For a (compact) Kahler manifold $M$, the **Ricci tensor** is the symmetric $2$-form
$$
r(u,v) = \text{tr}\big( w \mapsto (D_wD_u - D_uD_w - D_{[u,w]})v\big).
$$
The **Ricci curvature** is the $2$-form
$$
r(u,v) := r(I(u),v).
$$

So the question! The first Chern class of the Levi--Civita connection for $\Omega^{(0,1)}$ is a $(1,1)$-form. The Ricci curvature of $M$ is also a $(1,1)$-form. Are these two forms related in any way? For example, might one be a scalar multiple of the other?