It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness, one needs to introduce so called the Ricci form for a Kahler manifold $(X,g)$ defined by $$ Ric(\omega)=-i\partial \overline{\partial}\log(g_{i,\overline{j}}), $$ where $g=(g_{i,\overline{j}})$ is a Kahler metric. A Kahler manifold$(X,g)$ is called Ricci flat if $Ric(\omega)$ vanishes.

I wonder if the complex one coincides with real one for a Kahler manifold $(X,g)$. How Ricci curvature and Ricci form are related?

It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness, one needs to introduce so called the Ricci form for a Kahler manifold $(X,g)$ defined by $$ Ric(\omega)=-i\partial \overline{\partial}\log\det(g_{i,\overline{j}}), $$ where $g=(g_{i,\overline{j}})$ is a Kahler metric. A Kahler manifold$(X,g)$ is called Ricci flat if $Ric(\omega)$ vanishes.

I wonder if the complex one coincides with real one for a Kahler manifold $(X,g)$. How Ricci curvature and Ricci form are related?