Minimal sets of foliations on complex projective spaces

Dedicated to René Thom with admiration.     

Minimal singular Riemannian foliations

We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Minimal entropy rigidity for foliations of compact spaces

We formulate and prove a foliated version of a theorem of Besson, Courtois, and Gallot establishing the minimal entropy rigidity of negatively curved locally symmetric spaces. One corollary is a foliated version of Mostow’s rigidity theorem.     

Minimal, rigid foliations by curves on ℂℙ n

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space ℂℙ ⁿ for every dimension n≥2 and every degree d≥2. Precisely, we construct a foliation ℱ which is induced by a homogeneous vector field of degree d, has a finite singular set and all the regular leaves are dense in the whole of ℂℙ ⁿ . Moreover, ℱ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if ℱ is conjugate to another holomorphic foliation by a homeomorphism sufficiently close to the identity, then these foliations are also conjugate by a projective transformation. Finally, all these properties are persistent for small perturbations of ℱ.?This is done by considering pseudo-groups generated on the unit ball 𝔹 ⁿ ⊆ℂ ⁿ by small perturbations of elements in Diff(ℂ ⁿ ,0). Under open conditions on the generators, we prove the existence of many pseudo-flows in their closure (for the C ⁰-topology) acting transitively on the ball. Dynamical features as minimality, ergodicity, positive entropy and rigidity may easily be derived from this approach. Finally, some of these pseudo-groups are realized in the transverse dynamics of polynomial vector fields in ℂℙ ⁿ . Received March 7, 2002 / final version received November 26, 2002?Published online February 7, 2003 Most of this work has been carried out during a visit of the first author to IMPA/RJ and a visit of the second author to the University of Lille 1. We would like to thank these institutes for hospitality and express our gratitude to CNPq-Brazil and CNRS-France for the financial support which made these visits possible. We are also indebted to Paulo Sad, Marcel Nicolau and the referee whose comments helped us to improve on the preliminary version. Finally, the second author has partially conducted this research for the Clay Mathematics Institute.     

Minimal invariant varieties and first integrals for algebraic foliations

Let X be an irreducible algebraic variety over ℂ, endowed with an algebraic foliation . In this paper, we introduce the notion of minimal invariant variety V( , Y) with respect to ( , Y), where Y is a subvariety of X. If Y = {x} is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through x. First we prove that for very generic x, the varieties V( , x) have the same dimension p. Second we generalize a result due to X. Gomez- Mont (see [G-M]). More precisely, we prove the existence of a dominant rational map F : X → Z, where Z has dimension (n − p), such that for very generic x, the Zariski closure of F⁻¹(F(x)) is one and only one minimal invariant variety of a point. We end up with an example illustrating both results.     

Minimal sets and varieties

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.

     

Dicritical singularities in foliations with algebraic limit sets

Given a holomorphic foliation of the 2-dimensional projective space, sufficient conditions are given in order to be the pull back of a linear foliation by a rational map.     

Minimal covers of finite sets

We enumerate the minimal covers of a finite set S, classifying such covers by their cardinality, and also by the number of elements in S which they cover uniquely.     

Minimal sets in finite rings

We describe how to calculate the (, )-minimal sets in any finite ring.     

Minimal representation of convex polyhedral sets

A system of linear inequality and equality constraints determines a convex polyhedral set of feasible solutionsS. We consider the relation of all individual constraints toS, paying special attention to redundancy and implicit equalities. The main theorem derived here states that the total number of constraints together determiningS is minimal if and only if the system contains no redundant constraints and/or implicit equalities. It is shown that the existing theory on the representation of convex polyhedral sets is a special case of the theory developed here.The author is indebted to Dr. A. C. F. Vorst (Erasmus University, Rotterdam, Holland) for stimulating discussions and comments, which led to considerable improvements in many proofs. Most of the material in this paper originally appeared in the author's dissertation (Ref. 1). The present form was prepared with partial support from a NATO Science Fellowship for the Netherlands Organization for the Advancement of Pure Research (ZWO) and a CORE Research Fellowship.     

Minimal separating sets of maximum size

A connected graph G can be disconnected or reduced to a single vertex by removing an appropriate subset of the vertex set V(G), and can be disconnected by removing a suitable subset of the edge set E(G). Attention has usually been centered on separating sets having minimum cardinality, and parameters called the vertex connectivity and the edge connectivity defined. These classical concepts are generalized by using separating sets which are minimal. By considering the maximum as well as the minimum cardinality of such sets, one defines vertex and edge connectivity parameters. Sharp upper bounds are established for these numbers and their values computed for certain classes of graphs. An analogue of Whitney's theorem on connectivity is obtained. Parameters are also defined for minimal separating sets consisting of a mixture of vertices and edges, and these are shown to depend on the maximum and minimum values of the vertex and edge connectivity parameters.     

Minimal blocking sets in finite planes

A result on enclosed (k,n)-arcs in PG(2,q) is used to characterise all minimal blocking sets of sizes =q +n in PG(2,q) with at least twon-secants. This also permits a slight extension of the bounds on blocking sets established by Blokhuis and Brouwer.     

Minimal balanced sets and finite geometries

A balanced set is a collection of subsets of a finite set that can be weighted so as to cover the whole set uniformly. Minimal balanced sets are of interest in the theory of n-person games, in particular for the existence of outcomes that cannot be improved upon by any coalition (core of the game).The object of this paper is to determine the finite geometries which are minimal balanced sets. We prove that the dual of any t-design with t ? 2 is a minimal balanced set. In particular symmetrical 2-designs (as projective spaces, biplanes, etc.) are always minimal balanced sets. For 1-designs the problem becomes much more difficult, but it is for instance easy to prove that any partial geometry which is not the dual of a 2-Steiner system is never a minimal balanced set; in particular generalized quadrangles are never minimal balanced sets. For linear graphs the problem is completely solved: the dual of a connected linear graph is a minimal balanced set if and only if this linear graph is not bichromatic.