Choose a prime $p$. Let $X$ be a smooth proper scheme over $\mathbb{Q}$. Does there exist a smooth proper scheme $X'$ over $\mathbb{Z}_{(p)}$ such that $X\otimes_{\mathbb{Q}} \mathbb{C}$ is diffeomorphic to $X'\otimes_{\mathbb{Z}_{(p)}}\mathbb{C}$?
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1$\begingroup$ Yes, if $X$ is an abelian variety. And probably no in general. Here is a vague idea: Let $X$ be a smooth proper geometrically connected scheme over $\mathbb{Q}$ with no moduli (e.g., $\mathrm{H}^1(X,T_X)=0$) and no twists. It seems likely to me that such a variety won't have good reduction everywhere, so let's say $p$ is a prime of bad reduction. Then there won't be any $X'$ as in your question which is algebraically deformation equivalent to $X$ over $\mathbb{C}$. Now, there might still be an $X'$ which is diffeomorphic to $X$, though.... $\endgroup$– Ariyan JavanpeykarCommented Jun 10, 2020 at 16:11
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