Darboux theorem for generalized complex structures on transitive Courant algebroids

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.