Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space $(\mathbb{P}^N,\omega\_{FS})$ for some $N,$ via the Plucker embedding.

What can we say about the Ricci curvature of the Grassmannian? or even the scalar curvature? (any bounds?)