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arXiv:2105.15019 [math.DG]AbstractReferencesReviewsResources

The Standard cohomology of regular Courant algebroids

Xiongwei Cai, Zhuo Chen, Maosong Xiang

Published 2021-05-31Version 1

For any Courant algebroid $E$ over a smooth manifold $M$ with characteristic distribution $F$ which is regular, we study the standard cohomology $H^\bullet_{\operatorname{st}}(E)$ by using a special spectral sequence. We prove a theorem which tells how a natural transgression map $[\mathcal{T}]$ together with the Chevalley-Eilenberg cohomology of the ample Lie algebroid $A_E$ of $E$ with coefficient in the symmetric tensor product ${S(TM/F)}$ of the normal bundle $TM/F$ determines $H^\bullet_{\operatorname{st}}(E)$, thereby significantly reducing the range of generators of the standard cohomology. We can also recover an earlier result by Ginot and Grutzmann (J.Symplectic Geom.7(2009), no.3, 311-335) in the situation that the base manifold $M$ splits. In addition, we apply the theorem to two special types of regular Courant algebroids: generalized exact ones and those arising from regular Lie algebroids.

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