Global structure of holomorphic webs on surfaces

Vincent Cavalier; Daniel Lehmann

Displaying similar documents to “Global structure of holomorphic webs on surfaces”

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let $Ω$ an open set in $C^{4}$ near $z_{0} \in \partial Ω$ , $λ$ a suitable holomorphic function near $z_{0}$ . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\overline{\partial} u = λ f$ , ( $f$ is a $(0, 1)$ form, $\overline{\partial}$ closed in $U (z_{0})$ in $U (z_{0})$ with supp $(u) \subset \overline{Ω} \cap U (z_{0})$ , then we deduce an extension result for $C . R .$ functions on $\partial Ω \cap U (z_{0})$ , as holomorphic fonctions in $Ω \cap V (z_{0})$ .

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let $D_{j} \subset ℂ^{k_{j}}$ be a pseudoconvex domain and let $A_{j} \subset D_{j}$ be a locally pluriregular set, j = 1,...,N. Put $X : = ⋃_{j = 1}^{N} A ₁ \times . . . \times A_{j - 1} \times D_{j} \times A_{j + 1} \times . . . \times A_{N} \subset ℂ^{k ₁ + . . . + k_{N}}$ . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f ̂ |}_{X ∖ M} = f$ . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

The non-pluripolarity of compact sets in complex spaces and the property $(L B^{\infty})$ for the space of germs of holomorphic functions

Le Mau Hai, Tang Van Long (2002)

Studia Mathematica

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The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $(L B^{\infty})$ for the dual space of the space of germs of holomorphic functions on that compact set.

On the geometry of the space of m-pairs of holomorphic subspaces of the n-dimensional projective space $P_{n}$ ove r an albebra $𝒜$

Е.М. Кузнецова (1986)

Trudy Geometriceskogo Seminara

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On spaces of holomorphic functions in ℂⁿ

Diana D. Jiménez S., Lino F. Reséndis O., Luis M. Tovar S. (2014)

Banach Center Publications

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Following the line of Ouyang et al. (1998) to study the $_{p}$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among $_{p} (ₙ)$ , the α-Bloch, the Dirichlet $_{p}$ and the little $_{p, 0}$ spaces.

Flat 3-webs of degree one on the projective plane

A. Beltrán, M. Falla Luza, D. Marín (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The aim of this work is to study global $3$ -webs with vanishing curvature. We wish to investigate degree $3$ foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree $3$ foliations whose Legendre transform are webs with zero curvature.

Proper holomorphic self-mappings of the minimal ball

Nabil Ourimi (2002)

Annales Polonici Mathematici

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The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f: D → Ω with branch locus $V_{f}$ is factored by automorphisms if and only if $f_{*} (π ₁ (D ∖ f^{- 1} (f (V_{f})), x))$ is a normal subgroup of $π ₁ (Ω ∖ f (V_{f}), b)$ for some $b \in Ω ∖ f (V_{f})$ and $x \in f^{- 1} (b)$ .

Quasi-completeness on the Spaces of Holomorphic Germs

Roberto Luiz Soraggi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Sia $E$ uno spazio $D F$ riflessivo e sia $K$ un compatto di $E$ . Si dimostra che lo spazio dei germi olomorfi su $K$ , con la topologia naturale, è un limite induttivo regolare e quasi completo purché lo spazio dei germi olomorfi all'origine sia un limite induttivo regolare.

On L₁-subspaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2010)

Studia Mathematica

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We study the spaces $H_{μ} (Ω) = f : Ω \to ℂ h o l o m o r p h i c : \int_{0}^{R} \int_{0}^{2 π} | f (r e^{i φ}) | d φ d μ (r) < \infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{μ} (Ω)$ is either isomorphic to l₁ or to ${(\sum \oplus A ₙ)}_{(1)}$ . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

A set on which the local Łojasiewicz exponent is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F = (f ₁, . . ., f ₘ) : U \to ℂ^{m}$ , F(0) = 0, the Łojasiewicz exponent ₀(F) is attained on the set z ∈ U: f₁(z)·...·fₘ(z) = 0.

Biholomorphic maps determined on the boundary

Nozomu Mochizuki (1977)

Annales de l'institut Fourier

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Let $D$ be a bounded domain in $C^{n}$ such that the boundary $b D$ is topologically $S^{2 n - 1}$ in $R^{2 n}$ with a regular point; let $f : \tilde{D} \to C^{n}$ be a holomorphic map where $\tilde{D}$ is a neighborhood of $\overline{D}$ . If $f$ is one-to-one when restricted to $b D$ , then $f : D \to f (D)$ is biholomorphic.

Approximation of sets defined by polynomials with holomorphic coefficients

Marcin Bilski (2012)

Annales Polonici Mathematici

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Let X be an analytic set defined by polynomials whose coefficients $a ₁, . . ., a_{s}$ are holomorphic functions. We formulate conditions on sequences $a_{1, ν}, . . ., a_{s, ν}$ of holomorphic functions converging locally uniformly to $a ₁, . . ., a_{s}$ , respectively, such that the sequence $X_{ν}$ of sets obtained by replacing $a_{j}$ ’s by $a_{j, ν}$ ’s in the polynomials converges to X.

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...

On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in $𝐂^{n}$

Jean Erik Björk (1974)

Annales de l'institut Fourier

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Let $V$ be an algebraic variety in $C^{n}$ and when $k \geq 0$ is an integer then $Pol (V, k)$ denotes all holomorphic functions $f (z)$ on $V$ satisfying $| f (z) | \leq C_{f} (1 + | z |)^{k}$ for all $z \in V$ and some constant $C_{f}$ . We estimate the least integer $ϵ (V, k)$ such that every $f \in Pol (V, k)$ admits an extension from $V$ into $C^{n}$ by a polynomial $P (z_{1}, ..., z_{n})$ , of degree $k + ϵ (V, k)$ at most. In particular ${lim}_{k > \infty} ϵ (V, k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$ . The existence of the finite integer $ϵ (V, k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem. ...

Remarks on absolutely regular and regular $K^{'} M_{p}$ -distributions

S. Pilipović (1991)

Annales Polonici Mathematici

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nbspAbstract. We give the definition of $K^{'} M_{P}$ regular distributions and some characterizations of absolutely regular and regular $K^{'} M_{p}$ -distributions.

The space $N_{*}$ of holomorphic functions on bounded symmetric domains

Manfred Stoll (1976)

Annales Polonici Mathematici

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The genera, reflexibility and simplicity of regular maps

Marston Conder, Jozef Širáň, Thomas Tucker (2010)

Journal of the European Mathematical Society

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This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to $g - 1$ , where $g$ is the genus, all orientably-regular maps of genus $p + 1$ for $p$ prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and...