Comptes Rendus
no. 7-8
Geometry
On almost complex structures which are not compatible with symplectic forms
[Sur les structures presque complexes qui ne sont pas compatibles avec des formes symplectiques]

Présenté par : Jean-Pierre Demailly

Luigi Vezzoni1
1 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 429-431.

Dans cette Note on démontre que la structure presque complexe sous-jacente à une structure presque hermitienne non kälérienne admettant une connexion compatible avec une torsion antisymétrique ne peut pas, même localement, être calibrée par une forme symplectique.

In this Note we prove that the underlying almost complex structure to a non-Kähler almost Hermitian structure admitting a compatible connection with skew-symmetric torsion cannot be calibrated by a symplectic form even locally.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.01.002

Luigi Vezzoni 1

1 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy 
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Luigi Vezzoni. On almost complex structures which are not compatible with symplectic forms. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 429-431. doi : 10.1016/j.crma.2011.01.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.002/

[1] R.L. Bryant On the geometry of almost complex 6-manifolds, Asian J. Math., Volume 10 (2006) no. 3, pp. 561-605

[2] R.L. Bryant Submanifolds and special structures on the octonions, J. Differential Geom., Volume 17 (1982) no. 2, pp. 185-232

[3] Th. Friedrich; S. Ivanov Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math., Volume 6 (2002), pp. 303-335

[4] M. Lejmi Strictly nearly Kähler 6-manifolds are not compatible with symplectic forms, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006) no. 11–12, pp. 759-762

[5] A. Newlander; L. Nirenberg Complex analytic coordinates in almost complex manifolds, Ann. Math. (2), Volume 65 (1957), pp. 391-404

[6] T. Rivière; G. Tian The singular set of J-holomorphic maps into projective algebraic varieties, J. Reine Angew. Math., Volume 570 (2004), pp. 47-87

[7] T. Rivière; G. Tian The singular set of 1−1 integral currents, Ann. Math. (2), Volume 169 (2009) no. 3, pp. 741-794

[8] J.-C. Sikorav Holomorphic Curves in Symplectic Geometry (F. Audin et al., eds.), Progress in Math., vol. 117, Birkhäuser, 1994

[9] A. Tomassini Some examples of non calibrable almost complex structures, Forum Math., Volume 14 (2002), pp. 869-876

Cité par Sources :

☆ This work was supported by the Project M.I.U.R. “Riemannian Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M.

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