Darboux theorem for generalized complex structures on transitive Courant algebroids

TL;DR

This paper extends the Darboux theorem to generalized complex structures on transitive Courant algebroids, providing a local normal form under certain conditions, generalizing previous results on manifolds and Lie groups.

Contribution

It establishes a Darboux-type theorem for generalized complex structures on transitive Courant algebroids, broadening the scope of local normal form results.

Findings

Derived local form of generalized complex structures around regular points

Extended Gualtieri's Darboux theorem to Courant algebroids

Connected the structure to invariant complex structures on Lie groups

Abstract

Let E be a transitive Courant algebroid with scalar product of neutral signature. A generalized almost complex structure \mathcal J on E is a skew-symmetric smooth field of endomorphisms of E which squares to minus the identity. We say that \mathcal J is integrable (or is a generalized complex structure) if the space of sections of its (1,0) bundle is closed under the Dorfman bracket of E. In this paper we determine, under certain natural conditions, the local form of \mathcal J around regular points. This result is analogous to Gualtieri's Darboux theorem for generalized complex structures on manifolds and extends Wang's description of skew-symmetric left-invariant complex structures on compact semisimple Lie groups.