Mathematical Research Letters

Volume 21 (2014)

Number 1

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

Pages: 121 – 125



Baohua Fu (Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China)

Jun-Muk Hwang (Korea Institute for Advanced Study, Seoul, Korea)


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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