A splitting theorem for the Kupka component of a foliation of ${\bf C} {\bf P}^n,\,n\ge 6$. Addendum to a paper by O. Calvo-Andrade and N. Soares

Edoardo Ballico

Displaying similar documents to “A splitting theorem for the Kupka component of a foliation of $𝐂 𝐏^{n}, n \geq 6$ . Addendum to a paper by O. Calvo-Andrade and N. Soares”

A splitting theorem for the Kupka component of a foliation of $ℂ ℙ^{n}, n \geq 6$ . Addendum to an addendum to a paper by Calvo-Andrade and Soares

Edoardo Ballico (1999)

Annales de l'institut Fourier

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Here we show that a Kupka component $K$ of a codimension 1 singular foliation $F$ of $ℂ ℙ^{n}, n \geq 6$ is a complete intersection. The result implies the existence of a meromorphic first integral of $F$ . The result was previously known if $\deg (K)$ was assumed to be not a square.

Natural liftings of foliations to the $r$ -tangent bunde

Mikulski, Włodzimierz M.

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Let $F$ be a $p$ -dimensional foliation on an $n$ -manifold $M$ , and $T^{r} M$ the $r$ -tangent bundle of $M$ . The purpose of this paper is to present some reltionship between the foliation $F$ and a natural lifting of $F$ to the bundle $T^{r} M$ . Let $L_{q}^{r} (F)$ $(q = 0, 1, \dots, r)$ be a foliation on $T^{r} M$ projectable onto $F$ and $L_{q}^{r} = {L_{q}^{r} (F)}$ a natural lifting of foliations to $T^{r} M$ . The author proves the following theorem: Any natural lifting of foliations to the $r$ -tangent bundle is equal to one of the liftings $L_{0}^{r}, L_{1}^{r}, \dots, L_{n}^{r}$ . The exposition is clear and well organized. ...

Lifts of Foliated Linear Connectionsto the Second Order Transverse Bundles

Vadim V. Shurygin, Svetlana K. Zubkova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The second order transverse bundle $T_{}^{2} M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra $𝔻^{2}$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general $𝔻^{2}$ -smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a $𝔻^{2}$ -smooth foliated diffeomorphism between two second order transverse...

Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki (2012)

Annales Polonici Mathematici

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It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation $ℱ_{G}$ . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing $C^{r}$ diffeomorphism group G is simple iff the foliation $ℱ_{[G, G]}$ defined by [G,G] admits no proper...

Warped compact foliations

Szymon M. Walczak (2008)

Annales Polonici Mathematici

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The notion of the Hausdorffized leaf space $\tilde{}$ of a foliation is introduced. A sufficient condition for warped compact foliations to converge to $\tilde{}$ is given. Moreover, a necessary condition for warped compact Hausdorff foliations to converge to $\tilde{}$ is shown. Finally, some examples are examined.

Minimal, rigid foliations by curves on $ℂ ℙ^{n}$

Frank Loray, Julio C. Rebelo (2003)

Journal of the European Mathematical Society

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We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $ℂ ℙ^{n}$ for every dimension $n \geq 2$ and every degree $d \geq 2$ . 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 $ℂ ℙ^{n}$ . Moreover, $ℱ$ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if $ℱ$ is conjugate to another holomorphic...

Foliations by complex manifolds involving the complex Hessian

Julian Ławrynowicz, Jerzy Kalina, Masami Okada

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SummaryIn 1979 the second named author proved, in a joint paper with J. Ławrynowicz, the existence of a foliation of a bounded domain in $ℂ^{n}$ by complex submanifolds of codimension k+p-1, connected in some sense with a real (1,1) C³-form of rank k and the pth power of the complex Hessian of a C³-function u with im u plurisubharmonic and the property that for every leaf of this foliation the restricted functions im u, re u and $(\partial / \partial z_{j}) i m u$ , $(\partial / \partial z_{j}) r e u$ are pluriharmonic and holomorphic, respectively.Now the...

The generic dimension of the first derived system

Robert P. Buemi (1978)

Annales de l'institut Fourier

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Any $r$ -dimensional subbundle of the cotangent bundle on an $n$ -dimensional manifold $M$ partitions $M$ into subsets $M_{0}, ..., M_{m}$ ( $m$ being the minimum of $r$ and $C (n - r, 2)$ , the combinations of $n - r$ things taken 2 at a time). $M_{i}$ is the set on which the first derived systems of the subbundle has codimension $i$ . In this paper we prove the following: Theorem. Let $s \geq 2$ and let $Q$ be a generic $C^{s}$ $r$ -dimensional subbundle of the cotangent bundle of an $n$ -dimensional manifold $M$ . The codimension...

Extending regular foliations

J. W. Smith (1969)

Annales de l'institut Fourier

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A $p$ -dimensional foliation $F$ on a differentiable manifold $M$ is said to extend provided there exists a $(p + 1)$ -dimensional foliation $F^{'}$ on $M$ with $F \subset F^{'}$ . Our main result asserts that if $M$ and $F$ extends over relatively compact subsets of $M$ .

The natural operators lifting projectable vector fields to some fiber product preserving bundles

W. M. Mikulski (2003)

Annales Polonici Mathematici

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Admissible fiber product preserving bundle functors F on $ℱ ℳ_{m}$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ ℳ_{m}$ all natural operators $B : T_{p r o j | ℱ ℳ_{m, n}} \to T F$ lifting projectable vector fields to F are classified.

Linear liftings of affinors to Weil bundles

Jacek Dębecki (2003)

Colloquium Mathematicae

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We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A} M$ , where $T^{A}$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

Displaying similar documents to “A splitting theorem for the Kupka component of a foliation of 𝐂 𝐏 n , n ≥ 6 . Addendum to a paper by O. Calvo-Andrade and N. Soares”

Displaying similar documents to “A splitting theorem for the Kupka component of a foliation of $𝐂 𝐏^{n}, n \geq 6$ . Addendum to a paper by O. Calvo-Andrade and N. Soares”