2025/12/11 by Vicente Cortés, Cortés, Vicente, Liana David +1
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2512.10482
Generalised almost complex structures \mathcal J on transitive Courant algebroids E are studied in terms of their components with respect to a splitting E≅ TM ⊕ T^*M ⊕ \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 (ω, ρ) consisting of a symplectic structure ω on M and a representation ρ: π1(M) → Aut(\mathfrak g, ⟨ ⋅ ,⋅ ⟩_\mathfrakg, J_\mathfrakg) by automorphism of a quadratic Lie algebra (\mathfrak g, ⟨ ⋅ ,⋅ ⟩_\mathfrakg) commuting with an integrable (in the sense of Lie algebras) skew-symmetric complex structure J_\mathfrakg. 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.