The modular class of a Poisson map

We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.Comment: Final version accepted for publication in Annales de l'Institut Fourier. Several changes made to the manuscript, based on referees' remark

Integrability of Lie brackets

In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.Comment: 46 pages, published versio

Integrability of Poisson brackets

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas. As an application of these results, we show that a Poisson manifold admits a complete symplectic realization if, and only if, it is integrable. We discuss also the integration of submanifolds and Morita equivalence of Poisson manifolds.Comment: 43 pages, 1 figur