On the zero locus of normal functions and the étale Abel-Jacobi map

阿贝尔 - 雅可比映射的（非）可扩展性

(Non-)Extendability of Abel-Jacobi Maps

研究阿贝尔 - 雅可比映射的“自然”定义轨迹，表明对于恰当几何约化曲线\(C\)，其从光滑轨迹\(C^{sm}\)到雅可比簇的映射，除特定情况外不扩展到更大曲线，由此得出该映射在特定情况外有闭像，对非\(\mathbb{P}^1\)不可约曲线是闭嵌入。

An Arakelov Inequality in Characteristic p and Upper Bound of p-Rank Zero Locus

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $g\geq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.

Gauss map on the theta divisor and Green's functions

In an earlier paper we constructed a Cartier divisor on the theta divisor of a principally polarised abelian variety whose support is precisely the ramification locus of the Gauss map. In this note we discuss a Green's function associated to this locus. For jacobians we relate this Green's function to the canonical Green's function of the corresponding Riemann surface.

Ekedahl-Oort层与超奇异轨迹的交会

Ekedahl-Oort strata meeting the supersingular locus

本文给出了特征p>2时主极化阿贝尔簇模栈中EO层与超奇异轨迹相交的显式组合判据，并证明了渐近几乎所有p-秩为零的EO层均相交，同时应用于酉PEL Shimura簇给出下界。

The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves

Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.

拓扑Abel-Jacobi映射与混合Hodge结构

Topological Abel-Jacobi Map and Mixed Hodge Structures

本文证明了Zhao定义的拓扑Abel-Jacobi映射与Schnell基于混合Hodge结构实分裂性质提出的替代构造相一致，回答了Schnell的问题。

关于尖点曲线上正规函数的秩

On the Rank of Normal Functions on Pointed Curves

本文研究尖点曲线模空间上正规函数的秩，证明其在阿贝尔微分分层外取极大值，在该分层上恰降一，并由此描述双分歧轨迹的局部几何。

Abel-Jacobi isomorphism for one cycles on Kirwan's resolution of the moduli space SU_C(2,O_C)

In this paper, we consider the moduli space $\cSU_C(r,\cO_C)$ of rank $r$ semistable vector bundles with trivial determinant on a smooth projective curve $C$ of genus $g$. When the rank $r=2$, F. Kirwan constructed a smooth log resolution $\ov{X}\rar \cSU_C(2,\cO_C)$. Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.to the normal crossing divisor), and a subsequent Abel-Jacobi map on the relative Chow group of null-homologous one cycles (tensored with $\Q$). This map takes values in the intermediate Jacobian of the compactly…

超奇异Kottwitz-Rapoport地层和Deligne-Lusztig变种

Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties

研究了具有Iwahori水平结构的Siegel模簇的正特征.在这些空间上，我们有牛顿分层和科特维茨-拉波波特分层;人们想知道这些分层是如何相互关联的。我们给出了一个简单的描述，所有KR地层完全包含在超奇异轨迹作为不相交工会的Deligne-Lusztig品种。我们还给出了一个明确的数值描述的KR分层的阿贝尔簇。