Compactifications of C^n and the complex projective space
Thomas Peternell
TL;DR
The paper tackles the problem of classifying smooth compactifications $X$ of $\mathbb{C}^n$ with a smooth divisor at infinity, proving that for even $n$ the only such compactification is $X\simeq \mathbb{P}_n$ with a hyperplane as the divisor, under a Kähler assumption and a cohomology-bijectivity condition up to degree $2n-2$. It derives this through cohomology-ring identifications $H^*(X)\cong H^*(\mathbb{P}_n)$ and $H^*(Y)\cong H^*(\mathbb{P}_{n-1})$, and by analyzing Chern-number identities tied to the tangent sequence, yielding a quadratic with solutions $n=r-1$ (projective space) or $n=2r-1$ (an obstacle in odd dimensions). When $h^0(X,\mathcal{O}_X(1))\ge 2$, the odd-dimension analysis collapses to $X\simeq \mathbb{P}_n$; otherwise the authors develop a program based on a system of Chern-class equations and $A_k$-invariants to rule out non-projective solutions. For odd $n$, the paper provides partial results, including reductions to Fano geometry with index $(n+1)/2$ and partial verifications (e.g., for $n\equiv 1 \mod 4$ by Ping Li), and outlines a path toward a full resolution via arithmetic constraints on Chern data.
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.