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The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic surfaces.

Recall how to construct a moduli space $\mathcal{M}$ containing a curve $X$ using deformation theory (reference is Hartshorne's book):

  1. Representability. To construct the formal neighbourhood $D_{\mathcal{M},X}$ of $X$ in $\mathcal{M}$, note that if $\text{Spec}A$ is a fat point (i.e. $A$ is Artinian) then maps into $\mathcal{M}$ correspond to flat families of schemes over $\text{Spec}A$ with central fibre $X$. Then use Artin representability to show (under conditions) that this functor $$F\ :\ \text{Artin rings} \ \to \ \text{Sets/Groupoids}$$ sending $$A\ \mapsto\ \{\text{flat families of schemes over }\text{Spec}A \text{with central fibre }X\}$$ is represented by a formal scheme/stack $D_{\mathcal{M},X}$.
  2. Algebraicity. To construct an open neighbourhood of $X$, we need to show that $D_{\mathcal{M},X}$ can be “algebraised” to a scheme/stack $U_{\mathcal{M},X}$. This can be done using Artin algebraicity, under conditions. A stronger condition is that $H^2(X,\mathcal{O}_X)=0$.

Question: (edit: answered in comments below) Let $X$ be an algebraic surface. I have not seen a general definition of a moduli stack of algebraic surfaces containing $X$. So I assume either 1 or 2 does not work—which is it, and why? If $F$ is not represented by the formal completion of some stack, is there no hope of building a moduli stack for $X$?

tl;dr what (if anything) fails when you try to use deformation theory to build a moduli stack of surfaces near $X$?

Edit:

Edited question: What is the obstruction to the existence of the analytic moduli stacks $\mathcal{M}$? Do they always exist?

There are other questions, e.g. Results about moduli of surfaces, but it is unclear how many problems there come from not working with stacks, and also it is unclear what is expected to be true.

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    $\begingroup$ What about K3 surfaces? The general deformation (in the Kodaira-Spencer sense) of an algebraic K3 is not algebraic, so you need some extra structure in order to build a moduli space in the algebraic category (for instance, you can fix a polarization). $\endgroup$
    – Francesco Polizzi
    Commented Jun 7, 2022 at 14:44
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    $\begingroup$ Pulcinella, you are wrong. $\endgroup$
    – Jason Starr
    Commented Jun 7, 2022 at 18:07
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    $\begingroup$ About what part? $\endgroup$
    – Pulcinella
    Commented Jun 7, 2022 at 19:36
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    $\begingroup$ A flat, locally finitely presented morphism of complex analytic spaces to Spec of a local Artin $\mathbb{C}$-algebra (with residue field equal to $\mathbb{C}$) is the morphism of complex analytic spaces associated to a flat, locally finitely presented morphism of schemes if and only if the base change to $\text{Spec}\ \mathbb{C}$ is the complex analytic space associated to a finitely presented $\mathbb{C}$-scheme. So there is no "intrinstic" way to distinguish "algebraic" infinitesimal deformations of K3 surfaces, Abelian varieties, etc., from the "analytic" infinitesimal deformations. $\endgroup$
    – Jason Starr
    Commented Jun 7, 2022 at 20:12
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    $\begingroup$ Thanks both very much, that was an embarrassing mistake by me. Your comments answer my question (for readers- mathoverflow.net/a/16000/119012 also seems relevant). As far as I can tell, you could ask my question in the analytic category- in that case, is the (obstruction to the) existence of an analytic moduli stack $\mathcal{M}$ well known? If not maybe I should edit my question to be about that. $\endgroup$
    – Pulcinella
    Commented Jun 8, 2022 at 0:19

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