Components of generalised complex structures on transitive Courant algebroids

Vicente Cortés; Liana David

Paper

Components of generalised complex structures on transitive Courant algebroids

Abstract

Generalised almost complex structures

on transitive Courant algebroids

are studied in terms of their components with respect to a splitting

, where

denotes the base of

and

its bundle of quadratic Lie algebras. Necessary and sufficient integrability equations for

are established in this formalism. As an application, it is shown that the integrability of

implies that one of the components defines a Poisson structure on

. 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

consisting of a symplectic structure

and a representation

by automorphism of a quadratic Lie algebra

commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure

. 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.