ScalingStacks

In this section we recall some background on the asymptotic behavior of the relative Calabi-Yau volume forms on a polarized Calabi-Yau degeneration family, and set up some notation for later use.

As in the introduction, we assume we have a polarized Calabi-Yau family $\pi:X\to\Delta^{*}$ with relative polarization $L$, and we fix a trivializing section $\Omega$ of $K_{X}$ and define trivializations $\Omega_{t}$ of $K_{X_{t}}$ by $\Omega=dt\wedge\Omega_{t}$ along $X_{t}$. Up to passing to a finite base change, we may assume that $X$ admits a semistable model, where $\mathfrak{X}$ is smooth and $X_{0}=\sum_{i\in I}E_{i}$ is reduced and has simple normal crossings. In this case we have

$$K_{\mathfrak{X}/\Delta}=\sum_{i\in I}a_{i}E_{i},\quad a_{i}\in\mathbb{Z},$$

and letting $\kappa=\min_{i\in I}a_{i}$, up to replacing $\Omega_{t}$ by $t^{-\kappa}\Omega_{t}$, we may assume without loss that $\kappa=0$.

Recall that $\omega_{t}$ denotes the Calabi-Yau metric on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)|_{X_{t}}$, and that the dimension $m$ of the essential skeleton of $X$ is assumed to be stricty positive (and necessarily $m\leqslant n$). Denote by $\mu_{t}$ the Calabi-Yau volume form on $X_{t}$ normalized to be a probability measure, i.e.

(2.1)

$$\mu_{t}=\frac{\omega_{t}^{n}}{\int_{X_{t}}\omega_{t}^{n}}=\frac{i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}}{\int_{X_{t}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}}.$$

Let us now recall the asymptotic behavior of the integrals $\int_{X_{t}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}$, largely following [3]. For any $J\subset I$ we denote by $E_{J}=\bigcap_{j\in J}E_{j}$. As in [13, 14], we fix a Kähler metric on $\mathfrak{X}$ and for $\varepsilon>0$ small and $J\subset I$ with $E_{J}\neq\emptyset$ we define

$$E^{0}_{J}=\{x\in X_{t}\ |\ d(x,E_{J})<\varepsilon\}\backslash\{x\in X_{t}\ |\ d(x,E_{J^{\prime}})<\varepsilon\text{ for some }J^{\prime}\supsetneq J\}.$$

For any given $x\in E_{J}\subset\mathfrak{X}$ let $p=|J|-1$ and pick local coordinates $z_{0},\dots,z_{n}$ on $x\in V\subset\mathfrak{X}$, defined in the unit polydisc, such that $z_{0},\dots,z_{p}$ are defining equations for $E_{j},j\in J$, so that in these coordinates we have $t=z_{0}\cdots z_{p}$. We shall call these adapted coordinates. We can then write

$$\Omega=f_{J}\prod_{i=0}^{p}z_{i}^{a_{i}}dz_{i}\wedge\prod_{j=p+1}^{n}dz_{j},$$

where $f_{J}$ is a local non-vanishing holomorphic function. Since $dt\wedge\Omega_{t}=\Omega$ along $X_{t}$, on $E^{0}_{J}$ we get

$$\Omega_{t}=f_{J}z_{0}^{a_{0}}\cdots z_{p}^{a_{p}}\prod_{j=1}^{p}\frac{dz_{j}}{z_{j}}\wedge\prod_{k=p+1}^{n}dz_{k},$$

$$i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}=|f_{J}|^{2}|z_{0}|^{2a_{0}}\cdots|z_{p}|^{2a_{p}}\prod_{j=1}^{p}i\frac{dz_{j}}{z_{j}}\wedge\frac{d\overline{z_{j}}}{\overline{z_{j}}}\wedge\prod_{k=p+1}^{n}idz_{k}\wedge d\overline{z_{k}},$$

from which using polar coordinates $z_{j}=\exp(x_{j}\log|t|+i\theta_{j}),j\in J,$ one can easily see as in [3] that

$$\int_{E^{0}_{J}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}\sim|\log|t||^{m_{J}},$$

where

$$m_{J}=|\{j\in J\ |\ a_{j}=0\}|-1,$$

while

$$\int_{X_{t}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}\sim|\log|t||^{m},$$

where

(2.2)

$$m=\max\{|J|-1\ |\ E_{J}\neq\emptyset,a_{j}=0\text{ for all }j\in J\}=\dim\mathrm{Sk}(X).$$

The local logarithmic variables $x_{j}=\frac{\log|z_{j}|}{\log|t|}$ vary in the standard simplex

$$\Delta_{J}=\left\{0\leqslant x_{j}\leqslant 1\ |\ \sum_{j=0}^{p}x_{j}=1\right\},$$

and in this way one obtains a map $\mathrm{Log}_{t}:V\backslash\bigcup_{j}E_{j}\to\Delta_{J}$, see [3].

For each $i\in I$ we fix now a defining section $\sigma_{i}\in H^{0}(\mathfrak{X},\mathcal{O}(E_{i}))$ and a Hermitian metric $h_{i}$ on $\mathcal{O}(E_{i})$, so that $r_{i}:=|\sigma_{i}|^{2}_{h_{i}}$ is a smooth nonnegative function of $\mathfrak{X}$ which vanishes precisely along $E_{i}$ and is uniformly comparable to $|z_{i}|^{2}$ in any adapted coordinate chart as above where $z_{i}$ is the local defining equation of $E_{i}$. In particular,

$$\tilde{x}_{i}:=\frac{\log r_{i}}{\log|t|},$$

is now defined on the whole $\mathfrak{X}\backslash\bigcup_{j}E_{j}$, and in the adapted coordinates as above it is equal to $2x_{i}$ up to very small errors (as $t$ approaches $0$). It follows that on $X_{t}$ (for $|t|$ sufficiently small) in an adapted coordinate chart near a point of $E^{0}_{J}$ as above, the point $\left(\frac{1}{2}\tilde{x}_{0},\dots,\frac{1}{2}\tilde{x}_{p}\right)$ will lie very close to $\Delta_{J}$.