ScalingStacks

The measure also singles out a distinguished subcomplex $Sk(\mathcal{X})$, called the essential skeleton, consisting of the simplices in $\Delta_{\mathcal{X}}$ whose vertices correspond to $E_{i}$ with $a_{i}=0$. This is where the limit of the normalised CY measure is supported. The dimension of $Sk(\mathcal{X})$ is a measurement of how transcendental the degeneration $X$ is; it is reflected by the growth order of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. In the case of a maximal degeneration, $\dim_{\mathbb{R}}Sk(\mathcal{X})=n$. Let us analyze the CY measure more explicitly for maximal degenerations, in a semistable snc model. For $E_{J}$ corresponding to an $n$-dimensional simplex in $Sk(\mathcal{X})$, on $E_{J}^{0}$

$$\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|u_{J}|^{2}\prod_{1}^{n}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}.$$

(4)

Here $u_{J}$ limits to its value $u_{J}(E_{J})$ at the point stratum $E_{J}$, which is called the Poincaré residue of $\Omega$, and is easily seen to be independent of the choice of coordinates $z_{i}$. It is a consequence of the residue theorem on Riemann surfaces that $|u_{J}(E_{J})|^{2}$ is independent of such $J$ [3, Thm. 7.1]. Thus the pushforward to $\Delta_{\mathcal{X}}$ of the normalised CY measure (1) converges smoothly in the interior of $\Delta_{J}$ to a constant multiple of the Lebesgue measure:

$$\text{Log}_{\mathcal{X}*}d\mu_{t}=\text{Log}_{\mathcal{X}*}\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}\xrightarrow{t\to 0}d\mu_{0}:=\text{Const}\cdot dx_{1}\ldots dx_{n}.$$

(5)

Notice $dx_{1}\ldots dx_{n}$ is canonically defined due to the presence of an integral affine structure on $\Delta_{J}$. Viewed as a measure on $\Delta_{\mathcal{X}}$, the limit $d\mu_{0}$ has null measure on the complement of the $n$-dimensional faces of $Sk(\mathcal{X})$, as the integral of $d\mu_{t}$ in the corresponding region is $O(\frac{1}{|\log|t||})$. The constant in (5) is independent of $J$ and its sole purpose is to make $d\mu_{0}$ a probability measure.