A new necessary condition for analytic varieties satisfying the local Phragmén-Lindelöf condition

Tobias Heinrich

Displaying similar documents to “A new necessary condition for analytic varieties satisfying the local Phragmén-Lindelöf condition”

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety $X$ over a field $K$ is of Hilbert type if $X (K)$ is not thin. We prove that if $f : X \to S$ is a dominant morphism of $K$ -varieties and both $S$ and all fibers $f^{- 1} (s)$ , $s \in S (K)$ , are of Hilbert type, then so is $X$ . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

Local analytic solutions of the functional equation $Φ (z) = H (z; Φ [f_{1} (z)], . . ., Φ [f_{m} (z)])$

Wilhelmina Smajdor (1970)

Annales Polonici Mathematici

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Divisors in global analytic sets

Francesca Acquistapace, A. Díaz-Cano (2011)

Journal of the European Mathematical Society

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We prove that any divisor $Y$ of a global analytic set $X \subset ℝ^{n}$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$ . We also prove that there are functions with arbitrary multiplicities along $Y$ . The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X / Y$ is not connected and $Y$ represents the zero class in $H_{q - 1}^{\infty} (X, ℤ_{2})$ then the divisor $Y$ is globally principal.

Bruhat-Tits theory from Berkovich’s point of view. I. Realizations and compactifications of buildings

Bertrand Rémy, Amaury Thuillier, Annette Werner (2010)

Annales scientifiques de l'École Normale Supérieure

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We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $G$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $ℬ (G, k)$ to the Berkovich analytic space $G^{an}$ associated with $G$ . Composing this map with the projection of $G^{an}$ to its flag varieties, we define a family of compactifications of $ℬ (G, k)$ . This generalizes results by Berkovich in the case of split groups. Moreover,...

Radii of p-valence of certain analytic functions of order $α$ with negative coefficients

M. K. Aouf (1989)

Matematički Vesnik

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Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of $C^{n}$

J. Siciak (1969)

Annales Polonici Mathematici

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The Brauer group and the Brauer–Manin set of products of varieties

Alexei N. Skorobogatov, Yuri G. Zahrin (2014)

Journal of the European Mathematical Society

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Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$ , and let $\bar{X}$ and $\bar{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$ , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $(\bar{X}) \oplus$ Br( $\bar{Y})$ has finite index in the Galois invariant subgroup of Br $(\bar{X} \times \bar{Y})$ . This implies that the cokernel of the natural map Br $(X) \oplus$ Br $(Y) \to$ Br $(X \times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the...

Construction of Mendelsohn designs by using quasigroups of $(2, q)$ -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $q$ be a positive integer. An algebra is said to have the property $(2, q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2, q)$ if every member of $𝒱$ has the property $(2, q)$ . Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2, q)$ , $2 < q$ , blocks of Steiner system of type $(2, q)$ are obtained. The stated correspondence...

Asymptotics of eigensections on toric varieties

A. Huckleberry, H. Sebert (2013)

Annales de l’institut Fourier

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Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities $| ϕ_{n} |^{2} = | s_{N} |^{2} / | | s_{N} {| |}_{L^{2}}^{2}$ for eigensections $s_{N} \in Γ (X, L^{N})$ approaching a semiclassical ray. Here $X$ is a normal compact toric variety and $L$ is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate...

Polarizations of Prym varieties for Weyl groups via abelianization

Herbert Lange, Christian Pauly (2009)

Journal of the European Mathematical Society

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Let $π : Z \to X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group $G$ . For any dominant weight $λ$ consider the curve $Y = Z / Stab (λ)$ . The Kanev correspondence defines an abelian subvariety $P_{λ}$ of the Jacobian of $Y$ . We compute the type of the polarization of the restriction of the canonical principal polarization of $Jac (Y)$ to $P_{λ}$ in some cases. In particular, in the case of the group $E_{8}$ we obtain families of Prym-Tyurin varieties. The main idea is...

$J$ -invariant of linear algebraic groups

Viktor Petrov, Nikita Semenov, Kirill Zainoulline (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a semisimple linear algebraic group of inner type over a field $F$ , and let $X$ be a projective homogeneous $G$ -variety such that $G$ splits over the function field of $X$ . We introduce the $J$ -invariant of $G$ which characterizes the motivic behavior of $X$ , and generalizes the $J$ -invariant defined by A. Vishik in the context of quadratic forms. We use this $J$ -invariant to provide motivic decompositions of all generically split projective homogeneous $G$ -varieties, e.g. Severi-Brauer varieties,...

Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, $W \subset k^{n + 1}$ , and $F : k ⁿ \to k^{n + 1}$ such that ${F |}_{V} = f$ and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, $W \subset k^{m}$ with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

Properties of functions concerned with Caratheodory functions

Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $𝒫_{n}$ denote the class of analytic functions $p (z)$ of the form $p (z) = 1 + c_{n} z^{n} + c_{n + 1} z^{n + 1} + \dots$ in the open unit disc $𝕌$ . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p (z)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

On a problem concerning quasianalytic local rings

Hassan Sfouli (2014)

Annales Polonici Mathematici

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Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f \in_{m}$ and f̂ be its Taylor series at $0 \in ℝ^{m}$ . Split the set $ℕ^{m}$ of exponents into two disjoint subsets A and B, $ℕ^{m} = A \cup B$ , and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g, h \in_{m}$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some...

Non-supersingular hyperelliptic jacobians

Yuri G. Zarhin (2004)

Bulletin de la Société Mathématique de France

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Let $K$ be a field of odd characteristic $p$ , let $f (x)$ be an irreducible separable polynomial of degree $n \geq 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^{2} = f (x)$ and $J (C)$ its jacobian. We prove that $J (C)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n \geq 7$ or $p \neq 3$ .

The Briançon-Skoda number of analytic irreducible planar curves

Jacob Sznajdman (2014)

Annales de l’institut Fourier

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The Briançon-Skoda number of a ring $R$ is defined as the smallest integer k, such that for any ideal $I \subset R$ and $l \geq 1$ , the integral closure of $I^{k + l - 1}$ is contained in $I^{l}$ . We compute the Briançon-Skoda number of the local ring of any analytic irreducible planar curve in terms of its Puiseux characteristics. It turns out that this number is closely related to the Milnor number.

On certain subclasses of analytic functions associated with the Carlson–Shaffer operator

Jagannath Patel, Ashok Kumar Sahoo (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class $R^{λ} (a, c, A, B)$ of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass ${\tilde{R}}^{λ} (a, c, A, B)$ of $R^{λ} (a, c, A, B)$ and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

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We take up the study of the Brill-Noether loci $W^{r} (L, X) : = {η \in {Pic}^{0} (X) | h^{0} (L \otimes η) \geq r + 1}$ , where $X$ is a smooth projective variety of dimension $> 1$ , $L \in Pic (X)$ , and $r \geq 0$ is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for $h^{0} (K_{D})$ , where $D$ is a divisor that moves linearly on a smooth projective variety $X$ of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension $> 2$ . In the $2$ -dimensional case...

Elements of large order on varieties over prime finite fields

Mei-Chu Chang, Bryce Kerr, Igor E. Shparlinski, Umberto Zannier (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C (𝒱)$ such that for almost all primes $p$ for all but at most $C (𝒱)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

On Hattori spaces

A. Bouziad, E. Sukhacheva (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a subset $A$ of the real line $ℝ$ , Hattori space $H (A)$ is a topological space whose underlying point set is the reals $ℝ$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H (A)$ to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated...