Is there a geometric realization of $\mathbf{C}((t))$-varieties?

Question

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.

Answer 1

I know nothing about $\mathbf{A}^1$-homotopy. I will describe how one can get a functor from smooth varieties over $\mathbf{C}((t))$ to ${\rm Top}_{/\mathbf{S}^1}$ using log smooth proper models and Kato-Nakayama spaces of their special fibers.

Review of Kato–Nakayama spaces

If $X$ is an fs (finite and saturated) log scheme such that the underlying scheme $\underline X$ is of finite type over $\mathbf{C}$, one can associate to it in a functorial way a topological space $X^{\rm log}$, called the Kato–Nakayama space of $X$, together with a proper map $\tau:X^{\rm log}\to X$. The points of $X^{\rm log}$ correspond to pairs $(x, h)$ of a point $x\in\underline X$ and a map $h:\mathscr{M}_{X, x}\to \mathbf{S}^1$ such that for $f\in \mathscr{O}^\times_{X, x}$, $h(f) = f(x)/|f(x)|$.

In the special case when $X=\mathbf{A}^1$ with the usual log structure given by the open immersion $j:\mathbf{G}_m\hookrightarrow\mathbf{A}^1$, $X^{\rm log}$ is the `real blow-up' $$ X^{\rm log} = \mathbf{R}_{\geq 0}\times \mathbf{S}^1 $$ and the map $\tau$ sends $(r, \theta)$ to $r\cdot\theta\in \mathbf{C}=\mathbf{A}^1(\mathbf{C})$. The open immersion $j$ factors through $\tau$, and the fiber $\tau^{-1}(0)\cong \mathbf{S}^1$.

In general, the Kato-Nakayama space has the following nice properties:

1. If $f:X\to Y$ is strict (i.e., the induced $f^*\mathscr{M}_Y\to \mathscr{M}_X$ is an isomorphism), then the square $\require{AMScd}$ \begin{CD} X^{\rm log} @>>> Y^{\rm log} \\ @VVV @VVV \\ X @>>> Y \\ \end{CD} is cartesian.

   2. If $X=\mathbf{A}_P$ for an fs monoid $P$ (by definition, $\mathbf{A}_P={\rm Spec}(P\to \mathbf{C}[P])$), then $$ X^{\rm log} = {\rm Hom}(P, \mathbf{R}_{\geq 0}\times \mathbf{S}^1 ) $$ and the map $\tau$ is induced by the multiplication $\mathbf{R}_{\geq 0}\times \mathbf{S}^1 \to \mathbf{C}$. In particular, $X^{\rm log}$ is a manifold with boundary. Since an fs log scheme is by definition one which locally admits a strict map to some $\mathbf{A}_P$, this together with 1. gives a complete local description of $X^{\rm log}$.

   3. The fiber over $x\in \underline X$ is a torsor under ${\rm Hom}(\mathscr{M}_{X, x}, \mathbf{S}^1)$ in a natural way.

   4. If $X$ is log regular (say, log smooth over $\mathbf{C}$ with the trivial log structure), then $X^{\rm log}$ is a manifold with boundary. If $X_{\rm tr}$ denotes the biggest open of $\underline X$ where the log structure is trivial, then the inclusion $j:X_{\rm tr}\hookrightarrow \underline X$ lifts to an open immersion $\bar j : X_{\rm tr}\hookrightarrow X^{\rm log}$ along $\tau$. Moreover, $\bar j$ is a homotopy equivalence.

   5. (Nakayama–Ogus) If $f:X\to Y$ is log smooth and exact, then the induced $f^{\rm log}:X^{\rm log}\to Y^{\rm log}$ is a topological submersion (i.e., locally on the source looks like a projection), and its fibers are manifolds with boundary. If $\underline f$ is proper, then $f^{\rm log}$ is a locally trivial fibration.

Notation. $V=\mathbf{C}[[t]]$, $K = {\rm Frac}(V)= \mathbf{C}((t))$, $\bar K = \bigcup_n \mathbf{C}((t^{1/n}))$ the algebraic closure of $K$, $k=V/m = \mathbf{C}$, $S={\rm Spec}\, V$, $\eta = {\rm Spec}\, K$, $\bar\eta = {\rm Spec}\, \bar K$, $s={\rm Spec}\, k$. We give $S$ the natural log structure coming from the open immersion $\eta\hookrightarrow S$.

Let $C$ be the category of smooth schemes over $\eta$, and let $\bar C$ be the category of proper log smooth fs log schemes over $S$. We have two crucial functors: $$ X\mapsto X_{\rm tr} : \bar C \to C $$ $$ X\mapsto X_s^{\rm log} : \bar C \to {\rm Top}_{/\mathbf{S}^1} $$ The first one associates to a log scheme $X/S$ the locus $X_{\rm tr}$ where the log structure is trivial. This maps to $S_{\rm tr} = \eta$, so is an $\eta$-scheme, and it is smooth when $X$ is log smooth. The second functor is the Kato–Nakayama space of the special fiber $X_s$ with the induced log structure, endowed with the natural map $X_s^{\rm log} \to s^{\rm log} \cong \mathbf{S}^1$. It is a locally trivial fibration whose fibers are manifolds with boundary (by Nakayama-Ogus).

The functors have the following nice properties:

1. Suppose that an object $X$ of $\bar C$ arises as a base change of a proper log smooth analytic space $X$ over an open disc $\Delta$ with log structure induced by $\Delta\setminus\{0\}$. Then $Y^{\rm log}\to U^{\rm log}$ is a locally trivial fibration (by Nakayama–Ogus) and $X^{\rm log}_s\to s^{\rm log}$ is its base change to the base point $s=0$. So in this `convergent' case the construction recovers what you want.

2. The functor $X\mapsto X_{\rm tr}$ is essentially surjective. This is easily proved using Hironaka: embed a given smooth $Y/\eta$ into a proper $\bar Y_0/S$ by Nagata, then resolve the singularities to get a regular $\bar Y/S$ such that $\bar Y\setminus Y$ is a divisor with normal crossings. Finally, endow $\bar Y$ with the log structure associated to the inclusion $Y\hookrightarrow \bar Y$. Then $\bar Y/S$ is log smooth (this fails if $k$ has positive characteristic). It may not be semistable, and the morphism to $S$ might not be saturated, but I guess we don't care.

Target theorem. The functor $X\mapsto X_s^{\rm log}$ factors through $X\mapsto X_{\rm tr}$ (at least if we change the target category into the homotopy category).

I will sketch how I would prove it. Let $Y$ be an object of $C$. The second functor is essentially surjective; let $X$ be an object of $\bar C$ with $X_{\rm tr} \cong Y$. We want to check that $Y\mapsto X_s^{\rm log}$ is independent of $X$. Given two such $X, X'$, we can find a third one related to both by log blow-ups $X''\to X$, $X''\to X'$. Log blow-ups are log etale and should not change the Kato-Nakayama space of the special fiber, so the induced $(X'')^{\rm log}_s\to X^{\rm log}$ are isomorphisms. For functoriality, given a map $f:Y'\to Y$ in $C$ one can using Hironaka find a `model' $X'\to X$ in $\bar C$ and look at the associated map $(X')^{\rm log}_s\to X^{\rm log}_s$.

End note. We could even work with proper formal schemes over $S$ and the theory should be the same.

Below is my initial answer. Let $X/\mathbb{C}((t))$ be a smooth proper variety. I will define the bundle over $\mathbb{S}^1$ you want up to a finite covering $\mathbb{S}^1\to\mathbb{S}^1$.

After a finite ramified base change, $X$ has a log smooth (semistable) model $\mathscr{X}/\mathbb{C}[[t]]$ where $\mathbb{C}[[t]]$ is given the natural log structure. Let $X_0/S$ be the log special fiber over the log point $S={\rm Spec}(\mathbb{N}\to\mathbb{C})$. Consider the associated map on Kato–Nakayama spaces $$ X_0^{\rm log} \to S^{\log}\cong \mathbb{S}^1. $$ This is a locally trivial bundle over $\mathbb{S}^1$ (because $X_0/S$ is log smooth and proper), and if the model $\mathscr{X}/\mathbb{C}[[t]]$ arises as the base change of an algebraic family, it agrees with what happens topologically on the generic fiber.

If $X$ is non-smooth or non-proper, you can probably use simplicial methods to reduce to the smooth proper case, adding some horizontal log structure.

P.S. If I find the time, I will try to add some more details later today.

Answer 2

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.

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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.