Sectional curvatures and the separation set of the complex projective space in its Plücker model

The Plücker embedding of the complex projective space $\mathbb CP^{k-1}$ in the Grassmannian $G^+_{2,2k}$ of bivectors is used for proving several theorems on the relationship between the complex structure of $\mathbb CP^{k-1}$ and its Riemannian geometry. It is shown that the separation set of $\mathbb CP^{k-1}$ in the Plücker model is a face of $G^+_{2,2k}$ for a certain calibration.