ScalingStacks

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.