Components of generalised complex structures on transitive Courant algebroids

Abstract

Generalised almost complex structures $\mathcal J$ on transitive Courant algebroids $E$ are studied in terms of their components with respect to a splitting $E\cong TM \oplus T^*M \oplus \mathcal G$, where $M$ denotes the base of $E$ and $\mathcal G$ its bundle of quadratic Lie algebras. Necessary and sufficient integrability equations for $\mathcal J$ are established in this formalism. As an application, it is shown that the integrability of $\mathcal J$ implies that one of the 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 $(\omega , \rho )$ consisting of a symplectic structure $\omega$ on $M$ and a representation $\rho : \pi_1(M) \to \mathrm{Aut}(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}}, J_{\mathfrak{g}})$ by automorphism of a quadratic Lie algebra $(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}})$ commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure $J_{\mathfrak{g}}$. 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.