ScalingStacks

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.