no. 5
Linearization of transition functions of a semi-positive line bundle along a certain submanifold
[Linéarisation des fonctions de transition d’un fibré en droites semi-positif le long d’une certaine sous-variété]
Koike, Takayuki1
1 Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku Osaka, 558-8585, (Japan)
Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2237-2271.

Soit X une variété complexe et L un fibré en droites holomorphe sur X. Supposons que L est semi-positive, à savoir L admet une métrique hermitienne lisse avec une courbure de Chern semi-positif. Soit Y une sous-variété kählérienne compacte de X telle que la restriction de L à Y est topologiquement triviale. Nous examinons l’obstruction pour que L soit plat unitaire sur un voisinage de Y. Comme application, par exemple, nous prouvons l’existence d’un fibré en droites nef, grand et non semi-positif sur une surface projective non singulière.

Let X be a complex manifold and L be a holomorphic line bundle on X. Assume that L is semi-positive, namely L admits a smooth Hermitian metric with semi-positive Chern curvature. Let Y be a compact Kähler submanifold of X such that the restriction of L to Y is topologically trivial. We investigate the obstruction for L to be unitary flat on a neighborhood of Y in X. As an application, for example, we show the existence of nef, big, and non semi-positive line bundle on a non-singular projective surface.

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DOI : 10.5802/aif.3439
Classification : 32J25, 14C20
Keywords: Hermitian metrics, neighborhoods of subvarieties, Ueda theory
Mot clés : métrique hermitienne, voisinage de sous-variété, théorie d’Ueda
Koike, Takayuki 1

1 Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku Osaka, 558-8585, (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Koike, Takayuki. Linearization of transition functions of a semi-positive line bundle along a certain submanifold. Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2237-2271. doi : 10.5802/aif.3439. https://aif.centre-mersenne.org/articles/10.5802/aif.3439/

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