ON THE JULIA SETS OF COMPLEX CODIMENSION-ONE TRANSVERSALLY HOLOMORPHIC FOLIATIONS

2009 ◽  
Author(s):  
Taro Asuke
Keyword(s):  
Julia Sets ◽  
Holomorphic Foliations ◽  
Codimension One

scholarly journals Flags of holomorphic foliations

2011 ◽  
Vol 83 (3) ◽  
pp. 775-786 ◽  
Author(s):  
Rogério S. Mol
Keyword(s):  
Finite Number ◽  
Complex Manifold ◽  
Necessary Conditions ◽  
Holomorphic Foliations ◽  
Lower Dimension ◽  
Higher Dimension ◽  
Basic Properties ◽  
Codimension One ◽  
Tangent Sheaf ◽  
Polar Classes

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties oft hese objects and, in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, n > 3, we establish some necessary conditions for a foliation, we find bounds of lower dimension to leave invariant foliations of codimension one. Finally, still in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" /> involving the degrees of polar classes of foliations in a flag.

scholarly journals On holomorphic one-forms transverse to closed hypersurfaces

2003 ◽  
Vol 75 (3) ◽  
pp. 265-269
Author(s):  
Toshikazu Ito ◽  
Bruno Scárdua
Keyword(s):  
Affine Space ◽  
Smooth Boundary ◽  
Singular Points ◽  
Real Hypersurfaces ◽  
Holomorphic Foliations ◽  
Codimension One ◽  
Closed Hypersurfaces ◽  
The One

In this note we announce some achievements in the study of holomorphic distributions admitting transverse closed real hypersurfaces. We consider a domain with smooth boundary in the complex affine space of dimension two or greater. Assume that the domain satisfies some cohomology triviality hypothesis (for instance, if the domain is a ball). We prove that if a holomorphic one form in a neighborhood of the domain is such that the corresponding holomorphic distribution is transverse to the boundary of the domain then the Euler-Poincaré-Hopf characteristic of the domain is equal to the sum of indexes of the one-form at its singular points inside the domain. This result has several consequences and applies, for instance, to the study of codimension one holomorphic foliations transverse to spheres.

scholarly journals Higher codimensional foliations with Kupka singularities

2017 ◽  
Vol 28 (03) ◽  
pp. 1750019
Author(s):  
O. Calvo-Andrade ◽  
M. Corrêa ◽  
A. Fernández-Pérez
Keyword(s):  
Normal Form ◽  
Positive Integer ◽  
Projective Space ◽  
Complete Intersection ◽  
Holomorphic Foliations ◽  
Connected Component ◽  
Codimension One ◽  
Dimension Formula ◽  
Rational Fibration

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].

ON THE SINGULAR SCHEME OF CODIMENSION ONE HOLOMORPHIC FOLIATIONS IN ℙ3

2010 ◽  
Vol 21 (07) ◽  
pp. 843-858 ◽  
Author(s):  
LUIS GIRALDO ◽  
ANTONIO J. PAN-COLLANTES
Keyword(s):  
Singular Locus ◽  
The Other ◽  
Holomorphic Foliation ◽  
Holomorphic Foliations ◽  
Codimension One ◽  
Other Hand ◽  
Tangent Sheaf ◽  
Torsion Free

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion-free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in ℙ3 is locally free if and only if the singular scheme is a curve, and that it splits if and only if it is arithmetically Cohen–Macaulay. On the other hand, we discuss when a split foliation in ℙ3 is determined by its singular scheme.

Degrees of spaces of holomorphic foliations of codimension one inPn

2017 ◽  
Vol 221 (11) ◽  
pp. 2791-2804 ◽  
Author(s):  
Daniel Leite ◽  
Israel Vainsencher
Keyword(s):  
Holomorphic Foliations ◽  
Codimension One

scholarly journals A NON-EXISTENCE THEOREM FOR MORSE TYPE HOLOMORPHIC FOLIATIONS OF CODIMENSION ONE TRANSVERSE TO SPHERES

2010 ◽  
Vol 21 (04) ◽  
pp. 435-452 ◽  
Author(s):  
TOSHIKAZU ITO ◽  
BRUNO SCARDUA
Keyword(s):  
Existence Theorem ◽  
Affine Space ◽  
Holomorphic Foliation ◽  
Holomorphic Foliations ◽  
Codimension One ◽  
Concentric Spheres ◽  
Linear Foliation

We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.

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