Displaying similar documents to “Twistor transforms of quaternionic functions and orthogonal complex structures”

Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

Annales Polonici Mathematici

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Let D j be a bounded hyperconvex domain in n j and set D = D × × D s , j=1,...,s, s ≥ 3. Also let Ω π be the image of D under the proper holomorphic map π. We characterize those continuous functions f : Ω π that can be extended to a real-valued pluriharmonic function in Ω π .

Equidistribution towards the Green current for holomorphic maps

Tien-Cuong Dinh, Nessim Sibony (2008)

Annales scientifiques de l'École Normale Supérieure

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Let f be a non-invertible holomorphic endomorphism of a projective space and f n its iterate of order n . We prove that the pull-back by f n of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to f when n tends to infinity. We also give an analogous result for the pull-back of positive closed ( 1 , 1 ) -currents and a similar result for regular polynomial automorphisms of  k .

Spaces of geometrically generic configurations

Yoel Feler (2008)

Journal of the European Mathematical Society

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Let X denote either ℂℙ m or m . We study certain analytic properties of the space n ( X , g p ) of ordered geometrically generic n -point configurations in X . This space consists of all q = ( q 1 , , q n ) X n such that no m + 1 of the points q 1 , , q n belong to a hyperplane in X . In particular, we show that for a big enough n any holomorphic map f : n ( ℂℙ m , g p ) n ( ℂℙ m , g p ) commuting with the natural action of the symmetric group 𝐒 ( n ) in n ( ℂℙ m , g p ) is of the form f ( q ) = τ ( q ) q = ( τ ( q ) q 1 , , τ ( q ) q n ) , q n ( ℂℙ m , g p ) , where τ : n ( ℂℙ m , g p ) 𝐏𝐒𝐋 ( m + 1 , ) is an 𝐒 ( n ) -invariant holomorphic map. A similar result holds true for mappings of the configuration...

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of P κ ( X ) , where κ ⊆ X. Suppose μ < κ, μ < μ = μ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in [ H ( λ ) ] μ which are internally club but not internally approachable.

On surfaces with p 𝑔 = q = 1 and non-ruled bicanonical involution

Carlos Rito (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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This paper classifies surfaces S of general type with p g = q = 1 having an involution i such that S / i has non-negative Kodaira dimension and that the bicanonical map of S factors through the double cover induced by i . It is shown that S / i is regular and either: a) the Albanese fibration of S is of genus 2 or b) S has no genus 2 fibration and S / i is birational to a K 3 surface. For case a) a list of possibilities and examples are given. An example for case b) with K 2 = 6 is also constructed.

Steinness of bundles with fiber a Reinhardt bounded domain

Karl Oeljeklaus, Dan Zaffran (2006)

Bulletin de la Société Mathématique de France

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Let E denote a holomorphic bundle with fiber D and with basis B . Both D and B are assumed to be Stein. For D a Reinhardt bounded domain of dimension d = 2 or 3 , we give a necessary and sufficient condition on D for the existence of a non-Stein such E (Theorem 1 ); for d = 2 , we give necessary and sufficient criteria for E to be Stein (Theorem 2 ). For D a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for E to be Stein (Theorem...

Regular coordinates and reduction of deformation equations for Fuchsian systems

Yoshishige Haraoka (2012)

Banach Center Publications

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For a Fuchsian system d Y / d x = ( j = p ( A j ) / ( x - t j ) ) Y , (F) t , t , . . . , t p being distinct points in ℂ and A , A , . . . , A p M ( n × n ; ) , the number α of accessory parameters is determined by the spectral types s ( A ) , s ( A ) , . . . , s ( A p ) , where A = - j = 1 p A j . We call the set z = ( z , z , . . . , z α ) of α parameters a regular coordinate if all entries of the A j are rational functions in z. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Equidistribution towards the Green current

Vincent Guedj (2003)

Bulletin de la Société Mathématique de France

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Let f : k k be a dominating rational mapping of first algebraic degree λ 2 . If S is a positive closed current of bidegree ( 1 , 1 ) on k with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks λ - n ( f n ) * S converge to the Green current T f . For some families of mappings, we get finer convergence results which allow us to characterize all f * -invariant currents.

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

Kuzman Adzievski (2006)

Annales Polonici Mathematici

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We study questions related to exceptional sets of pluri-Green potentials V μ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials V μ are defined by V μ ( z ) = B l o g ( 1 / | ϕ z ( w ) | ) d μ ( w ) , where for a fixed z ∈ B, ϕ z denotes the holomorphic automorphism of B satisfying ϕ z ( 0 ) = z , ϕ z ( z ) = 0 and ( ϕ z ϕ z ) ( w ) = w for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets

Juan Rivera-Letelier (2001)

Fundamenta Mathematicae

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Given d ≥ 2 consider the family of polynomials P c ( z ) = z d + c for c ∈ ℂ. Denote by J c the Julia set of P c and let d = c | J c i s c o n n e c t e d be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters c d : those for which the critical point 0 is not recurrent by P c and without parabolic cycles. The Hausdorff dimension of J c , denoted by H D ( J c ) , does not depend continuously on c at such c d ; on the other hand the function c H D ( J c ) is analytic in - d . Our first result asserts that there is still some...

On the n -fold symmetric product of a space with a σ - ( P ) -property c n -network ( c k -network)

Luong Q. Tuyen, Ong V. Tuyen (2020)

Commentationes Mathematicae Universitatis Carolinae

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We study the relation between a space X satisfying certain generalized metric properties and its n -fold symmetric product n ( X ) satisfying the same properties. We prove that X has a σ - ( P ) -property c n -network if and only if so does n ( X ) . Moreover, if X is regular then X has a σ - ( P ) -property c k -network if and only if so does n ( X ) . By these results, we obtain that X is strict σ -space (strict -space) if and only if so is n ( X ) .

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

Pointed k -surfaces

Graham Smith (2006)

Bulletin de la Société Mathématique de France

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Let S be a Riemann surface. Let 3 be the 3 -dimensional hyperbolic space and let 3 be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping ϕ : S 3 = ^ . If i : S 3 is a convex immersion, and if N is its exterior normal vector field, we define the Gauss lifting, ı ^ , of i by ı ^ = N . Let n : U 3 3 be the Gauss-Minkowski mapping. A solution to the Plateau problem ( S , ϕ ) is a convex immersion i of constant Gaussian curvature equal to k ( 0 , 1 ) such that the Gauss lifting ( S , ı ^ ) is complete and n ı ^ = ϕ . In this...

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let D be an open set of a Stein manifold X of dimension n such that H k ( D , 𝒪 ) = 0 for 2 k n - 1 . We prove that D is Stein if and only if every topologically trivial holomorphic line bundle L on D is associated to some Cartier divisor 𝔡 on D .

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let f m be the category of m -dimensional manifolds and local diffeomorphisms and  let T be the tangent functor on f m . Let 𝒱 be the category of real vector spaces and linear maps and let 𝒱 m be the category of m -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors F : 𝒱 m 𝒱 admitting f m -natural operators J ˜ transforming classical linear connections on m -dimensional manifolds M into almost complex structures J ˜ ( ) on F ( T ) M = x M F ( T x M ) .

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

On the asymptotics of counting functions for Ahlfors regular sets

Dušan Pokorný, Marc Rauch (2022)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the so-called Ahlfors regular sets (also known as s -regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit lim ε 0 + ε s N ( ε , K ) exist, where K is an s -regular set and N ( ε , K ) is for instance the ε -packing number of K ?

When C p ( X ) is domain representable

William Fleissner, Lynne Yengulalp (2013)

Fundamenta Mathematicae

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Let M be a metrizable group. Let G be a dense subgroup of M X . We prove that if G is domain representable, then G = M X . The following corollaries answer open questions. If X is completely regular and C p ( X ) is domain representable, then X is discrete. If X is zero-dimensional, T₂, and C p ( X , ) is subcompact, then X is discrete.

L p , q spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the L p , q spaces................................. 114. Integral representation of bounded linear functionals on L p , q ( B ) ........ 235. Examples in L p , q theory...................................................................................

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let  X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in  X . We consider the image 𝒩 1 ( D , X ) of  𝒩 1 ( D ) in  𝒩 1 ( X ) under the natural push-forward of 1 -cycles. We show that ρ X - ρ D codim 𝒩 1 ( D , X ) 8 . Moreover if codim 𝒩 1 ( D , X ) 3 , then either X S × T where S is a Del Pezzo surface, or codim 𝒩 1 ( D , X ) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X - ρ T = 4 .