ScalingStacks

Notice at this stage we still have the freedom to adjust $h_{\mathcal{L}}$ and $r_{i}$ which enter the construction in section 5.2. This information controls the small scale metric, and is not directly visible to the NA MA solution. For each $x\in\text{Int}(\Delta_{J})$, let $\phi_{x}$ be the solution to the complex MA equation on $E_{J}$:

$$\frac{\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}\right)^{n-m}}{\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}}=\frac{\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}{\int_{E_{J}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}.$$

(19)

If we wish to completely determine $\phi_{x}$ we need to fix a normalisation, but this choice is not essential. Under our hypotheses the dependence of $\phi_{x}$ on $x$ can be made smooth.