Canonical blow-ups of Grassmann manifolds

2020/07/13 by Fang, Hanlong
#14M99 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.06200

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 s is given, and open questions are raised.

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