  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 $\hat A(X)=0$.
