Mall bundles and flat connections
[Fibrés de Mall et connexions plates sur les variétés de Hopf]
Ornea, Liviu1, 2 ; Verbitsky, Misha3, 4
1 University of Bucharest, Faculty of Mathematics and Informatics, 14 Academiei str., 70109 Bucharest (Romania)
2 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str.010702-Bucharest (Romania)
3 Instituto Nacional de Matemática Pura e Aplicada (IMPA) Estrada Dona Castorina, 110 Jardim Botânico, CEP 22460-320 Rio de Janeiro, RJ (Brasil)
4 Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str. Moscow, (Russia)
Annales de l'Institut Fourier, Online first, 28 p.

Un fibré de Mall sur une variété de Hopf H=ℂ n ∖0 ℤ est un fibré vectoriel holomorphe dont le tiré en arrière sur ℂ n ∖0 est trivial. Nous définissons des fibrés de Mall résonants et non-résonants en généralisant la notion de résonance dans les EDO et nous démontrons qu’un fibré de Mall non-résonant admet une connexion holomorphe plate. Nous employons cette observation pour démontrer une version du théorème de linéarisation de Poincaré–Dulac, en montrant que chaque contraction holomorphe bijective et non-résonante de ℂ n est linéaire dans certaines coordonnées holomorphes adaptées. Nous définissons la notion de résonance pour les variétés de Hopf et nous montrons que chaque variété de Hopf non-résonante est linéaire : ce résultat avait été déjà obtenu par Kodaira au moyen du théorème de Poincaré–Dulac.

A Mall bundle on a Hopf manifold H=ℂ n ∖0 ℤ is a holomorphic vector bundle whose pullback to ℂ n ∖0 is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincaré–Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of ℂ n is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-resonant Hopf manifolds are linear; previously, this result was obtained by Kodaira using the Poincaré–Dulac theorem.

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DOI : 10.5802/aif.3647
Classification : 14F06, 32L05, 32L10, 53C07, 34C20
Keywords: Holomorphic contraction, Hopf manifold, holomorphic bundle, holomorphic connection, affine manifold, resonance, coherent sheaf, Dolbeault cohomology.
Mot clés : Contraction holomorphe, variété de Hopf, fibré holomorphe, connexion holomorphe, variété affine, résonance, faisceau cohérent, cohomologie de Dolbeault.
Ornea, Liviu 1, 2 ; Verbitsky, Misha 3, 4

1 University of Bucharest, Faculty of Mathematics and Informatics, 14 Academiei str., 70109 Bucharest (Romania)
2 Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str.010702-Bucharest (Romania)
3 Instituto Nacional de Matemática Pura e Aplicada (IMPA) Estrada Dona Castorina, 110 Jardim Botânico, CEP 22460-320 Rio de Janeiro, RJ (Brasil)
4 Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva Str. Moscow, (Russia) 
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Ornea, Liviu; Verbitsky, Misha. Mall bundles and flat connections. Annales de l'Institut Fourier, Online first, 28 p.

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