Scalar curvature on compact complex manifolds

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. Székelyhidi, V. Tosatti, and B. Weinkove.