Proposition 23.2.711EBB7AD1165

Proof. On a chart where $\omega $ is conformally Kähler,

Let $*$ be the real Hodge star of the Riemannian metric associated with $\omega $, and let $\mathrm d^*$ be the formal adjoint of the exterior derivative. By the pointwise Hodge-star identities for Hermitian metrics, $\lambda ^{\mathrm c}\wedge \omega ^{n-1}$ is a fixed nonzero constant multiple of $*\lambda $. Consequently, the Gauduchon condition $\mathrm{dd}^{\mathrm{c}}(\omega ^{n-1})=0$ is equivalent to $\mathrm{d}*\lambda =0$, that is, to $\mathrm{d}^*\lambda =0$. Since $\mathrm{d}\lambda =0$, co-closedness is equivalent to harmonicity. The Gauduchon representative exists by theorem 11.4.2, and conformal changes preserve the LCK condition.

Stokes' theorem and $\mathrm{d}(\omega ^{n-1})=(n-1)\lambda \wedge \omega ^{n-1}$ give

For a real one-form, $\lambda \wedge \lambda ^{\mathrm c} =\mathrm i\lambda ^{1,0}\wedge \lambda ^{0,1}/(2\pi )$ is semipositive, and its trace with respect to $\omega $ is positive wherever $\lambda \ne 0$. This proves the last assertion. $\square$