PomeD1CB266917DAReturn to the Hopf threefold and the forms $s$, $\alpha $, $\beta $ and $\omega =\alpha +\beta $ of section 23.1; recall that $s=\log |z|^2$ on the covering $\mathbb C^3\setminus \{ 0\} $. The radial description makes the Lee form explicit and shows that this Gauduchon metric is also Vaisman.
Proposition 23.2.24.32084430C8AD The Gauduchon form $\omega =\alpha +\beta $ of section 23.1 is Vaisman, with Lee form $\lambda =-\mathrm{d}s$, and
In particular, the bounded mass property of definition 13.14.14 fails for the ordinary potential theory of a background that is simultaneously Gauduchon and Vaisman.
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. $\square$
The unbounded masses arise in the ordinary family $\omega +\mathrm{dd}^{\mathrm{c}}u$, whose members need not satisfy the Lee equation. The volume problem within the class of Vaisman metrics imposes this additional geometric constraint.
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