Hamiltonian actions of compact type from a Poisson point of view

Abstract

This thesis concerns the study of Hamiltonian actions and momentum maps in the Poisson geometric framework introduced by Mikami and Weinstein. More precisely, we study Hamiltonian actions of proper symplectic groupoids, focusing on two topics. The main body of this text is divided accordingly, into two parts. In the first part we focus on the orbit spaces of such actions. The content of this part is closely related to the so-called symplectic reduction theorems, due to Marsden and Weinstein, Meyer, and Lerman and Sjamaar. Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle-Guillemin-Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids. In the second part we study toric Hamiltonian actions by regular and proper symplectic groupoids. Examples of these include toric manifolds, proper Lagrangian fibrations and proper isotropic realizations of Poisson manifolds of compact types. Our main results concern the classification of such Hamiltonian actions in terms of the image of the momentum map and in terms of a new invariant, that we call the ext-invariant of the toric action. In particular, we give a generalization of Delzant’s classification theorem, that unifies Delzant’s theorem and the classification of Lagrangian fibrations appearing (implicitly) in the work of Duistermaat. The theory of regular Poisson manifolds of compact types, and in particular the integral affine orbifold structure on the leaf space, plays a fundamental role here. The image of the momentum map of a toric action turns out to be what we call a Delzant subspace of the leaf space – a generalization of the notion of Delzant polytope appearing in Delzant’s classification of toric manifolds. Furthermore, our classification involves the cohomology of orbifold sheaves for orbifold versions of the sheaves in the papers of Duistermaat and Dazord-Delzant on Lagrangian and isotropic fibrations. As is the case for proper isotropic realizations of Poisson manifolds of compact types, the symplectic gerbe and the Lagrangian Dixmier-Douady class turn out to encode the obstruction to the existence of toric actions of regular and proper symplectic groupoids with momentum image equal to a prescribed Delzant subspace.