Hyperplanes In General Position Research Articles

A uniqueness theorem for holomorphic curves on annulus sharing hypersurfaces

Recently, some authors have given the uniqueness theorems for the holomorphic curves on an annulus in hyperplanes in general position and hypersurfaces in general position for Veronese embedding. In this paper, we will give a uniqueness theorem in hypersurfaces in general position.

On Absolute and Quantitative Subspace Theorems

Abstract The Absolute Subspace Theorem, a vast generalization and a quantitative improvement of Schmidt’s Subspace Theorem, was first established by Evertse and Schlickewei and then strengthened remarkably by Evertse and Ferretti. We study quantitative generalizations and extensions of subspace theorems in various contexts. We establish a generalization of Evertse and Ferretti’s Absolute Subspace Theorem for hyperplanes in general position. We obtain improved (non-absolute) Quantitative Subspace Theorems for hyperplanes in general position and in subgeneral position. We show a Semi-quantitative Subspace Theorem for hyperplanes in non-subdegenerate position.

On the finiteness of meromorphic mappings sharing few hyperplanes without multiplicities

In this paper, we show that the set of meromorphic mappings from Cm to Pn(C) which share 2n+1 hyperplanes in general position regardless of multiplicities has at most two elements, provided n≥12.

Degeneracy and finiteness problems for holomorphic curves from a disc into $\mathbf{P}^n(C)$ with finite growth index

Let $f^1,f^2,f^3$ are three holomorphic curves from a complex disc $\Delta (R)$ into $\mathbf{P}^n(\mathbf{C})\ (n\ge 2)$ with finite growth indexes $c_{f^1},c_{f^2},c_{f^3}$ and sharing $q (q \ge 2n+2)$ hyperplanes in general position regardless of multiplicity. In this paper, we will show the above bounds for the sum $c_{f^1}+c_{f^2}+c_{f^3}$ to ensure that $f^1\wedge f^2\wedge f^3=0$ or there are two curves among $\{f^1,f^2,f^3\}$ coincide to each other. Our results are generalizations of the previous degeneracy and finiteness results for linearly non-degenerate meromorphic mappings from $\mathbf{C}^m$ into $\mathbf{P}^n(\mathbf{C})$ sharing $(2n+2)$ hyperplanes regardless of multiplicities.

On the Weyl–Ahlfors theory of derived curves

For derived curves intersecting a family of decomposable hyperplanes in subgeneral position, we obtain an analog of the Cartan–Nochka Second Main Theorem, generalizing a classical result of Fujimoto about decomposable hyperplanes in general position.

Algebraic dependences and finiteness of meromorphic mappings sharing $2n+1$ hyperplanes with truncated multiplicities

In this paper, we give some results on the algebraic dependence and finiteness of meromorphic mappings from $\mathbf{C}^m$ into $\mathbf{P}^n(\mathbf{C})$ sharing $2n+1$ hyperplanes in general position with truncated multiplicities to level $n$, where all zeros with multiplicities more than certain values do not need to be counted.

Results on Uniqueness Problem for Meromorphic Mappings Sharing Moving Hyperplanes in General Position Under More General and Weak Conditions

The aim of the paper is to deal with the algebraic dependence and uniqueness problem for meromorphic mappings by using the new second main theorem with different weights involved the truncated counting functions, and some interesting uniqueness results are obtained under more general and weak conditions where the moving hyperplanes in general position are partly shared by mappings from ℂn into ℙN (ℂ), which can be seen as the improvements of previous well-known results.

Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space

Abstract In this article, we prove a new generalization of uniqueness theorems for meromorphic mappings of a complete Kähler manifold M into ℙ n (ℂ) sharing hyperplanes in general position with a general condition on the intersections of the inverse images of these hyperplanes.

On uniqueness of meromorphic mappings from complete Kähler manifold into ℙn(ℂ)

In this paper, we prove a uniqueness theorem for meromorphic mappings of a complete Kähler manifold [Formula: see text] into [Formula: see text] sharing hyperplanes in general position under a general condition that the codimension of the intersection of inverse images of any [Formula: see text] hyperplanes is at least two.

Meromorphic mappings of a complete connected Kähler manifold into a projective space sharing hyperplanes

ABSTRACT Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball in (). In this article, we will show that if three meromorphic mappings of M into satisfying the condition and sharing hyperplanes in general position regardless of multiplicity with certain positive constants K and (explicitly estimated), then or or . Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to level n + 1 then Our results generalize the finiteness and uniqueness theorems for meromorphic mappings of into sharing 2n + 2 hyperplanes in general position with truncated multiplicity.

Uniqueness theorem of meromorphic mappings of a complete Kähler manifold into a projective space

In this article, we prove a uniqueness theorem for meromorphic mappings of a complete Kähler manifold $M$ into $\mathbb{P}^n(\mathbb{C})$ sharing hyperplanes in general position with a general condition on the intersections of the inverse images of these hyperplanes.

On the Nevanlinna-Cartan Second Main Theorem for non-Archimedean Holomorphic Curves

Recenty, J. M. Anderson and A. Hinkkanen ([2]) introduced the integrated reduced counting functions for holomorphic curves and proved an improved version of second main theorem for holomorphic curves with integrated reduced counting functions in the complex case. In this paper, we will prove a version of second main theorem for non-Archimedean holomorphic curves intersecting hyperplanes in general position with integrated reduced counting functions.

A Uniqueness Theorem of Meromorphic Maps and a Generalization of the Borel’s Lemma

This paper gives a generalization of the classical Borel’s Lemma. Then as an application of this generalized Borel’s Lemma, a uniqueness theorem for two linearly non-degenerate meromorphic maps of ℂm into ℙn(ℂ) (n ≥ 2) sharing 2n + 2 hyperplanes in general position is proved.

A Difference Analogue of Cartan’s Second Main Theorem for Meromorphic Mappings

In this paper, we prove a difference analogue of Cartan’s second main theorem for a meromorphic mapping on ℂm intersecting a finite set of fixed hyperplanes in general position on ℙn(ℂ). As an application, we prove a uniqueness theorem for a class of holomorphic curves by inverse images of n + 4 hyperplanes. This result is so far the best result about the uniqueness problem for holomorphic curves by inverse images of hyperplanes.

Degeneracy and Finiteness Theorems for Meromorphic Mappings in Several Complex Variables

In this article, we prove that there are at most two meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 2)$ sharing $2n+2$ hyperplanes in general position regardless of multiplicity, where all zeros with multiplicities more than certain values do not need to be counted. We also show that if three meromorphic mappings $f^1,f^2,f^3$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 5)$ share $2n+1$ hyperplanes in general position with truncated multiplicity then the map $f^1\times f^2\times f^3$ is linearly degenerate.

THE CRAMÉR–WOLD THEOREM ON QUADRATIC SURFACES AND HEISENBERG UNIQUENESS PAIRS

Two measurable sets $S,\unicode[STIX]{x1D6EC}\subseteq \mathbb{R}^{d}$ form a Heisenberg uniqueness pair, if every bounded measure $\unicode[STIX]{x1D707}$ with support in $S$ whose Fourier transform vanishes on $\unicode[STIX]{x1D6EC}$ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathbb{R}^{d}$. As a corollary we obtain a new, surprising version of the classical Cramér–Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients.

Value distribution of leafwise holomorphic maps on complex laminations by hyperbolic Riemann surfaces

We discuss the value distribution of Borel measurable maps which are holomorphic along leaves of complex laminations. In the case of complex lamination by hyperbolic Riemann surfaces with an ergodic harmonic measure, we have a defect relation appearing in Nevanlinna theory. It gives a bound of the number of omitted hyperplanes in general position by those maps.

Second Main Theorem and Unicity of Meromorphic Mappings for Hypersurfaces in Projective Varieties

Let V be a projective subvariety of \(\mathbb P^{n}(\mathbb C)\). A family of hypersurfaces \(\{Q_{i}\}_{i=1}^{q}\) in \(\mathbb P^{n}(\mathbb C)\) is said to be in N-subgeneral position with respect to V if for any 1≤i1<⋯<iN+1≤q, \( V\cap (\bigcap _{j=1}^{N+1}Q_{i_{j}})=\varnothing \). In this paper, we will prove a second main theorem for meromorphic mappings of \(\mathbb C^{m}\) into V intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give a uniqueness theorem for meromorphic mappings of \(\mathbb C^{m}\) into V sharing a few hypersurfaces without counting multiplicity. In particular, we extend the uniqueness theorem for linearly nondegenerate meromorphic mappings of \(\mathbb C^{m}\) into \(\mathbb P^{n}(\mathbb C)\) sharing 2n+3 hyperplanes in general position to the case where the mappings may be linearly degenerated.

A modification of the Nevanlinna–Cartan theory for holomorphic curve

In this paper, we prove some fundamental theorems for holomorphic curves on intersecting a finite set of fixed hyperplanes in general position in with modified counting and characteristic functions.

Motivic structures on higher homotopy groups of hyperplane arrangements

In this note, we show the existence of motivic structures on certain objects arising from the higher (rational) homotopy groups of non-nilpotent spaces. Examples of such spaces include several families of hyperplane arrangements. In particular, we construct an object in Nori’s category of motives whose realization is a certain completion of \(\pi _{n}({\mathbb P}^{n} {\setminus } \{L_{1}, \ldots , L_{n+2}\})\) where the \(L_{i}\) are hyperplanes in general position. Similar results are shown to hold in Vovoedsky’s setting of mixed motives.