Components of generalised complex structures on transitive Courant algebroids

Generalised almost complex structures J J on transitive Courant algebroids E are studied in terms of their components with respect to a splitting E TM ⊕ T^*M ⊕ G E ≅ T M ⊕ T ∗ M ⊕ G , where M denotes the base of E and G G its bundle of quadratic Lie algebras. Necessary and sufficient integrability equations for J J are established in this formalism. As an application, it is shown that the integrability of J J implies that one of its components defines a Poisson structure on M . Then the structure (normal form) of generalised complex structures for which the Poisson structure is non-degenerate is determined. It is shown that it is fully encoded in a pair (ω , ρ ) ( ω , ρ ) consisting of a symplectic structure ω ω on M and a representation ρ : π ₁(M) → Aut( g, · ,· g, Jg) ρ : π 1 ( M ) → Aut ( g , ⟨ · , · ⟩ g , J g ) by automorphisms of a quadratic Lie algebra ( g, · ,· g) ( g , ⟨ · , · ⟩ g ) commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure Jg J g . Examples of such representations and obstructions for the existence of non-degenerate generalised complex structures are discussed. Finally, a construction of generalised complex structures on transitive Courant algebroids over complex manifolds for which the Poisson structure degenerates along a complex analytic hypersurface is presented.