We argue as in the last part of the proof of Theorem B.3. Let $\mathcal{A}\in\Pic(\mathcal{X})$ be sufficiently ample to guarantee:

• (i)

  $\pi^{*}\mathcal{A}-cD$ is nef.

• (ii)

  $\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)$ is globally generated on $\mathcal{X}$.

• (iii)

  $H^{p}\left(\mathcal{X},\mathcal{A}\otimes R^{m}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\right)=0\,\,\text{ for all }p\geq 1\text{ and }m\geq 0$.

Note that the first condition can be achieved since $-D$ is $\pi$-globally generated. The degeneration of the Leray spectral sequence shows that

$$H^{0}\left(\mathcal{X},\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\right)=H^{q}\left(\mathcal{X}^{\prime},\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)\otimes\pi^{*}\mathcal{A}\right),$$

which vanishes by Theorem B.3. It follows that $\mathcal{A}\otimes R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)=0$ by global generation, whence the result. ∎