Poisson Manifolds of Compact Types (PMCT 1)

TL;DR

This paper introduces Poisson manifolds of compact types (PMCTs), showing they share key properties with compact symplectic manifolds and revealing unexpected links to symplectic topology and other geometric theories.

Contribution

It establishes fundamental properties of PMCTs, demonstrating their analogy with compact symplectic manifolds and connecting them to symplectic topology and Lie theory.

Findings

PMCTs exhibit Poisson cohomology similar to de Rham cohomology of compact symplectic manifolds

The Moser trick can be adapted to PMCTs

Construction of a nontrivial PMCT related to symplectic circle actions

Abstract

This 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 of PMCTs, which already show that they are the analogues of compact symplectic manifolds, thus placing them in a prominent position among all Poisson manifolds. For instance, their Poisson cohomology behaves very much like the de Rham cohomology of compact symplectic manifolds (Hodge decomposition, non-degenerate Poincar\'e duality pairing, etc.) and 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 nontrivial 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, Lie theory and symplectic gerbes.