On the sum of the Hermitian scalar curvatures of a compact Hermitian manifold (Q1081853)

In this brief paper we prove that if the sum of the two Hermitian scalar curvatures is positive at every point of a compact Hermitian manifold, then all the plurigenera of the manifold vanish. As an application we show that if a compact Hermitian manifold of complex dimension 2 or 3 has positive holomorphic sectional curvature at every point then here also all the plurigenera of the manifold vanish. Weaker results are obtained if the quantities are assumed to be only nonnegative.