The local structure of generalized complex branes

We show (modulo a parity condition) that, a generalized complex brane in a generalized complex manifold is locally equivalent to a holomorphic coisotropic submanifold of a holomorphic Poisson structure, with higher-rank branes corresponding to holomorphic Poisson modules. We describe (but do not prove here) the global version of this holomorphicity result. Finally, we use the "local holomorphic gauges" to give examples, in the Hopf surface with nonstandard generalized complex structure, of branes which are neither Lagrangian nor complex.