Using the AKSZ prescription we construct 2D and 3D topological field theories associated to generalized complex manifolds. These models can be thought of as 2D and 3D generalizations of A-and B-models. Within the BV framework we show that the 3D model on a two-manifold cross an interval can be reduced to the 2D model.

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized K\"ahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define \emph{extended} actions and a generalized notion of moment map. Key examples of generalized K\"ahler reduced spaces include new explicit bi-Hermitian metrics on $\CC P^2$.Comment: 34 pages. Presentation greatly improved, one subsection added, errors corrected, references added. v3: a few changes in the presentation, material slightly reorganized, final version to appear in Adv. in Mat

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 nonholomorphic 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 noncommutative algebraic geometry.

Generalized Kähler geometry is the natural analogue of Kähler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.A striking feature of the Courant bracket is that it has symmetries fixing the underlying space M. Any closed 2-form B ∈ Ω 2,cl (M) acts on TM, preserving the Courant bracket, via the bundle map:(1.2)Because of this symmetry, we may "twist" or modify the global structure of TM as an orthogonal bundle with a Courant bracket, keeping its local structure unchanged. These twisted structures are therefore classified by a characteristic class in H 1 (Ω 2,cl (M)) and are called exact Courant algebroids [13,18,19].

We show that all 6-dimensional nilmanifolds admit generalized complex structures. This includes the five classes of nilmanifold which admit no known complex or symplectic structure. Furthermore, we classify all 6-dimensional nilmanifolds according to which of the four types of left-invariant generalized complex structure they admit. We also show that the two components of the left-invariant complex moduli space for the Iwasawa manifold are no longer disjoint when they are viewed in the generalized complex moduli space. Finally, we provide an 8-dimensional nilmanifold which admits no left-invariant generalized complex structure.