  We introduce the weighted orthogonal Ricci curvature -- a two-parameter
version of Ni--Zheng's orthogonal Ricci curvature. This curvature serves as a
very natural object in the study of the relationship between the Ricci
curvature(s) and the holomorphic sectional curvature. In particular, in
determining optimal curvature constraints for a compact K\"ahler manifold to be
projective. In this direction, we prove a number of vanishing theorems using
the weighted orthogonal Ricci curvature(s) in both the K\"ahler and Hermitian
category.
