In this section we translate our problem to a family of degenerating complex Monge-Ampère equations, and we state our estimates.

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.

In this setting we look at the Kähler forms $\omega_{t}=\omega_{0}+t\omega_{X}$ for $0<t\leq 1$, which are cohomologous to the Ricci-flat metrics $\tilde{\omega}_{t}$. We then define a smooth function $E$ by

$$\mathrm{Ric}(\omega_{X})=\sqrt{-1}\partial\overline{\partial}E,\quad\int_{X}e^{E}\omega_{X}^{n}=\int_{X}\omega_{1}^{n},$$

which is possible thanks to the $\partial\overline{\partial}$-lemma. Then the equation $\mathrm{Ric}(\tilde{\omega}_{t})=0$ is equivalent to

$$\tilde{\omega}_{t}^{n}=a_{t}e^{E}\omega_{X}^{n},$$

where

$$a_{t}=\frac{\int_{X}\omega_{t}^{n}}{\int_{X}\omega_{1}^{n}}.$$

Using the $\partial\overline{\partial}$-lemma again, we can find smooth functions $\varphi_{t}$ for $0<t\leq 1$ so that $\tilde{\omega}_{t}=\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$, $\sup_{X}\varphi_{t}=0$ and we have

(2.5)

$$(\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}=a_{t}e^{E}\omega_{X}^{n}.$$

Notice that as $t$ approaches zero, the constants $a_{t}$ behave like

(2.6)

$$\binom{n}{m}\frac{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}{\int_{X}\omega_{1}^{n}}t^{n-m}+O(t^{n-m+1}).$$

We can then write (2.5) as

(2.7)

$$(\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t})^{n}=c_{t}t^{n-m}e^{E}\omega_{X}^{n},$$

where the constant $c_{t}$ is bounded away from zero and infinity as $t$ goes to zero. Equation (2.7) has been studied for example in [KT] where a uniform $L^{\infty}$ bound on $\varphi_{t}$ was conjectured. When $m=1$ such a bound can be easily proved using the Moser iteration method (see [ST1]). The bound in the general case was then proved independently by Demailly and Pali [DP] and by Eyssidieux, Guedj and Zeriahi [EGZ2]:

Our goal is to show higher order estimates for $\varphi_{t}$ which are uniform on compact sets of $X\backslash S$. Notice that since

$$0<\textrm{tr}_{\omega_{X}}\tilde{\omega}_{t}=\textrm{tr}_{\omega_{X}}\omega_{t}+\Delta_{\omega_{X}}\varphi_{t},$$

and since $\textrm{tr}_{\omega_{X}}\omega_{t}$ is uniformly bounded, we always have a uniform lower bound for $\Delta_{\omega_{X}}\varphi_{t}$.

The following are our main results, and together they imply Theorem 1.2.