Let $(X,L)$ be an algebraic degeneration family with an ample polarization, and $(\mathcal{X},\mathcal{L})$ be a semistable snc model. This is not required to be CY, nor do we impose any maximal degeneration condition. It induces a formal model by base change. Consider the metric $\left\lVert\cdot\right\rVert=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}$ on $(X_{K},L)$, where we assume $\phi=\phi\circ r_{\mathcal{X}}$ and $\phi$ is ‘sufficiently smooth’ on $\Delta_{\mathcal{X}}$. Our plan is to use the heuristic logic at the end of section 3.4 to guess a formula for the NA measure over the interior of any face $\Delta_{J}\subset\Delta_{\mathcal{X}}$, in terms of differential operators. The answer will involve an interesting correction factor to the real MA measure.

Let $\Delta_{J}$ correspond to $E_{J}=\cap_{i\in J}E_{i}$ as usual. We first explain how to associate a class in $H^{1,1}(E_{J})$ to each $x\in\text{Int}(\Delta_{J})$. Let $x_{i}$ be the local affine coordinate corresponding to $E_{i}$ for any $i\in I$; we will only need $E_{i}\cap E_{J}\neq\emptyset$. We introduce an overparametrisation: let $u$ be a function of all $\{x_{i}\}_{i\in I}$, such that $u$ agrees with $\phi$ at least on a neighbourhood of $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$, and we assume $u$ is smooth. Here $x_{i}$ are treated as independent variables for $u$, even though on $\Delta_{\mathcal{X}}$ they satisfy various linear constraints. Consider the class in $H^{1,1}(E_{J})$ defined by the affine linear combination of derivatives:

$$\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)=c_{1}(\mathcal{L})-\sum_{I}\frac{\partial u}{\partial x_{i}}c_{1}(\mathcal{O}(E_{i}))$$

(12)

which depends only on $\phi$ because in $H^{1,1}(E_{J})$

$$c_{1}(\mathcal{O}(\sum_{I}E_{i}))=c_{1}(\text{div}(dt))=0,\quad c_{1}(\mathcal{O}(E_{i}))=0\quad\forall E_{i}\cap E_{J}=\emptyset.$$

Notice $\left\lVert\cdot\right\rVert=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}$ and $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ are invariant under the change

$$\mathcal{L}\to\mathcal{L}+\sum_{I}d_{i}E_{i},\quad\phi\to\phi+\sum_{I}d_{i}\phi_{E_{i}}$$

where $d_{i}\in\mathbb{R}$ and $\phi_{E_{i}}$ denote the model functions associated to $E_{i}$. Thus the function $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ on $\text{Int}(\Delta_{J})$ is intrinsically associated to $\left\lVert\cdot\right\rVert$.

Our strategy is to consider 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 on $X\sqcup\Delta_{\mathcal{X}}$, and take the limit of the complex MA measures associated to the curvature forms of $h_{t}$. Introduce smooth Hermitian metrics $h_{\mathcal{L}}$ and $h_{E_{i}}$ on $\mathcal{L}$ and $\mathcal{O}(E_{i})$ for $i\in I$. Use these to produce smooth functions $r_{i}$ on $\mathcal{X}$ for $i\in I$, such that near $E_{i}\subset\mathcal{X}$ the singularity is governed by $r_{i}\sim|z_{i}|e^{-\phi_{i}}$ for smooth local $\phi_{i}$, where $z_{i}$ are local defining equations for $E_{i}$, and away from $E_{i}$ the function $r_{i}$ is smooth and bounded positively from below. In order for $h_{t}$ to have the appropriate convergence behaviour as $t\to 0$ around $\text{Int}(\Delta_{J})\subset X\sqcup\Delta_{\mathcal{X}}$, we use the ansatz

$$h_{t}\sim h_{\mathcal{L}}\exp\left(2\log|t|u(\frac{\log r_{i}}{\log|t|},i\in I)\right)$$

where $r_{i}$ are regarded as smooth functions on $X_{t}$. The curvature form in $(X_{t},c_{1}(L))$ is

$$-dd^{c}\log h_{t}^{1/2}\sim-dd^{c}\log h_{\mathcal{L}}^{1/2}-\log|t|dd^{c}u,$$

$$dd^{c}u=\sum_{i,j\in I}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}d\log r_{i}\wedge d^{c}\log r_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}.$$

Our goal is to extract the limiting contribution of $(-dd^{c}\log h_{t}^{1/2})^{n}$ to $\text{Int}(\Delta_{J})$ as $t\to 0$. We need to separate this contribution from $\Delta_{J^{\prime}}$ with $J^{\prime}\supsetneq J$; algebraically this means taking the measure contribution near $E_{J}\subset\mathcal{X}$, but away from deeper strata $E_{J^{\prime}}$. To formalize this, we consider the quantitative statum on $X_{t}$ (cf. section 3.1)

$$E_{J,t}^{\epsilon}=\{q\in X_{t}|d(q,E_{J})<\epsilon\}\setminus\bigcup_{J^{\prime}\supsetneq J}\{q\in X_{t}|d(q,E_{J^{\prime}})<\epsilon\},\quad\epsilon\ll 1.$$

We need to compute

$$\lim_{\epsilon\to 0}\lim_{t\to 0}\text{Log}_{\mathcal{X}*}\left((-dd^{c}\log h_{t}^{1/2})^{n}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{J,t}^{\epsilon}\right)$$

where the order of limit is important.

For fixed $\epsilon$, after deleting terms suppressed by order $O(\frac{1}{|\log|t||})$, we have the asymptote on $E_{J,t}^{\epsilon}\subset X_{t}$ as $t\to 0$:

$$dd^{c}u\sim\sum_{0}^{p}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i},$$

where $z_{0},\ldots z_{p}$ are the local defining functions of the divisors $E_{i}\subset\mathcal{X}$, and $E_{J}=\cap_{0}^{p}E_{i}$. Using that $t=z_{0}\ldots z_{p}$ locally around $E_{J}$, we can eliminate the $z_{0}$ variable to write

$$dd^{c}u\sim\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i},$$

hence

$$\begin{split}-dd^{c}\log h_{t}^{1/2}\sim&\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}\\ &-dd^{c}\log h_{\mathcal{L}}^{1/2}-\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}.\end{split}$$

(13)

Notice that $dd^{c}\log r_{i}\sim-dd^{c}\phi_{i}$ has a smooth extension to the central fibre as $t\to 0$; the singular effect is eliminated by $dd^{c}\log|z_{i}|=0$ on $X_{t}$. The term $\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}$ dominates in the directions transverse to $E_{J}$, and the term $-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}$ dominates in the directions tangential to $E_{J}$.

The measure asymptote is now

$$\begin{split}(-dd^{c}\log h_{t}^{1/2})^{n}\sim&p!\det(D^{2}\phi)\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\right)^{n-p}\wedge\\ &\prod_{i=1}^{p}\frac{1}{2\pi|\log|t||}d\log|z_{i}|\wedge d\arg{z}_{i},\end{split}$$

(14)

where $D^{2}\phi=(\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}})_{p\times p}$ is the Hessian matrix of $\phi$. Thus the pushforward measure has the limit as $t\to 0$:

$$\begin{split}\lim_{t\to 0}\text{Log}_{\mathcal{X}*}\left((-dd^{c}\log h_{t}^{1/2})^{n}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{J,t}^{\epsilon}\right)=p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|\\ \times\int_{E_{J,0}^{\epsilon}}\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\right)^{n-p}.\end{split}$$

Taking $\epsilon\to 0$,

$$\begin{split}\lim_{\epsilon\to 0}\lim_{t\to 0}\text{Log}_{\mathcal{X}*}\left((-dd^{c}\log h_{t}^{1/2})^{n}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{J,t}^{\epsilon}\right)=p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|\\ \times\int_{E_{J}}\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\right)^{n-p}.\end{split}$$

Here an interesting topological effect takes place. Even though $dd^{c}\log r_{i}$ starts life as an exact form on $X_{t}$, it acquires a first Chern class in the process of smooth extension to the central fibre $\mathcal{X}_{0}$, because we are removing the distributional contribution $dd^{c}\log|z_{i}|=[E_{i}]$. Thus on $E_{J}$, the smooth closed (1,1)-form

$$-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}$$

lies in the $H^{1,1}$ class

$$c_{1}(\mathcal{L})-\sum_{I}\frac{\partial u}{\partial x_{i}}c_{1}(\mathcal{O}(E_{i})),$$

which is exactly the class $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ we introduced earlier (cf. (12)). We have thus obtained a formula for the double limit:

$$p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)^{n-p}\cdot E_{J}).$$

Notice all auxiliary choices are eliminated at this stage. According to our heuristic logic that the NA MA measure should be the limit of the corresponding complex MA measures on $X_{t}$, we conclude the heuristic formula for the NA MA measure over $\text{Int}(\Delta_{J})$

$$r_{\mathcal{X}*}MA(\left\lVert\cdot\right\rVert)=p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)^{n-p}\cdot E_{J}).$$

(15)

Notice the RHS is a differential operator in the potential $\phi$, because the intersection theoretic term $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ is affine linear in the first order derivatives of $\phi$. The formula exhibits a curious mixture of intersection theory with real MA operator.