Generalized Moser lemma

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 (⊗ⁱE) ⊗ (⊗^jE^*) 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.