Compactifications of C^n and the complex projective space

TL;DR

This paper investigates the unique ways complex projective space can compactify complex Euclidean space with smooth hypersurfaces, providing complete results for even dimensions and partial insights for odd dimensions.

Contribution

It proves that for even dimensions, complex projective space is the only such compactification, extending understanding of complex geometric structures.

Findings

Unique compactification for even n

Partial results for odd n

Confirmed case for n ≡ 1 mod 4 by Ping Li

Abstract

We show that the complex projective space is the only projective manifold compactifying $\mathbb{C}^n$ by a smooth connected hypersurface, provided $n$ is even. In the odd dimensional case we give some partial results. The case when $n \equiv 1 $ mod $4$ has now been settled by Ping Li.