Ricci cohomology classes

Question

Let $(M,\omega)$ be a Kaehler manifold with Ricci curvature $Ric$. I define 2-forms $\rho_k$ by the following equations:

$$\rho_k (X,Y)=\omega (Ric^k (X),Y)$$

then we have $d\rho_k=0$, the 2-forms $\rho_k$ define Ricci cohomology classes:

$$\dot{\rho_k} \in H^2 (M,{\bf R})$$

How are the Ricci cohomology classes related with the topology of the manifold?