Adelic Chern Forms and Applications

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。

一种关于全实代数数域上Kudla-Millson提升的adele方法

An adelic approach to the Kudla-Millson lifts over totally real number fields

本文通过adele方法计算全实域上正交Shimura簇的Kudla-Millson theta提升的Fourier展开，证明其单射性并给出上同调维数下界。

环面丛的 Arakelov 几何：Okounkov 体与 BKK

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。

A Chern-Weil formula for the Chern character of a perfect curved module

Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN_0^{II}(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.

Chern forms of hermitian metrics with analytic singularities on vector bundles

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ with analytic singularities on a holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Ampère products on the projectivization of $E$. The Chern and Segre currents represent the Chern and Segre classes of $E$, respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where $h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases wh…

Chern obstructions for collections of 1-forms on singular varieties

We introduce a certain index of a collection of germs of 1-forms on a germ of a singular variety which is a generalization of the local Euler obstruction corresponding to Chern numbers different from the top one.

The Multiplicity Polar Theorem, collections of 1-forms and Chern numbers

In this work we show how the Multiplicity Polar Theorem can be used to calculate Chern numbers for a collection of 1-forms.

Chern classes of reductive groups and an adjunction formula

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. In the case where the complete intersection is a curve, this formula gives an explicit answer for the Euler characteristic and the genus of the curve.

Symmetric roots and admissible pairing

Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant. We provide a simple geometric interpretation for this invariant, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant introduced in this paper coincides with an invariant introduced in a recent paper by S.-W. Zhang.

纽结、素数与类域论

Knots, primes and class field theory

本文对类域论进行几何推广，通过阿代尔构造搭建关联概形与整体域整数环的几何框架，构造出将有理数域有限阿贝尔扩张映射到阿代尔类空间扇区有限覆叠的函子，建立了周期轨道单值群与素数处弗罗贝尼乌斯伽罗瓦作用的对应关系。

复 Bott-Chern 上同调中的广义 Lelong–Poincaré 公式

Generalised Lelong-Poincaré formula in complex Bott-Chern cohomology

本文用拓扑方法证明广义 Lelong–Poincaré 公式，表明在零点轨迹纯余维等于全纯向量丛秩时，最高陈类等于其概形零点轨迹在复 Bott-Chern 上同调中的闭链类。

Bott-Chern Forms and Arithmetic Intersections

Let \E : 0 --> S --> E --> Q --> 0 be a short exact sequence of hermitian vector bundles with metrics on S and Q induced from that on E. We compute the Bott-Chern form of \E corresponding to any characteristic class, assuming E is projectively flat. The result is used to obtain a new presentation of the Arakelov Chow ring of the arithmetic grassmannian.

环簇丛的陈类

Chern Classes of Toric Variety Bundles

本文解决了Khovanskii--Monin猜想，给出环簇丛切丛全陈类用基空间与环簇纤维陈类表示的公式，并推广至稳定复拟环流形，应用于环面螺旋球簇及Masuda公式的新证明。

Representability of Chern-Weil forms

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Griffiths by producing metrics whose Chern forms are positive. The second scenario deals with a particular rank-2 bundle (related to the vortex equations) over a product of a Riemann surface and the sphere.