Integrable Systems and Foliations

Dans les articles [B-F-L] et [B-L] que j’ai écrits avec P. Foulon et F. Labourie, nous décrivons, sur toute variété compacte, les flots d’Anosov de contact et les difféomorphismes d’Anosov symplectiques qui ont leurs feuilletages stable et instable de classe C∞. Pour cela, nous utilisons de façon essentielle la proposition suivante due à M. Gromov (voir 1.3 pour la définition des objets qui interviennent dans cet énoncé).

The work of Golod and Šafarevič on class field towers motivated the conjecture that b₂ > b² ₁/4 for nilpotent Lie algebras of dimension at least 3, where b _i denotes the i ^th Betti number. Using a new lower bound for b ₂ and a characterization of Lie algebras of the form g/Z(g), we prove this conjecture for 2-step algebras. We also give the Betti numbers of nilpotent Lie algebras of dimension at most 7 and use them to establish the conjecture for all nilpotent Lie algebras whose centres have codimension ≤ 7.

Nous montrons que les pseudogroupes de génération compacte quasiparallélisés méromorphes de dimension complexe un, classés dans [Cav], apparaissent tous comme pseudogroupe d’holonomie de feuilletages transversalement holomorphes sur des variétés compactes.

Let K be a connected compact Lie group that acts on the connected compact symplectic manifold (X, ω) preserving the symplectic form. If for every ξ in the Lie algebra p of K, the vector field ξx(x) = d/dt| _t=o exp(tξ) · x is Hamiltonian relative to a function, say Hξ, then the map Φ : X → p* (to the dual p* of p) defined by is called momentum mapping for the action of K. If the momentum mapping is equivariant relative to the given action of K on X and the coadjoint action of K on p*, the action is called Hamiltonian. A remarkable property of this map was discovered by Guillemin-Sternberg [GS1,GS2] and Kirwan [Ki2]. It asserts that if T is a maximal torus of K and t ₊ ^* is a positive Weyl chamber, then Φ(X) ∩ t ₊ ^* is a convex polytope. This theorem was first proved as follows. The image Φ(X) ∩ t ₊ ^* was shown to be a finite union of compact convex polytopes in [GS1], and a convex polytope for X a Kähler manifold [GS2]. From the partial result in [GS1], Kirwan [Ki2]deduced convexity by appealing to her Morse theory (developed in [Ki1]) for||Φ||²

Dans cet article, nous nous proposons de discuter de la possibilité éventuelle de généraliser les théorèmes fondamentaux d’uniformisation des surfaces de Riemann aux feuilletages et laminations. Nous commençons par rappeler quelques énoncés extrêmement classiques.

On sait que dans la quantification par déformations sur un espace de phase, introduite en 1978 par Flato, Fronsdal, Sternheimer et moi-même, apparaît d’une manière fondamentale un 2-cocyle de Chevalley, noté classiquement S _Γ ³ , de l’algèbre de Lie de Poisson qui n’est jamais un cobord.

Following C. Ehresmann, to every foliated manifold one associates a pseudogroup of transformations that represents the transverse structure of the foliation. Conversely, every pseudogroup comes from a foliated manifold.

Etale groupoids play a central role in the theory of foliations. Well-known examples include the Haefliger groupoid Γ ^q which classifies C ^∞-foliations of codimension q [H71] and the holonomy groupoid of any foliation [W83]. In particular, invariants of leaf spaces of foliations are usually defined in terms of the classifying space or the C*-algebra associated to this holonomy groupoid (see [C, H84, Mo, BN] and many others).

In this work we study the global models of a r-form ω, on the sphere S ⁵, which is transitive, i.e. we assume that for each p Є S ⁵ and each υ Є T _P S ⁵ there exists a vector field X on S ⁵ such that L _x ω = 0 and X( _p ) = υ. For volume forms (Moser theorem), closed 3-forms, non closed 2-forms and some non closed 3-forms, one explicitly obtains all their global models.

This is an expository paper. In it, the Poisson—Nijenhuis structures are motivated and defined in the general algebraic framework of Gel’fand and Dorfman. Then, in the particular case of Lie algebroids and differentiable manifolds, the Poisson—Nijenhuis structures are related to the notion of a complementary 2-form, that has been introduced and studied by the author in [20], and several examples of complementary forms and Poisson—Nijenhuis manifolds are given.

In this paper we show how there is associated an integrable Hamiltonian system to a certain set of algebraic-geometric data. Roughly speaking these data consist of a family of algebraic curves, parametrized by an affine algebraic variety B, a subalgebra C of O(B) and a polynomial φ(x, y) in two variables. The phase space is constructed geometrically from the family of curves and has a natural projection onto B; the regular functions on B lead to an algebra of functions in involution and the level sets of the moment map are symmetric products of algebraic curves.

While completely transparant from the geometrical point of view, a slight change of these integrable Hamiltonian systems is needed in order to explicitly realize these integrable Hamiltonian systems. Thus, we associate to the same data another integrable Hamiltonian system and show how they relate to the first one: there is a birational map between them (which is regular in one direction) which is (in the regular direction) a morphism of integrable Hamiltonian systems. Both the Poisson structure and the functions in involution are found by performing an Euclidean division of two polynomials, so that when the data are explicitly given, all ingredients of the integrable Hamiltonian system can be easily computed from it in an explicit way.