Poisson manifolds of compact types (PMCT 1) (Q2332784)

The authors has announced a series of papers concerning the Poisson manifolds of compact type (PMCTs). Some important classes of examples are discussed and some of the main properties are explored, namely: (i) Poisson cohomology admits a Hodge decomposition and Poincaré duality holds. (ii) There are natural operations such as fusion product, Hamiltonian quotients, gauge equivalence, etc., which preserve the PMCT nature. (iii) Leaves are embedded submanifolds and have finite holonomy. (iv) There exist local linear models around leaves. The above properties belong to a larger set of properties that ``distinguish PMCTs from general Poisson manifolds, showing that they are very rigid objects and placing them in a singular position in Poisson geometry. These properties reflect the deep connections of the theory of PMCTs with other subjects.'' In Section 2 a background on Poisson structures is given. In Section 3 all six classes of PMCTs are defined. In order to enlarge the list of fundamental examples the more general setting of Dirac geometry is mentioned. In Section 4 many basic examples of PMCTs that reveal some of their remarkable properties are presented. In Section 5 general constructions of PMCTs are pointed out. Section 6 deals with Poisson homotopy groups and strong compactness. In Section 7 some basic Poisson topological properties of the symplectic foliation of PMCTs as well as their Poisson cohomology, the Hodge decomposition, and Poincaré duality are discussed. In Section 8 the existence of a local linear model for PMCTs around leaves is proved. This model can be seen as a version of Moser's stability at the groupoid level.