Which projective varieties are étale over affine space? In Liu's answer to this MO question there is a characterization of smooth _affine_ varieties which are étale over affine space. I was wondering if one can give a similar characterization for projective varieties. Let $X$ be a smooth _projective_ variety of dimension $n$. When is $X$ étale over $\mathbb A^n$? How can one construct an étale map $X\to \mathbb A^n$? I should maybe point out that I'm primarily interested in the case when the base field is $\mathbb C$. Thank you for any help!

The image of a projective variety under any morphism is proper, and the only subvarieties of $\mathbb{A}^n$ that are proper are those of dimension $0$. So a projective variety of dimension $n$ is etale over $\mathbb{A}^n$ iff $n=0$.