Components of generalised complex structures on transitive Courant algebroids

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 $(ω, ρ)$ consisting of a symplectic structure $ω$ on $M$ and a representation $ρ: π_1(M) \to \mathrm{Aut}(\mathfrak g, \langle \cdot ,\cdot \rangle_{\mathfrak{g}}, J_{\mathfrak{g}})$ 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.

💡 Research Summary

This paper conducts a comprehensive study of generalized almost complex structures (GACS) and generalized complex structures (GCS) on transitive Courant algebroids. A transitive Courant algebroid E over a manifold M incorporates an additional bundle of quadratic Lie algebras G, leading to a splitting E ≅ TM ⊕ T*M ⊕ G, which generalizes the exact Courant algebroid (where G=0) central to Hitchin’s original formulation.

The authors’ primary methodology involves decomposing a GACS J into six tensor component fields relative to this splitting: J: TM→TM, A: G→G, B: TM→TM, C: TM→TM, μ: TM→G, ν: T*M→G. Lemma 2 establishes the algebraic relations these components must satisfy for J to be an almost complex structure (J² = -Id). The core technical achievement is Theorem 28, which provides the full set of necessary and sufficient integrability conditions (Nijenhuis tensor = 0) for J as a system of partial differential equations involving these components and the defining data (∇, R, H) of the standard Courant algebroid.

A significant and elegant consequence (Proposition 3) derived from just one of these integrability equations is that for any integrable GCS, the component B is always a Poisson bivector on M, generalizing a known result from the exact case.

The paper then focuses on the “non-degenerate” case where B is invertible. Using a dictionary (Theorem 4) to translate between the component formalism and an alternative data set (W, D, σ, ε) describing the (1,0)-bundle, the authors achieve a complete structure theorem. Theorem 12 shows that, up to isomorphism, the underlying Courant algebroid must be “untwisted” (R=0, H=0, so ∇ is flat), and the GCS takes the form J = J_ω ⊕ A. Here, J_ω is the standard GCS associated to the symplectic structure ω = -B⁻¹ on TM ⊕ T*M, and A is a field of skew-symmetric complex structures on the bundle G that is parallel with respect to the flat connection ∇.

This leads to the main classification result in Theorem 14. There is a natural bijection between isomorphism classes of non-degenerate GCS with symplectic structure ω on transitive Courant algebroids with fiber type (𝔤, ⟨·,·⟩), and isomorphism classes of pairs (J_𝔤, ρ). Here, J_𝔤 is an integrable (in the Lie algebra sense) skew-symmetric complex structure on the quadratic Lie algebra 𝔤, and ρ: π₁(M) → Aut(𝔤, ⟨·,·⟩, J_𝔤) is a representation by J_𝔤-linear automorphisms. This result deeply ties the existence of such geometric structures to topological data (representations of the fundamental group).

The theory is illustrated with examples of both heterotic and non-heterotic type, and obstructions are discussed, showing that not every quadratic Lie algebra (e.g., 𝔰𝔬(3)) can serve as the fiber type for a bundle supporting a non-degenerate GCS. Finally, the paper extends beyond the non-degenerate regime by presenting a construction (Proposition 25) of GCS on transitive Courant algebroids over complex manifolds for which the Poisson structure degenerates along a complex analytic hypersurface, demonstrating the breadth and applicability of the component-based analysis.