Lemma B.6 (Local vanishing) . [01I5]

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Lemma B.6 (Local vanishing).

Let $\mathcal{X}$ be a regular model, let $\mathfrak{a}$ be a vertical ideal sheaf on $\mathcal{X}$ and let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be an SNC model such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(-D)$ with $D\in\Div_{0}(\mathcal{X}^{\prime})$ . Then we have

R^{q}\pi_{*}\omega_{\mathcal{X}^{\prime}}\left(-\lfloor c\,D\rfloor\right)=0\,\,\text{ for all }q\geq 1.