Siegel尖点形式的斜率与紧化Kuga簇的几何

Universal abelian variety and Siegel modular forms

We prove that the ring of Siegel modular forms of weight divisible by g+n+1 is isomorphic to the ring of (log) pluricanonical forms on the n-fold Kuga family of abelian varieties and its certain compactifications, for every arithmetic group for a symplectic form of rank 2g>2. We also give applications to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety has canonical singularities.

全纯微分形式的 Siegel 算子

Siegel operators for holomorphic differential forms

本文利用环面紧化与全纯 Leray 滤过，几何解释 Siegel 模簇上微分形式的 Siegel 算子，并建立多种边界消失概念的等价性。

The Kodaira dimension of Siegel modular varieties of genus 3 or higher

We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\geq3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an explicit lower bound on $n$ that is sufficient for the compactified moduli space to be of general type if one further explicit condition is satisfied.

Siegel 旗簇在 Chow 环中的前推

Pushforward of Siegel flag varieties in the Chow ring

本文计算了 Siegel 情形下子旗簇在 Chow 环中的类，并指出其对应于阿贝尔簇紧化模空间 tautological 环中支撑在环面边界上的类生成元，猜想该对应关系并非巧合。

The geometry of Siegel modular varieties

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties; classification of the compactified varieties by Kodaira dimension, etc.; moduli of abelian surfaces and especially applications of the lifting of Jacobi forms to modular forms; projective models of some special Siegel modular 3-folds; non-principally polarized abelian surfaces; and constructing degenerating families of abelian varieties.

论 Siegel 模簇的上同调维数与形式 Siegel 模形式的模性

On the cohomological dimension of Siegel modular varieties and the modularity of formal Siegel modular forms

本文证明亏格g≥2时Siegel模簇的凝聚上同调维数至多为g(g+1)/2-2，由此推出其紧化边界满足Grothendieck-Lefschetz条件，从而形式Siegel模形式自动成为经典模形式，推广了Bruinier和Raum的工作。