TITLE: Is there a geometric realization of $\mathbf{C}((t))$-varieties?
QUESTION [28 upvotes]: Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle.
When $F = \mathbf{C}$, there's a Betti realization functor $MV_{\mathbf{C}} \to H$.  Is there a similar realization functor $MV_{\mathbf{C}((t))} \to H_{/S^1}$?
Etale homotopy theory gives a realization in profinite spaces over a profinite circle.  I'm looking for a refinement of that.
If a variety over $\mathbf{C}((t))$ is defined using only convergent power series, I think it is clear which bundle over $S^1$ I want.  If the variety is defined using finitely many nonconvergent power series, I can imagine some tricks for defining the bundle over S^1 that I want, even up to homeomorphism.  But if a map between varieties has some nonconvergent power series in it, I bet there is no way to write down a map between those bundles, not functorially.  I hope that it can be done up to homotopy.

REPLY [7 votes]: I don't know about $\mathbb A^1$-homotopy categories. Let me try to do something that I think will work just for the category of quasiprojective varieties.
The key lemma is this:

There is a canonical way to associate to a variety $X$ over $\mathbb C$ and a $\mathbb C((t))$-point of $X$ a homotopy class of maps from the circle to $X$ that is functorial for maps of varieties over $\mathbb C$.

To do this, by a stratification of $X$ reduce to the case when $X$ is smooth, then by resolution of singularities we may assume that $X$ has a smooth normal crossings compactification. Then extend the $\mathbb C((t))$-point to a $\mathbb C[[t]]$-point of the compactification. Locally near that point, $X$ looks like $\mathbb R^n$ times a torus, so a homotopy class of circle maps is a tuple of integers - you just take the $t$-adic valuations of the coordinates in the obvious way.
Could this depend on the compactification? No. It only depends on the coordinates up to some higher power of $t$, i.e. it is a locally constant function in the $t$-adic topology on $X(\mathbb C((t)))$. Because we may approximate any solution by a solution with algebraic (and hence convergent) power series using Hensel's lemma for the ring of algebraic power series, it is sufficient to check this independence for convergent power series, where it is obvious, because our definition recovers the usual definition given by evaluating the power series on a small circle.
An identical reduction argument proves functorially.

We view a quasiprojective variety $X - Z \subseteq \mathbb P^n$ over $\mathbb C((t))$ as a $\mathbb C((t))$-point of the flag Hilbert scheme parameterizing pairs of closed subscheme $Z \subseteq X \subseteq \mathbb P^n$. This Hilbert scheme admits a stratification into strata where the fibers have constant topological type. For each stratum $M$ in the Hilbert scheme, we have a fibration of manifolds (or more general spaces) over $M(\mathbb C)$, so $\mathbb C(t))$-point of $M$ defines a homotopy class of circle embeddings into $M$ and hence a homeomorphism class of manifolds over the circle.
For functoriality, we use a more intricate moduli space parameterizing ordered triples of two quasiprojective varieties and a map between them. Again there will be a stratification of constant topological types, and a homotopy class of circle embeddings into a stratum, giving a class of pairs of manifolds w/ a map between them, up to, I believe, homotopy.
You should be able to check that this agrees with composition of maps bby passing to the moduli space of triples of varieties and pairs of morphisms. To check that the homotopies defined their satisfy the appropriate coherence condition, you would have pass to the moduli space of quadruples of varieties with triples of morphisms, and so on.