Higher arithmetic K-theory
Yuichiro Takeda
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 a concrete definition of higher K-theory in Arakelov geometry is given. The K-thoery defined in this paper is a higher extension of the arithmetic K_0 group of an arithmetic variety defined by Gillet and Soule. Products and direct images in this K-theory are discussed.
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Arakelov theory of noncommutative arithmetic surfaces
本文的目的是在一个非对易的设置启动Arakelov理论。更确切地说，我们关心的是非交换算术曲面。我们介绍了一个版本的算术相交理论的非交换算术曲面，我们证明了算术黎曼-罗克定理在此设置。
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.
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.
In the early 2000's Levine and Morel have given a geometric construction of an algebraic cobordism group defined for all smooth quasi projective varieties over a field. We show how we can refine their construction to build an Arakelov version of this group for Arakelov varieties over a number field, and how this integrates well in the general philosophy of Arakelov geometry.
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.
Epsilon constants and equivariant Arakelov Euler characteristics
研究了支持有限群G的驯服作用的算术簇上厄米层的等变Arakelov-欧拉示性数.群作用的驯服使我们能够在一个特别精细的“射影”算术类群中产生一个等变的阿拉克洛夫-欧拉示性数。然后，我们表明，等变的Arakelov-欧拉示性数的各种复杂的微分确定的L-功能的动机从算术品种使用辛表示的群G的的常数。我们的结果可以被视为第一个高维版本的Cassou-Noguès泰勒辛Artin根数的厄米结构的整数环的特征，其次作为一个有符号的等变版本的布洛赫导体公式。
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.
We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from $BGL$ to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic $K$-theory and arithmetic Chow groups. Finally, we give a conceptual explanation of the height pairing: it is given by the natural pairing of motivic homology and Arakelov motivic cohomology.
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。
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.
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic Schubert calculus on these spaces. We also give a method to compute the natural arithmetic intersection numbers which arise, and prove that they are all rational numbers.
Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they are all rational numbers.
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。
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。
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result an…
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.
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。
New perspectives in Arakelov geometry
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。
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。
We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves. We show that for families of manifolds with ample canonical bundle this invariant is uniformly bounded. As a consequence we establish that such families over a base of arbitrary dimension verify the aforementioned Arakelov inequality, answering a question of Viehweg.
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。
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. …
Stability of Arakelov bundles and tensor products without global sections
本文研究算术曲线上的Arakelov向量丛，即数域的位集上的Arakelov向量丛。主要结果是，对于每个半稳定丛E，存在一个丛F，使得$E \otimes F$至少有一定的斜率，但没有整体瓣。它的动机是一个类似的定理的Faltings向量丛的代数曲线，并包含闵可夫斯基-Hlawka定理的球包装作为一个特殊情况。这个证明使用了Siegel平均值公式的一个Adelic版本。
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。
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。
For an arithmetic surface $X\to B=\operatorname{Spec} O_K$ the Deligne pairing $\left <\,,\,\right > \colon \operatorname{Pic}(X) \times \operatorname{Pic}(X) \to \operatorname{Pic}(B)$ gives the "schematic contribution" to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach we show that the Deligne pairing can be lifted to a pairing $\left<\,,\,\right>_i:\ker(d^1_\times)\times \ker(d^1_\times)\to\operatorna…
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…
Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $$ \det(Δ)=\exp(-ζ'(0)) $$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking the logarithm it is asymptotically upper bounded by $g$ for $g>1$. The construction of a cohomology theory for arithmetic surfaces in Arakelov theory has long been an open problem. In particular, it is not known if $h^{1}(X,L)\ge 0$. We view this as an indirect piece of evidence th…
Arithmetic Fujita approximation over adelic curves
本文在adelic曲线的Arakelov理论框架下，证明了Fujita近似定理的类比结果，验证了Chen Huayi和Moriwaki Atsushi的猜想。
Semi-stable extensions on arithmetic surfaces
在给定的算术曲面上，受Miyaoka工作的启发，我们考虑了向量丛，它是线丛的另一个线丛的扩张。给出了它们对一般纤维的限制是半稳定的充分条件。然后，我们应用Arakelov理论中Bogomolov不等式的算术模拟，并由此导出这些线丛之间的扩张类格中某些连续极小值的下界。
Segal's Gamma-rings provide a natural framework for absolute algebraic geometry. We use Almkvist's global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt functor-ring of Almkvist for the simplest Gamma-ring S. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between Lambda-rings and the Arithmetic site. Then, we concentrate on the Arakelov compactification of Z which acquires a structure sheaf of S-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a d…
We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.
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。
We give two kinds of generalizations of Arakelov type inequalities for higher dimensional families. These results give higher dimensional generalizations (in both fibers and bases) of the weakly boundedness in Paršin-Arakelov's reformulation of the geometric Shafarevich conjecture. As a consequence, we deduce the following results. Hyperbolicity: We give an alternative proof (using the theory of degeneration of Hodge structure) to the hyperbolicity (in Paršin-Arakelov's reformulation, i.e. Viehweg's hyperbolicity conjecture) of the base of a family with maximal variation whose general fibers a…
A map from the higher arithmetic $K$-group defined by the author to the higher arithmetic Chow group constructed by Burgos and Feliu is given. It is a higher extension of the arithmetic Chern character established by Gillet and Soulé, and it can also be regarded as an analogue of Beilinson's regulator in Arakelov geometry. It is shown that this map is compatible with the pull-back map and the product structure.
By using the $\mathbb R$-filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic projective varieties.
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。
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.
Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action of certain diagonalisable group scheme. This equivariant arithmetic K-theory is defined by means of a natural extension of Burgos-Wang's simplicial description of Beilinson's regulator map to the equivariant case. As a byproduct of this work, we give an analytic refinement of the Riemann-Roch theorem for higher equivariant algebraic K-theory. And as an appl…
Arithmetic invariants of Euclidean lattice
本文研究了欧几里得格式的算术不变量，探讨了其在阿雷科洛夫几何中的应用，包括算术类比、有限性定理及梅林变换的积分表示。
In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic conce…
Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers
我们定义和调查的范围内的Arakelov几何的扩张群。我们引入的“算术扩张群”是算术概型$X$上的$Ø_X$ -模上的通常扩张群的解析型群的扩张。在本文中，我们专注于第一算术扩张群-其元素可以描述在$X$上的厄米向量丛的容许短正合序列-我们特别考虑的情况下，$X$是一个“算术曲线”，即频谱$\Spec Ø_K$的整数环在一些数域$K$。然后研究$X$上的算术扩张与格和数的几何的新老问题有关。
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.
Polylogarithms, regulators and Arakelov motivic complexes
构造了正则复射影代数簇X的重n Bloch Higher Chow群复形到重n Deligne复形的一个显式调节器映射。我们定义的Arakelovweight n motivic复杂的圆锥这个地图移动了一个。它的最后一个上同调群是由H.吉列和C. Soule。本文将Grassmannian n-代数（定义见[G5]）与GL_n（C）的对称空间的几何联系起来。当n=2时，我们通过双对数恢复了理想测地四面体体积的罗巴切夫斯基公式。利用它与对称空间的关系，我们通过Grassmannian n-代数构造了K_{2n-1}（C）上的Borel调节子.本文研究了Chow双对数，证明了一个倒易律，它加强了曲线上Milnor K_3的Suslin倒易律。
Holography principle and arithmetic of algebraic curves
根据全息原理（由于G.特胡夫特湖Susskind，J. Maldacena等人），在某些有边界流形上的量子引力弦论可以用边界上的共形场论来研究。只有少数数学上精确的结果证实了这个令人兴奋的计划。在本文中，我们解释从这个角度来看，最初出现在算术几何代数曲线的几个建设。我们证明了在算术无穷远处的双曲几何和Arakelov几何之间的关系包含了与黑洞的欧几里德AdS_3全息图完全相同的几何数据。此外，在欧氏AdS_2全息的情况下，我们给出了边界为非对易空间的体/边界对应的一些结果。
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 bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic $\mathbb{R}$-Cartier divisor $\overline{D}$ are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized $\mathbb{R}$-divisors on toric varie…
We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, we study the Arakelov geometry of toric varieties. In particular, we consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. We show that these notions can be translated in terms of convex analysis, and are closely related to objects like polyhedral complexes, concave functions, real Monge-Ampère measures, and Legendre-Fen…
Numerical bounds for semi-stable families of curves or of certain higher dimensional manifolds
给定射影曲线Y的开子集U和光滑曲线族f：V->U，在Y上具有半稳定约化，我们证明了对于秩>2的Hodge结构的子变差，Arakelov不等式必是严格的.对于家庭的n倍，我们证明了一个类似的结果的假设下，（n，0）组成部分的希格斯束定义纤维的双有理映射。
In this paper we show that the regulator defined by Goncharov from higher algebraic Chow groups to Deligne-Beilinson cohomology agrees with Beilinson's regulator. We give a direct comparison of Goncharov's regulator to the construction given by Burgos and Feliu. As a consequence, we show that the higher arithmetic Chow groups defined by Goncharov agree, for all projective arithmetic varieties over an arithmetic field, with the ones defined by Burgos and Feliu.
Arithmetic Hodge structure and higher Abel-Jacobi maps
在本文中，我们利用[作者：动机与代数de Rham上同调]中定义的高阶Abel-Jacobi映射，给出了它在代数圈中的一些应用.特别地，我们证明了数域上代数圈上的Beilinson猜想蕴含曲面上零圈上的Bloch猜想。此外，我们构造了一个零圈的产品曲线的Mumford不变量为零，但不是高Abel-Jacobi不变。
In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.
We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The definition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soule, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The definition relies on the chain morphism representing Adams operations in higher algebraic K-theory given previously by the author. In this paper it is shown that a slight modification of this chain morphism commutes …
We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over a number field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a height pairing between two higher algebraic cycles, of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of the higher arithmetic intersection…
We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck's trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed.
Arithmetic Mixed Sheaves
给出了算术混合层的一种形式，包括算术混合Hodge结构的情形，并证明了某些高阶扩张群的非零性，以及在非平凡整体2-形式存在的条件下，任意维光滑真复簇上零圈的第二Abel-Jacobi映射的非平凡性.对于两个余维圈，圈映射的内射性退化为数域上光滑投射簇的Abel-Jacobi映射的内射性。(This表明Asakura的附加假设是不必要的）。这里也可以在循环映射和阿贝尔-雅可比映射的定义中使用实现系统。一些结论可以推广到更高的Chow群上，并证明了C的一个猜想。关于不可分解高等圈的可数性。
Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $λ_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmeti…
We discuss the following topics: n-dimensional local fields and adelic groups; harmonic analysis on local fields and adelic groups for two-dimensional schemes (function spaces, Fourier transform, Poisson formula); representations of discrete Heisenberg groups; adelic Heisenberg groups and their representations arising from two-dimensional schemes.
We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_q$. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the $\ell$-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree $n$ in the limit as $q \to \infty$. The results of this paper can be v…
A high-codimensional Yuan's inequality and its application to higher arithmetic degrees
本文推广袁氏算术Siu不等式至高余维，证明高维算术度存在性并建立相对度公式，给出Dang等人提出的广义Kawaguchi-Silverman猜想的反例。
Divisibility and torsion in higher Chow groups over arithmetic fields
本文研究算术域上光滑概形的高阶Chow群的可除性与无挠性，并应用于有限域、局部域和整体域。
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