Uniqueness of equivariant compactifications of $\mathbb{C}^n$ by a Fano manifold of Picard number 1

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 $\mathbb{C}^n$ equivariantly. Hassett and Tschinkel showed that when $X = \mathbb{P}^n$ with $n \geq 2$, there are many distinct ways that $X$ can be realized as equivariant compactifications of $\mathbb{C}^n$. 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 $\mathbb{C}^n$ equivariantly in more than one ways. This answers questions raised by Hassett-Tschinkel and Arzhantsev-Sharoyko.

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Published 25 July 2014