Corollary 23.2.10F6950D5704C9

Proof. If there were a big class, the modification criterion recalled in section 10.10 would give a modification $\pi :Y\to X$ with $Y$ compact Kähler. The form $\pi ^*\kappa _\lambda $ is closed and de Rham exact, so the $\partial \bar\partial $-lemma on $Y$ gives a smooth real function $h$ on $Y$ with $\pi ^*\kappa _\lambda =\mathrm{dd}^{\mathrm{c}}h$. Since $\pi $ has degree one, pushing forward gives the current identity $\kappa _\lambda =\mathrm{dd}^{\mathrm{c}}(\pi _*h)$. By Stokes' theorem, the pairing of this current with a Gauduchon form is zero, which contradicts (23.2.8) for the Gauduchon representative of the LCK structure. $\square$