Moser Lemma in Generalized Complex Geometry

TL;DR

This paper extends Moser's lemma from symplectic geometry to generalized complex structures on Courant algebroids, providing new tools for understanding their deformations and local triviality.

Contribution

It generalizes Moser's lemma to GCS on Courant algebroids, introduces a cohomological framework, and offers a criterion for local triviality of GCS.

Findings

Extended Lie derivative to Courant algebroid sections.

Provided a cohomological interpretation of GCS deformations.

Established a criterion for local triviality generalizing Darboux-Weinstein theorem.

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 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.