scholarly journals A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Bruno Scardua
Keyword(s):  
Periodic Orbits ◽  
Complex Manifold ◽  
Stability Theorem ◽  
Holomorphic Foliation ◽  
Holomorphic Foliations ◽  
Finitely Generated ◽  
Zero Measure

We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset.

HOLOMORPHIC FOLIATIONS AND CHARACTERISTIC NUMBERS

2005 ◽  
Vol 07 (05) ◽  
pp. 583-596 ◽  
Author(s):  
MARCIO G. SOARES
Keyword(s):  
Projective Space ◽  
Complex Manifold ◽  
Complex Projective Space ◽  
Holomorphic Foliation ◽  
Compact Complex Manifold ◽  
Holomorphic Foliations ◽  
Singular Set ◽  
One Dimensional ◽  
Characteristic Numbers ◽  
Complex Projective

We relate the characteristic numbers of the normal sheaf of a k-dimensional holomorphic foliation [Formula: see text] of a compact complex manifold Mn, to the characteristic numbers of the normal sheaf of a one-dimensional holomorphic foliation associated to [Formula: see text]. In case M is a complex projective space, we also obtain bounds for the degrees of the components of codimension k - 1 of the singular set of [Formula: see text].

scholarly journals Basic Cohomology of Canonical Holomorphic Foliations on Complex Moment-Angle Manifolds

2020 ◽  
Author(s):  
Hiroaki Ishida ◽  
Roman Krutowski ◽  
Taras Panov
Keyword(s):  
Complex Manifold ◽  
Toric Variety ◽  
Cohomology Ring ◽  
Betti Numbers ◽  
Holomorphic Foliation ◽  
Torus Action ◽  
Holomorphic Foliations ◽  
Basic Cohomology ◽  
Arbitrary Complex ◽  
Foliated Manifolds

Abstract We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

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.

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.

scholarly journals Takahasi semigroups

2017 ◽  
Vol 29 (5) ◽  
pp. 1145-1161
Author(s):  
Mário J. J. Branco ◽  
Gracinda M. S. Gomes ◽  
Pedro V. Silva
Keyword(s):  
Periodic Orbits ◽  
Free Group ◽  
Periodic Points ◽  
Finitely Generated ◽  
Completely Simple Semigroups ◽  
Clifford Semigroups ◽  
Completely Simple ◽  
Bounded Rank

AbstractTakahasi’s Theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations.

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.

scholarly journals Directed harmonic currents near hyperbolic singularities

2017 ◽  
Vol 38 (8) ◽  
pp. 3170-3187 ◽  
Author(s):  
VIÊT-ANH NGUYÊN
Keyword(s):  
Mass Distribution ◽  
Holomorphic Foliation ◽  
Holomorphic Foliations ◽  
Harmonic Current ◽  
Lelong Number ◽  
Harmonic Currents ◽  
Hyperbolic Singularity ◽  
Compact Complex Surfaces ◽  
Global Result ◽  
Local Result

Let $\mathscr{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^{2}$ having $0$ as a hyperbolic singularity. Let $T$ be a harmonic current directed by $\mathscr{F}$ which does not give mass to any of the two separatrices. We show that the Lelong number of $T$ at $0$ vanishes. Then we apply this local result to investigate the global mass distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.

scholarly journals HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

2018 ◽  
Vol 238 ◽  
pp. 104-136
Author(s):  
TAKAYUKI KOIKE
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Normal Bundle ◽  
Holomorphic Foliation ◽  
Compact Complex Manifold ◽  
Sufficient Condition ◽  
Existence Problem ◽  
Hermitian Metric ◽  
Flat Normal Bundle ◽  
Compact Submanifold

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

Adaptive Control of Transient Dynamics to Periodic Orbits

2014 ◽  
Vol 2 ◽  
pp. 82-85
Author(s):  
Hiroyasu Ando ◽  
Kazuyuki Aihara
Keyword(s):  
Adaptive Control ◽  
Periodic Orbits ◽  
Transient Dynamics

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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