Components of generalised complex structures on transitive Courant algebroids

Abstract

Abstract Generalised almost complex structures $$\mathcal {J}$$ J on transitive Courant algebroids E are studied in terms of their components with respect to a splitting $$E\cong TM \oplus T^*M \oplus \mathcal {G}$$ E ≅ T M ⊕ T ∗ M ⊕ G , where M denotes the base of E and $$\mathcal {G}$$ G its bundle of quadratic Lie algebras. Necessary and sufficient integrability equations for $$\mathcal {J}$$ J are established in this formalism. As an application, it is shown that the integrability of $$\mathcal {J}$$ 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 $$(\omega , \rho )$$ ( ω , ρ ) consisting of a symplectic structure $$\omega $$ ω on M and a representation $$\rho : \pi _1(M) \rightarrow \textrm{Aut}(\mathfrak g, \langle \cdot ,\cdot \rangle _{\mathfrak {g}}, J_{\mathfrak {g}})$$ ρ : π 1 ( M ) → Aut ( g , ⟨ · , · ⟩ g , J g ) by automorphisms of a quadratic Lie algebra $$(\mathfrak g, \langle \cdot ,\cdot \rangle _{\mathfrak {g}})$$ ( g , ⟨ · , · ⟩ g ) commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure $$J_{\mathfrak {g}}$$ 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.

The authors' abstract, as published at the source. Annals of Global Analysis and Geometry, 2026 · DOI ↗