AbstractThis is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses
several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental
properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the
de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.)
and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs
are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial
PMCT related to a classical question on the topology of the orbits of a free symplectic circle action.
In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry,
Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.
De Rham Cohomology and Hodge Decomposition For Quantum Groups