Smoothness of the "Archimedean special fiber" in Arakelov geometry

Question

If $X$ is a scheme over, let's say, $\mathbb{Z}_p$, one can consider its special fiber obtained by reduction modulo $p$ ans it certainly makes sense to ask if this special fiber is smooth or not.

The ring of p-adic integers $\mathbb{Z}_p$ gives a local (formal) picture near $p$ in $Spec(\mathbb{Z})$. One basic intuition at the source of Arakelov theory is that a similar picture should exist near the "point at infinity" of an hypothetical compactification of $Spec(\mathbb{Z})$ obtained in some sense by adding the Archimedean place. In particular, a variety over $\mathbb{C}$ should be an analogue of a variety over $\mathbb{Q}_p$ (or maybe $\mathbb{C}_p$, anyway) and there should be an Archimedean analogue of a variety over $\mathbb{Z}_p$. Arakelov theory gives a proposal for such an analogue: a variety over $\mathbb{C}$ endowed with a Hermitian metric. There should also exist an analogue of the special fiber but I don't think that such thing exists yet (it should be something over something of characteristic one).

Without knowing what is the "Archimedean special fiber", my question is: is there an expectation for the notion of smoothness of this special fiber? Given a variety over $\mathbb{C}$ endowed with an Hermitian metric, is there a well-defined yes/no criterion for the smoothness of the hypothetical Archimedean special fiber?

This question is motivated by the fact that the notions of good/bad reductions play a key role in p-adic theory and I would like to understand what is the analogue story at the Archimedean place.

Answer 1

In Arakelov geometry, the conventional wisdom is that the ``closed fibre at $\infty$'' should be viewed as totally degenerate. This is the extreme opposite of smoothness. A visualization in the case of a curve is proposed in:

Yu. Manin: Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry, Inventiones, 1991.

This was inspired by Mumford's work on $p$-adic Schottky groups, extending to higher genus the rigid analytic picture of the Tate curve.

This work of Consani and Marcolli could be also helpful:

C. Consani, M. Marcolli: Noncommutative geometry, dynamics, and $\infty$-adic Arakelov geometry, Selecta Math. 2004.

The Tate curve may be enough to convince you of such a point of view. Analytically, over $\mathbb{C}_p$ or $\mathbb{C}$, a $\mathbb{G}_m$-reduction elliptic curve or a complex elliptic curve are, respectively, both a quotient of $\mathbb{G}_m$ by a one-dimensional group of periods. There is little similarity with an elliptic scheme over $\mathbb{Z}_p$. In the Tate curve, making increasingly ramified extension of the base field, the closed fibre of the minimal desingularization is a Neron polygon of projective lines, whose group of irreducible components converges to a circle $\mathbb{R}/\mathbb{Z}$: a tropical elliptic curve, and the skeleton of the Berkovich analytification. In contrast, for an elliptic curve over $\mathbb{Z}_p$ the analytification is contractible, giving the wrong Betti number in comparison to what happens Archimedeanly. This consideration extends to curves of higher genus $g$: the first Betti number of the analytification is $\leq g$, with equality if and only if the curve is totally degenerating.

The dynamics of rational maps illuminate another, though related, point. If the map has a good reduction at $p$ then its dynamics is completely predictable: the Julia set has no $\mathbb{C}_p$-points. But the Julia sets of complex dynamics are always non-empty, and they are typically quite intricate.

Finally, we may compare the geometry or dynamics over $\mathbb{Q}$ with the corresponding situation over the rational function field $k(t)$. If in the function field model we have a rational map over $k(t)$ with everywhere potential good reduction, then the map is isotrivial. The irrelevance of isotriviality for algebraic varieties or rational maps over number fields (as in Bogomolov's problem, Lehmer's problem, the $abc$ conjecture) leads us to admit the degeneration of at least some of the fibres: the Archimedean ones. If, for instance, we took the map $z \mapsto z^2$ over $\mathbb{Q}$ to be modeled by the squaring map over $k(t)$, which is isotrivial, the Lehmer problem (on the spectral gap of the dynamical Mahler measure) has a trivial, affirmative answer. Not so if we admit a non-isotrivial map: conceivably, this will have the same answer and comparable difficulty as the classical Lehmer problem. In the case of $z \mapsto z^2$, a more faithful model for the Lehmer question is not the squaring map of $\mathbb{F}_q(t)$, but the map $c(z) = tz + z^q$, which is the isogeny of multiplication by $t$ on the Carlitz module over $\mathbb{F}_q(t)$. In either case we have exactly the same progress on Lehmer's question, namely, Dobrowolski's bound; and the map $c$ has a bad reduction at the ``infinite place.''

Added. Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: Formes differentielles reelles et courants sur les espaces de Berkovich. Arakelov geometry considers line bundles $\bar{L} = (L,\| \cdot\|)$ with a smooth hermitian metric. The Chern (curvature) form $c_1( \bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is either zero or a full support signed measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. For a $p$-adic metric $\|\cdot\|_p$ in $L$ there are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$. But if this metric arises from an integral model $(\mathfrak{X},\mathfrak{L})$ over $\mathbb{Z}_p$, where $\mathfrak{X}_{/\mathbb{F}_p}$ is regular, then the obtained (signed) measure charges a single point of the Berkovich space: the unique point reducing to the generic point of the scheme $\mathfrak{X}_{/\mathbb{F}_p}$.