Paper detail

Integration of generalized complex structures

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a weakly holomorphic symplectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only on the associated stack, i.e., 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. Crucial to our solution are several 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. Finally, we implement our generalized complex integration procedure in several concrete examples.