Projective compactifications of hypersurface complements

Let $U$ be a smooth affine variety of the form $U=\mathrm{P}^n\setminus H$, where $H\subset\mathrm{P}^n$ is a smooth connected hypersurface. We prove that every smooth projective compactification of the complex manifold $U$, by adding a smooth connected divisor, is isomorphic to $\mathrm{P}^n$. The proof uses the rational Reeb index of the normal circle bundle and positive $S^1$-equivariant symplectic homology. This paper is motivated by the work of Chi Li and Zhengyi Zhou, and makes a generalization to their result.