Submanifolds of generalized complex manifolds

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds. For generalized complex manifolds that arise from complex (resp., symplectic) manifolds, all three classes specialize to complex (resp., symplectic) submanifolds. In general, however, all three classes are distinct. We discuss some interesting features of our theory of submanifolds, and illustrate them with a few nontrivial examples. We then support our ``symplectic/Lagrangian viewpoint'' on the submanifolds introduced by Gualtieri and Hitchin by defining the ``generalized complex category'', modelled on the constructions of Guillemin-Sternberg and Weinstein. We argue that our approach may be useful for the quantization of generalized complex manifolds.