The map $M^{-}\to\{z_{3}z_{4}=1-z_{1}-z_{2}\}$ is a holomorphic open embedding. The $S^{1}$-action is identified as

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

and the holomorphic volume form is $\Omega^{(1)}=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}$.