ScalingStacks

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]:

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  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.

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  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 $n!$.