(Holomorphic structure) The map

$$M^{-}_{\nu}\to\{Z_{3}Z_{4}=1-z_{1}-z_{2}\}\subset\mathbb{C}^{2}_{Z_{3},Z_{4}}\times(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}$$

extends continuously over the singular locus $S\cap M^{-}_{\nu}$ and defines a holomorphic open embedding under the complex structure $J$. The $S^{1}$-action is identified as

$$e^{i\theta}\cdot(z_{1},z_{2},Z_{3},Z_{4})=(z_{1},z_{2},e^{i\theta}Z_{3},e^{-i\theta}Z_{4}),$$

and the holomorphic volume form is $\Omega=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dZ_{3}\wedge dZ_{4}.$ The Kähler structure is $C^{1,\alpha}$-regular near $S$.