Lemma 12.22.351797AE0DE1F

Proof. Use the factorization $p=\pi \circ F$ above and lemma 12.22.1. By compactness there is $a>0$ with $h_ Z\geq aF^*h_ Y$. Pulling back the semipositive form $\Psi $ and integrating, we obtain

Here the changes of variables use that $F$ and $p$ have degree one, and the Poincaré–Lelong formula, together with $\mathrm{dd}^{\mathrm{c}}p^*\Psi =0$, replaces the pairing with $\eta _ D$ by the integral over $D$. This last integral vanishes: on the regular locus of each component of $D$, the restriction of $p$ has rank at most $n-2$, so the restriction of the $(n-1,n-1)$-form $p^*\Psi $ is zero. Taking $K=C_0/a$ proves the assertion. $\square$