Let $k$ be an algebraically closed field, and $Z,U$ be smooth varieties over $k$. A variety is by definition integral (irreducible + connected).

Can we classify all smooth projective varieties $X$ over $k$ with a closed subvariety isomorphic to $Z$ and the complement isomorphic to $U$ (assume there is one)? Of course they are all birational and are the same in the Grothendieck group. In general, it seems too difficult. I hope there is a simple condition for uniqueness, and classification in low dimension case (for curve it's trivial).

Examples:

• $Z$ is a point, this can be thought as whether the one-point compactification is unique.
• $Z=\mathbb P^{n-1}$, $U=\mathbb A^n$, $X=\mathbb P^n$. Are there any other examples?
• $Z=\mathbb P^{n-1}$, $U=\mathbb P^1 \times \mathbb A^{n-1}$, then consider any rank $2$ projective bundle over $\mathbb P^{n-1}$. Are there any other examples?
• $X_0$ is an abelian variety, $Z$ an abelian subvariety, $U=X_0-Z$.
• The general type case.