Poisson manifolds of compact types (PMCT 1)

Abstract

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 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 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, orbifolds, Lie Theory and symplectic gerbes.

Resolutions of Proper Riemannian Lie Groupoids

“…Hence there are no codimension-one strata. Now Lemma 3.3 implies property (4). The final property is now easily realized by the denseness of S m .…”

“…In the general case of an bundle of abelian groups, Moerdijk in [15] associates to such extensions a cohomology class δ( G) ∈ H 2 ( M /E; K), where K is the sheaf of smooth sections of the bundle K. This class is invariant under Morita equivalence and therefore we conclude: In case the band of the gerbe K is trivial with fiber T, one can use the exponential sequence to get the cohomology class in H 3 (M/E, Z), the so-called Dixmier-Douady class. It would be interesting to study proper Lie groupoids whose desingularization is a central extension of a properétale groupoid, which is related to [4]. For nonabelian stabilizers, when the band of the gerbe is trivial, there is a natural class in H 2 ( M /E, Z(K)), where Z(K) is the center subgroup of K. It is an interesting open question what properties of the original groupoid G this cohomology class exactly measures.…”

Measures on differentiable stacks

“…We would also like to stress that the original motivation for this paper comes from the study of Poisson manifolds of compact types [8,9,10]; there, the space of symplectic leaves comes with two interesting measures: the integral affine and the Duistermaat-Heckman measures. Although the leaf space is an orbifold, the measures are of a "stacky" nature that goes beyond Haefliger's framework.…”

“…This case presents several particularities, due to the presence of symplectic/Poisson geometry. In particular, we recast the measures from [9]: Theorem 1.3. Any regular proper symplectic groupoid (G, Ω) over M carries a canonical transverse density.…”

“…We also revisit the notion of modular class and of Haar systems (and the existence of cut-off functions). Our original motivation comes from the study of Poisson manifolds of compact types [8,9,10], which provide two important examples of such measures: the affine and the Duistermaat-Heckman measures.…”