Proposition 23.2.44A642AD47FF61

Proof. The radial complex line is orthogonal to $\ker \mathrm dr\cap \ker \mathrm d^{\mathrm c}r$. On a radial complex line, use a logarithmic coordinate $w=x+\mathrm i y$, so that $r=2x+\text{constant}$ and $\beta =\pi ^{-1}\mathrm dx\wedge \mathrm dy$. The Riemannian metric of $\omega $ restricts there to $D\pi ^{-1}(\mathrm dx^2+\mathrm dy^2)$, whence

A parallel Lee form has constant norm. Since $k\ne 0$, the Vaisman condition would therefore imply $D=Q C^{-2m}$ for a constant $Q>0$. At a maximum and at a minimum of $C$, where $C'=0$, the Gauduchon identity would then give $kC^{m+1}+Q=0$. This equation has only one positive root, so the maximum and the minimum of $C$ coincide. Thus $C$, $D$ and $F=C^ mD$ would all be constant, a contradiction. $\square$