Submanifolds of generalized complex manifolds

Abstract

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 in \cite{Gua}, \cite{H3} (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds and their relationships. 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. Along the way, we obtain a complete and explicit classification of all linear generalized complex structures. We then support our ``symplectic/Lagrangian viewpoint'' on the submanifolds introduced in \cite{Gua}, \cite{H3} by defining the ``generalized complex category'', modelled on the constructions of Guillemin-Sternberg \cite{GS} and Weinstein \cite{Wei2}. We argue that our approach may be useful for the quantization of generalized complex manifolds.