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Let $R$ be a complete DVR with quotient field $K$ and let $f:\mathfrak{X} \to \mathrm{Spf}(R)$ be a smooth proper formal scheme.

If the (rigid analytic) generic fibre of $f$ is (the analytification of) a projective variety $X$ over $\mathrm{Spec}(K)$, then is $f$ the formal completion of a smooth projective scheme over $\mathrm{Spec}(R)$?

One could also drop the projectivity hypothesis and just assume that the generic fibre "is" a (proper) variety over $K$.

It seems reasonable to expect this to be true at least when one knows that $X$ has a unique smooth proper model over $\mathrm{Spec}(R)$ (which is a scheme).

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  • $\begingroup$ Guess: there could be non-projective toric varieties which deform to projective ones. This would answer the first question negatively. $\endgroup$
    – Piotr Achinger
    Commented Nov 14, 2017 at 9:51
  • $\begingroup$ @PiotrAchinger: I agree that something like this could perhaps happen, so one should probably also assume that the special fibre is projective. $\endgroup$
    – naf
    Commented Nov 14, 2017 at 10:36

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