Now we return to the higher dimensional situation. One interesting type of singularities is locally modeled on the product of the above 2 dimensional model with a flat space $\mathbb{C}^{n-3}$. Then the space $Q\equiv\mathbb{C}^{n-3}\oplus Q_{0}$ is still smooth and the singular set of $\tilde{\omega}$ and $h$ is the real codimension 3 subspace $P=\mathbb{C}^{n-1}\oplus\{0\}\subset Q$, and they both have transversal Dirac type singularities along $P$. In our applications, we need to consider the non-linear situation. So $D$ is an $n-1$ dimensional complex manifold and $H\subset D$ is a smooth complex hypersurface, and we want our solution $(\tilde{\omega},h)$ to the equation (2.26) to satisfy a distributional equation on $Q$ of the form

(2.58)

$$(\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}\frac{2^{n-1}\tilde{\omega}^{n-1}}{(\sqrt{-1})^{(n-1)^{2}}\Omega_{D}\wedge\bar{\Omega}_{D}})\wedge dz=2\pi\cdot\delta_{P}$$

where $P\equiv H\times\{0\}$ and $\delta_{P}$ is a degree $3$ current given by integration along $P$. This equation has appeared in the literature [Zha04] in a slightly different form. A solution to this equation, with suitable regularity, will give rise to a Calabi-Yau metric with an $S^{1}$ action whose fixed point locus is a complex codimension two submanifold and transverse to which the action is modeled on the above standard $S^{1}$ action on $\mathbb{C}^{2}$. This is exactly what we are motivated to search for from the algebro-geometric discussion at the beginning of this section.

Unfortunately, solving the non-linear equation together with distribution (2.58) in general seems very difficult. Motivated by recent results in the study of adiabatic limits of $G_{2}$ manifolds [Don17, FHN17], we attempt to study the equation when the $S^{1}$ orbit is very small. Again suppose $(D,\omega_{D},\Omega_{D})$ is $n-1$ dimensional Calabi-Yau, then for $T$ large we know there are trivial constant solutions with $\tilde{\omega}=T\omega_{D}$ and $h=T^{n-1}$. Now we look for a perturbation $\tilde{\omega}=T\omega+\psi$ for $T$ large. To the first order we know $\psi$ must satisfy the linearized equation at $T\omega$, hence

(2.59)

$$\partial_{z}^{2}\psi+T^{n-2}d_{D}d_{D}^{c}\Tr_{\omega_{D}}\psi=0,$$

which by Kähler identities is equivalent to

(2.60)

$$\partial_{z}^{2}\psi-T^{n-2}d_{D}d_{D}^{*}\psi=0.$$

Up to a scaling of the $z$ variable this is equivalent to the equation

(2.61)

$$\partial_{z}^{2}\psi-d_{D}d_{D}^{*}\psi=0.$$

If we can at the same time achieve $d_{D}\psi=0$, then this is equivalent to that $\psi\wedge dz$ being a harmonic 3-form on the product $Q=D\times\mathbb{R}_{z}$. Again the interesting case is when $\psi$ has singularities, and we want to study the case when the singular set is of the form $H\times\{0\}\subset Q$ for $H$ a smooth hypersurface in $D$, and correspondingly $\psi$ satisfies

(2.62)

$$\Delta_{Q}\psi=2\pi\cdot\delta_{P}$$

This is a generalization of Green’s function to 3-forms and we shall call it a Green’s current, which is our main object of study in Section 3.

When $n=2$, the above Green’s current is simply the Green’s function and this has been used in [HSVZ18] to obtain exact solutions to a family of incomplete Calabi-Yau metric by Gibbons-Hawking construction. In higher dimension using Green’s current we can apply (2.19) to define a family of approximately Calabi-Yau metrics. This is our main object of study in Section 4.

For our geometric application in this paper the family of incomplete approximately Calabi-Yau metrics will be sufficient, see Section 7. On the other hand, one can also perturb these to genuine Calabi-Yau metrics. This will be discussed in Section 6.