Branes on Poisson varieties

We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kähler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is non-holomorphic in nature. Finally we show an equivalence between certain configurations of branes on Poisson varieties and generalized Kähler structures, and use this to construct explicitly new families of generalized Kähler structures on compact holomorphic Poisson manifolds equipped with positive Poisson line bundles (e.g. Fano manifolds). We end with some speculations concerning the connection to non-commutative algebraic geometry.