Recall from Section 2.1 the additional $U(1)$-symmetry

$$\mu_{i}\mapsto\mu_{i},\quad\eta\mapsto e^{i\theta}\eta,$$

acting on the base, which lifts to some $T^{2}$-equivariant action on $M$ preserving the metric and rotating $\Omega$. Properly speaking, the continuous symmetry group fits naturally into an extension sequence

$$1\to T^{2}\to U(1)^{3}\to U(1)\to 1,$$

and we are making a non-unique choice to split the extension. A particular choice can be identified complex geometrically as

$$(z_{1},z_{2},z_{0})\mapsto(z_{1},z_{2},e^{i\theta}z_{0}).$$