Uniqueness of equivariant compactifications of C^n by a Fano manifold of Picard number 1

TL;DR

This paper proves that among Fano manifolds of Picard number 1 with smooth VMRT, only projective space can compactify the complex vector group C^n in multiple equivariant ways, establishing its uniqueness.

Contribution

It establishes the uniqueness of projective space as the only Fano manifold of Picard number 1 with smooth VMRT that admits multiple equivariant compactifications of C^n.

Findings

Projective space uniquely admits multiple equivariant compactifications of C^n.

Other Fano manifolds of Picard number 1 with smooth VMRT do not admit such multiple compactifications.

Answers longstanding questions by Hassett-Tschinkel and Arzhantsev-Sharoyko.

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