Motivation: Let $Y_i$ be two smooth projective varieties over $\mathbb R$ which are isomorphic over $\mathbb C$, although $Y_i(\mathbb R)$ (under analytic topology) can be different, they do have some common topological invariants, for instance their $\mathbb F_2$ cohomology must be the same, see Etale cohomology of and forms of algebraic groups.

Now let $Y_i$ be two integral schemes flat and projective over over $\mathbb C[[t]]$ which are isomorphic and smooth over $\mathbb C((t))$, what common topological invariants do $Y_i(\mathbb C)$ (base change along $t=0$) have? For simplity, we assume the special fibers are normal (one can assume they are normal crossing divisors in $Y_i$ if necessary). So in the curve case, the special fibers are of same topology.

  • $\begingroup$ The statement claimed in your Motivation is incorrect. If $Y_1$ and $Y_2$ are real conics with (resp. without) real points, they are isomorphic over $\mathbb{C}$, but their real loci do not have the same cohomology. $\endgroup$ – Olivier Benoist Oct 10 at 6:30
  • $\begingroup$ @OlivierBenoist thank you, how to correct it? $\endgroup$ – sawdada Oct 12 at 16:03

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