ScalingStacks

We mentioned in section 3.2 that for a given polarized algebraic degeneration, the choice of snc models is highly non-unique. There are two viewpoints on extracting invariant information:

• •

  In birational geometry, one aims to find optimal representatives via the minimal model program. Typically, this will leave the snc world, and require divisorial log terminal models [62][63], but the minimal models may still be non-unique.

• •

  In non-archimedean geometry, one looks simultaneously at the tower of all snc models, and take the formal limit of their dual complexes, known as the Berkovich space.

Good references can be found in [49, A] [3, Appendix][6, chapter 2,3].

An insight of Berkovich is that by thinking of points as multiplicative seminorms, one obtains a kind of geometry analogous to complex manifolds. Let $K\simeq\mathbb{C}(\!(t)\!)$ be equipped with its standard absolute value $|\cdot|_{0}=e^{-ord_{t}}$ where $ord_{t}$ is the valuation defined by the vanishing order. Its ultrametric property

gives the name ‘non-archimedean’ to the subject. Let $X_{K}$ be a smooth, geometrically connected, projective scheme over $\text{Spec}(K)$; the main examples come from base changing an algebraic degeneration family $X$ over a punctured curve. Choose a finite cover of $X_{K}$ by affine open sets of the form $U=\text{Spec}(A)$, where $A$ is a finitely generated $K$-algebra. The space $U^{an}$ is defined as the set of all multiplicative seminorms $|\cdot|_{x}:A\to\mathbb{R}_{\geq 0}$ extending the absolute value of $K$, endowed with the weakest topology so that the function $x\mapsto|f|_{x}$ is continuous for any $f\in A$. The Berkovich space $X_{K}^{an}$ is then obtained by gluing together $U^{an}$; the notation stands for ‘analytification’. As a topological space $X_{K}^{an}$ is compact and Hausdorff. In the CY case, the point-set description of $X_{K}^{an}$ is meant to encode information about the base of the SYZ fibration; there is also a natural structure sheaf which encodes information about the complex structure [49].

Let $R\simeq\mathbb{C}[\![t]\!]$. The concept of models over $\text{Spec}(R)$ is entirely analogous to the case over algebraic curves. The dual intersection complexes $\Delta_{\mathcal{X}}$ for snc models over $\text{Spec}(R)$ can be compared with $X_{K}^{an}$ through two natural maps:

• •

  There is a continuous embedding map $emb:\Delta_{\mathcal{X}}\to X_{K}^{an}$. Writing $\mathcal{X}_{0}=\sum b_{i}E_{i}$, each divisor $E_{i}$ defines $val_{E_{i}}=\frac{ord_{E_{i}}}{b_{i}}$ through the vanishing order $ord_{E_{i}}$, so that $e^{-val_{E_{i}}}$ is a point in $X_{K}^{an}$, called a divisorial point. More generally, given a point $x=(x_{0},\ldots x_{p})$ in the interior of a face $\Delta_{J}\subset\Delta_{\mathcal{X}}$ corresponding to $E_{J}=\cap_{0}^{p}E_{i}$, we can associate a quasi-monomial valuation: expanding any local function $f$ around $E_{J}$ in Taylor series,

  $$f=\sum_{\alpha\in\mathbb{N}^{p+1}}f_{\alpha}z_{0}^{\alpha_{0}}\ldots z_{p}^{\alpha_{p}},\quad f_{\alpha}\in K(E_{J})$$

  then the quasi-monomial valuation is

  $$val_{x}(f)=\min\{\sum_{0}^{p}\alpha_{i}x_{i}|f_{\alpha}\neq 0\}.$$

  Thus $x$ gives rise to a point $e^{-val_{x}}\in X_{K}^{an}$. We shall regard $\Delta_{\mathcal{X}}$ as a subset of $X_{K}^{an}$. In particular, the essential skeleton embeds into $X_{K}^{an}$.

• •

  There is a continuous retraction map $r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}$, which restricts to the identity on $\Delta_{\mathcal{X}}\subset X^{an}$. Any point $e^{-v}\in X_{K}^{an}$ admits a center on $\mathcal{X}$. This is the unique scheme theoretic point $\xi\in X_{0}$ such that $|f|_{x}\leq 1$ for $f\in\mathcal{O}_{\mathcal{X},\xi}$ and $|f|_{x}<1$ for $f\in m_{\mathcal{X},\xi}$. Let $J\subset I$ be the maximal subset such that $\xi\in E_{J}$. Then $r_{\mathcal{X}}(x)\in\Delta_{\mathcal{X}}$ corresponds to the quasi-monomial valuation with the same value for $-\log|z_{j}|_{x},j\in J$. Concretely, one should think the retraction map is about reading off logarithmic coordinates.

  For another perspective, if $\mathcal{X}^{\prime}$ is a blow up of $\mathcal{X}$, then there is a natural simplicial map $\Delta_{\mathcal{X}^{\prime}}\to\Delta_{\mathcal{X}}$, which is identity on $\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{X}^{\prime}}$. The retraction map $X_{K}^{an}\to\Delta_{\mathcal{X}}$ can be viewed as a formal limit for very large $\mathcal{X}^{\prime}$.

With these comparison maps, the Berkovich space $X_{K}^{an}$ is homeomorphic to the inverse limit of the dual intersection complexes of the snc models:

Conceptually, an snc model gives a finite approximation of the Berkovich space.

We now indicate how NA geometry is unified with complex geometry. Consider an algebraic degeneration $X$ over a punctured curve. Let $|\cdot|$ denote the usual absolute value for complex numbers. Given a $\mathbb{C}$-point $z\in X_{t}$ for $0<|t|\ll 1$, inside some affine chart $U=\text{Spec}(A)$ of $X$, we can define a multiplicative seminorm $A\to\mathbb{R}_{\geq 0}$ (not non-archimedean!)

As a sequence of points $z$ move towards $t\to 0$, for any given meromorphic function $f=\sum a_{k}t^{k}$ on the base, $\lim_{t\to 0}\log|f(z)|/\log|t|=ord_{0}(f)$ which is the standard NA valuation on $K$. Thus the points on $X_{K}^{an}$ are natural limits of the multiplicative seminorms defined by $\mathbb{C}$-points on $X_{t}$. One can formalize this notion by introducing a hybrid topology on $X\sqcup X_{K}^{an}$, so that $X_{K}^{an}$ takes the place of the central fibre [3, Appendix]. The functions $f\in A$ then induce local continuous functions on $X\sqcup X_{K}^{an}$.

The ‘hybrid’ space $X\sqcup\Delta_{\mathcal{X}}$ discussed in section 3.1 can be understood as a finite approximation. Given an snc model $\mathcal{X}$, and take a sequence of $\mathbb{C}$-points $q_{k}$ tending to $e^{-v}\in X^{an}$, whose image under the retraction map $r_{\mathcal{X}}$ is $x=(x_{0},\ldots x_{p})\in\Delta_{J}\subset\Delta_{\mathcal{X}}$. Tautologically $q_{k}$ concentrate near $E_{J}$, and in the local coordinates $z_{0},\ldots z_{p}$, we have $\log|z_{i}(q_{k})|/\log|t|\to v(z_{i})=x_{i}$, which is equivalent to $\text{Log}_{\mathcal{X}}(z_{k})\to x=(x_{0},\ldots x_{p})\in\Delta_{J}\subset\Delta_{\mathcal{X}}$. Formally, the topology on $X\sqcup X_{K}^{an}$ is the inverse limit of $X\sqcup\Delta_{\mathcal{X}}$ by taking all snc models.

We now discuss functions, line bundles, and metrics on $X_{K}^{an}$ [6]. Given a model $\mathcal{X}$ over $\text{Spec}(R)$ and a Cartier divisor $D$ supported on the central fibre $\mathcal{X}_{0}$, we can associate a continuous function on $X_{K}^{an}$ by setting

The association $D\mapsto\phi_{D}$ extends by $\mathbb{Q}$-linearity. Functions obtained in the $\mathbb{Q}$-span using all such choices of models and divisors are called model functions on $X_{K}^{an}$, which form a dense subset of $C^{0}(X_{K}^{an})$. The restrictions of such functions to dual intersection complexes are piecewise affine.

To understand the complex geometric meaning, we think of models base changed from snc models over an algebraic curve $S$. The divisor $D$ prescribes a class of functions $\phi$ on the total space of the snc model with analytic singularities:

where $f$ is a local defining function of $D$. When we consider the rescaling of the restrictions to $X_{t}$

only the singular term is relevant in the limit $t\to 0$, and $\phi_{t}$ converge to $-\phi_{D}$ in the hybrid topology.

We think about line bundles on $X_{K}^{an}$ via the GAGA principle: the line bundles on $X_{K}^{an}$ correspond to the line bundles $L$ on the scheme $X_{K}$. A continuous metric on $L$ assigns to each local section $s$ a nonnegative continuous local function $\left\lVert s\right\rVert$ on open subsets of $X_{K}^{an}$, compatible with the sheaf structure, such that $\left\lVert fs\right\rVert(x)=|f|_{x}\left\lVert s\right\rVert(x)$, and $\left\lVert s\right\rVert>0$ if $s$ is a local frame of $L$. Given a continuous metric, any other continuous metric on $L$ is of the form $\left\lVert\cdot\right\rVert e^{-\phi}$ for some $\phi\in C^{0}(X^{an})$, analogous to the usual relation between Hermitian metrics and Kähler potentials. As such $\phi$ is referred to as a potential function.

Given a model $\mathcal{X}$ for $X_{K}$, a model $\mathcal{L}$ of $L$ is a line bundle $\mathcal{L}\to\mathcal{X}$ with $\mathcal{L}|_{X}=L$. To this data we can associate a unique metric $\left\lVert\cdot\right\rVert_{\mathcal{L}}$ on $L$ with the following property: if $s$ is a nowhere vanishing local section of $\mathcal{L}$ on an open set $\mathcal{U}\subset\mathcal{X}$, then $\left\lVert s\right\rVert_{\mathcal{L}}\equiv 1$ on $\mathcal{U}\cap X_{K}$. This is well defined because any two such sections differ by the multiplication of an invertible function, whose NA absolute value equals the constant one. One can extend this construction to $\mathbb{Q}$-line bundles, and the metrics arising this way are called model metrics. They are dense within the continuous metrics.

To see the complex geometric interpretation, we imagine a line bundle $\mathcal{L}$ on some snc models $\mathcal{X}$ over an algebraic curve. Equip $\mathcal{L}$ with any smooth Hermitian metric $h$. Given a local section $s$ of $\mathcal{L}$, the prescription compatible with (13) is to consider the local functions on $X_{t}$

Taking the limit as $t\to 0$, we precisely get the model metric. Notice the ambiguity in the choice of the Hermitian metric is obliterated in the limit.

A paramount notion in Kähler geometry is the positivity of the metric, usually phrased in terms of psh properties of the potential. The above discussion suggests that in the NA setting, namely $t\to 0$, such a notion should be expressible as a numerical property of the line bundle.

In Kähler geometry the definition of psh function is local in the complex charts. Since the dual intersection complexes are simplicial objects, one would expect the NA analogous notion to be related to convex functions. This intuition is partially valid:

The picture is that general $\theta$-psh functions define convex functions on the faces of $\Delta_{\mathcal{X}}$, and among them the $\theta$-psh model functions give piecewise affine approximations with finer and finer grids.

A fundamental result in Kähler geometry is that any Kähler metric in an integral class can be approximated by Fubini-Study metrics associated with projective embeddings. While the usual Fubini-Study metric depends on a choice of a Hermitian inner product on the $\mathbb{C}$-vector space of global sections, the NA analgoue depends on a NA norm on the $K$-vector space $V=H^{0}(X_{K},mL)$ for $m\gg 1$, with the ultrametric property $\left\lVert x+y\right\rVert_{V}\leq\max\{\left\lVert x\right\rVert_{V},\left\lVert y\right\rVert_{V}\}$. In our case $K=\mathbb{C}(\!(t)\!)$ one can select a $K$-basis $s_{0},s_{1},\ldots s_{N}$ for $V$, such that

The NA Fubini-Study metric on $L\to X_{K}$ can be defined as

Concretely in the orthogonal basis, written in a local trivialisation,

A NA analogue of the Fubini-Study approximation theorem gives an alternative view on semipositive metrics:

For the complex geometric interpretation, we assume as usual $X_{K}$ is the base change of an algebraic degeneration family $X$, with an ample polarisation line bundle $L$. For any given NA Fubini-Study metric (14), we can associate a family of Fubini-Study metrics on $(X_{t},L)$:

Here $s_{j}$ make sense for finite $t$ because they are selected as finite Laurent polynomials in $t$. As $t\to 0$, the Fubini-Study metrics converge to the NA analogue, or more precisely $\left\lVert s\right\rVert_{FS,t}^{1/|\log|t||}$ converges to (14) in the hybrid topology on $X\sqcup X_{K}^{an}$.

The NA Monge-Ampère measure [11] is defined through intersection theory in a somewhat counterintuitive manner. As a motivation, we consider the complex analytic setting of an snc model $\mathcal{X}$ over an algebraic curve, equipped with a Hermitian line bundle $(\mathcal{L},h)$ with curvature form $\theta$ in the class $c_{1}(\mathcal{L})$. Then $\theta^{n}$ defines a family of $n$-forms on $X_{t}$, such that $\int_{X_{t}}\theta^{n}$ equals the intersection number $(L^{n})$. The question is to describe the limit of these $n$-forms, when we view $X_{t}$ as converging to the dual intersection complex $\Delta_{\mathcal{X}}$ (cf. section 3.1).

We write $X_{0}=\sum_{i\in I}b_{i}E_{i}$. Recall that the regions on $X_{t}$ corresponding to the faces in the dual intersection complex are from the algebraic perspective only small neighbourhoods of $E_{J}$. Thus the limit of $\theta^{n}|_{X_{t}}$ can only be supported at the vertices of $\Delta_{\mathcal{X}}$, which correspond to the components $E_{i}$. The amount of delta masses concentrated at the vertices are

where $b_{i}$ appears due to the multiplicity of the sheets. Reassuringly,

gives the correct total mass.

Back to the NA setting, given a model $\mathbb{Q}$-line bundle $\mathcal{L}\to\mathcal{X}$ for $L\to X_{K}$, we write $\mathcal{X}_{0}=\sum_{i}b_{i}E_{i}$, and denote the divisorial points associated to $E_{i}$ as $q_{i}$. We can then define the NA Monge-Ampère measure for the model metric $\left\lVert\cdot\right\rVert_{\mathcal{L}}$ as the following signed atomic measure supported at $q_{i}\in X_{K}^{an}$:

This definition is compatible with pullback of line bundles by the projection formula, and ensures the total mass is the intersection number $(L^{n})$. If $\left\lVert\cdot\right\rVert_{\mathcal{L}}$ is furthermore semipositive, then the intersection numbers are non-negative, so $MA(\left\lVert\cdot\right\rVert_{\mathcal{L}})$ is a measure.

The theory of NA MA measures bears strong resemblance to the complex MA measures [4][5]:

• •

  In the complex analytic world, one first define the complex MA for smooth potentials. A general continuous semipositive potential in a Kähler class is the uniform limit of smooth potentials, and its complex MA measure is then determined by the weak continuity under $C^{0}$-convergence.

• •

  In the NA world, one first define the NA MA measure for the model metrics. A general continuous semipositive metric on $L$ is the uniform limit of a sequence of continuous semipositive model metrics [6, Cor. 8.8], and its NA MA measure can be defined as the unique limiting Radon measure of the NA MA measures for the sequence [4, Cor. 3.5].

Their main difference lies in the highly nonlocal appearance of the NA MA measure. The recent result of Vilsmeier [76] offers a more concrete perspective:

The rigorous proof of this comparison uses intersection theory, and the following is a heuristic explanation. Consider an snc model $\mathcal{X}$ over an algbebraic curve as in the motivation, and assume furthermore that it is semistable. Recall our heuristic dictionary that a metric $\left\lVert\cdot\right\rVert$ on $L\to X_{K}$ should encode a family of Hermitian metrics $h_{t}$ on $L\to X_{t}$, such that $h_{t}^{1/|\log|t||}\to\left\lVert\cdot\right\rVert^{2}$ in the hybrid topology, and the NA MA measure of $\left\lVert\cdot\right\rVert$ should be the limit of the measures associated to the curvature forms of $h|_{X_{t}}$. We now focus on the neighbourhood of an $n$-dimensional open face $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}\subset\Delta_{\mathcal{X}}\sqcup X$, where we have local coordinates $z_{0},\ldots z_{n}$ with $\prod_{0}^{n}z_{i}=t$, and $x_{i}=\frac{\log|z_{i}|}{\log|t|}$. In the local picture we identify metrics with potentials, so $\left\lVert\cdot\right\rVert\sim e^{-\phi}$, and after ignoring $C^{0}$-fluctuation effects $h_{t}^{1/|\log|t||}\sim e^{-2\phi\circ\text{Log}_{\mathcal{X}}}$. Imposing more smoothness assumptions, the curvature form of $h_{t}$ is approximately

The NA MA measure should agree with the limiting pushforward measure

which equals the real MA measure up to the factor $n!$.

The central result of NA pluripotential theory is the solution to the NA analogue of the Calabi conjecture. A good survey is [5].

Their strategy uses a variational method. There is a concave energy functional $\mathcal{E}$ on the space of continuous semipositive metrics on $L$ (equivalently viewed as continuous $\theta$-psh potentials $\phi$), whose first variation is given by the NA MA measure. One seeks a maximizer of the functional

by first enlarging the space of $\phi$ to a function space $PSH(X_{K}^{an},\theta)$ which is compact modulo the addition of a real constant; this is analogous to the $L^{1}$-compactness of $PSH(X,\omega)/\mathbb{R}$ in the Kähler setting. The notions of the NA MA measure and the energy functional extend naturally to the energy class functions inside $PSH(X_{K}^{an},\theta)$, much like in the complex pluripotential theory setting. One then shows the maximizer is in fact a critical point, namely a weak solution to the NA MA equation. This is subtle since small perturbations of functions in $PSH(X_{K}^{an},\theta)$ may fall outside of the class by losing positivity. One proves the continuity of the weak solution using analogues of Kolodziej’s estimates. The uniqueness of the solution again relies on the concavity of $\mathcal{E}$.

While this strategy shares a very similar logical structure with the complex analytic setting, the technical foundations are built upon intersection theory and vanishing theorems in birational geometry, instead of differential operators.

The main case of interest to us is when $X_{K}$ arises from a large complex structure limit. Then NA pluripotential theory provides a unique solution to

where $d\mu_{0}$ is the Lebesgue measure supported on the essential skeleton $Sk(X)\subset X_{K}^{an}$ (cf. section 3.1). We call $\left\lVert\cdot\right\rVert_{CY}$ the non-archimedean Calabi-Yau metric.

Very little is proven about the non-archimedean CY metric $\left\lVert\cdot\right\rVert_{CY}$ beyond existence and continuity. We now discuss the meaning of the following conjectural NA MA-real MA comparison property.

Notice $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$ inherits a natural integral affine structures. Since the restriction of $\phi_{0}$ is convex on these faces by Prop. 5.2, its real MA measure makes sense, and by Prop. 5.4 it satisfies the real MA equation on $\text{Int}(\Delta_{J})$

A few comments are in order:

• •

  The comparison property is a conjecture in algebraic/non-archimedean geometry, and does not a priori involve PDE concepts. Its PDE implications come a posteriori.

• •

  The Kontsevich-Soibelman picture (cf. section 3.3) expects that there is a solution of the real MA equation on the essential skeleton, away from some singular locus. The NA-MA equation via the comparison property is the only known systematic method to produce solutions.

• •

  In the context of toric invariant metrics on toric varieties, there are comparison results between NA MA measure and real MA equation, cf. [28, Prop. 4.4.4].

• •

  The reader may feel that NA pluripotential theory is a very long-winded way to solve the real MA equation. However, surprisingly enough, it is not even known how to formulate the real MA equation globally on $Sk(X)$ in general, not just on the $n$-dimensional faces but also on the lower dimensional faces.

  One problem is that $Sk(X)$ does not come with an obvious preferred affine structure, but only a piecewise affine structure, so there is no obvious coordinate independent definition of the real MA measure. It seems that the affine structure conjectured by Kontsevich and Soibelman would need to be solved simultaneously with the real MA equation, rather like free boundary PDE problems.

  Another problem is that solving the real MA equation requires first specifying the class of convex functions to be admitted as potentials, just like solving the complex Monge-Ampère equation requires first specifying the meaning of Kähler potentials. We do not currently know any direct way of defining the class of convex potentials on piecewise affine manifolds such as $Sk(X)$. The semipositive metrics on the Berkovich space $X_{K}^{an}$, abstract as it may be, is our only available substitute.

• •

  If one believes the NA Calabi-Yau metric $\left\lVert\cdot\right\rVert_{CY}$ is the potential theoretic limit of the Calabi-Yau metrics on $X_{t}$ in the hybrid topology, and that the information of $\left\lVert\cdot\right\rVert_{CY}$ can be recovered from data on $Sk(X)$, then one may be inclined to think that the potential of $\left\lVert\cdot\right\rVert_{CY}$ factors globally through some retraction map $X_{K}^{an}\to Sk(X)$, defined perhaps through some divisorial log terminal minimal model.

  Such a statement would need to confront the difficulty that the divisorial log terminal model is not necessarily unique, and in principle the retraction map depends on the choice of the model. It seems highly nontrivial how the NA MA equation would select a preferred retraction map.

  The formulation of the comparison property is more cautious than this. We allow the dual intersection complex $\Delta_{\mathcal{X}}$ to be strictly bigger than $Sk(X)$, and there is no assumption on the complement of the $n$-dimensional faces of $Sk(X)$. Regarding the problem above, if one is undecided between a finite number of candidate retraction maps, then one can pass to a common snc resolution (and perhaps pass to finite base change, to find a semistable snc resolution). Of course, the more we blow up the model $\mathcal{X}$, the weaker is the comparison property.

  003C

  Remark 13. The very recent work of Pille-Schneider and Mazzon [59] proposes gluing the retraction maps associated to several divisorial log terminal models to obtain a map $X_{K}^{an}\to Sk(X)$. Their map still factors through the dual intersection complex of some larger snc model, hence is compatible with the comparison property.

• •

  Without the comparison property, it seems hard to give any differential geometric interpretation to $\left\lVert\cdot\right\rVert_{CY}$ at all, since $X_{K}^{an}$ contains arbitrarily large dual intersection complexes, and thus is highly complicated. The heuristic intuition of this hypothetical scenario, is that the potential theoretic limit of the Calabi-Yau metrics would require infinitely many blow ups to describe. This is not yet ruled out by a theorem; we leave the reader to judge its plausibility.

The open question for algebraic geometers is