On Volumes of Arithmetic Line Bundles II

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The 'arithmetic extension groups' we introduce are extensions by groups of analytic types of the usual extension groups attached to $Ø_X$-modules over an arithmetic scheme $X$. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over $X$ - and we especially consider the case when $X$ is an 'arithmetic curve', namely the spectrum $\Spec Ø_K$ of the ring of integers in some number field $K$. T…

Very stable extensions on arithmetic surfaces

Given a line bundle L on a smooth projective curve over the complex numbers, we show that a general extension E of L by the trivial line bundle is very stable: line bundles contained in E have degree much less than half the degree of E. From this result we deduce new inequalities for the successive minima of the euclidean lattice H^1(X,L^{-1}), where L is an hermitian line bundle on the arithmetic surface X.

Effective Bounds of Linear Series on Algebraic Varieties and Arithmetic Varieties

In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert--Samuel formula and the arithmetic Hilbert--Samuel formula.

Arithmetically nef line bundles

Let $L$ be a line bundle on a scheme $X$, proper over a field. The property of $L$ being nef can sometimes be "thickened", allowing reductions to positive characteristic. We call such line bundles arithmetically nef. It is known that a line bundle $L$ may be nef, but not arithmetically nef. We show that $L$ is arithmetically nef if and only if its restriction to its stable base locus is arithmetically nef. Consequently, if $L$ is nef and its stable base locus has dimension $1$ or less, then $L$ is arithmetically nef.

An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms. As an application, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.

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.

On the Irreducibility and Distribution of Arithmetic Divisors

We introduce the notion of $ε$-irreducibility for arithmetic cycles meaning that the degree of its analytic part is small compared to the degree of its irreducible classical part. We will show that for every $ε>0$ any sufficiently high tensor power of an arithmetically ample hermitian line bundle can be represented by an $ε$-irreducible arithmetic divisor. Our methods of proof also allow us to study the distribution of divisors of small sections of an arithmetically ample hermitian line bundle $\overline{\mathcal{L}}$. We will prove that for increasing tensor powers $\overline{\mathcal{L}}^{\o…

Heights for line bundles on arithmetic surfaces

For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by the Theta divisor.

Chern invariants of some flat bundles in the arithmetic Deligne cohomology

In this note, we investigate the Chern classes of flat bundles in the arithmetic Deligne Cohomology, introduced by Green-Griffiths, Asakura-Saito. We show nontriviality of the Chern classes in some cases and the proof also indicates that generic flat bundles (in its moduli space) can be expected to have nontrivial classes, which is in contrast with some Gauss-Manin systems.

On arithmetic intersection numbers on self-products of curves

We give a close formula for the Néron-Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number $\hatω^2$ of the dualizing sheaf of a curve in terms of Zhang's invariant $φ$. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the cu…

Arithmetic ampleness and an arithmetic Bertini theorem

Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal L}^{\otimes n}$ that define an irreducible divisor on $\mathcal X$ tends to $1$ as $n$ tends to $\infty$. We prove variants of this statement for schemes mapping to such an $\mathcal X$. On the way to these results, we discuss some general properties of arithmetic ampleness, including restriction theorems, and upper bounds for the number of effective sections o…

算术大性与一致的Bogomolov型结果

Arithmetic bigness and a uniform Bogomolov-type result

本文证明了曲线泛族的容许典范丛是大的adelic线丛，并基于此证明了所有特征global fields上曲线的一致Bogomolov型定理，为一致Mordell-Lang型结果提供了新方法。

Continuity of volumes on arithmetic varieties

We introduce the volume function for hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef hermitian invertible sheaf. We also give another several applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.

Stability and Arithmetic

Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for such studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using semi-stable parabolic bundles & for p-adic number fields using what we call semi-stable filtered (phi,N;omega)-modules; and non-abelian zeta functions for function fields over finite fields using semi-stable bundles & for number fields using semi-stable lattices.

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.

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 result of this paper is a generalized Hodge index theorem on arithmetic toric varieties.

Continuous extension of arithmetic volumes

This paper is the sequel of the paper "Continuity of volumes on arithmetic varieties", in which we established the arithmetic volume function of smooth hermitian Q-invertible sheaves and proved its continuity. The continuity of the volume function has a lot of applications as treated in the paper as above. In this paper, we would like to consider its continuous extension over R.

Arithmetic volumes of unitary Shimura curves

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.

Higgs Bundles in Geometry and Arithmetic

We introduce the notions of deformation Higgs bundle and Riemann-Finsler metric on the moduli space of polarized varieties. We also use the Higgs-de Rham flow in the p-adic setting. These are the key novelties in our program.

The arithmetic volume of the moduli space of abelian surfaces

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 $g=2$.

Arithmetic aspects of moduli of sheaves on curves

We describe recent work on the arithmetic properties of moduli spaces of stable vector bundles and stable parabolic bundles on a curve over a global field. In particular, we describe a connection between the period-index problem for Brauer classes over the function field of the curve and the Hasse principle for rational points on etale forms of such moduli spaces, refining classical results of Artin and Tate. Submitted to proceedings of a Clay workshop on vector bundles on curves in honor of Newstead.

adelic线丛的整体多重位势理论

Global pluripotential theory for adelic line bundles

本文将拟射影算术簇上的adelic线丛理论与整体Berkovich空间的多重位势理论联系起来，建立了adelic线丛子范畴与解析化上配备连续多次下调和度量的线丛的等价关系，并推广了Monge-Ampère测度的构造与应用。

环面簇中超曲面的算术体积与Mahler测度

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。

Semi-stable extensions on arithmetic surfaces

On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for some successive minima in the lattice of extension classes between these line bundles.

Majorations explicites de fonctions de Hilbert-Samuel géométrique et arithmétique

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.

模堆栈的算术体积

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。

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.

酉 Shimura 簇的算术体积

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。

Remarks on the arithmetic restricted volumes and the arithmetic base loci

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.

凹变换与adelic线丛的体积

Concave transforms and volumes of adelic line bundles

研究adelic线丛的凹变换积分与特征体积关系，建立一维恒等式和高维不等式，给出Yuan猜想反例。

On the concavity of the arithmetic volumes

In this note, we study the differentiability of the arithmetic volumes along arithmetic R-divisors, and give some equality conditions for the Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big arithmetic R-divisors.

Modular embeddings and automorphic Higgs bundles

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings and apply our criteria to study the effect of abstract field automorphisms of $\mathbb{C}$ on modular embeddings. Finally we derive that the absolute Galois group of $\mathbb{Q}$ operates on the dessins d'enfants defined by principal congruence subgroups of (arithmetic or non-ar…

Vector bundles and Arakelov geometry on the projective line over the integers

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.

On the arithmetic Chern character

We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supports on the finite fibers. Next, we compute these classes in the situation encountered by the second author when proving a "Kodaira vanishing theorem" for arithmetic surfaces.

Arithmetic behaviour of Frobenius semistability of syzygy bundles for plane trinomial curves

Here we consider the set of bundles $\{V_n\}_{n\geq 1}$ associated to the plane trinomial curves $k[x,y,z]/(h)$. We prove that the Frobenius semistability behaviour of the reduction mod $p$ of $V_n$ is a function of the congruence class of $p$ modulo $2λ_h$ (an integer invariant associated to $h$). As one of the consequences of this, we prove that if $V_n$ is semistable in characteristic 0, then its reduction mod $p$ is strongly semistable, for $p$ in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles $V_n$, there is a common Zariski dense set of such …

Arithmetic statistics on cubic surfaces

In this paper we compute the distributions of various markings on smooth cubic surfaces defined over the finite field $\mathbb{F}_q$, for example the distribution of pairs of points, `tritangents' or `double sixes'. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers.

关于曲线算术亏格为一的纤维化canonical bundle公式

On canonical bundle formula for fibrations of curves with arithmetic genus one

本文在特征p>0下研究相对维数为一的纤维化canonical bundle公式，区分分离与非分离情况，并证明klt对在特定条件下为A的纤维空间。

线丛的算术变形

Arithmetic Deformation of Line Bundles

本文提出了一种研究线丛混合特征变形的新方法，通过构造特定子方案，证明正特征纤维中的线丛在该子方案外可提升至特征零。

Differentiability of the arithmetic volume function along the base conditions

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.

Secant varieties and successive minima

Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C.Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A.Granville on the divisibility properties of binomial coefficients in a given line of Pascal's triangle.

On the Bertini regularity theorem for arithmetic varieties

Let $\mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $\overline{\mathcal{L}}$. We prove that the proportion of global sections $σ$ with $\left\lVert σ\right\rVert_{\infty}<1$ of $\overline{\mathcal{L}}^{\otimes d}$ whose divisor does not have a singular point on the fiber $\mathcal{X}_p$ over any prime $p<e^{\varepsilon d}$ tends to $ζ_{\mathcal{X}}(1+\dim \mathcal{X})^{-1}$ as $d\rightarrow \infty$.

算术Hodge指标定理在拟射影簇上的局部版本

A local version of the arithmetic Hodge index theorem over quasiprojective varieties

本文提出拟射影簇上度量线丛的局部相交数，证明局部算术Hodge指标定理，并由此得到Monge-Ampère方程在阿基米德与非阿基米德情形下的唯一性。

Shukas的算术体积与兰格兰兹对偶性

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。

Computations of Bott-Chern classes on P(E)

We compute the Bott-Chern classes of the metric Euler sequence describing the relative tangent bundle of the variety P(E) of hyperplans of a holomorphic hermitian vector bundle (E,h) on a complex manifold. We give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle and to the computation of the height of P(E) with respect to the tautological quotient bundle O_E(1).

Geometric and arithmetic aspects of rational elliptic surfaces

PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem regarding the set of fibers whose Mordell-Weil ranks are at least 3 units higher than the generic rank of the elliptic fibration.

Stability of Arakelov bundles and tensor products without global sections

This paper deals with Arakelov vector bundles over an arithmetic curve, i.e. over the set of places of a number field. The main result is that for each semistable bundle E, there is a bundle F such that $E \otimes F$ has at least a certain slope, but no global sections. It is motivated by an analogous theorem of Faltings for vector bundles over algebraic curves and contains the Minkowski-Hlawka theorem on sphere packings as a special case. The proof uses an adelic version of Siegel's mean value formula.

Second Chern numbers of vector bundles and higher adeles

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic $K$-theory and depends on the canonical $\mathbb{Z}$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.

On the reductive Borel-Serre compactification: $L^p$-cohomology of arithmetic groups (for large $p$)

The $L^2$-cohomology of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the $L^p$-cohomology of an arithmetic quotient, for $p$ finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are …

Arithmetic positivity on toric varieties

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…

算术θ不变量与算术平衡测度

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。

Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety -- provided this coherent sheaf is a subquotient of a hermitian vector bundle -- and using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of t…

Arithmetic Bogomolov-Gieseker's inequality

Let f : X --> Spec(Z) be an arithmetic variety of dimension d >= 2 and (H, k) an arithmetically ample Hermitian line bundle on X. Let (E, h) be a rank r vector bundle on X. In this paper, we will prove that if E is semistable with respect to H on each connected component of the infinite fiber of X, then { c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 } c_1(H, k)^{d-2} >= 0. Moreover, if the equality of the above inequality holds, then E is projectively flat and h is a weakly Einstein-Hermitian metric.

Periodicity of Hitchin's uniformizing Higgs bundles

We link the periodicity of Hitchin's uniformizing Higgs bundle with the arithmetic geometry of its underlying curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.

Arithmetic geometry of toric varieties. Metrics, measures and heights

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…