\kappa(X,J)=-\infty$. We also introduce the complex Yamabe number $\lambda_c(X)$ for compact complex manifold $X$, and show that if $\lambda_c(X)>0$, then $\kappa(X)=-\infty$; moreover, if $X$ is also spin, then the Hirzebruch $A$-hat genus $\hat A(X)=0$.