Integration of generalized complex structures

TL;DR

This paper develops a method to integrate generalized complex manifolds into weakly holomorphic symplectic groupoids, linking their integrability to the integrability of underlying Poisson structures, and introduces new technical tools for Lie groupoid actions.

Contribution

It introduces a novel integration approach for generalized complex manifolds using weakly holomorphic symplectic groupoids and develops new tools for Lie groupoid actions on Courant algebroids.

Findings

Generalized complex manifolds are integrable if and only if their underlying Poisson structures are integrable.

The paper constructs weakly holomorphic symplectic groupoids as integrations.

New reduction procedures for Lie groupoid actions on Courant algebroids are developed.

Abstract

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.