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I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable.

If possible, the more examples involving abelian varieties, the merrier.

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    $\begingroup$ The 'generic' deformation to $\mathbb{Z}_p$ of an abelian variety over $\mathbb{F}_p$ of dim $g \ge 2$ will fail to be algebraizable: the deformation space is $g^2$-dimensional, while the algebraizable locus is a countable union of formal subschemes of dimension $\frac{g(g-1)}{2}$. A similar thing is true for K3 surfaces: the algebraizable locus in this case is a union of formal subschemes of codimension 1. $\endgroup$
    – Keerthi Madapusi
    Commented May 18, 2018 at 18:53
  • $\begingroup$ @KeerthiMadapusiPera: Should that be $g(g+1)/2$? BTW, there is of course a relationship between these two examples; the family of Kummer K3s associated to a family of non-algebraizable Abelian surfaces is not algebraizable. $\endgroup$
    – Daniel Litt
    Commented May 18, 2018 at 20:52
  • $\begingroup$ @DanielLitt Yes, of course. Thanks for noticing that. $\endgroup$
    – Keerthi Madapusi
    Commented May 18, 2018 at 21:21
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    $\begingroup$ A rather disorienting example is a non-algebraizable formal abelian scheme of relative dimension $g>1$ admitting "CM" by the ring of integers of a CM field $K$ of dimension $2g$ (its rigid-analytic generic fiber is "CM abeloid" yet non-algebraizable in the sense of rigid-analytic GAGA, making one more appreciative of automatic algebraizability of complex-analytic tori with CM!). For examples (with $g=2$) and more, see Theorem 2.2.3 and the discussion between its statement and proof in the book Complex Multiplication and Lifting Problems by Chai-Conrad-Oort. $\endgroup$
    – nfdc23
    Commented May 19, 2018 at 2:57
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    $\begingroup$ One example in the $p$-adic situation that I used to find quite dizzying: take a K3 $X_0$ over $k$ with Picard number one and a non-algebraic formal deformation $X$ over $W(k)$. Then no line bundle on $X_0$ lifts to $X$, and so $X$ has trivial Picard group. On the other hand, there is the amusing fact that if $L$ is a line bundle on $X_0$, then $L^{p^n}$ lifts to $X_n = X\otimes W_{n+1}(k)$. So the relative Pic of $X/W(k)$ is the disjoint union of ${\rm Spf}\, W(k)$ (zero section) and infinitely many copies of ${\rm Spf}\, W_n(k)$ for various $n$. $\endgroup$
    – Piotr Achinger
    Commented May 19, 2018 at 8:45

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