Let $X$ be a smooth irreducible projective variety over $\mathbb C$, $Y_1, Y_2$ are two closed smooth subvarieties. Assume $X-Y_1 \cong X-Y_2$, what do $Y_1$ and $Y_2$ have in common (at least topologically)?
For example, if $X$ is a curve then we must have $Y_1 \cong Y_2$. If $X$ is a projective space and the complement is an affine space, then again $Y_1 \cong Y_2$. It seems that $Y_1$ and $Y_2$ may not be isomorphic in general but do have some common invariants (e.g one can consider the excision sequence for compactly supported cohomology), is there any interesting example?
Moreover, fix $X$ and a open subset $U$ of $X$, can we describe the isomorphism classes of $Y$ such that $X-Y \cong U$ (can there be infinitely many)?
In general, if one only requires $Y_i$ be normal and reduced, then are the singularities of $Y_i$ comparable (e.g $Y_1$ is smooth then $Y_2$ is also smooth)?