Related papers: On Bott-Morse Foliations and their Poisson Structu…

We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable…

It is known that there exist singular Poisson structures in $4$-manifolds, whose symplectic foliation is given by singular fibrations over surfaces. In this work, we describe the effect of the monodromy of the fibration on the Poisson…

We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely…

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson…

Some Poisson structures do admit resolutions by symplectic manifolds of the same dimension. We give examples and simple conditions under which such resolutions can not exist.

We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson…

We establish existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. Namely, we show that with enough weighted blowups, one can reduce the singularities of such Poisson…

We classify all the quadratic Poisson structures on $so^*(4)$ and $e^*(3)$, which have the same foliation by symplectic leaves as the canonical Lie-Poisson tensors. The separated variables for the some of the corresponding bi-integrable…

We classify all of the 4-dimensional linear Poisson structures of which the corresponding Lie algebras can be considered as the extension by a derivation of 3-dimensional unimodular Lie algebras. The affine Poisson structures on R^3 are…

A Bott manifold is a smooth projective toric variety having an iterated $\mathbb{C} P^1$-bundle structure. A certain family of Bott manifolds is used to understand the structure of Bott--Samelson varieties (or…

Symmetrical top is a special case of a general top. The canonical Poisson structure on T*SE(3) is the common method of its description. This Poisson structure is invariant under the right action of SO(3). However the Hamiltonian of the…

We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a…

We study codimension one (transversally oriented) foliations $\fa$ on oriented closed manifolds $M$ having non-empty compact singular set $\sing(\fa)$ which is locally defined by Bott-Morse functions. We prove that if the transverse type of…

We study codimension one smooth foliations with Morse type singularities on closed ma-nifolds. We obtain a description of the manifold in case the number of centers in greater then the number of saddles. This result relies on and extends…

We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled…

In this work we classify foliations on $\mathbb{CP}^3$ of codimension 1 and degree $2$ that have a line as singular set. To achieve this, we do a complete description of the components. We prove that the boundary of the exceptional…

We are interested in analytic singular Poisson structures with a non zero linear part at the singularity. Using recent work of the author about holomorphic normalization of commutative familly of singular vector fields, we obtain results…

A Poisson structure is represented by a bivector whose Schouten bracket vanishes. We study a global Poisson structure on $S^4$ associated with a holomorphic Poisson structure on $\mathbb{CP}^3$. The space of the Poisson structures on $S^4$…

Let $\Pi$ be a rank $2$ Poisson Structure in the Projective Space defined by the dimension $2$ foliation $\mathcal{F}$ in the pull-back component. We prove that for a generic choice of $\mathcal{F}$, the irreducible component of the Poisson…

A geometric description of the first Poisson cohomology groups is given in the semilocal context, around (possibly singular) symplectic leaves. This result is based on the splitting theorems for infinitesimal automorphisms of coupling…