Irréductibilité de la première équation de Painlevé

A definition of reducibility for algebraic codimension two foliations given by closed 2-forms is proposed. Reducible foliations are characterised on theirs Galois groupoids. We apply this to the foliation given by the first Painlevé equation. Its Galois groupoid is computed and this proves its irreducibility. To cite this article: G. Casale, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On deformations of transversely homogeneous foliations

A transversely homogeneous foliation is a foliation whose transverse model is a homogeneous space G/H. In this paper we consider the class of transversely homogeneous foliations on a manifold M which can be defined by a family of 1-forms on M fulfilling the Maurer–Cartan equation of the Lie group G. This class includes as particular cases Lie foliations and certain homogeneous spaces foliated by points. We develop, for the foliations belonging to this class, a deformation theory for which both the foliation and the model homogeneous space G/H are allowed to change. As the main result we show that, under some cohomological assumptions, there exist a versal space of deformations of finite dimension for the foliations of the class and when the manifold M is compact. Some concrete examples are discussed.     

Twisted index theory for foliations

For any Lie groupoid with a twisting, we define an analytic index morphism using the Connes tangent groupoid. This morphism agrees with the one of the Lie groupoid when the twisting is trivial. We discuss a longitudinal index theorem, geometric cycles, push-forward maps and Baum–Connes assembly maps for foliations with a twisting on the space of leaves.     

Minimal sets of Cartan foliations

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ?, there exists one more foliation (M, \(\mathcal{O}\)), which is generally singular and is called an aureole foliation; moreover, the foliations ? and \(\mathcal{O}\) have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ?/? with a compactly embedded Lie subalgebra ? in ?, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ?). We prove that for such foliations, there exists a unique minimal set ?, and ? is contained in the closure of any leaf. If the foliation (M, ?) is proper, then ? is a unique closed leaf of this foliation.     

The automorphism groups of foliations with transverse linear connection

The category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.     

Holonomy transformations for singular foliations

In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations cannot be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation.     

Groupoïde d'holonomie et géométrie globale

The purpose of this Note is to show that global geometry requires Lie groupoid theory. In this way we show that holonomy groupoid of a regular foliation leads to a satisfactory solution of many global existence problems. Three examples are given: Palais'theorem on local group actions, a characteristics method and links between Lie groupoid and algebroid morphisms. Complete proofs will be given elsewere.     

The wave trace of the basic Laplacian of a Riemannian foliation near a non-zero period

Given a compact boundaryless Riemannian manifold that admits a Riemannian foliation, recall that the space of leaf closures is a singular stratified space. Associated to this space is an operator called the basic Laplacian defined on the space of smooth functions that are constant on the leaves (and, hence, the closures of the leaves of the foliation). The corresponding basic spectrum is, under certain assumptions, an infinite subset of the spectrum of the ordinary laplacian. If the metric is bundle-like with respect to the foliation, the trace of the basic wave operator can be analyzed, and invariants of the basic spectrum can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the leaf closures, and from them, basic wave trace asymptotic expansions are derived. Using the connection between Riemannian foliations and manifolds being acted upon by a compact Lie group of isometries, $G$ , the wave trace for the $G$ -invariant spectrum of a $G$ -manifold is sketched out as a related result.     

On the integrability of Lie subalgebroids

Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible. We also consider the problem of integrability by closed subgroupoids, and we give conditions under which the closure of a subgroupoid is again a subgroupoid.     

Hierarchies of holonomy groupoids for foliated bundles

We give a new construction of the holonomy groupoid of a regular foliation in terms of a partial connection on a diffeological principal bundle of germs of transverse parametrisations, which may be viewed as a systematisation of Winkelnkemper’s original construction using ideas from gauge theory. We extend these ideas to construct a novel holonomy groupoid for any foliated bundle, which we prove sits at the top of a hierarchy of diffeological jet holonomy groupoids associated with the foliated bundle. This shows that while the Winkelnkemper holonomy groupoid is the smallest Lie groupoid that integrates a foliation, it is far from the smallest diffeological groupoid that does so.

     

Germes de feuilletages présentables du plan complexe

Let \(\mathcal{F}\) be a germ of a singular foliation of the complex plane. Assuming that \(\mathcal{F}\) is a generalized curve D. Marín and J.–F. Mattei proved the incompressibility of the foliation in a neighborhood from which a finite set of analytic curves is removed. We showin the present work that this hypothesis cannot be eluded, by buildingexamples of foliations, reduced after one blow–up, for which the property does not hold. Even if we manage to prove that the individual saddle–node foliation is incompressible, their leaves not retracting tangentially on all the components of the definition domain boundary forbids a generalization of Marín–Mattei’s construction.We finally characterize a near–complete class of foliations,called stronglypresentable, forwhich the construction of Marín–Mattei’smonodromy can be carried out.     

Integration of complex polynomial differential equations in higher dimension,a first step: Lie transverse structure

We prove that a one-dimensional holomorphic foliationswith generic singularities on a complex projective space ?P ^m+1, m ≥ 2, exhibiting a Lie group transverse structure in some Zariski open subset, is logarithmic. That is, it is given by a system of m closed rational one-forms with simple poles. The foliation is given by a linear vector field in some affine space ? ^m+1 ? ?P ^m+1 if, and only if, it exhibits only one singularity in this affine space. An application to foliations invariant under Lie group transverse actions is given.     

The Noether–Fano inequalities for codimension one singular holomorphic foliations

The idea of the proof of the classical Noether–Fano inequalities can be adapted to the domain of codimension one singular holomorphic foliations of the projective space. We obtained criteria for proving that the degree of a foliation on the plane is minimal in the birational class of the foliation and for the non-existence of birational symmetries of generic foliations (except automorphisms). Moreover, we give several examples of birational symmetries of special foliations illustrating our results.      

Smooth automorphisms and path-connectedness in Borel dynamics

Let Aut(X, B) be the group of all Borel automorphisms of a standard Borel space (X, B). We study topological properties of Aut(X, B) with respect to the uniform and weak topologies, τ and p, defined in [Bezuglyi S., Dooley A.H., Kwiatkowski J., Topologies on the group of Borel automorphisms of a standard Borel space, Preprint 2003]. It is proved that the class of smooth automorphisms is dense in (Aut(X, B), p). Let Ctbl(X) denote the group of Borel automorphisms with countable support. It is shown that the topological group Aut₀(X, B) = Aut(X, B)/Ctbl(X) is path-connected with respect to the quotient topology τ₀. It is also proved that Aut₀(X, B) has the Rokhlin property in the quotient topology p₀, i.e., the action of Aut₀(X,B) on itself by conjugation is topologically transitive.     

Leaf closures of Riemannian foliations: A survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply-connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino’s structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations, including the recent proof of Molino’s conjecture, and discuss singular Killing foliations.     

On the developability of Lie subalgebroids

For a Lie groupoid G with algebroid g, one says that a subalgebroid h⊂g is developable if it can be integrated to a closed Lie subgroupoid of the universal covering groupoid of G. Under some additional hypotheses, we construct an algebroid b, depending on G and h, and prove that the developability of h is equivalent to the integrability of b. This result extends the Almeida-Molino obstruction to developability of foliations.     

Several Cohomology Algebras Connected with the Poisson Structure

The structure of a Lie superalgebra is defined on the space of multiderivations of a commutative algebra. This structure is used to define some cohomology algebra of Poisson structure. It is shown that when a commutative algebra is an algebra of C -functions on the C -manifold, the cohomology algebra of Poisson structure is isomorphic to an algebra of vertical cohomologies of the foliation corresponding to the Poisson structure.     

On realizing measured foliations via quadratic differentials of harmonic maps toR-trees

We give a brief, elementary and analytic proof of the theorem of Hubbard and Masur [HM] (see also [K], [G]) that every class of measured foliations on a compact Riemann surfaceR of genusg can be uniquely represented by the vertical measured foliation of a holomorphic quadratic differential onR. The theorem of Thurston [Th] that the space of classes of projective measured foliations is a 6g—7 dimensional sphere follows immediately by Riemann-Roch. Our argument involves relating each representative of a class of measured foliations to an equivariant map from to anR-tree, and then finding an energy minimizing such map by the direct method in the calculus of variations. The normalized Hopf differential of this harmonic map is then the desired differential. Partially supported by NSF grant DMS9300001; Alfred P. Sloan Research Fellow.     

Nontoric foliations by Lagrangian tori of toric Fano varieties

A construction of a foliation of a toric Fano variety by Lagrangian tori is presented; it is based on linear subsystems of divisor systems of various degrees invariant under the Hamiltonian action of distinguished function-symbols. It is shown that known examples of foliations (such as the Clifford foliation and D. Auroux’s example) are special cases of this construction. As an application, nontoric Lagrangian foliations by tori of two-dimensional quadrics and projective space are constructed.