Use of theory of Lie algebroids in (better) understanding of generalised complex structures

Question

Let $M$ be a smooth manifold. A Lie algebroid over $M$ is a vector bundle $E\rightarrow M$ over $M$, with a Lie bracket on $\Gamma(M,E)$, a morphism of vector bundles $\rho:E\rightarrow TM$, such that, the following conditions are satisfied:

1. the map $\rho:E\rightarrow TM$ induce a morphism of Lie algebras $\Gamma(M,E)\rightarrow \Gamma(M,TM)$,
2. the Lie algebra structure on $\Gamma(M,E)$ is “compatible” with the $C^\infty(M)$-algebra structure on $\Gamma(M,E)$, upto a correction; this goes by the name Leibniz condition.

A generalized complex structure on a manifold $M$ is a morphism of vector bundles $J:TM\oplus TM^*\rightarrow TM\oplus TM^*$ such that it is compatible with some bracket operation and some ''inner product'' on $\Gamma(M,TM\oplus TM^*)$.

In most of the references about generalized complex structures, they introduce the notion of Lie algebroid. I could not see detailed justification of introducing Lie algebroid over $M$ when discussing generalized complex structure on $M$. So, I am thinking of the following question:

How is the theory of Lie algebroids useful in (better) understanding of generalized complex structures?

Any pointers are welcome. I have done google search for ''Lie algebroids and generalised complex structure" but could not find anything specific.

Answer 1

The compatibility conditions that you mention in the definition of a generalized complex structure are equivalent to the statement that the $+i$-eigenbundle $L$ of $J$ is a complex Dirac structure:

1. compatibility of $J$ with the inner product is equivalent to $L$ being 'isotropic', and
2. compatibility of $J$ with the Courant bracket is equivalent to the sections of $L$ being closed under this bracket.

A Dirac structure automatically inherits the structure of a Lie algebroid over $M$. So $L$ in this case is naturally a Lie algebroid. Many of the properties and constructions for $J$ can be stated in terms of the Lie algebroid $L$. For example:

1. The Lie algebroid cohomology of $L$ is a fundamental invariant of $J$. When $J$ comes from a symplectic form, it is the usual de Rham cohomology, whereas when $J$ comes from a complex structure, it is a sum of Dolbeault cohomology groups.
2. A generalized holomorphic bundle for $J$ can be defined to be a representation of $L$. When $J$ comes from a symplectic form, this is a bundle with flat connection, and when $J$ comes from a complex structure, this is a co-higgs bundle.

Note that the Lie algebroid structure on $L$ is somewhat less information than $J$. On the other hand, if you view $L$ as a Dirac structure in $(TM \oplus T^{*}M)\otimes \mathbb{C}$, then it is equivalent data to $J$. So it's often better to work with it in this way, and many constructions are very naturally stated using the formalism of Dirac geometry.

Answer 2

In the introduction chapter of Marco Gualtieri's thesis he says the following:

--- describe and study the Courant bracket, which, while It is not a Lie bracket, does restrict, on involute maximal isotropic-sub bundles, to be a Lie bracket, and thus endows the bundle L with the structure of a Lie algebroid.

I could not find a result that says exactly this, but I think above observation follows from Proposition 3.27 in the thesis which says that for a maximal isotropic sub-bundle $L$ of $TM\oplus T^*M$ (or its complexification) the conditions $L$ being involutive is equivalent to the condition $\text{Jac}|_L=0$.

So, if we start with a generalised complex structure, the Jacobiator restricted $L$ vanishes. So, the Courant bracket satisfies the Jacobi identity.

Thus, the map $L\subseteq TM\oplus T^*M\rightarrow TM$ is a Lie algebroid over the manifold $M$.

So, every generalised complex structure on $M$ is a Lie algebroid over $M$.