Theorem 12.25.83F58257FA940

Proof. Let $B_\theta $ be the sum of the forms $\kappa _ i$ over the factors in which $H_ i$ occurs and of the forms $Q_ i$ over the factors in which only $Q_ i$ occurs. It is closed and semipositive of constant rank $R(\theta )$. Since the coefficients are bounded and $H_ i,Q_ i\leq \kappa _ i$, we have $\theta \leq CB_\theta $ for some $C>0$.

For the lower bound, let $s_ i=\min _ X(a_ i+b_ if_ i)>0$ for each blown-up factor in which $H_ i$ occurs, and put

By (12.25.5), $s_ iH_ i+\mathrm{dd}^{\mathrm{c}}(s_ i\psi _{E_ i}/2)=(s_ i/2)(\kappa _ i+[E_ i])\geq (s_ i/2)\kappa _ i$. The remaining terms $(a_ i+b_ if_ i-s_ i)H_ i$ and all terms involving $Q_ i$ are semipositive. On plain factors in which $H_ i$ occurs and on factors in which only $Q_ i$ occurs, the relevant coefficient is bounded below by a positive constant. Taking the minimum of these constants gives $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq cB_\theta $ for some $c>0$. For $\theta =0$ take $\varphi =0$ and any $c>0$. Now apply lemma 12.25.1 with $\Omega =\sum _ i\kappa _ i$. $\square$