Canonical blow-ups of Grassmann manifolds

We introduce certain canonical blow-ups $\mathcal T_{s,p,n}$, as well as their distinct submanifolds $\mathcal M_{s,p,n}$, of Grassmann manifolds $G(p,n)$ by partitioning the Pl\"ucker coordinates with respect to a parameter $s$. Various geometric aspects of $\mathcal T_{s,p,n}$ and $\mathcal M_{s,p,n}$ are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of K\"ahler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which $\mathcal T_{s,p,n}$ are examples, as a generalization of the wonderful compactification. Lastly, a generalization of $\mathcal T_{s,p,n}$ according to vector-valued parameters $\overline s$ is given, and open questions are raised.