Components of generalised complex structures on transitive Courant algebroids

TL;DR

This paper analyzes the components and integrability conditions of generalized complex structures on transitive Courant algebroids, revealing their relation to Poisson and symplectic structures and providing explicit normal forms and examples.

Contribution

It establishes integrability equations for generalized complex structures on transitive Courant algebroids and characterizes structures with non-degenerate Poisson components, including explicit normal forms.

Findings

Integrability of structures implies a Poisson structure on the base manifold.

Normal form characterized by a symplectic structure and a quadratic Lie algebra representation.

Construction of structures over complex manifolds with degenerating Poisson structures.

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.