# Subvarieties with isomorphic complements

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)?

• 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. – R. van Dobben de Bruyn Oct 5 at 7:11
• @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 ? – sawdada Oct 6 at 3:53