Scalar curvature on compact complex manifolds

Abstract

In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if $K_X$ (resp. $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Sz\'{e}kelyhidi, V. Tosatti and B. Weinkove.