Assume that $M$ is a Riemannian manifold which is equipped with  symplectic structure $\omega$.

Inspired by the definition of "Scalar curvature", one can define the quantity $tr_{\omega} Ric$ where $Ric$ is the Ricci curvature tensor associated to the metric but the trace is computed with respect to $\omega$ not with respect to the metric.

I am not sure whether this quantity is equal to the scalar curvature when $M$ is a Kahlar manifold.