ScalingStacks

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:

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

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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 $\left\lVert\cdot\right\rVert$.