In general it is not easy to directly solve the equation (2.26), but we can easily see some special solutions, which will be important for us.

Suppose $(D,\Omega)$ is an $n-1$ dimensional compact Calabi-Yau manifold, and $\omega_{D}$ is a Calabi-Yau metric on $D$ with $[\omega_{D}]\in 2\pi H^{2}(D;\mathbb{Z})$, satisfying

(2.29)

$$\frac{\omega_{D}^{n-1}}{(n-1)!}=\frac{(\sqrt{-1})^{(n-1)^{2}}}{2^{n-1}}\Omega_{D}\wedge\bar{\Omega}_{D}.$$

If we set

(2.30)

$$\begin{cases}\tilde{\omega}(z)=z\cdot\omega_{D};\\ h=z^{n-1}\end{cases}$$

Then as long as $z>0$, $(\tilde{\omega},h)$ clearly satisfy (2.26) and the integrality condition is also achieved, so we get (incomplete) Calabi-Yau metrics in $n$ dimension.

This metric has already appeared in Kähler geometry, which is usually expressed in terms of a Kähler potential. To explain this, we fix a holomorphic line bundle $L_{D}$ with first Chern class $\frac{1}{2\pi}\omega_{D}$, and also fix a hermitian metric on $L_{D}$ whose curvature form is $-\sqrt{-1}\omega_{D}$. Then we consider the subset $\mathcal{C}$ of the total space of $L_{D}$ consisting of all elements $\xi$ with $0<|\xi|<1$. It is endowed with a nowhere vanishing holomorphic volume form $\Omega_{\mathcal{C}}$ and a Ricci-flat Kähler metric $\omega_{\mathcal{C}}$ which is incomplete as $|\xi|\to 1$ and complete as $|\xi|\to 0$. The holomorphic volume form $\Omega_{\mathcal{C}}$ is given by (as in Section 4.2)

(2.31)

$$\Omega_{\mathcal{C}}=\sqrt{-1}\frac{d\xi}{\xi}\wedge\Omega_{D}$$

The metric $\omega_{\mathcal{C}}$ is given by the Calabi ansatz

(2.32)

$$\omega_{\mathcal{C}}=\frac{n}{n+1}\sqrt{-1}\partial\bar{\partial}(-{\log|\xi|^{2}})^{\frac{n+1}{n}}.$$

It is straightforward to check that

(2.33)

$$\omega_{\mathcal{C}}^{n}=\frac{1}{n2^{n-1}}(\sqrt{-1})^{n^{2}}\Omega_{\mathcal{C}}\wedge\overline{\Omega}_{\mathcal{C}},$$

Clearly the Calabi-Yau structure $(\omega_{\mathcal{C}},\Omega_{\mathcal{C}})$ is invariant under the natural $S^{1}$ action on $L_{D}$. Applying the $S^{1}$ reduction as in Section 2.1, we get that the moment map is given by

(2.34)

$$z=(-{\log|\xi|^{2}})^{1/n},$$

and the reduced family of Kähler metrics on $D$ is given by

(2.35)

$$\tilde{\omega}=z\cdot\omega_{D}.$$

The function $h$ is

(2.36)

$$h=\frac{2}{n}z^{n}.$$

So we see this gives rise to the above solution to (2.30) (up to a multiplicative constant on $h$), We call the space $(\mathcal{C},\omega_{\mathcal{C}},\Omega_{\mathcal{C}})$ a Calabi model space. In Section 4.2, Remark 4.12.2 we shall see the formula (2.32) can also be recovered from (2.30), and this works in a more general situation.

Now from the second construction the connection 1-form $\Theta$ is given by the Chern connection 1-form on $L_{D}$. We claim that by varying the holomorphic structures on $L_{D}$ we obtain all the gauge equivalence classes of $\Theta$. This follows from the fact that there is a natural isomorphism between the group $\mathcal{S}_{h}$ of the isomorphism classes of holomorphic line bundles with $c_{1}=0\in H^{2}(D;\mathbb{R})$ and the group $\mathcal{S}_{f}$ of gauge equivalence classes of flat $U(1)$ connections on $D$. Abstractly, we know the first group fits into an exact sequence

(2.37)

$$0\rightarrow\frac{H^{1}(D;\mathcal{O})}{H^{1}(D;\mathbb{Z})}\rightarrow\mathcal{S}_{h}\rightarrow H^{2}_{\text{tor}}\rightarrow 0,$$

where $H^{2}_{tor}$ denotes the torsion subgroup in $H^{2}(D;\mathbb{Z})$, and the second group fits into a short exact sequence

(2.38)

$$0\rightarrow\frac{H^{1}(D;\mathbb{R})}{H^{1}(D;\mathbb{Z})}\rightarrow\mathcal{S}_{f}\rightarrow\text{Hom}(H_{1,\text{tor}},S^{1})\rightarrow 0$$

where $H_{1,tor}$ is the torsion subgroup in $H_{1}(D;\mathbb{Z})$. The isomorphism between $\mathcal{S}_{h}$ and $\mathcal{S}_{f}$ induces an isomorphism on the torsion quotients, which coincides with the isomorphism

(2.39)

$$H^{2}_{\text{tor}}\simeq\text{Ext}(H_{1}(D;\mathbb{Z}),\mathbb{Z})\simeq\text{Hom}(H_{1,tor},S^{1})$$

given by the universal coefficient theorem.

We mentioned in the above that gauge equivalent choices of the connection 1-form $\Theta$ yield isomorphic Calabi-Yau structures on $\mathcal{C}$. Now we observe that for different choices of gauge equivalence classes which differ only by an element in the identity component of $\mathcal{S}_{f}$, the resulting Calabi-Yau structures are also isomorphic, via a diffeomorphism that covers a holomorphic isometry on $D$. For this we fix a choice of $\Theta$, then given any vector field $V$ on $D$, let $\hat{V}$ be the horizontal lift of $V$ to the $U(1)$ bundle with respect to the connection $\Theta$. The infinitesmal variation of $\Theta$ along the flow of $\hat{V}$ is given by

(2.40)

$$\mathcal{L}_{\hat{V}}\Theta=d(\hat{V}\lrcorner\Theta)+\hat{V}\lrcorner d\Theta=V\lrcorner\omega_{D}.$$

Since $\omega_{D}$ is Ricci-flat, every harmonic 1-form on $D$ is parallel, so by Bochner’s theorem, the map $V\mapsto V\lrcorner\omega_{D}$ defines an isomorphism between the space of parallel vector fields on $D$ and the space of harmonic 1-forms on $D$. A parallel vector field is automatically holomorphic and Killing, we see if $\Theta^{\prime}$ differs from $\Theta$ by a harmonic 1-form, then they are related by the flow of some $\hat{V}$ for a parallel vector field $V$.