ScalingStacks

Recall that the volume of $X_{y}$ is a homological constant independent of $y$, and that we assume that it is equal to $1$. Since $c_{1}(X_{y})=0$, there is a smooth function $F_{y}$ such that $\mathrm{Ric}(\omega_{y})=\sqrt{-1}\partial\overline{\partial}F_{y}$ and $\int_{X_{y}}(e^{F_{y}}-1)\omega_{y}^{n-m}=0$. The functions $F_{y}$ vary smoothly in $y$, since so do the Kähler forms $\omega_{y}$. By Yau’s theorem there is a unique Ricci-flat Kähler metric $\omega_{SF,y}$ on $X_{y}$ cohomologous to $\omega_{y}$, given by the solution of

(4.1)

$$\omega_{SF,y}^{n-m}=e^{F_{y}}\omega_{y}^{n-m}.$$

If we write $\omega_{SF,y}=\omega_{y}+\sqrt{-1}\partial\overline{\partial}\zeta_{y}$, the functions $\zeta_{y}$ vary smoothly in $y$ and so they define a smooth function $\zeta$ on $X\backslash S$. We then define a real closed $(1,1)$-form $\omega_{SF}$ on $X\backslash S$ by $\omega_{SF}=\omega_{X}+\sqrt{-1}\partial\overline{\partial}\zeta$, and call it the semi-flat form. Notice that $\omega_{SF}$ is not necessarily nonnegative (it is Kähler only in the fiber directions), but on $X\backslash S$ the $(n,n)$-form $\omega_{SF}^{n-m}\wedge\omega_{0}^{m}$ is strictly positive, and so we can define a smooth positive function $F$ on $X\backslash S$ by

(4.2)

$$F=\frac{\Omega}{\omega_{SF}^{n-m}\wedge\omega_{0}^{m}}.$$

We claim that $F$ is actually constant on each fiber $X_{y}$, and so it is the pullback of a function on $Y\backslash f(S)$. To see this, fix a point $y\in Y\backslash f(S)$ and choose local coordinates $z^{1},\dots,z^{n-m}$ on the fiber $X_{y}$, which extend locally to coordinates in a ball in $X$. Then take local coordinates $w^{n-m+1},\dots,w^{n}$ near $y\in Y\backslash f(S)$, so that $z^{1},\dots,z^{n-m},z^{n-m+1}=f^{*}(w^{n-m+1}),\dots,z^{n}=f^{*}(w^{n})$ give local holomorphic coordinates on $X$. In these coordinates write

$$\omega_{0}=\sqrt{-1}\sum_{i,j=n-m+1}^{n}g^{0}_{i\overline{j}}dz^{i}\wedge d\overline{z}^{j},$$

$$\omega_{SF,y}=\sqrt{-1}\sum_{i,j=1}^{n-m}g^{SF}_{i\overline{j}}dz^{i}\wedge d\overline{z}^{j},$$

$$\Omega=G(\sqrt{-1})^{n}dz^{1}\wedge\dots\wedge d\overline{z}^{n}.$$

Then locally

$$F=\frac{G}{\det(g^{0}_{i\overline{j}})\det(g^{SF}_{i\overline{j}})},$$

and so on the fiber $X_{y}$ we have

$$\sqrt{-1}\partial\overline{\partial}\log F=-\mathrm{Ric}(\tilde{\omega}_{1})+\mathrm{Ric}(\omega_{SF,y})=0,$$

because $\omega_{0}$ is the pullback of a metric from $Y$, and so $F$ is indeed constant on $X_{y}$. Moreover, it is easy to check [ST2, Lemma 3.3] that on $Y\backslash f(S)$ we have

$$F=\frac{f_{*}\Omega}{\omega_{Y}^{m}},$$

and so

$$\int_{Y}F\omega_{Y}^{m}=\int_{X}\Omega=\int_{X}\omega_{1}^{n}$$

is finite. In fact there is a positive $\varepsilon$ so that $\int_{Y}F^{1+\varepsilon}\omega_{Y}^{m}$ [ST2, Proposition 3.2]. Then we apply [ST2, Theorem 3.2], which relies on the seminal work of Kołodziej [K] and further generalizations [EGZ1, Z], to solve (uniquely) the complex Monge-Ampère equation

(4.3)

$$(\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\psi)^{m}=\frac{\int_{X}\omega_{0}^{m}\wedge\omega_{X}^{n-m}}{\int_{X}\omega_{1}^{n}}F\omega_{Y}^{m},$$

with $\psi\in L^{\infty}(Y)$ and moreover $\psi$ is smooth on $Y\backslash f(S)$ (the proof of this follows the arguments of Yau in [Y1]). We will call $\omega=\omega_{Y}+\sqrt{-1}\partial\overline{\partial}\psi$ the Kähler metric on $Y\backslash f(S)$ that we’ve just constructed. Its Ricci curvature is the Weil-Petersson metric that we are about to define. Recall that the fibers $X_{y}$ have torsion canonical bundle, so that there is a number $k$ such that $K_{X_{y}}^{\otimes k}$ is trivial for all $y\in Y\backslash f(S)$. The Weil-Petersson metric is a smooth nonnegative $(1,1)$-form on $Y\backslash f(S)$ defined as the curvature form of a pseudonorm on the relative canonical line bundle $f_{*}(\Omega^{n-m}_{X/Y})^{\otimes k}$: if $\Psi_{y}$ is a local nonzero holomorphic section of $f_{*}(\Omega^{n-m}_{X/Y})^{\otimes k}$, which means that $\Psi_{y}$ is a nonzero holomorphic $k$-pluricanonical form on $X_{y}$ that varies holomorphically in $y$, then we let its length be

$$|\Psi_{y}|^{2}_{h_{WP}}=\int_{X_{y}}(\Psi_{y}\wedge\overline{\Psi_{y}})^{\frac{1}{k}}.$$

For $k>1$ this is not a Hermitian metric, but just a pseudonorm. The Weil-Petersson metric $\omega_{WP}$ on $Y\backslash f(S)$ is just formally the curvature of $h_{WP}$, that is locally we set

$$\omega_{WP}=-\sqrt{-1}\partial\overline{\partial}\log|\Psi_{y}|^{2}_{h_{WP}},$$

and this is well-defined because the bundle $K_{X_{y}}^{\otimes k}$ is trivial. It is a classical fact (see [FS]) that $\omega_{WP}$ is pointwise nonnegative. As an aside, we note here that one can realize $\omega_{WP}$ as the actual curvature form of an honest Hermitian metric on a relative canonical bundle if one takes a finite unramified $k$-sheeted cyclic cover $\tilde{X}\to X$ so that the smooth fibers of $\tilde{X}\to Y$ now have trivial canonical bundle.