Given a $(\sigma,\tau)$-type Gibbons-Hawking solution on a domain $B$, the total space of the torus bundle $M\to B^{\circ}$ can be compactified to the fibration $\bar{M}\to B$ such that $\bar{M}$ is biholomorphic (in the orbifold sense) to $Z_{\sigma,\tau}(B)$ in a manner which respects the fibration

$$\pi:Z_{\sigma,\tau}\to T_{\tau}.$$

In particular, such a solution defines a Ricci-flat Kähler (orbifold) metric on $Z_{\sigma,\tau}(B)$ with the holomorphic volume form $\Omega=\Omega_{CY}$.