Differentiability of the arithmetic volume function
Huayi Chen
在Arakelov几何中引入了正交积，证明了算术体函数是连续可微的。作为应用，我们计算了厄米特线丛的渐近测度的分布函数和其他几个算术不变量。
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we show that the arithmetic volume function defined on the space of pairs of adelic R-Cartier divisors and base conditions is differentiable at a big pair, and that its derivative is given by an arithmetic restricted positive intersection number defined for the pair.
Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge--Ampère equations, and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.
Differentiability of the $χ$-volume function over an adelic curve
研究adelic曲线上adelic线丛丰沛锥中$χ$-体积函数的可微性，借助Witt Nyström结果的非Archimedes版本加以证明，并得到对数等分布应用。
On the concavity of the arithmetic volumes
本文研究了等差体积沿等差R-因子的可微性，并给出了nef和大等差R-因子锥上等差体积的Brunn-Minkowski不等式成立的几个等价条件.
Let $X$ be a normal projective variety over a complete discretely valued field and $L$ a line bundle on $X$. We denote by $X^\textrm{an}$ the analytification of $X$ in the sense of Berkovich and equip the analytification $L^\textrm{an}$ of $L$ with a continuous metric $\| \ \|$. We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of $L$. We prove that the non-archimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Ampère measure. Such a …
Differentiability of volumes of divisors and a problem of Teissier
本文给出了Boucksom，Demailly，Paun和Peternell首次引入的伪有效类的正积的代数构造，并利用它们证明了投射簇的Neron-Severi空间上的体积函数是（一次）可微的.微分表示为一个积极的产品，我们还将其与Ein等人和高山介绍的限制卷。然后，我们应用我们的可微性结果证明了代数几何版本的凸几何中的不等式，使我们能够表征的平等情况下的Khovanskii-Teissier不等式的nef和大类。
Continuity of volumes on arithmetic varieties
我们引入体积函数的厄米可逆层的算术品种作为一个模拟的几何体积函数。本文的主要结果是算术体积函数的连续性。作为结果，我们有算术希尔伯特-塞缪尔公式的nef厄米可逆层。我们还给出了它的一些应用，如推广的Hodge指标定理，算术Bogomolov-Gieseker不等式等。
Équidistribution et différentiabilité
我们提出了一个等分布的准则，某些算术不变的可微性。结合斜率法和渐近测度，该准则对原来通过变分原理得到的等分布结果给出了一个新的“概念性”证明。
On Volumes of Arithmetic Line Bundles
我们显示了一个算术推广的最近工作的Lazarsove-Mustata使用Okounkov机构研究线性系列的线丛。作为应用，我们得到了算术线丛体积的一个对数不等式和大线丛的一个算术Fujita逼近定理。
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 result of this paper is a generalized Hodge index theorem on arithmetic toric varieties.
Continuous extension of arithmetic volumes
本文是“算术簇上体积的连续性”一文的续篇，文中建立了光滑Hermitian Q-可逆层的算术体积函数，并证明了它的连续性。体积函数的连续性有许多应用，本文已作了讨论。在本文中，我们将考虑它在R上的连续扩张。
We compute the arithmetic volumes of integral models of unitary Shimura curves. This establishes the base case of an inductive argument to compute the arithmetic volumes of unitary Shimura varieties of higher dimension, to appear in subsequent work of Bruinier and the author.
Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized Hodge bundle $\overlineω_g$ on $\overline{\mathcal{A}}_g$. In 1999, Kühn proved a formula for the arithmetic volume of $\overline{\mathcal{A}}_1$ in terms of special values of the Riemann zeta function. In this article, we generalize his result to the case $g=2$.
The arithmetic volume of hypersurfaces in toric varieties and Mahler measures
该研究确定了光滑射影环面簇中超曲面的典范算术体积，并由此证明了这类超曲面上的广义Hodge指标定理，为算术代数几何领域相关问题提供了新结果。
On Volumes of Arithmetic Line Bundles II
对于算术簇上的埃尔米特线丛，我们构造了一个与线丛的泛纤维相关联的Okounkov体上的凸连续函数。连续函数的积分给出了厄米线丛的欧拉示性数的增长。这是Nystrom最近工作的全球版本。
Arithmetic volumes of moduli stacks of Shtukas
本文研究模堆栈中tautological classes的算术体积及其与Artin L函数导数的关系，并探讨phantom tautological ring的结构及其在函数域Colmez猜想中的应用。
Arithmetic volumes of unitary Shimura varieties
本文研究酉 Shimura 簇上度量线丛的算术体积，将其表示为 Dirichlet L-函数对数导数，并确定 Kudla-Rapoport 除子的算术体积及其与 Eisenstein 级数系数的关系。
In this paper, we collect some fundamental properties of the arithmetic restricted volumes (or the arithmetic multiplicities) of the adelically metrized line bundles. The arithmetic restricted volume has the concavity property and characterizes the arithmetic augmented base locus as the null locus. We also show a generalized Fujita approximation for the arithmetic restricted volumes.
Positive degree and arithmetic bigness
对于一般大的Hermitian线丛，我们建立了截断Harder-Narasimhan多边形的收敛性和极限的一致连续性。作为应用，我们证明了Moriwaki的一个猜想，即算术体积函数实际上是一个极限而不是一个上极限，并给出了如何利用算术体积函数计算Hermitian线丛的渐近多边形.
We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor, we give formulae for its arithmetic volume and its chi-arithmetic volume, and we characterize when it is arithmetically ample, nef, big or pseudo-effective, in terms of combinatorial data. As an application, we prove a Dirichlet's unit theorem on toric varieties, we give a characterization for the existence of a Zariski decomposition of a toric metrized R-divisor, and we prove a toric arithmetic F…
Arithmetic volume of Shtukas and Langlands duality
本文扩展了Feng-Yun-Zhang的工作，将Shukas的算术体积与ζ函数的导数联系起来，允许任意共权重，并首次在L-函数导数的公式中赋予对偶群结构作用。
Arithmetic theta invariants and arithmetic equilibrium measures
本研究引入算术簇上埃尔米特线丛的两个新算术不变量，导出内蕴算术平衡测度，证明其可检测算术正性、支配算术体积变分，为相关问题提供新的分析框架。
Existential closedness of $\overline{\mathbb{Q}}$ as a globally valued field via Arakelov geometry
本研究利用算术体积函数可微性与算术Bertini型定理分类闭点高度逼近问题，证明了$\overline{\mathbb{Q}}$作为全局赋值域（GVF）的存在封闭性，引入GVF泛函并给出Zhang不等式等号成立的新判据。
On an arithmetic inequality on $\mathbb{P}^1_{\mathbb{Q}}$
建立了$\mathbb{P}^1_{\mathbb{Q}}$上复曲面度量化因子的高度与$χ$ -算术体积的不等式.这部分地回答了布尔戈斯、森胁、菲利蓬和桑布拉的一个问题（[5，注释5.13]）。
We study arithmetic intersections on twisted (quaternionic) Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic twisted Hilbert modular surface. This quantity is then related to the arithmetic volume of a Shimura curve, via the arithmetic Grothendieck-Riemann-Roch theorem and the Jacquet-Langlands correspondence.
Transforming metrics on a line bundle to the Okounkov body
设$L$是复射影流形$X.$上的一个大的全纯线丛我们证明了如何将$L$的Okounkov体上的凸函数与$L.$上的任何连续度量$ψ$联系起来我们称之为$ψ,$的Chebyshev变换记为$c[ψ].$我们的主要定理指出$L$关于两个度量的度量体积之差，这是Berman-Boucksom引入的一个概念，等于度量的切比雪夫变换的差在Okounkov体上的积分。当度规具有正曲率时，度规体积与Monge-Ampère能量一致，这是Kähler-Einstein几何和Arakelov几何中的一个著名泛函。我们表明，这可以被看作是一个推广的经典结果的Chebyshev常数和勒让德变换的不变度量环面流形。作为应用，我们证明了度量体积在大度量$\mathbb{R}$ -因子锥中的可微性。这推广了Boucksom-Favre-Jonsson关于大$\mathbb{R}$ -因子的普通体积的可微性的结果和Berman-Boucksom关于当基础线丛固定时度量体积的可微性的结果。
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 of limit of it. Moreover, we show that the arithmetic volume is continuous and log concave.
Arithmetic linear series with base conditions
本文研究了具有基条件的算术线性级数的体积。作为一个应用，我们考虑了算术簇上的Zagliki分解问题。
In the setting of Arakelov geometry over adelic curves, we introduce the $χ$-volume function and show some general properties. This article is dedicated to talk about the continuity of $χ$-volume function. By discussing its relationship with volume function, we prove its continuity around adelic $\mathbb{Q}$-ample $\mathbb{Q}$-Cartier divisors and its continuity in the trivially valued case. The study of the variation of arithmetic Okounkov bodies leads us to its continuous extension on arithmetic surfaces.
The Siegel-Weil formula in geometry and arithmetic
本文从经典、几何和算术三个视角审视西格尔 - 韦伊公式，回顾正定格椭圆theta级数的公式，讨论任意符号格高亏格情形，给出几何西格尔 - 韦伊公式，还提及（部分推测性的）算术西格尔 - 韦伊公式，揭示特殊圈次数与艾森斯坦级数中心导数的联系。
In this paper, we introduce positivity notions for pairs of adelic R-Cartier divisors and R-base conditions, and study fundamental properties of the arithmetic volumes defined for such pairs. We show that the Gateaux derivatives of the arithmetic volume function at big pairs along the directions of adelic R-Cartier divisors are given by suitable arithmetic positive intersection numbers. As a corollary, we obtain an Arakelov theoretic analogue of the Bonnesen-Diskant inequality in convex geometry.
Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.
Low Genus Curves on Compact Shimura Surfaces
本文研究双曲2-球紧算术商上的曲线，利用体积估计及特殊子簇与奇点几何，证明大判别式时不存在任意固定亏格的复曲线。
Successive minima of toric height functions
给定整体域上复曲面簇上的复曲面度量化R-因子，我们给出了相应高度函数的本质极小值的一个公式。在适当的正性条件下，我们还给出了所有连续极小值的计算公式。我们应用这些结果的研究，在复曲面设置，连续极小值和其他算术不变量，如高度和算术体积之间的关系。我们还应用我们的公式计算连续极小的几个家庭的例子，包括加权射影空间，环面丛和子环面的平移。
Real Cubic Surfaces and Real Hyperbolic Geometry
稳定实三次曲面的模空间是实双曲四维空间与离散非算术群的商。在常曲率度量-1下，模空间的体积为37π^2/1080。模空间的五个连通分支中的每一个都可以用一个特定的算术群来描述为实双曲四维空间的商。我们计算这些组件的体积。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。