We plan to construct an approximate Calabi-Yau metric using the generalised Gibbons-Hawking ansatz, on a singular $T^{2}$-bundle $M^{+}$ over an open neighbourhood of the origin inside the real 4-dimensional base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$, whose discriminant locus is

Here $\eta=x+\sqrt{-1}y$ is a complex variable with period 1. The topological situation is described in Section 1.1.3, Example 1.9 and the expected complex structure can be found in Section 1.1.6.

This situation has very strong similarity with the Taub-NUT type metric on $\mathbb{C}^{3}$ in Chapter 2, the only difference being the periodicity condition on $\eta$. The basic heuristic idea is to perturb the constant solution (cf. Example 1.6) after incorporating the topology. The information in the constant solution is encoded by the base metric

with $a_{ij}$ being a real symmetric positive definite matrix and $A=\det a$, analogous to Section 2.1. We call $a_{ij}$ the coupling constants and emphasize that $a_{ij}$ are parameters we would like to vary. We impose the scale invariant ellipticity bound

The $A\gg 1$ assumption is essential for the perturbative way of thinking to be effective; this assumption was absent in the $\mathbb{C}^{3}$ case because there was no intrinsic scale provided by periodicity. The appearance of the gluing parameter $A$ means we need to carefully track down $A$-dependence in our estimates; in this Chapter all constants in estimates depend on $a_{ij}$ only through the above scale invariant ellipticity constant unless stated otherwise.

Exactly the same discussions as in Section 2.1 lead us to consider the linearised equations (2.2)(2.3)(2.4), which describe the first order corrections we need to make to the constant solution. The key difference is the periodicity requirement. The principle of superposition allows us to immediately produce the solution from Proposition 2.2. We recall from there the functions $\alpha_{1},\alpha_{2},\alpha_{3}$.

We obtain by the generalised Gibbons-Hawking construction a Kähler ansatz $(\tilde{g}^{(1)},\tilde{\omega}^{(1)},\tilde{J}^{(1)},\tilde{\Omega}^{(1)})$ associated to

A subtlety here is that the connection $\vartheta=(\vartheta_{1},\vartheta_{2})$ can be twisted by a flat connection. This choice is parametrised by $H^{1}(\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times S^{1}\times\mathbb{R}\setminus\mathfrak{D},T^{2})=H^{1}(\mathbb{R}^{2}\times S^{1}\times\mathbb{R},T^{2})=T^{2}$, since the codimension 3 subset $\mathfrak{D}$ inside the base does not affect the fundamental group. We sometimes suppress mentioning this choice since it does not have a strong impact on the geometry, especially because we will exclusively work with $T^{2}$-invariant tensors, which are rarely sensitive to the flat connection.

The Kähler structure is well defined over the region where the matrix $\tilde{V}_{(1)}^{ij}$ is positive definite and $\tilde{W}_{(1)}$ is positive, except at the singular point $(\mu_{1},\mu_{2},\eta)=0$. A sufficient condition for positive definiteness will be given in (3.9). The Kähler structure extends smoothly across $\mathfrak{D}_{i}$, where the local structure is modelled on the Taub-NUT fibration described by $g_{\text{Taub}}$ for $|\vec{\mu}|_{a}\gtrsim A^{-1/4}$ (cf. Section 2.3).