New perspectives in Arakelov geometry
Caterina Consani, Matilde Marcolli
在这次调查中，写的第七次会议的CNTA于2002年5月在蒙特利尔举行的会议记录，我们描述了如何康纳斯的光谱三重理论提供了一个统一的观点，通过非交换几何，阿基米德和完全分裂退化纤维的算术表面。
In this survey, written for the proceedings of the VII meeting of the CNTA held in May 2002 in Montreal, we describe how Connes' theory of spectral triples provides a unified view, via noncommutative geometry, of the archimedean and the totally split degenerate fibers of an arithmetic surface.
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
We study locally free sheaves of rank two on the projective line over the integers, especially indecomposable ones. Subsequently we apply various concepts of Arakelov geometry to these sheaves. We compute for example the arithmetic Chern classes and use the arithmetic Riemann-Roch theorem.
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.
Arakelov geometry of Cuntz-Pimsner algebras
利用C*-代数的K理论研究Arakelov几何，证明V的Picard群与关联的Cuntz-Pimsner代数的K0群同构，并应用于数域上代数簇的有限性问题。
A Hirzebruch proportionality principle in Arakelov geometry
证明了Arakelov几何中一个不动点公式的代数推广蕴含了Abel概型基的算术Chow环中重言子环的结果.其中的结果是一个阿拉克洛夫版本的Hirzebruch比例原则和公式的临界功率$\hat c_1$的霍奇束。
We study the semistability of the tensor product of hermitian vector bundles by using the $\varepsilon$-tensor product and the geometric (semi)stability of vector subspaces in the tensor product of two vector spaces.
In this paper we prove a concentration theorem for arithmetic $K_0$-theory, this theorem can be viewed as an analog of R. Thomason's result in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. Köhler and D. Roessler and they also gave a correct route of its proof. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.
We present an Arakelov theoretic version of the deformation to the normal cone. In particular, the geometric data is enriched with a deformation of a Hermitian line bundle. We introduce numerical invariants called arithmetic Hilbert invariants and prove the conservation of these invariants along the deformation. In a following article, this conservation of number theorem will allow a demonstration of the arithmetic Hilbert-Samuel theorem.
A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula
这是第二个一系列的文件处理一个类似的Arakelov几何的全纯Lefschetz不动点公式。我们使用的主要结果的第一篇论文证明剩余公式“`一拉博特”的算术特征类生活在算术品种采取行动的diagonalisable环面;最近的结果Bismut-Goette的等变（雷-辛格）分析扭转发挥了关键作用的证明。
A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties
给出了Spec Z上Chevalley格式的整表示的Jantzen和公式的一个新的证明.这是通过将Arakelov几何中Lefschetz型的不动点公式应用于广义旗簇来实现的。我们的证明涉及到复齐性空间上所有等变丛的等变Ray-Singer挠率的计算。此外，我们找到了几个显式公式的任何广义旗簇的整体高度。
Arakelov intersection indices of linear cycles and the geometry of buildings and symmetric spaces
本文将Manin的方法推广到射影空间上的线性周期无穷远处的Arakelov理论的几何解释。我们展示了如何解释某些非阿基米德Arakelov相交数的线性圈$P^{n-1}$与组合几何的Bruhat-Tits建设相关的PGL（n）。这种几何设置有一个阿基米德的类似物，即与SL（n，C）相关联的黎曼对称空间，我们用它来解释类似的阿基米德相交数的线性循环以类似的方式。
New Approach to Arakelov Geometry
这项工作是致力于一个新的完全代数的方法来阿拉克洛夫几何，这并不要求正在考虑的品种是一般光滑或投影。为了构造这样一种方法，我们发展了一种广义环和概型理论，它包括经典环和概型以及“奇异”对象，如F_1（“一元域”），Z_ \infty（“实整数”），T（热带数）等，从而提供了一种研究这些物体的系统方法。这种广义环和概型理论发展到代数K理论、相交理论和Chern类的构造。然后证明了Q上代数簇的Arakelov模型的存在性，并将我们的一般结果应用于这类模型。
The $θ$-density in Arakelov geometry
本文为射影算术簇上丰富Hermitian线丛构造了$θ$-密度，证明其行为与Arakelov几何中仅考虑范数小于1的整体截面的通常密度相似，并由此证明了关于不可约性和正则性的两个Bertini型定理的类似结论。
Geometrie d'Arakelov et hauteurs canoniques sur des varietes semi-abeliennes
半阿贝尔簇上的典范高度和Arakelov几何。本文在Arakelov几何的框架下，给出了交换簇的乘法群扩张上的典范高度的构造。这些规范高度是来自于阿贝尔簇的高度和我们称之为相对高度的总和。最后对相对高度为零的点给出了一些补充。
We study two kinds of push-forwards of $δ$-forms and define the pull-backs of $δ$-forms. As a generalization of Gubler-Künnemann, we prove the projection formula and the tropical Poincaré-Lelong formula. As an application, we follow the idea of Gubler-Künnemann and generalize the notion of $δ$-forms on algebraic varieties, this allows us to define the first Chern forms for any piecewise smooth metrics.
Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry
在阿拉克洛夫理论中，算术曲面的完备化是通过“无穷远处的闭纤维”的正式线性组合扩大因子群来实现的。马宁描述了任何这样的闭纤维的对偶图在一个无限纠缠的有界测地线在一个双曲双曲体赋予一个肖特基均匀化。本文研究数域中整数环上的算术曲面，其纤维亏格为$g\geq 2$。我们利用Connes的谱三重数理论将双曲几何的双曲体与Deninger的阿基米德上同调和本文第一作者所介绍的局部单值$N$在算术无穷远处的锥的上同调联系起来。
On the Arakelov Geometry of Moduli Spaces of Curves
在本文中，我们计算的度量的渐近线丛上的模空间的曲线，试图计算阿基米德高度的代数圈$C-C^-$的雅可比的光滑射影曲线属$g$时出现。表达结果的一种方法是说度量延伸（或多或少）到Moriwaki发现的线丛，该线丛在$\Mbar_g$中的每个完整曲线上都具有非负度，而不包含在边界中。
A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of C.M. abelian varieties
本文给出了P. Colmez定理的一个稍弱形式的新证明。这个定理给出了阿贝尔簇的Faltings高度的一个公式，它与一个CM作复乘。域，其在$\bf Q$上的伽罗瓦群是阿贝尔的;在椭圆曲线的情况下，它简化为Chowla和Selberg的公式。我们表明，该公式可以推导出的算术不动点公式证明了该系列的第一篇论文。我们的证明是内在的，因为它不依赖于任何特定的阿贝尔簇的周期的计算。
Arakelov geometry of toric bundles: Okounkov bodies and BKK
本文以 adelic 纤维丛观点研究环面丛，分析 Okounkov 体与 Boucksom-Chen 变换，并证明相交数公式的算术类比，用于计算半 Abel 簇环面紧化的高度和逐次最小值。
In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. …
Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion …
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。
Arithmetic Fujita approximation
利用斜率方法和与$\mathbb R$ -滤子有关的测度，证明了Moriwaki在Arakelov几何中Fujita逼近定理的一个算术模拟.
An arithmetic Riemann-Roch theorem in higher degrees
我们证明了一个类似的Arakelov几何的Grothendieck-Riemann-Roch定理。
This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surf…
Arakelov geometry on flag varieties over function fields and related topics
本文计算了函数域旗簇上高度函数的高度滤过、逐次极小值与凹变换，将高度视为逐次极小值加权平均以推广Zhang不等式，并揭示了增广基轨迹与高度滤过的等价性及旗丛的可动锥。
Differentiability of the arithmetic volume function
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 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.
Higher arithmetic K-theory
本文给出了Arakelov几何中高阶K理论的一个具体定义。本文定义的K-理论是Gillet和Soule定义的算术簇的算术K_0群的一个更高的推广。讨论了K-理论中的乘积和直接象。
In this paper, we build the global determinant method of Salberger by Arakelov geometry explicitly. As an application, we study the dependence on the degree of the number of rational points of bounded height in plane curves. We will also explain why some constants will be more explicit if we admit the Generalized Riemann Hypothesis.
A short survey on applications of algebraic geometry in topological data analysis.
We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.
Computer Algebra and Algebraic Geometry - Achievements and Perspectives
在这次调查中，我想介绍一些概念的代数几何，并试图证明富有成效的互动之间的代数几何和计算机代数，更普遍的是，数学和计算机科学。本文的目的之一是通过实例说明计算机代数对数学研究的有用性。
A Perspective on the Foundations of Derived Analytic Geometry
该研究提出在正合范畴中针对单纯对象范畴开展代数几何的方法，由此得到导出解析几何理论，为相关领域提供了新的理论视角。
The clear understanding of the non-convex landscape of neural network is a complex incomplete problem. This paper studies the landscape of linear (residual) network, the simplified version of the nonlinear network. By treating the gradient equations as polynomial equations, we use algebraic geometry tools to solve it over the complex number field, the attained solution can be decomposed into different irreducible complex geometry objects. Then three hypotheses are proposed, involving how to calculate the loss on each irreducible geometry object, the losses of critical points have a certain ran…
Logical Berkovich Geometry: A Point-free Perspective
本文通过无点技术强化了伯科维奇几何的基础结果，证明在非阿基米德域上收敛幂级数环的伯科维奇谱点对应$R$-好滤子，且无需域具有非平凡赋值，并探讨了拓扑与逻辑的关系。
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe …
A categorical perspective on non-abelian localization
本文从非阿贝尔局部化视角出发，为具有 $Θ$-分层的代数导出栈建立了虚拟 $K$-理论局部化公式及其范畴升级，并据此证明了普适穿墙公式与一维层栈上同调的有限性定理。
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical v…
Anabelian perspectives in Galois-Teichmüller theory
本文探讨了非交换几何在伽罗瓦-蒂希米歇尔理论中的应用，通过算法框架重构绝对伽罗瓦群。
The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- and Pin-structures of Fukaya-Oh-Ohta-Ono and Solomon, establishes properties of these structures in both perspectives, and again shows them to be suitably equivalent. The second part uses the intrinsic perspective on (relative) Spin- and Pin-structures to detail constructions of orientations on the determinants of real Cauchy-Riemann operator…
Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses
本文通过正性、可分性、量子幻方、非局部博弈和张量网络等例子，展示量子信息论与自由半代数几何的共同对象及结合双视角的价值。
What is... Random Algebraic Geometry?
本文综述随机代数几何，该领域将概率视角引入实代数几何，为希尔伯特第十六问题等经典难题提供现代解决途径。
A Bornological Perspective on the Representability of Derived Moduli Stacks of Solutions to PDEs
本文通过Bornological理论展示导出模栈的表示性无需复杂分析工具，提出新的微分几何模型以嵌入已有的Bornological几何理论。
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。