Generalized geometry of Norden manifolds

Let (M,J,g,D) be a Norden manifold with the natural canonical connection D and let Ĵ be the generalized complex structure on M defined by g and J. We prove that Ĵ is D-integrable and we find conditions on the curvature of D under which the ±i-eigenbundles of Ĵ, EĴ1,0, EĴ0,1, are complex Lie algebroids. Moreover we proove that EĴ1,0 and (EĴ1,0)∗ are canonically isomorphic and this allow us to define the concept of generalized ∂¯Ĵ-operator of  (M,J,g,D). Also we describe some generalized holomorphic sections. The class of Kähler-Norden manifolds plays an important role in this paper because for these manifolds EĴ1,0 and EĴ0,1 are complex Lie algebroids.