adelic曲线上$χ$-体积函数的可微性

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

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.

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。

Adelic Cartier divisors with base conditions and the Bonnesen-Diskant-type inequalities

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.

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.

Adelic divisors on arithmetic varieties

In this article, we generalize several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors and Zariski decompositions for arithmetic divisors on arithmetic surfaces, to the case of the adelic arithmetic divisors.

Differentiability of relative volumes over an arbitrary non-Archimedean field

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.

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。

Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)

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

We give an algebraic construction of the positive products of pseudo-effective classes first introduced by Boucksom, Demailly, Paun and Peternell, and use them to prove that the volume function on the Neron-Severi space of a projective variety is (once) differentiable. The differential is expressed as a positive product; we also relate it to the restricted volumes introduced by Ein et al and by Takayama. Then we apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-T…

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.

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。

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.

Transforming metrics on a line bundle to the Okounkov body

Let $L$ be a big holomorphic line bundle on a complex projective manifold $X.$ We show how to associate a convex function on the Okounkov body of $L$ to any continuous metric $ψ$ on $L.$ We will call this the Chebyshev transform of $ψ,$ denoted by $c[ψ].$ Our main theorem states that the difference of metric volume of $L$ with respect to two metrics, a notion introduced by Berman-Boucksom, is equal to the integral over the Okounkov body of the difference of the Chebyshev transforms of the metrics. When the metrics have positive curvature the metric volume coincides with the Monge-Ampère energy…