Compactifications of C^n and the complex projective space

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.

updated and enlarged version, minor explanation added