Proposition 23.2.2432084430C8AD

Proof. Using $\alpha =\mathrm{dd}^{\mathrm{c}}s$ and $\beta =\mathrm{d}s\wedge \mathrm d^{\mathrm c}s$, direct differentiation gives

Although $s$ is not a function on the quotient, $\mathrm{d}s$ descends to it. Its integral along the loop induced by $z\mapsto \mathrm{e}^{-1}z$ is $-2$, so the Lee form is not exact.

Let $g_{S^5}$ be the round metric on the unit sphere. Up to a constant factor, the Riemannian metric associated with $\omega $ on the cover is

The one-form $\mathrm{d}t$ is parallel for this product metric, which proves the Vaisman condition. Finally, $\kappa _\lambda =\mathrm{dd}^{\mathrm{c}}s=\alpha $ and $\lambda \wedge \lambda ^{\mathrm c}=\beta $, so $D_\lambda 1=\omega $ by (23.2.22). The unbounded masses are given by theorem 23.1.1. $\square$