ScalingStacks

We now assign a a collection of charts to $\partial\Delta_{\lambda}^{\vee}$, whose transition functions are piecewise linear. (Some authors prefer the terminology ‘piecewise affine’.) These are closely related to the holomorphic charts on $X_{s}$ in section 3.1.

Let $w\in N$ be the primitive integral outward normal vector to a facet $F(w)$ of $\Delta$, and choose an integral basis $m_{1},\ldots m_{n}$ for $\{m\in M|\langle w,m\rangle=0\}$, suitably oriented to be compatible with (12). On the open subset of $\partial\Delta_{\lambda}^{\vee}$,

we regard $m_{1},\ldots m_{n}$ as the affine linear coordinates, also written as $x^{m_{1}},\ldots x^{m_{n}}$. Such charts cover $\partial\Delta_{\lambda}^{\vee}$. We denote $\widetilde{Sing}$ as the subset of points on $\partial\Delta_{\lambda}^{\vee}$ which do not lie on the interior of the top dimensional faces. It is easy to check that the transition functions on overlapping charts in $\partial\Delta_{\lambda}^{\vee}\setminus\widetilde{Sing}$ lie in $SL(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$, so the volume form $dx^{m_{1}}\wedge\ldots dx^{m_{n}}$ is defined independent of the choice of charts. We call the associated measure $d\mu_{\infty}$ the Lebesgue measure on $\partial\Delta_{\lambda}^{\vee}$, with respect to which $\widetilde{Sing}$ is a null set. The set $\widetilde{Sing}$ has real codimension 1, and the transition functions are in general only piecewise linear.

We now examine the normalised canonical measure on $X_{s}$