Scalar curvature, Kodaira dimension and $${{\widehat{A}}}$$-genus

Abstract

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 Kodaira dimension of (X, J) is equal to $$-\infty $$ and the canonical bundle $$K_X$$ is not pseudo-effective. We also introduce the complex Yamabe number $$\lambda _c(X)$$ for compact complex manifold X, and show that if $$\lambda _c(X)$$ is greater than 0, then $$\kappa (X)$$ is equal to $$-\infty $$ ; moreover, if X is also spin, then the Hirzebruch A-hat genus $${{\widehat{A}}}(X)$$ is zero.