Integration of generalized complex structures

Abstract

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.