ScalingStacks

Our next goal is to assign good holomorphic charts to $X_{s}$ related to the stratification structure. We first consider the toric region, which shall be covered by $(\mathbb{C}^{*})^{n}$-charts. Let $w\in N$ be the primitive integral outward normal vector to a facet $F(w)=\{m\in\Delta|\langle w,m\rangle=1\}$ of $\Delta$. The chart parametrised by $w$ is contained inside the region

$$U_{w}^{s,o}=\{z\in X_{s}|e^{s\lambda(m)}|z^{m}|\ll 1,\quad\forall m\in\Delta_{\mathbb{Z}}\setminus(F(w)\cup\{0\})\}.$$

(11)

Let $m_{0}\in F(w)\cap\Delta_{\mathbb{Z}}$, and choose an integral basis $m_{1},\ldots,m_{n}$ for $\{m\in M|\langle w,m\rangle=0\}$. Then the monomials $z^{m_{1}},\ldots,z^{m_{n}}$ provide the local $(\mathbb{C}^{*})^{n}$-coordinates on the chart, since by the implicit function theorem $X_{s}$ is locally a graph $\{z^{m_{0}}=f(z^{m_{1}},\ldots,z^{m_{n}})\}$. In fact by the defining equation (8) of the hypersurface

$$z^{-m_{0}}\approx-\sum_{m\in F(w)}a_{m}e^{s\lambda(m)}z^{m-m_{0}},$$

whence the holomorphic volume form is (cf. (9))

$$\Omega_{s}=\pm\frac{d\log z^{m_{0}}\wedge\ldots d\log z^{m_{n}}}{dF_{s}}\approx d\log z^{m_{1}}\wedge\ldots d\log z^{m_{n}}.$$

(12)

(Here $m_{i}$ are suitably oriented to take care of $\pm 1$.) We regard the above region as an open subset of $(\mathbb{C}^{*})^{n}$, and denote the chart $U^{s}_{w}$ as the largest $T^{n}$-invariant subset, delineated by a collection of affine linear inequalities on the variables $\log|z^{m_{i}}|$.