Generalized Moser lemma

Stiénon, Mathieu

doi:10.1090/S0002-9947-10-04965-2

Generalized Moser lemma
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Trans. Amer. Math. Soc. 362 (2010), 5107-5123 Request permission

Abstract:

We show how the classical Moser lemma from symplectic geometry extends to generalized complex structures (GCS) on arbitrary Courant algebroids. For this, we extend the notion of a Lie derivative to sections of the tensor bundle $(\otimes ^i E)\otimes (\otimes ^j E^*)$ with respect to sections of the Courant algebroid $E$ using the Dorfman bracket. We then give a cohomological interpretation of the existence of one-parameter families of GCS on $E$ and of flows of automorphims of $E$ identifying all GCS of such a family. In the particular case of symplectic manifolds, we recover the results of Moser. Finally, we give a criterion to detect the local triviality of arbitrary GCS which generalizes the Darboux-Weinstein theorem.

References

Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
Vestislav Apostolov and Marco Gualtieri, Generalized Kähler manifolds, commuting complex structures, and split tangent bundles, Comm. Math. Phys. 271 (2007), no. 2, 561–575. MR 2287917, DOI 10.1007/s00220-007-0196-4
James Barton, Generalized complex structures on Courant algebroids, Ph.D. thesis, 2007.
Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri, Generalized Kähler and hyper-Kähler quotients, Poisson geometry in mathematics and physics, Contemp. Math., vol. 450, Amer. Math. Soc., Providence, RI, 2008, pp. 61–77. MR 2397619, DOI 10.1090/conm/450/08734
Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726–765. MR 2323543, DOI 10.1016/j.aim.2006.09.008
Ana Cannas da Silva and Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, vol. 10, American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999. MR 1747916
Gil R. Cavalcanti, Formality in generalized Kähler geometry, Topology Appl. 154 (2007), no. 6, 1119–1125. MR 2298627, DOI 10.1016/j.topol.2006.11.002
—, New aspects of the ddc-lemma. arXiv:math/0501406
Gil R. Cavalcanti and Marco Gualtieri, Generalized complex structures on nilmanifolds, J. Symplectic Geom. 2 (2004), no. 3, 393–410. MR 2131642, DOI 10.4310/JSG.2004.v2.n3.a5
Gil R. Cavalcanti and Marco Gualtieri, A surgery for generalized complex structures on 4-manifolds, J. Differential Geom. 76 (2007), no. 1, 35–43. MR 2312048
Theodore James Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661. MR 998124, DOI 10.1090/S0002-9947-1990-0998124-1
Marius Crainic, Generalized complex structures and Lie brackets. arXiv:math/0412097
Marco Gualtieri, Generalized complex geometry, DPhil thesis, 2003, arXiv:math/0401221
—, Generalized complex geometry. arXiv:math/0703298
—, Generalized geometry and the Hodge decomposition. arXiv:math/0409093
Nigel Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308. MR 2013140, DOI 10.1093/qjmath/54.3.281
Shengda Hu, Hamiltonian symmetries and reduction in generalized geometry, Houston J. Math. 35 (2009), no. 3, 787–811. MR 2534281
Shengda Hu, Reduction and duality in generalized geometry, J. Symplectic Geom. 5 (2007), no. 4, 439–473. MR 2413310
Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Reprint of the 1986 English edition, Classics in Mathematics, Springer-Verlag, Berlin, 2005. Translated from the 1981 Japanese original by Kazuo Akao. MR 2109686, DOI 10.1007/b138372
J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I, Ann. of Math. (2) 78 (1963), 112–148. MR 153030, DOI 10.2307/1970506
Camille Laurent-Gengoux, Mathieu Stiénon, and Ping Xu, Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 088, 46. MR 2439547, DOI 10.1093/imrn/rnn088
Zhang-Ju Liu, Alan Weinstein, and Ping Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), no. 3, 547–574. MR 1472888
Kirill C. H. Mackenzie and Ping Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), no. 2, 415–452. MR 1262213, DOI 10.1215/S0012-7094-94-07318-3
Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
Ryszard Nest and Boris Tsygan, Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems, Asian J. Math. 5 (2001), no. 4, 599–635. MR 1913813, DOI 10.4310/AJM.2001.v5.n4.a2
A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. MR 88770, DOI 10.2307/1970051
Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, Ph.D. thesis, 1999. arXiv:math/9910078
Pavol Ševera, Unpublished Letter to Alan Weinstein.
Alan Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), 329–346 (1971). MR 286137, DOI 10.1016/0001-8708(71)90020-X