ScalingStacks

Proof. Differentiating (4.3) we see that

$$\mathrm{Ric}(\omega)=\mathrm{Ric}(\omega_{Y})-\sqrt{-1}\partial\overline{\partial}\log F.$$

If we fix $y\in Y\backslash f(S)$ and choose $\Psi$ a local never vanishing holomorphic section of $f_{*}(\Omega^{n-m}_{X/Y})^{\otimes k}$, then we can define a local function $u=\frac{(\Psi\wedge\overline{\Psi})^{1/k}}{\omega_{SF}^{n-m}}$ on $X\backslash S$, which is constant on each fiber $X_{y}$. Since $\int_{X_{y}}\omega_{SF}^{n-m}=1$, we see that

$$-\sqrt{-1}\partial\overline{\partial}\log u=\omega_{WP}.$$

Then

(4.5)

$$\mathrm{Ric}(\omega)=\mathrm{Ric}(\omega_{Y})-\sqrt{-1}\partial\overline{\partial}\log\frac{u\Omega}{(\Psi\wedge\overline{\Psi})^{\frac{1}{k}}\wedge\omega_{0}^{m}}.$$

Picking local coordinates $z^{i}$ as above, and writing

$$\Psi=K[(\sqrt{-1})^{n-m}dz^{1}\wedge\dots\wedge dz^{n-m}]^{\otimes k},$$

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

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

we see that

$$\frac{u\Omega}{(\Psi\wedge\overline{\Psi})^{\frac{1}{k}}\wedge\omega_{0}^{m}}=\frac{uG}{|K|^{\frac{2}{k}}\det(g^{0}_{i\overline{j}})},$$

and since $K$ is holomorphic and $\Omega$ is Ricci-flat we see that

$$-\sqrt{-1}\partial\overline{\partial}\log\frac{uG}{|K|^{\frac{2}{k}}\det(g^{0}_{i\overline{j}})}=\omega_{WP}-\mathrm{Ric}(\omega_{Y}),$$

which together with (4.5) gives (4.4). ∎