ScalingStacks

We begin with the interior of $m$-dimensional faces of $Sk(X)$, which can be viewed as the generic region. Comparing with the heuristic formula (15), and making regularity assumptions on $\phi_{0}$, we deduce a second order PDE

$$\det(D^{2}\phi_{0})(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\cdot E_{J})=\frac{(L^{n})}{m!}\int_{E_{J}}\sqrt{-1}^{(n-m)^{2}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)},$$

(18)

where $(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\cdot E_{J})$ defines a polynomial in the gradient of $\phi_{0}$.