Related papers: On the local structure of Dirac manifolds
A $2n$-dimensional Poisson manifold $(M ,\Pi)$ is said to be $b^m$-symplectic if it is symplectic on the complement of a hypersurface $Z$ and has a simple Darboux canonical form at points of $Z$ which we will describe below. In this paper…
In this paper we study the modular classes of Dirac manifolds and of Dirac maps, and we discuss their basic properties. We apply these results to explain the relationship between the modular classes of the various structures involved in the…
We examine shifted symplectic and Poisson structures on spaces of framed maps. We prove some results about shifted Poisson structures analogous to those in existing ones about symplectic structures. Then, we consider the space Map(X,D,Y) of…
Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. K\"ahler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we…
A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.
On a foliated manifold equipped with an action of a compact Lie group $G$, we study a class of almost-coupling Poisson and Dirac structures, in the context of deformation theory and the method of averaging.
We provide local formul{\ae} for Poisson bivectors and symplectic forms on the leaves of Poisson structures associated to wrinkled fibrations on smooth $4$--manifolds.
We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as…
The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth…
We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not…
In this paper, we solve the problem of giving a gauge-theoretic description of the natural Dirac structure on a Lie Group which plays a prominent role in the theory of D- branes for the Wess-Zumino-Witten model as well as the theory of…
Let X(\Sigma) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure \Pi_\Sigma is constructed on the complex manifold X(\Sigma), the symplectic leaves of which are the…
There have been several attempts in recent years to extend the notions of symplectic and Poisson structures in order to create a suitable geometrical framework for classical field theories, trying to achieve a success similar to the use of…
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold…
Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic…
We construct a smooth codimension-one foliation on the five-sphere in which every leaf is a symplectic four-manifold and such that the symplectic structure varies smoothly. Our construction implies the existence of a complete regular…
We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the…
This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such…
In recent years, $b$-symplectic manifolds have become important structures in the study of symplectic geometry, serving as Poisson manifolds that retain symplectic properties away from a hypersurface. Inspired by this rich landscape,…
In this work, we initiate the study of rigidity and non-rigidity phenomena for Poisson homeomorphisms, defined as uniform $C^0$-limits of Poisson diffeomorphisms. First, we prove that Poisson homeomorphisms preserve the singular symplectic…