Theorem B.3 (Kawamata-Viehweg vanishing) . [01HZ]

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Theorem B.3 (Kawamata-Viehweg vanishing).

Let $\mathcal{X}$ be an SNC model of $X$ . Let $\mathcal{L}\in\Pic(\mathcal{X})$ be a line bundle whose restriction to the generic fiber $X$ is ample and such that $\mathcal{L}-D$ is nef for some $D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ with coefficients in $[0,1[$ . Then we have

H^{q}(\mathcal{X},\omega_{\mathcal{X}}\otimes\mathcal{L})=0\,\,\text{ for all }q\geq 1.