Foliation Of Dimension Research Articles

Uniform rational polytopes of foliated threefolds and the global ACC

AbstractIn this paper, we show the existence of uniform rational lc polytopes for foliations with functional boundaries in dimension . As an application, we prove the global ACC for foliated threefolds with arbitrary DCC coefficients. We also provide applications on the accumulation points of lc thresholds of foliations in dimension .

Leafwise quasigeodesic foliations in dimension three and the funnel property

AbstractWe construct one-dimensional foliations which are subfoliations of two-dimensional foliations in$3$-manifolds. The subfoliation is by quasigeodesics in each two-dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves.

On the resolution of singularities of one-dimensional foliations on three-manifolds

This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.

Invariant surfaces for toric type foliations in dimension three

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0) without saddle-nodes has invariant surface. We extend the argument of Cano–Cerveau for the nondicritical caseto the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to closed irreducible curves. We build the invariant surface as a germ along the singular locus and those closed irreducible invariant curves. The result of OrtizBobadilla–Rosales-Gonzalez–Voronin about the distribution of invariant branches indimension two is a key argument in our proof.

Holomorphic vector fields tangent to foliations in dimension three.

We consider a singular holomorphic vector field in a neighborhood of 0 ∈ C 3 and suppose that there is a singular holomorphic foliation of codimension one (outside its singular set, given by a holomorphic decomposition of this neighborhood into complex surfaces, called leaves) to which it is tangent. This means that, when both objects are non-singular, the orbits of the vector field are contained in the leaves of the foliation. First we consider the desingularizations of both objects, trying to relate their final models. Then we analyse the situation where the vector field is tangent to three independent foliations.

Stable Singularities and Stable Leaves of Holomorphic Foliations in Dimension Two

Stable Singularities and Stable Leaves of Holomorphic Foliations in Dimension Two

Higher codimensional foliations with Kupka singularities

We consider holomorphic foliations of dimension [Formula: see text] and codimension [Formula: see text] in the projective space [Formula: see text], with a compact connected component of the Kupka set. We prove that if the transversal type is linear with positive integer eigenvalues, then the foliation consists of the fibers of a rational fibration [Formula: see text]. As a corollary, if [Formula: see text] and has a transversal type diagonal with different eigenvalues, then the Kupka component [Formula: see text] is a complete intersection and the leaves of the foliation define a rational fibration. The same conclusion holds if the Kupka set has a radial transversal type. Finally, as an application, we find a normal form for non-integrable codimension-one distributions on [Formula: see text].

Simple dynamics and integrability for singularities of holomorphic foliations in dimension two

Simple dynamics and integrability for singularities of holomorphic foliations in dimension two

Germs of complex two dimensional foliations

The purpose of this paper is to show how some results about codimension one foliations in dimension three can be generalized to dimension two foliations in dimension n ≥ 4.

Elimination of resonances in codimension one foliations

The problem of reduction of singularities for germs of codimension one foliations in dimension three has been solved by Cano in [3]. The author divides the proof in two steps. The first one consists in getting pre-simple points and the second one is the passage from pre-simple to simple points. In arbitrary dimension of the ambient space the problem is open. In this paper we solve the second step of the problem.

On totally geodesic foliations with bundle-like metric

Let $$\mathcal{F}$$ be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for $$\mathcal{F}$$ and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed.

Reduction of the singularities of codimension one singular foliations in dimension three

Reduction of the singularities of codimension one singular foliations in dimension three

Riemannian foliationin N = 1,D = 11 supergravity

We propose a G-foliated model of the seven-dimensional internal manifoldM7 resulting from compactifying theN = 1,D = 11 model of supergravity of Cremmer, Julia and Scherk. This gives a generalisation of the usualG/H homogeneous-space approach. As a result, there is a natural way of expressing the internal spinor bundle as a graded tensor product. Under coordinate transformations mixing the tangential geometry with the transversal geometry of the foliation, spinors in this bundle with different internal helicities transform differently. We also argue that the ground-state compactifieation of the 11-dimensional manifoldM11 intoM7 ×M4 is the result of a global submersion ofM11 intoM4, giving rise to a foliation of dimension 7 and codimension 4. With these additional structures in place, we show that it is possible to incorporate physical chiral spinors into the original model while retaining the Riemannian geometry. There is also a canonical way of defining a Dirac operator on the internal spinor bundle, so that the mass spectrum of the internal Dirac operator is the sum of a tangential spectrum and a transversal spectrum.

Construction of singular holomorphic vector fields and foliations in dimension two

Construction of singular holomorphic vector fields and foliations in dimension two