Canonical blow-ups of Grassmann manifolds

TL;DR

This paper introduces canonical blow-ups of Grassmann manifolds, studies their geometric properties, and generalizes these constructions, contributing new insights into their structure and compactification methods.

Contribution

It defines and analyzes canonical blow-ups of Grassmann manifolds, explores their geometric features, and introduces the concept of homeward compactification as a generalization.

Findings

$ ext{T}_{s,p,n}$ are smooth and have rich symmetry properties.

The anti-canonical bundles of $ ext{T}_{s,p,n}$ are semi-positive.

Existence of K"ahler-Einstein metrics on these blow-ups is established.

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 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.