Positivity and the Kodaira embedding theorem

In his recent work arXiv:1708.06713, X. Yang proved a conjecture raised by Yau in 1982, which states that any compact K\"ahler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity. In this note, we prove a generalization to this statement by showing that any compact K\"ahler manifold with positive 2nd scalar curvature (which is the average of holomorphic sectional curvature over $2$-dimensional subspaces of the tangent space) must be projective. In view of generic 2-tori being non-Abelian, this condition is sharp in some sense. Vanishing theorems are also proved for the Hodge numbers when the condition is replaced by the positivity of the weaker interpolating $k$-scalar curvature.