Non locally trivializable CR line bundles over compact Lorentzian CR manifolds
[Fibrés en droites CR non localement triviaux sur des variétés CR Lorentziennes compactes]
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 101-108.

On considère une variété CR compacte de codimension CR quelconque qui vérifie certaines conditions géométriques en terme de sa forme de Levi. Sur ces variétés CR compactes, on construit une déformation du fibré en droites CR trivial sur M qui est topologiquement trivial sur M mais qui n’admet même pas de trivialization CR locale sur un ouvert arbitraire de M. En particulier, nos résultats s’appliquent au cas de variétés CR compactes Lorentziennes du type hypersurface.

We consider compact CR manifolds of arbitrary CR codimension that satisfy certain geometric conditions in terms of their Levi form. Over these compact CR manifolds, we construct a deformation of the trivial CR line bundle over M which is topologically trivial over M but fails to be even locally CR trivializable over any open subset of M. In particular, our results apply to compact Lorentzian CR manifolds of hypersurface type.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3152
Classification : 32V05,  32G07
Mots clés : fibrés vectoriel CR, repère local, variétés CR Lorentziennes
@article{AIF_2018__68_1_101_0,
     author = {Brinkschulte, Judith and Hill, C. Denson},
     title = {Non locally trivializable $CR$ line bundles over compact Lorentzian $CR$ manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {101--108},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3152},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3152/}
}
Brinkschulte, Judith; Hill, C. Denson. Non locally trivializable $CR$ line bundles over compact Lorentzian $CR$ manifolds. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 101-108. doi : 10.5802/aif.3152. http://archive.numdam.org/articles/10.5802/aif.3152/

[1] Andreotti, Aldo; Fredricks, Gregory; Nacinovich, Mauro On the absence of Poincaré lemma in tangential Cauchy-Riemann complexes, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 8 (1981), pp. 365-404 | Zbl 0482.35061

[2] Brinkschulte, Judith; Hill, C. Denson Obstructions to finite dimensional cohomology of abstract Cauchy-Riemann complexes, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 15 (2016), pp. 343-354 | Zbl 1337.32047

[3] Brinkschulte, Judith; Hill, C. Denson; Nacinovich, Mauro On the nonvanishing of abstract Cauchy–Riemann cohomology groups, Math. Ann., Volume 363 (2016) no. 3-4, pp. 1701-1715 | Article | Zbl 06618545

[4] Folland, Gerald B.; Kohn, Joseph J. The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, 57, Princeton University Press, 1972, viii+146 pages | Zbl 0247.35093

[5] Hill, C. Denson Counterexamples to Newlander-Nirenberg up to the boundary, Several complex variables and complex geometry (Proceedings of Symposia in Pure Mathematics), Volume 52, American Mathematical Society, 1991, pp. 191-197 | Zbl 0751.53012

[6] Hill, C. Denson; Nacinovich, Mauro Pseudoconcave CR manifolds, Complex analysis and geometry (Lecture Notes in Pure and Applied Mathematics), Volume 173 (1996), pp. 275-297 | Zbl 0921.32004

[7] Hill, C. Denson; Nacinovich, Mauro A weak pseudoconcavity condition for abstract almost CR manifolds, Invent. Math., Volume 142 (2000) no. 2, pp. 251-283 | Article | Zbl 0973.32018

[8] Webster, Sidney M. The integrability problem for CR vector bundles, Several complex variables and complex geometry (Proceedings of Symposia in Pure Mathematics), Volume 52, American Mathematical Society, 1991, pp. 355-368 | Zbl 0744.32002