Scalar curvature, Kodaira dimension and $\widehat{A}$-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 $\kappa (X,J)=-\mathrm{\infty }$. We also introduce the complex Yamabe number ${\lambda }_c(X)$ for compact complex manifold $X$, and show that if ${\lambda }_c(X)>0$, then $\kappa (X)=-\mathrm{\infty }$; moreover, if $X$ is also spin, then the Hirzebruch $A$-hat genus $\widehat{A}(X)=0$.