Reduction of Courant algebroids and generalized complex structures

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 $\text{CC}{P}^2$.