Arakelov Geometry over Adelic Curves

  title={Arakelov Geometry over Adelic Curves},
  author={Huayi Chen and Atsushi Moriwaki},
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions. 
Arithmetic over trivially valued field and its applications
By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic
Volume function over a trivially valued field
We introduce an adelic Cartier divisor over a trivially valued field and discuss the bigness of it. For bigness, we give the integral representation of the arithmetic volume and prove the existence
The theta invariants and the volume function on arithmetic varieties
We introduce a new arithmetic invariant for hermitian line bundles on an arithmetic variety. We use this invariant to measure the variation of the volume function with respect to the metric. The main
A relative bigness inequality and equidistribution theorem over function fields
. — For any line bundle written as a subtraction of two ample line bundles, Siu’s inequality gives a criterion on its bigness. We generalize this inequality to a relative case. The arithmetic meaning
The continuity of $\chi$-volume functions over adelic curves
In the setting of Arakelov geometry over adelic curves, we introduce the $\chi$-volume function and show some general properties. This article is dedicated to talk about the continuity of
Arithmetic Okounkov bodies and positivity of adelic Cartier divisors
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov
Toward Fermat's conjecture over arithmetic function fields.
Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show
Infinite rank Hermitian Lattices and Loop Groups
— In this paper, we associate a family of infinite-rank pro-Hermitian Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac-Moody
Successive minima and asymptotic slopes in Arakelov Geometry
Let X be a normal and geometrically integral projective variety over a global field K and let D be an adelic Cartier divisor on X. We prove a conjecture of Chen, showing that the essential minimum
Categorification of Harder-Narasimhan Theory via slope functions
where the subquotients Ei/Ei−1 are semi-stable vector bundles with strictly decreasing slopes. Later Harder-Narasimhan filtration has been generalized to the setting of pure sheaves over higher


Zariski decompositions on arithmetic surfaces
In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of
On volumes of arithmetic line bundles
Abstract We show an arithmetic generalization of the recent work of Lazarsfeld–Mustaţǎ which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity
Adelic divisors on arithmetic varieties
theorie des diviseurs # groupe topologique # variete algebrique # theorie de l'approximation
An arithmetic Riemann-Roch theorem in higher degrees
Nous demontrons un analogue du theoreme de Grothendieck-Riemann-Roch en geometrie d'Arakelov.
Arithmetic height functions over finitely generated fields
Abstract.In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s
Stable vector bundles on algebraic surfaces
We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of a certain moduli space of stable bundles on a rational
The Bogomolov conjecture for totally degenerate abelian varieties
We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang’s proof of the number field case replacing the complex analytic tools by tropical
Semiample invertible sheaves with semipositive continuous hermitian metrics
Let (L, h) be a pair of a semiample invertible sheaf and a semipositive continuous hermitian metric on a proper algebraic variety. In this paper, we prove that (L, h) is semiample metrized, which is
The canonical arithmetic height of subvarieties of an abelian variety over a finetely generated field
This paper is the sequel of [2]. In [4], S. Zhang defined the canonical height of subvarieties of an abelian variety over a number field in terms of adelic metrics. In this paper, we generalize it to
Non-density of small points on divisors on abelian varieties and the Bogomolov conjecture
The Bogomolov conjecture for a curve claims finiteness of algebraic points on the curve which are small with respect to the canonical height. Ullmo has established this conjecture over number fields,