Simultaneous metric uniformization of foliations by Riemann surfaces

We consider a two-dimensional linear foliation on torus of arbitrary dimension. For any smooth family of complex structures on the leaves we prove existence of smooth family of uniformizing (conformal complete flat) metrics on the leaves. We extend this result to linear foliations on and families of complex structures with bounded derivatives C ³-close to the standard complex structure. We prove that the analogous statement for arbitrary C two-dimensional foliation on compact manifold is wrong in general, even for suspensions over in dimension 3 the uniformizing metric can be nondifferentiable at some points; in dimension 4 the uniformizing metric of each noncompact leaf can be unbounded.     

Hyperbolic uniformization of Fermat curves

The ground-breaking research on the uniformization of curves was conducted at the beginning of the last century. Nevertheless, there are few examples in the literature of algebraic curves for which an explicit uniformization is known. In this article we obtain an explicit uniformization of the Fermat curves F _N , for each . The results presented here are based in part on an earlier study of the second author [6] in which each Riemann surface F _N (ℂ) was described as a quotient of the complex disk by a Fuchsian group Γ. 2000 Mathematics Subject Classification Primary—11F03, 11F06; Secondary—11F30 This work was partially supported by MCYT BFM2000-0627 and BMF2003-01898.     

Duality for projective curves

This paper studies the strict dual of a projective algebraic curve, mainly in positive characteristic. Inclusion relations among the osculating spaces of the dual and the duals of those of the curve are obtained and shown to be optimal in several cases. As a consequence, a characterization of the non-reflexive curves that coincide with their bidual is obtained.     

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.     

Limits of dual curves via foliations

We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.     

Minimal sets of foliations on complex projective spaces

Dedicated to René Thom with admiration.     

Total rigidity of polynomial foliations on the complex projective plane

Polynomial foliations of the complex plane are topologically rigid. Roughly speaking, this means that the topological equivalence of two foliations implies their affine equivalence. There exist various nonequivalent formalizations of the notion of topological rigidity. Generic polynomial foliations of fixed degree have the so-called property of absolute rigidity, which is the weakest form of topological rigidity. This property was discovered by the author more than 30 years ago. The genericity conditions imposed at that time were very restrictive. Since then, this topic has been studied by Shcherbakov, Gómez-Mont, Nakai, Lins Neto-Sad-Scárdua, Loray-Rebelo, and others. They relaxed the genericity conditions and increased the dimension. The main conjecture in this field states that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations. The main result of this paper is weaker than this conjecture but also makes it possible to compare topological types of distant foliations.     

Total rigidity of polynomial foliations on the complex projective plane

Polynomial foliations of the complex plane are topologically rigid. Roughly speaking, this means that the topological equivalence of two foliations implies their affine equivalence. There exist various nonequivalent formalizations of the notion of topological rigidity. Generic polynomial foliations of fixed degree have the so-called property of absolute rigidity, which is the weakest form of topological rigidity. This property was discovered by the author more than 30 years ago. The genericity conditions imposed at that time were very restrictive. Since then, this topic has been studied by Shcherbakov, Gómez-Mont, Nakai, Lins Neto-Sad-Scárdua, Loray-Rebelo, and others. They relaxed the genericity conditions and increased the dimension. The main conjecture in this field states that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations. The main result of this paper is weaker than this conjecture but also makes it possible to compare topological types of distant foliations. Original Russian Text ? Yu. S. Ilyashenko, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 64–76. To Vladimir Igorevich Arnold with admiration and love     

On the order of projective curves

Here we prove the existence of several componentsW of the Hilbert scheme of curves inP ⁿ such that the generalC W has Hartshorne-Rao module with order equal to its diameter.     

Bounds on leaves of one-dimensional foliations

Let X be a variety over an algebraically closed field, a onedimensional singular foliation, and a projective leaf of . We prove that

Compact leaves of structurally stable foliations

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspension foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary compact manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in the sense of Ehresmann and Reeb.     

On the existence of stable compact leaves for transversely holomorphic foliations

We prove that a transversely holomorphic foliation on a compact manifold exhibits some compact leaf with finite holonomy group, provided that the set of compact leaves is not a zero measure set. A similar result is stated for groups of complex diffeomorphisms and periodic orbits.     

Noncompact leaves of foliations of Morse forms

In this paper foliations determined by Morse forms on compact manifolds are considered. An inequality involving the number of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 862–865, June, 1998.The author wishes to thank Professor A. S. Mishchenko for his interest in this work and stimulating discussions.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00276.     

Growth of leaves in transversely affine foliations

In this note we give estimates for the growth of leaves in transversely affine foliations which depend on the properties of the affine holonomy group.

     

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values .

Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality.

Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms.

The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.