Eva Miranda

Symplectic and Poisson geometry on b-manifolds

Abstract Let M 2 n be a Poisson manifold with Poisson bivector field Π. We say that M is b-Poisson if the map Π n : M → Λ 2 n ( T M ) intersects the zero section transversally on a codimension one submanifold Z ⊂ M . This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of ( M , Π ) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.

Codimension one symplectic foliations and regular Poisson structures

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9].

Toric Actions on b-Symplectic Manifolds

We study Hamiltonian actions on b-symplectic manifolds with a focus on the e ective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classi es these manifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus. At the end of the paper we suggest further avenues of study, including an example of a toric action on a b 2-manifold and applications of our ideas to integrable systems on b-manifolds

Action-angle variables and a KAM theorem for b-Poisson manifolds

In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.

Geometric quantization of integrable systems with hyperbolic singularities

We construct the geometric quantization of a compact surface using a singular real polarization coming from an integrable system. Such a polarization always has singularities, which we assume to be of nondegenerate type. In particular, we compute the effect of hyperbolic singularities, which make an infinite-dimensional contribution to the quantization, thus showing that this quantization depends strongly on polarization.

A singular Poincaré lemma

We prove a Poincare lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of sp(2r, R). This result has a natural interpretation in terms of the cohomology associated to the infinitesimal deformation of a completely integrable system.

Examples of integrable and non-integrable systems on singular symplectic manifolds

We present a collection of examples borrowed from celes- tial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization trans- formations, Appells transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lower- ing its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with sin- gularities which are mainly of two types: b m -symplectic and m -folded symplectic structures. These examples comprise the three body prob- lem as non-integrable exponent and some integrable reincarnations such as the two xed-center problem. Given that the geometrical and dy- namical properties of b m -symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL, GLPR, MO, S, GMW], we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint.

Cotangent Models for Integrable Systems

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on b-Poisson/b-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle theorems in these settings: the theorem of Liouville–Mineur–Arnold for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds (Laurent- Gengoux et al., IntMath Res Notices IMRN 8: 1839–1869, 2011). Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a b-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved by the authors and Scott (Math. Pures Appl. (9) 105(1):66–85, 2016). This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the b-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on b-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds.

Integrable systems and group actions

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.

On a Poincare lemma for foliations

In this paper we revisit a Poincare lemma for foliated forms, with respect to a regular foliation, and compute the foliated cohomology for local models of integrable systems with singularities of nondegenerate type. A key point in this computation is the use of some analytical tools for integrable systems with nondegenerate singularities, including a Poincare lemma for the deformation complex associated to this singular foliation.