Uniqueness of equivariant compactifications of Cⁿ by a Fano manifold of Picard number 1

2013/01/23 by Baohua Fu, Fu, Baohua, Jun-Muk Hwang +1
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Meromorphic and Entire Functions #math.AG

paper · pdf · doi:10.48550/arxiv.1301.5486

arxiv created 2013/01/23 · arxiv updated 2013/01/24

Abstract

Let X be an n-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group Cⁿ equivariantly. Hassett and Tschinkel showed that when X = Pⁿ with n ≥ 2, there are many distinct ways that X can be realized as equivariant compactifications of Cⁿ. Our result says that projective space is an exception: among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying Cⁿ equivariantly in more than one ways. This answers questions raised by Hassett-Tschinkel and Arzhantsev-Sharoyko.

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• Fano manifolds obtained by blowing up along curves with maximal Picard number
• On Fano manifolds of Picard number one with big automorphism groups
• Compactifications of Cⁿ and the complex projective space
• Endomorphisms of hypersurfaces of Fano manifolds of Picard number 1