# Topology of different special fibers of a smooth projective variety over $\mathbb C((t))$

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.

• 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. – Olivier Benoist Oct 10 at 6:30
• @OlivierBenoist thank you, how to correct it? – sawdada Oct 12 at 16:03