Searcharxiv⌕ Search

arXiv subjects

Huayi Chen

Publications and source records attributed to Huayi Chen.

At least 19 recordsLinked to original sources
arXiv2023

Harder-Narasimhan games

In this article, we study the notion of semi-stability and the Harder-Narasimhan filtration from a game-theoretic point of view. This allows us to provide a unified proof for the existence and uniqueness of the Harder-Narasimhan filtration in various settings and offer a conceptual interpretation for semi-stability conditions. As an application, we establish the existence and uniqueness of a coprimary filtration for modules of finite type over a commutative Noetherian ring and interpret it as a Harder-Narasimhan filtration.

math.AG↗
arXiv2022

Equidistribution theorem over an adelic curve

In this article, we introduce the notion of global adelic space of an arithmetic variety over an adelic curve and prove an equidistribution theorem for a generic sequence of subvarieties. As an application, we prove a Bogomolov type theorem for Abelian varieties over an adelic curve of characteristic 0.

math.NT↗
arXiv2022

Hilbert-Samuel formula and positivity over adelic curves

We establish, in the setting of Arakelov geometry over adelic curves, an arithmetic Hilbert-Samuel theorem describing the asymptotic behaviour of the metrized graded linear series of an adelic line bundle in terms of its arithmetic intersection number. We then study positivity conditions of adelic line bundles.

math.AG↗
arXiv2021

Arithmetic intersection theory over adelic curves

We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can be written as an integral of local intersection numbers along the adelic curve. The integrability of the local intersection number is justified by using the theory of resultants.

math.AG↗
arXiv2020

Arakelov theory on arithmetic surfaces over a trivially valued field

In this article, we consider an analogue of Arakelov theory of arithmetic surfaces over a trivially valued field. In particular, we establish an arithmetic Hilbert-Samuel theorem and studies the effectivity up to R-linear equivalence of pseudoeffective metrised R-divisors.

math.AG↗
arXiv2019

Arakelov geometry over adelic curves

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.

math.AG↗
arXiv2018

On subfiniteness of graded linear series

Hilbert's 14th problem studies the finite generation property of the intersection of an integral algebra of finite type with a subfield of the field of fractions of the algebra. It has a negative answer due to the counterexample of Nagata. We show that a subfinite version of Hilbert's 14th problem has a confirmative answer. We then establish a graded analogue of this result, which permits to show that the subfiniteness of graded linear series does not depend on the function field in which we consider it. Finally, we apply the subfiniteness result to the study of geometric and arithmetic graded linear series.

math.AG↗
arXiv2017

Sufficient conditions for the dirichlet property

The effectivity up to R-linear equivalence (Dirichlet property) of pseudoeffective adelic R-Cartier divisors is a subtle problem in arithmetic geometry. In this article, we propose sufficient conditions for the Dirichlet property by using the dynamic system in the classic Arakelov geometry setting. We also give a numerical criterion of the Dirichlet property for adelic R-Cartier divisors on curves over a trivially valued field.

math.AG↗
arXiv2014

Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties

In this paper, we study obstructions to the Dirichlet property by two approaches: density of non-positive points and functionals on adelic R-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic R-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of the Dirichlet property.

math.AG↗
arXiv2013

Distribution of logarithmic spectra of the equilibrium energy

Let $L$ be a big invertible sheaf on a complex projective variety, equipped with two continuous metrics. We prove that the distribution of the eigenvalues of the transition matrix between the $L^2$ norms on $H^0(X,nL)$ with respect to the two metriques converges (in law) as $n$ goes to infinity to a Borel probability measure on $\mathbb R$. This result can be thought of as a generalization of the existence of the energy at the equilibrium as a limit, or an extension of Berndtsson's results to the more general context of graded linear series and a more general class of line bundles. Our approach also enables us to obtain a $p$-adic analogue of our main result.

math.MG↗
arXiv2009

Okounkov bodies of filtered linear series

We associate to a filtration of a graded linear series of a big line bundle a concave function on the Okounkov body whose law with respect to Lebesgue's measure describes the asymptotic distribution of the jumps of the filtration. As a consequence we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to filtrations by minima in the usual context of Arakelov geometry, thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and the arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain by a variant of this construction a short proof of the existence of the sectional capacity.

math.AG↗
arXiv2008

Équidistribution et différentiabilité

We propose a criterion of equidistribution by the differentiability of certain arithmetic invariants. Combined with the slope method and the asymptotic measures, this criterion gives a new "conceptual" proof to equidistribution results originally obtained via the variation principle.

math.AG↗
arXiv2008

Differentiability of the arithmetic volume function

We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants.

math.AG↗
arXiv2008

Arithmetic Fujita approximation

We prove an arithmetic analogue of Fujita's approximation theorem in Arakelov geometry, conjectured by Moriwaki, by using slope method and measures associated to $\mathbb R$-filtrations.

math.AG↗
arXiv2008

Positive degree and arithmetic bigness

We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit. As applications, we prove a conjecture of Moriwaki asserting that the arithmetic volume function is actually a limit instead of a sup-limit, and we show how to compute the asymptotic polygon of a Hermitian line bundle, by using the arithmetic volume function.

math.AG↗