Rationality of the instability parabolic and related results

抛物丛模空间的有理性

Rationality of Moduli Spaces of Parabolic Bundles

证明了亏格大于1的光滑曲线上行列式固定的抛物丛的模空间是有理的，只要与拟抛物结构相关的重数之一等于1.证明了若秩与度互质，则向量丛的模空间是稳定有理的，并且在该水平上得到的界强到足以在许多情况下得出有理的结论。

关于抛物联络模空间的周群

A remark on the Moduli of parabolic connections

本文研究亏格至少为3的紧Riemann曲面上有理一般权重抛物联络的模空间，确定底层抛物丛稳定情形的周群，并讨论其有理性与有理连通性。

曲线上具有额外结构的向量丛的有理性与Poincare族

Rationality and Poincare families for vector bundles with extra structure on a curve

利用King和Schofield的方法，证明了曲线上向量丛模栈上的迭代Grassmannian丛是仿射空间乘以0次向量丛模栈的双有理丛.其应用包括某些Brill-Noether轨迹的双有理型，具有抛物结构或水平结构的向量丛的模格式，以及A. Schmitt修饰的向量丛进一步的结果涉及Poincare族在模格式的有限覆盖上的存在性。

Zeta functions for function fields

We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group, maximal parabolic subgroup). Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given. We end this paper with an explanation on why these zetas are non-abelian in nature, using our up-coming works on 'parabolic reduction, stability and the mass'. The constructions and results were anno…

Closed Orbits and uniform S-instability in Geometric Invariant Theory

In this paper we consider various problems involving the action of a reductive group $G$ on an affine variety $V$. We prove some general rationality results about the $G$-orbits in $V$. In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of $G$ for such general $G$-actions. We apply our general rationality results to answer a question of Serre concerning the behaviour of his notion of $G$-complete reducibility under separable field extensions. Applications of our new optimality results also include a construction which allows us…

主丛的正特征结构群的扩张

Extension of structure groups of principal bundles in positive characteristic

本文研究了光滑射影簇$X$上半稳定主$G$ -丛在结构群正特征扩张下的性质。我们推广了Ramanan-Ramanathan \cite{Ramanan-Ramanathan}关于不稳定标志合理性的一些结果，证明了通过$G$表示的相关向量丛不是太不稳定，并且不稳定性可以由一个与半稳定丛无关的常数有界.作为一个结果，这一系列的同构类的半稳定$G$ -丛与固定度和陈类的有界性被证明。