I have a very naive question.


Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with trivial canonical bundle. Such surfaces often come in useful when studying the moduli and deformations of K3 surfaces over $\mathbb{C}$.



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> Is there a notion of "non-algebraic" K3 surfaces over fields of positive characteristic?
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As I say above, I am well aware that this question is very naive. Possible guesses I had in mind were formal schemes / stacks / rigid analytic spaces / ...?



Non-algebraic K3 surfaces in characteristic $p$
Let me briefly expand on Jason's comment.


Actually, "formal scheme" is the right word here. 


For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T\_X)=0, \quad h^1(X, T\_X)=20, \quad \, h^2(X, T\_X)=0,$$
so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x\_1, \ldots, x\_{20}]],$$
where $x\_1, \ldots, x\_{20}$ are indeterminates. 


Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation *is not effective*, in the sense that *there exists no algebraic deformation* $\mathcal{X} \to \textrm{Spec}(A)$, where $ \mathcal X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that any algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.


Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies. 


More details on this topic can be found in the books *Deformations of algebraic schemes* (Sernesi) and *Deformation theory* (Hartshorne).


