Transcendental Liouville Inequalities on Projective Varieties

Abstract Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non-vanishing integral global section of a hermitian line bundle over $X$ is zero or it cannot be too small with respect to the $\sup $ norm of the section itself. We study inequalities similar to Liouville’s for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

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Arithmetic hyperbolicity: automorphisms and persistence

Mathematische Annalen ◽

10.1007/s00208-021-02155-0 ◽

2021 ◽

Author(s):

Ariyan Javanpeykar

Keyword(s):

Automorphism Group ◽

Number Field ◽

Projective Variety ◽

Rational Points ◽

General Criterion ◽

Finitely Generated ◽

Integral Points ◽

Lang's Conjecture ◽

Lang’S Conjecture

AbstractWe show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

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On the Negativity of Moduli Spaces for Polarized Manifolds

Vietnam Journal of Mathematics ◽

10.1007/s10013-021-00507-6 ◽

2021 ◽

Author(s):

Kang Zuo

Keyword(s):

Line Bundle ◽

Moduli Spaces ◽

Complex Analysis ◽

Projective Varieties ◽

Canonical Line Bundle ◽

Characteristic Varieties ◽

Almost All ◽

Complex Projective ◽

Made In

AbstractGiven a log base space (Y, S), parameterizing a smooth family of complex projective varieties with semi-ample canonical line bundle, we briefly recall the construction of the deformation Higgs sheaf and the comparison map on (Y, S) made in the work by Viehweg–Zuo. While almost all hyperbolicities in the sense of complex analysis such as Brody, Kobayashi, big Picard and Viehweg hyperbolicities of the base U = Y ∖ S (under some technical assumptions) follow from the negativity of the kernel of the deformation Higgs bundle we pose a conjecture on the topological hyperbolicity on U. In order to study the rigidity problem we then introduce the notions of the length and characteristic varieties of a family f : X → Y, which provide an infinitesimal characterization of products of sub log pairs in (Y, S) and an upper bound for the number of subvarieties appearing as factors in such a product. We formulate a conjecture on a characterization of non-rigid families of canonically polarized varieties.

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Special varieties in adjunction theory and ample vector bundles

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100004771 ◽

2001 ◽

Vol 130 (1) ◽

pp. 61-75 ◽

Cited By ~ 10

Author(s):

A. LANTERI ◽

H. MAEDA

Keyword(s):

Vector Bundles ◽

Projective Variety ◽

Global Section ◽

Smooth Projective Variety ◽

Ample Vector Bundle ◽

Projective Varieties ◽

Adjunction Theory ◽

Set Up ◽

Important Position ◽

Smooth Projective Varieties

In this paper varieties are always assumed to be defined over the field [Copf ] of complex numbers.Given a smooth projective variety Z, the classification of smooth projective varieties X containing Z as an ample divisor occupies an extremely important position in the theory of polarized varieties and it is well-known that the structure of Z imposes severe restrictions on that of X. Inspired by this philosophy, we set up the following condition ([midast ]) in [LM1] in order to generalize several results on ample divisors to ample vector bundles:([midast ]) [Escr ] is an ample vector bundle of rank r [ges ] 2 on a smooth projective variety X of dimension n such that there exists a global section s ∈ Γ([Escr ]) whose zero locus Z = (s)0 is a smooth subvariety of X of dimension n − r [ges ] 1.

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On the density of rational points on elliptic fibrations

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1999.511.87 ◽

1999 ◽

Vol 1999 (511) ◽

pp. 87-93 ◽

Cited By ~ 16

Author(s):

F. A Bogomolov ◽

Yu Tschinkel

Keyword(s):

Number Field ◽

Algebraic Variety ◽

General Type ◽

Finite Extension ◽

Rational Points ◽

Projective Varieties ◽

Elliptic Fibrations ◽

Geometric Invariants ◽

Smooth Projective Varieties

1. Introduction Let X be an algebraic variety defined over a number field F. We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

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LINEAR SYSTEMS ON IRREGULAR VARIETIES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748019000069 ◽

2019 ◽

Vol 19 (6) ◽

pp. 2087-2125 ◽

Cited By ~ 3

Author(s):

Miguel Ángel Barja ◽

Rita Pardini ◽

Lidia Stoppino

Keyword(s):

Continuous Function ◽

Line Bundle ◽

Linear Systems ◽

Projective Variety ◽

Geometrical Properties ◽

Complex Projective Variety ◽

Wide Range ◽

Image Position ◽

Irregular Varieties ◽

Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

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Galois criterion for torsion points of Drinfeld modules

International Journal of Number Theory ◽

10.1142/s1793042121500901 ◽

2021 ◽

pp. 1-12

Author(s):

Chien-Hua Chen

Keyword(s):

Maximal Ideal ◽

Abelian Varieties ◽

Number Fields ◽

Rational Points ◽

Drinfeld Module ◽

Torsion Point ◽

Drinfeld Modules ◽

Torsion Points ◽

Ideal Formula ◽

Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

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On the projective varieties associated with some subrings of the ring of Thetanullwerte

Nagoya Mathematical Journal ◽

10.1017/s002776300000475x ◽

1994 ◽

Vol 133 ◽

pp. 71-83 ◽

Cited By ~ 2

Author(s):

Riccardo Salvati Manni

Keyword(s):

Line Bundle ◽

Abelian Variety ◽

Torsion Points ◽

Projective Varieties ◽

Symmetric Line

Let (X, L) be a principally polarized abelian variety (ppav) of dimension g such that L is a symmetric line bundle, i.e. i*L ≃ L where i is the inversion map i(x) = — x. We shall denote by X[2] the two torsion points of X which are fixed by i. For any x in X[2] we have an isomorphism(1) .

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Newton–Okounkov Bodies of Flag Varieties and Combinatorial Mutations

International Mathematics Research Notices ◽

10.1093/imrn/rnaa276 ◽

2020 ◽

Author(s):

Naoki Fujita ◽

Akihiro Higashitani

Keyword(s):

Mirror Symmetry ◽

Projective Variety ◽

Fundamental Problem ◽

Flag Varieties ◽

Systematic Method ◽

Fano Manifolds ◽

Projective Varieties ◽

Combinatorial Properties ◽

Toric Degenerations ◽

Higher Rank

Abstract A Newton–Okounkov body is a convex body constructed from a projective variety with a globally generated line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of projective varieties. Its combinatorial properties heavily depend on the choice of a valuation, and it is a fundamental problem to relate Newton–Okounkov bodies associated with different kinds of valuations. In this paper, we address this problem for flag varieties using the framework of combinatorial mutations, which was introduced in the context of mirror symmetry for Fano manifolds. By applying iterated combinatorial mutations, we connect specific Newton–Okounkov bodies of flag varieties including string polytopes, Nakashima–Zelevinsky polytopes, and Feigin–Fourier–Littelmann–Vinberg polytopes.

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