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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)?

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    $\begingroup$ It implies $[Y_1] = [Y_2] \in K_0(\mathbf{Var}_{\mathbf C})$. For example, this implies (in the smooth projective case) that $H^{p,q}(Y_1) = H^{p,q}(Y_2)$ for all $p,q$ (Deligne) and that $Y_1$ is stably birational to $Y_2$ (Larsen–Lunts). A first example of non-isomorphic smooth projective varieties with the same class in $K_0$ is $\mathbf P^1 \times \mathbf P^1$ and $\operatorname{Bl}_p \mathbf P^2$. I don't know how to make these occur in an $X$ with isomorphic complement, but maybe people who understand toric varieties can easily write down such a thing. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 5, 2019 at 7:11
  • $\begingroup$ @R.vanDobbendeBruyn Thanks, so the case $X$ is a surface is also true as stably birational smooth projective curves must be isomorphic. For the case $X$ is a threefold, is there any restriction for two smooth projective surfaces to be embedded in a common threefold ? $\endgroup$
    – Zhiyu
    Commented Oct 6, 2019 at 3:53

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