The continuity of $χ$-volume functions over adelic curves

adelic曲线上$χ$-体积函数的可微性

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。

Convex Bodies associated to Linear series of Adelic Divisors on Quasi-Projective Varieties

In this article we define and study convex bodies associated to linear series of adelic divisors over quasi-projective varieties that have been introduced recently by Xinyi Yuan and Shou-Wu Zhang. Restricting our attention to big adelic divisors, we deduce properties of volumes obtained by Yuan and Zhang using different convex geometric arguments. We go on to define augmented base loci and restricted volumes of adelic divisors following the works of Michael Nakamaye and develop a similar study using convex bodies to obtain analogous properties for restricted volumes. We closely follow methods …

拟射影簇上的阿代尔线丛

Adelic line bundles on quasi-projective varieties

本书构建有限生成域上拟射影簇的阿代尔线丛理论，探讨其相交、体积与高度理论，并应用于拟射影簇代数点的高度研究。

Volume function over a trivially valued field

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 over trivially valued field and its applications

By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic $χ$-volume along semiample divisors. Moreover, we give a upper bound estimate of arithmetic Hilbert-Samuel function.

Zhang--Kawazumi的phi-不变量的渐近行为

Asymptotic Behavior of the Zhang--Kawazumi's phi-invariant

本文研究了退化图phi-不变量的连续性，证明其在Riemann曲面模空间上诱导adelic divisor，并利用Yuan--Zhang的adelic divisors给出了模空间边界附近Zhang--Kawazumi不变量的渐近表达式。

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.

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.

斜率有界性与等分布定理

Slope boundedness and Equidistribution theorem

本文证明了特征0函数域上adelic线丛极小斜率的有界性，并据此推导出一般点与小点的等分布定理，同时将方法扩展至数域有限位并证明了$χ$-体积的连续性。

拓扑阿德曲面：扎里斯基-里曼空间、代数覆盖、哈德纳-纳拉辛汉滤子和高度

Topological adelic curves: Zariski-Riemann spaces, algebraic coverings, Harder-Narsimhan filtrations and heights

本文引入拓扑阿德曲面，用于处理可能不可数域上的阿列克夫几何，并推广了迪奥菲勒特逼近与内万林纳理论的类比。通过伪绝对值概念，证明了拓扑阿德曲面的代数覆盖、哈德纳-纳拉辛汉滤子和体积函数的存在性，并定义了循环的高度及推广了内万林纳第一主定理。

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.

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.

On the semi-continuity problem of normalized volumes of singularities

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes can only jump down at countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Xu and the author, we show that K-semistability is a very generic or empty condition in any $\mathbb{Q}$-Gorenstein flat family of log Fano pairs.

关于射影直线上加权点的模空间Weil-Petersson体积的连续性

On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line

本文证明了射影直线上加权点的模空间中，从Fano几何趋近Calabi-Yau几何时，Weil-Petersson体积连续，且局部化计算的CM体积收敛到McMullen的几何体积。

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.