The inverse matrix $(V^{-1})^{ij}$ describes the metric restricted to the torus fibres, and the matrix $W^{p\bar{q}}$ describes the metric induced on the Kähler quotients. This viewpoint is taken by Pedersen and Poon [23], whose argument shows that Calabi-Yau manifolds with Hamiltonian torus symmetries necessarily arise from this construction locally. The local existence of the potential $\Phi$ is equivalent to the linear integrability condition

(1.10)

$$\frac{\partial V^{ij}}{\partial\mu_{k}}=\frac{\partial V^{ik}}{\partial\mu_{j}},\quad\frac{\partial W^{p\bar{q}}}{\partial\eta_{r}}=\frac{\partial W^{rq}}{\partial\eta_{p}},\quad\frac{\partial W^{pr}}{\partial\bar{\eta}_{q}}=\frac{\partial W^{p\bar{q}}}{\partial\bar{\eta}_{r}},$$

and

(1.11)

$$\frac{\partial^{2}W^{p\bar{q}}}{\partial\mu_{i}\partial\mu_{j}}+4\frac{\partial^{2}V^{ij}}{\partial\eta_{p}\partial\bar{\eta}_{q}}=0.$$

In particular when $N=2,\mathfrak{n}=1$, the Calabi-Yau condition is $V=W$ and we recover the usual Gibbons-Hawking equation from (1.11).