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

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.

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.

• Thank you. The first point about Lie algebroid cohomology is closely related to de rham or Dolbeault is believable but I did not see that before. Can you suggest some reference. The second point about generalised holomorphic bundle is out of my coverage area. I have absolutely no idea.. Can you point me to some reference. Jul 12, 2021 at 17:23
• Both of these facts are discussed in Section 3 of Marco Gualtieri. Generalized complex geometry. Ann. of Math. (2), 174(1):75–123, 2011. If you want more detail then I would recommend his PhD thesis. Jul 12, 2021 at 17:35
• I started reading his thesis. I did not reach there yetThank you:) Jul 13, 2021 at 1:14

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 $$M$$.