We can choose a different $g$-parallel complex structure $J_{1}$ on $\mathfrak{M}$ such that $J_{1}\theta=zdx$. With respect to $J_{1}$ we can view $\mathfrak{M}$ as a holomorphic elliptic fibration over a punctured disc in $\mathbb{C}$. The monodromy of this fibration is given by the matrix

(2.26)

$$\begin{bmatrix}1&b\\ 0&1\end{bmatrix}\in\SL(2,\mathbb{Z}).$$

See [Sco83] for more details. This gives a different compactified model for $\mathfrak{M}$ where the compactifying divisor is a singular fiber of Kodaira type $I_{b}$. The $J_{1}$-Kähler form of our hyperkähler model metric is then given by an appropriate semi-flat ansatz [GSVY90, GW00], and provides the model at infinity for the gravitational instantons with volume growth $\sim r^{4/3}$ and curvature decay $\sim r^{-2}$ constructed in [Hei12]. We will pursue this observation further in [HSVZ], connecting the complete hyperkähler metrics of [TY90] and of [Hei12] by global hyperkähler rotations.