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.
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.
Accepted:
Published online:
Masataka Iwai 1
CC-BY 4.0
@article{AFST_2021_6_30_1_63_0,
author = {Masataka Iwai},
title = {Nadel{\textendash}Nakano vanishing theorems of vector bundles with singular {Hermitian} metrics},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {63--81},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 30},
number = {1},
year = {2021},
doi = {10.5802/afst.1666},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1666/}
}
TY - JOUR AU - Masataka Iwai TI - Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 63 EP - 81 VL - 30 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1666/ DO - 10.5802/afst.1666 LA - en ID - AFST_2021_6_30_1_63_0 ER -
%0 Journal Article %A Masataka Iwai %T Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 63-81 %V 30 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1666/ %R 10.5802/afst.1666 %G en %F AFST_2021_6_30_1_63_0
Masataka Iwai. Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 63-81. doi : 10.5802/afst.1666. https://afst.centre-mersenne.org/articles/10.5802/afst.1666/
[1] On positivity and base loci of vector bundles, Eur. J. Math., Volume 1 (2015) no. 2, pp. 229-249 | DOI | MR | Zbl
[2] Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009) no. 2, pp. 531-560 | DOI | MR | Zbl
[3] Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., Volume 145 (2008) no. 2, pp. 341-378 | DOI | MR | Zbl
[4] Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. Math., Volume 207 (2017) no. 1, pp. 345-387 | MR | Zbl
[5] Singular Hermitian metrics on vector bundles, J. Reine Angew. Math., Volume 502 (1998), pp. 93-122 | MR | Zbl
[6] Complex analytic and differential geometry (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)
[7] Estimations pour l’opérateur d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982) no. 3, pp. 457-511 | DOI | Numdam | Zbl
[8] Analytic methods in algebraic geometry, Surveys of Modern Mathematics, 1, International Press; Higher Education Press, 2012, viii+231 pages | MR | Zbl
[9] A transcendental approach to Kollár’s injectivity theorem, Osaka J. Math., Volume 49 (2012) no. 3, pp. 833-852 | Zbl
[10] A transcendental approach to Kollár’s injectivity theorem II, J. Reine Angew. Math., Volume 681 (2013), pp. 149-174 | Zbl
[11] Injectivity theorem for pseudo-effective line bundles and its applications (2017) (https://arxiv.org/abs/1605.02284)
[12] Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften, 265, Springer, 1984, xviii+249 pages | MR | Zbl
[13] Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, 1969, pp. 185-251 | Zbl
[14] Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Puaun, Local and global methods in algebraic geometry (Contemporary Mathematics), Volume 712, American Mathematical Society, 2018, pp. 143-195 | Zbl
[15] Approximations and examples of singular Hermitian metrics on vector bundles, Ark. Mat., Volume 55 (2017) no. 1, pp. 131-153 | DOI | MR | Zbl
[16] A vanishing theorem of Kollár-Ohsawa type, Math. Ann., Volume 366 (2016) no. 3-4, pp. 1451-1465 | DOI | MR | Zbl
[17] Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 1, pp. 21-38 | DOI | Zbl
[18] Singular Hermitian metrics and positivity of direct images of pluricanonical bundles, Algebraic geometry: Salt Lake City 2015 (Proceedings of Symposia in Pure Mathematics), Volume 97, American Mathematical Society, 2018, pp. 519-553 | MR | Zbl
[19] Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebr. Geom., Volume 27 (2018) no. 2, pp. 211-272 | DOI | MR | Zbl
[20] Singular hermitian metrics on holomorphic vector bundles, Ark. Mat., Volume 53 (2015) no. 2, pp. 359-382 | DOI | MR | Zbl
Cited by Sources: