ScalingStacks

Consider an algebraic Calabi-Yau degeneration family $X\to S\setminus\{0\}$ as in the Introduction. We are given the holomorphic volume forms $\Omega_{t}$ on the fibres $X_{t}$, and let us follow [3] to consider the question of calculating the asymptote of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$ as $t\to 0$. Since we only care about small $t$, we are free to shrink $S$. For instance, we may assume $dt$ is nowhere vanishing on $S$.

A very useful tool is to fill in the central fibre by choosing an snc model (cf. Remark 1.2) $\mathcal{X}$ over $S$. The central fibre $\mathcal{X}_{0}$ is an snc divisor with components $E_{i}$ for $i\in I$, and we write $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$. In the special case of semistable snc models $b_{i}=1$ for $i\in I$; this can always be achieved after finite base change. The canonical divisor $K_{\mathcal{X}}$ is supported on $\mathcal{X}_{0}$ as $K_{X}$ has a trivialising section $\Omega$. We may write $K_{\mathcal{X}}=\sum_{i}(a_{i}+b_{i}-1)E_{i}$, so that the relative log canonical divisor

Shifting all $a_{i}$ by a constant $\kappa$ is equivalent to multiplying $\Omega$ by $t^{\kappa}$, which gives an elementary factor $|t|^{2\kappa}$ to $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. Thus we shall always assume $\min a_{i}=0$.

It is useful to introduce a quantitative stratification on $X_{t}$ according to the intersection pattern of $E_{i}$. Let $E_{J}=\cap_{i\in J}E_{i}$ for $J\subset I$, which is irreducible if nonempty. Using the distance function of a fixed smooth background Kähler metric on $\mathcal{X}$, we can write

Around $\emptyset\neq E_{J}\subset\mathcal{X}$, we denote $p=|J|-1$, and introduce local coordinates $z_{0},\ldots z_{n}$ on $\mathcal{X}$, such that $z_{0},z_{1},\ldots,z_{p}$ are the defining equations of $E_{i}$ for $i\in J$. The conditions on the divisors mean that away from deeper strata we may arrange $t=\prod_{0}^{p}z_{i}^{b_{i}}$, and

for some local nowhere vanishing holomorphic function $u_{J}$. By definition $\Omega=dt\wedge\Omega_{t}$ along $X_{t}$, so on $E_{J}^{0}$

Notice also that the local equation $t=\prod_{0}^{p}z_{i}^{b_{i}}$ has $b_{J}=\gcd_{i\in J}b_{i}$ sheets of solutions. Using the polar coordinates by $z_{i}=e^{x_{i}\log|t|+\sqrt{-1}\theta_{i}}$ for $i\in J$, ones sees that the magnitude of $\int_{E_{J}^{0}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}$ is $O(|\log|t||^{l})$ for $l=|\{j\in J:a_{j}=0\}|-1$.

The local logarithmic variables $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ lie on the simplex

These depend on the choice of $z_{i}$, but since the local defining equation of divisors differ by a nowhere vanishing holomorphic function, the ambiguity of $x_{i}$ is only $O(\frac{1}{|\log|t||})$ for $0<|t|\ll 1$. Taking a more global viewpoint, the combinatorial pattern of how these simplices fit together exactly reflects the intersection pattern of the divisors $E_{i}$. Formally, this information is encoded in the dual intersection complex $\Delta_{\mathcal{X}}$ for the snc model $\mathcal{X}$: this is the polyhedral complex whose vertices $v_{i}$ correspond to $E_{i}$, and we assign a simplex $\Delta_{J}$ with vertices $v_{i}$ for $i\in J$ if and only if $E_{J}\neq 0$. The coodinates $x_{j}$ then define a piecewise integral affine structure on $\Delta_{\mathcal{X}}$. Up to the above $O(\frac{1}{|\log|t||})$ ambiguity, we now have a logarithm map $\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$, locally described by $x_{i}=\frac{\log|z_{i}|}{\log|t|}$. Consequently, the ‘hybrid’ space $X\sqcup\Delta_{\mathcal{X}}$ is equipped with a natural topology, so that a sequence of points $z_{k}\in X_{t}$ converges to $x\in\Delta_{\mathcal{X}}$ iff $t\to 0$ and $\text{Log}_{\mathcal{X}}(z_{k})\to x$. The name ‘hybrid’ refers to the mixture of algebraic varieties with simplicial objects. The measure calculation above explains why $\Delta_{\mathcal{X}}$ (or rather the essential skeleton inside, see below) is a better candidate notion as a limit of $X_{t}$ than the algebraic limit $\mathcal{X}_{0}$. Intuitively, the Calabi-Yau measure in the limit becomes mutually orthogonal with the measure of any fixed Fubini-Study metric, and the region carrying most of CY measure looks ‘small’ from the algebraic perspective.

The measure also singles out a distinguished subcomplex $Sk(\mathcal{X})$, called the essential skeleton, consisting of the simplices in $\Delta_{\mathcal{X}}$ whose vertices correspond to $E_{i}$ with $a_{i}=0$. This is where the limit of the normalised CY measure is supported. The dimension of $Sk(\mathcal{X})$ is a measurement of how transcendental the degeneration $X$ is; it is reflected by the growth order of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. In the case of a maximal degeneration, $\dim_{\mathbb{R}}Sk(\mathcal{X})=n$. Let us analyze the CY measure more explicitly for maximal degenerations, in a semistable snc model. For $E_{J}$ corresponding to an $n$-dimensional simplex in $Sk(\mathcal{X})$, on $E_{J}^{0}$

Here $u_{J}$ limits to its value $u_{J}(E_{J})$ at the point stratum $E_{J}$, which is called the Poincaré residue of $\Omega$, and is easily seen to be independent of the choice of coordinates $z_{i}$. It is a consequence of the residue theorem on Riemann surfaces that $|u_{J}(E_{J})|^{2}$ is independent of such $J$ [3, Thm. 7.1]. Thus the pushforward to $\Delta_{\mathcal{X}}$ of the normalised CY measure (1) converges smoothly in the interior of $\Delta_{J}$ to a constant multiple of the Lebesgue measure:

Notice $dx_{1}\ldots dx_{n}$ is canonically defined due to the presence of an integral affine structure on $\Delta_{J}$. Viewed as a measure on $\Delta_{\mathcal{X}}$, the limit $d\mu_{0}$ has null measure on the complement of the $n$-dimensional faces of $Sk(\mathcal{X})$, as the integral of $d\mu_{t}$ in the corresponding region is $O(\frac{1}{|\log|t||})$. The constant in (5) is independent of $J$ and its sole purpose is to make $d\mu_{0}$ a probability measure.

One problem with the dual intersection complex is that it involves a choice of an snc model. The snc models are highly nonunique: we can keep blowing up to get a directed set of snc models. One would like to extract intrinsic information about the degeneration family. There are two general strategies. First, one can analyze the relation between different models and seek an optimal choice using the minimal model program [36][37]; this usually leaves the snc world, and even the optimal choices may still be nonunique. Alternatively we can consider all (snc) models simultaneously, by the language of NA geometry. Good references can be found in [29, A] [3, Appendix][2, 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 [29].

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 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 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 $X_{K}^{an}$.

• •

  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 monomial valuation with the same value for $-\log|z_{j}|_{x},j\in J$.

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.

While dual intersection complexes depend strongly on the model, in the CY case the embedding image of the essential skeleton $Sk(\mathcal{X})\subset X_{K}^{an}$ as a set is independent of the model, so can be written as $Sk(X)$. This can be expected as $Sk(\mathcal{X})$ should support the limiting normalised CY measure, a property independent of the model choice. However, as we blow up snc models, the essential skeleton as a simplical complex can be subdivided.

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 $X_{K}^{an}$ [2]. 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 $S$. 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 (6) 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.

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. Now a general continuous semipositive metric on $L$ is the uniform limit of a sequence of continuous semipositive model metrics [2, 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].

The theory of NA MA measures bears strong resemblance to the complex pluripotential theory for complex MA measures [4][5]. The difficulty of working with them lies in the highly nonlocal appearance of the above definition, ultimately caused by the tension between the algebraic limit $\mathcal{X}_{0}$ and the tropical limit $\Delta_{\mathcal{X}}$. The recent result of Vilsmeier [45] offers a more concrete perspective:

The rigorous proof of this comparison uses intersection theory. We present a heuristic calculation that hopefully makes the relation between NA MA measure and real MA measure more intuitive to differential geometers. 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 polarized algebraic maximal degeneration of CY manifolds $L\to X$. Then NA pluripotential theory provides a 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).

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. 3.5, its real MA measure makes sense, and by Prop. 3.7 it satisfies the real MA equation on $\text{Int}(\Delta_{J})$

Then the regularity theory of real MA equation (cf. section 2.5) will apply, so we may view the comparison property as a regularity assumption on $\left\lVert\cdot\right\rVert_{CY}$. Some subtleties are discussed in [21, Appendix].

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 $V$ (called an ‘orthogonal basis’ [17, section 2.1.2]), 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 (9), 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 $t$. By our dictionary, we should consider the limit of $\left\lVert s\right\rVert_{FS,t}^{1/|\log|t||}$ as $t\to 0$. Since

in the limit the difference between maximum and square length disappears, so $\left\lVert s\right\rVert_{FS,t}^{1/|\log|t||}$ converges to (9) in the hybrid topology on $X\sqcup X_{K}^{an}$.