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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 $\mathbb{C}$.

Is there a notion of "non-algebraic" K3 surfaces over fields of positive characteristic?

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 / ...?

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  • $\begingroup$ For every uncountable, algebraically closed field $k$, I believe there exist formally smooth, formal schemes over $\text{Spf}\ k[[t]]$ with closed fibers being K3 surfaces and that are not algebraizable. Is that what you are asking about? $\endgroup$
    – Jason Starr
    Commented Jun 15, 2015 at 13:00
  • $\begingroup$ Quite possibly; does every non-algebraic K3 surface over $\mathbb{C}$ arise this way? $\endgroup$
    – Daniel Loughran
    Commented Jun 15, 2015 at 13:12

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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).

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    $\begingroup$ To add a useful comment for beginners: The big difference between $\mathbb{C}$ and $\mathbb{F}_p$ here is that we can talk about convergence of power series over $\mathbb{C}$. Thus, we can take the formal power series in the $x_i$ and plug in nonzero complex numbers to get non-algebraic $K_3$'s. Over $\mathbb{F}_p$, the power series still exist, but it doesn't make sense to evaluate them anywhere except at $0$. $\endgroup$
    – David E Speyer
    Commented Jun 15, 2015 at 13:59
  • $\begingroup$ Right, good point. $\endgroup$
    – Francesco Polizzi
    Commented Jun 15, 2015 at 14:01
  • $\begingroup$ since the word "beginner" has been used, a link to a previous answer seems relevant (I didn't know beforehand that all complete regular local rings are pretty much all created equal) mathoverflow.net/a/191737/73972 $\endgroup$
    – pro
    Commented Jun 15, 2015 at 14:24

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