Moser Lemma in Generalized Complex Geometry

Details Moser Lemma in Generalized Complex Geometry Moser Lemma in Generalized Complex Geometry

2007-02-23

arxiv.org/abs/math/0702718v1

Trans. Amer. Math. Soc. 362 (2010), no. 10, 5107-5123

Mathematics

Differential Geometry

18 pages

Scientific paper

10.1090/S0002-9947-10-04965-2

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 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 cases of symplectic, 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.

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