Mall bundles and flat connections
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.

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.

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.

Received:
Revised:
Accepted:
Online First:
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) 
@unpublished{AIF_0__0_0_A104_0,
     author = {Ornea, Liviu and Verbitsky, Misha},
     title = {Mall bundles and flat connections},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3647},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Ornea, Liviu
AU  - Verbitsky, Misha
TI  - Mall bundles and flat connections
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3647
LA  - en
ID  - AIF_0__0_0_A104_0
ER  - 
%0 Unpublished Work
%A Ornea, Liviu
%A Verbitsky, Misha
%T Mall bundles and flat connections
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3647
%G en
%F AIF_0__0_0_A104_0
Ornea, Liviu; Verbitsky, Misha. Mall bundles and flat connections. Annales de l'Institut Fourier, Online first, 28 p.

[1] Abels, Herbert Properly discontinuous groups of affine transformations: a survey, Geom. Dedicata, Volume 87 (2001) no. 1-3, pp. 309-333 | DOI | MR | Zbl

[2] Arnolʼd, Vladimir I. Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, 250, Springer, 1988, xiv+351 pages | DOI | MR | Zbl

[3] Atiyah, Michael F. Complex analytic connections in fibre bundles, Trans. Am. Math. Soc., Volume 85 (1957), pp. 181-207 | DOI | MR | Zbl

[4] Auslander, Louis; Markus, Lawrence Holonomy of flat affinely connected manifolds, Ann. Math., Volume 62 (1955), pp. 139-151 | DOI | MR | Zbl

[5] Biswas, Indranil Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree, Proc. Am. Math. Soc., Volume 126 (1998) no. 10, pp. 2827-2834 | DOI | MR | Zbl

[6] Biswas, Indranil; Dumitrescu, Sorin Holomorphic affine connections on non-Kähler manifolds, Int. J. Math., Volume 27 (2016) no. 11, 1650094, 14 pages | DOI | MR | Zbl

[7] Biswas, Indranil; Dumitrescu, Sorin Holomorphic Riemannian metric and the fundamental group, Bull. Soc. Math. Fr., Volume 147 (2019) no. 3, pp. 455-468 | DOI | MR | Zbl

[8] Buchdahl, Nicholas P.; Harris, Adam Holomorphic connections and extension of complex vector bundles, Math. Nachr., Volume 204 (1999), pp. 29-39 | DOI | MR | Zbl

[9] Dolbeault cohomology of Hopf manifolds, https://mathoverflow.net/questions/25723/dolbeault-cohomology-of-hopf-manifolds

[10] Dulac, Henri Recherches sur les points singuliers des équations différentielles, J. de l’Éc. Pol., Volume 9 (1904) no. 2, pp. 5-125 | Zbl

[11] Forstnerič, Franc Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 56, Springer, 2011, xii+489 pages | DOI | MR | Zbl

[12] Friedman, Avner Foundations of modern analysis, Dover Publications, 2010

[13] Gan, Ning; Zhou, Xiang-Yu The cohomology of vector bundles on general non-primary Hopf manifolds, Recent progress on some problems in several complex variables and partial differential equations (Contemporary Mathematics), Volume 400, American Mathematical Society, 2006, pp. 107-115 | DOI | MR | Zbl

[14] Goldman, William Geometric Structures on Manifolds, AMS Open Math. Notes, 2021 (https://www.ams.org/open-math-notes/omn-view-listing?listingId=111282)

[15] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience, 1978, xii+813 pages | MR | Zbl

[16] Grothendieck, Alexander Sur quelques points d’algèbre homologique, Tôhoku Math. J., Volume 9 (1957), pp. 119-221 (English translation: http://www.math.mcgill.ca/barr/papers/gk.pdf) | DOI | MR | Zbl

[17] Gunning, Robert C.; Rossi, Hugo Analytic functions of several complex variables, AMS Chelsea Publishing, 2009, xiv+318 pages | DOI | MR | Zbl

[18] Haefliger, André Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds, Compos. Math., Volume 55 (1985) no. 2, pp. 241-251 | Numdam | MR | Zbl

[19] Ise, Mikio On the geometry of Hopf manifolds, Osaka J. Math., Volume 12 (1960), pp. 387-402 | MR | Zbl

[20] Kobayashi, Shoshichi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, 1987, xii+305 pages (Kanô Memorial Lectures, 5) | DOI | MR | Zbl

[21] Kodaira, Kunihiko On the structure of compact complex analytic surfaces. II, Am. J. Math., Volume 88 (1966), pp. 682-721 | DOI | MR | Zbl

[22] Kodaira, Kunihiko On the structure of compact complex analytic surfaces. III, Am. J. Math., Volume 90 (1968), pp. 55-83 | DOI | MR

[23] Koszul, Jean-Louis; Malgrange, Bernard Sur certaines structures fibrées complexes, Arch. Math., Volume 9 (1958), pp. 102-109 | DOI | MR | Zbl

[24] Lattès, Samuel Sur les formes réduites des transformations ponctuelles dans le domaine d’un point double, Bull. Soc. Math. Fr., Volume 39 (1911), pp. 309-345 | DOI | Numdam | MR | Zbl

[25] Libgober, Anatoly Cohomology of bundles on homological Hopf manifolds, Sci. China, Ser. A, Volume 52 (2009) no. 12, pp. 2688-2698 | DOI | MR | Zbl

[26] MacLane, Saunders Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer, 1971, ix+262 pages | MR | Zbl

[27] Mall, Daniel The cohomology of line bundles on Hopf manifolds, Osaka J. Math., Volume 28 (1991) no. 4, pp. 999-1015 | MR | Zbl

[28] Mall, Daniel Contractions, Fredholm operators and the cohomology of vector bundles on Hopf manifolds, Arch. Math., Volume 66 (1996) no. 1, pp. 71-76 | DOI | MR | Zbl

[29] Okonek, Christian; Schneider, Michael; Spindler, Heinz Vector bundles on complex projective spaces, Modern Birkhäuser Classics, Birkhäuser/Springer, 2011, viii+239 pages (corrected reprint of the 1988 edition, with an appendix by S. I. Gelfand) | DOI | MR | Zbl

[30] Ornea, Liviu; Verbitsky, Misha Embedding of LCK manifolds with potential into Hopf manifolds using Riesz–Schauder theorem, Complex and symplectic geometry (Springer INdAM Series), Volume 21, Springer, 2017, pp. 137-148 | DOI | MR | Zbl

[31] Ornea, Liviu; Verbitsky, Misha Non-linear Hopf manifolds are locally conformally Kähler, J. Geom. Anal., Volume 33 (2023) no. 7, 201, 10 pages | DOI | MR | Zbl

[32] Poincaré-Dulac theorem, Encyclopedia of Mathematics (https://encyclopediaofmath.org/wiki/Poincare-Dulac_theorem)

[33] Ramani, Vimala; Sankaran, Parameswaran Dolbeault cohomology of compact complex homogeneous manifolds, Proc. Indian Acad. Sci., Math. Sci., Volume 109 (1999) no. 1, pp. 11-21 | DOI | MR | Zbl

[34] Shima, Hirohiko The geometry of Hessian structures, World Scientific, 2007, xiv+246 pages | DOI | MR | Zbl

[35] Siu, Yum-tong Extension of locally free analytic sheaves, Math. Ann., Volume 179 (1969), pp. 285-294 | DOI | MR | Zbl

[36] Sternberg, Shlomo Local contractions and a theorem of Poincaré, Am. J. Math., Volume 79 (1957), pp. 809-824 | DOI | Zbl

[37] Wu, Hung Normal families of holomorphic mappings, Acta Math., Volume 119 (1967), pp. 193-233 | DOI | MR | Zbl

Cited by Sources: