Canonical blow-ups of Grassmann manifolds

Abstract

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 Plücker 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 Kähler-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.

With an appendix by Kexing Chen. Typos corrected and references updated. Comments are welcome

References in corpus (2)

• K-Stability of Fano spherical varieties
• Kähler-Einstein metrics on group compactifications