ScalingStacks

First of all let us recall our setup from the introduction: $(X,\omega_{X})$ is a compact Kähler manifold of complex dimension $n$ with $c_{1}(X)=0$, or in other words a Calabi-Yau manifold. We have a map holomorphic map $f:X\to Z$, where $(Z,\omega_{Z})$ is another compact Kähler manifold, with image $Y\subset Z$ and so that $f:X\to Y$ has connected fibers. $Y$ is assumed to be an irreducible normal subvariety of $Z$ of dimension $m$ with $0<m<n$, and we let $\omega_{Y}$ be the restriction of $\omega_{Z}$ to the regular part of $Y$. We also set $\omega_{0}=f^{*}\omega_{Z}$, which is a smooth nonnegative $(1,1)$ form on $X$ whose cohomology class lies on the boundary of the Kähler cone. There is a proper subvariety $S\subset X$ such that $Y\backslash f(S)$ is smooth and $f:X\backslash S\to Y\backslash f(S)$ is a smooth submersion. Yau’s theorem [Y1] says that in each Kähler class of $X$ there is a unique Kähler metric with Ricci curvature identically zero. For each $0<t\leq 1$ we call $\tilde{\omega}_{t}$ the Ricci-flat Kähler metric cohomologous to $[\omega_{0}]+t[\omega_{X}]$, and we wish to study the behaviour of these metrics when $t$ goes to zero. On $X$ we have

$$\omega_{0}^{k}\wedge\omega_{X}^{n-k}=0,$$

for $m+1\leq k\leq n$, and

(2.1)

$$\omega_{0}^{m}\wedge\omega_{X}^{n-m}=H\omega_{X}^{n},$$

where the smooth non-negative function $H$ vanishes precisely on $S$ and is such that $H^{-\gamma}$ is in $L^{1}$ for some small $\gamma>0$. This is because $H$ is locally comparable to a sum of squares of holomorphic functions (the minors of the Jacobian of $f$). In particular it follows that

$$\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}>0.$$

For later purposes we need the following construction. Let $\mathcal{I}$ be the ideal sheaf of $f(S)$ inside $Z$. We cover $Z$ by a finite number of open sets $U_{k}$ so that on each $U_{k}$ the ideal $\mathcal{I}$ is generated by holomorphic functions $h_{k,j}$, with $1\leq j\leq N_{k}$. We then fix $\eta_{k}$ a partition of unity subordinate to the covering $\{U_{k}\}$ and we let

(2.2)

$$\sigma=\sum_{k,j}\eta_{k}|h_{k,j}|^{2},$$

if $S\neq\emptyset$ and otherwise we just set $\sigma=1$. Then $\sigma$ is a smooth nonnegative function on $Z$ with zero locus precisely $f(S)$ and there is a constant $C$ so that on $Z$ we have

(2.3)

$$\sigma\leq C,\quad 0\leq\sqrt{-1}\partial\sigma\wedge\overline{\partial}\sigma\leq C\omega_{Z},\quad-C\omega_{Z}\leq\sqrt{-1}\partial\overline{\partial}\sigma\leq C\omega_{Z}.$$

Then for any $y\in Y\backslash f(S)$ we have the inequality

(2.4)

$$\sigma(y)^{\lambda}\leq C\inf_{X_{y}}H,$$

for some constants $C,\lambda$, and we are free to enlarge $\lambda$ if needed. This is because both of the function $H$ and $f^{*}\sigma$ on $X$ are locally comparable to a sum of squares of holomorphic functions and they both have zero set equal to $S$. By taking a log resolution of the ideal sheaf of $S$ inside $X$ and we can assume that $S$ is a divisor with simple normal crossings, and then the holomorphic functions have well defined vanishing orders along the irreducible components of $S$, and (2.4) follows.