Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics

Iwai, Masataka

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 1, pp. 63-81.

Nous étudions une métrique hermitienne singulière d’un fibré vectoriel. Premièrement, nous montrons que le faisceau de sections holomorphes localement carrées et holomorphes d’un faisceau vectoriel avec une métrique hermitienne singulière, qui est un analogue de rang supérieur d’un faisceau d’idéaux multiplicateurs, est cohérent sous certaines hypothèses. Deuxièmement, nous prouvons un théorème d’annulation de type Nadel–Nakano d’un faisceau de vecteurs avec une métrique hermitienne singulière Nous n’utilisons pas une technique d’approximation d’une métrique hermitienne singulière. Nous appliquons ces théorèmes à une métrique hermitienne singulière induite par des sections holomorphes et un fibré vectoriel gros, et nous obtenons une généralisation du théorème d’annulation de Griffiths. Enfin, nous montrons une généralisation du théorème d’annulation d’Ohsawa.

We study a singular Hermitian metric of a vector bundle. First, we prove that the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analog of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel–Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generalization of Griffiths’ vanishing theorem. Finally, we show a generalization of Ohsawa’s vanishing theorem.

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