关于Deligne-Simpson问题

某些簇的连通性与Deligne-Simpson问题

The connectedness of some varieties and the Deligne-Simpson problem

Deligne-Simpson问题（DSP）弱DSP）是这样公式化的：{ \em给出了选择共轭类$C_j\subset GL(n,{\bf C})$或$c_j\subset gl(n,{\bf C})$的充要条件，使得存在不可约的（分别为. $(p+1)$ -矩阵$M_j\in C_j$或$A_j\in c_j$的元组，满足等式$M_1... M_{p+1}=I$或$A_1+... +A_{p+1}=0$ }。矩阵$M_j$和$A_j$被解释为正则线性系统的单值算子和黎曼球面上Fuchsian算子的矩阵剩余。对于共轭类的$(p+1)$ -元组，其中一个共轭类具有不同的特征值：1）证明了若DSP对给定的共轭类正解，则簇$\{(M_1,..., M_{p+1})|M_j\in C_j,M_1... M_{p+1}=I\}$或$\{(A_1,..., A_{p+1})|A_j\in c_j,A_1+... +A_{p+1}=0\}$是连通的; 2）给出了弱DSP正解的充要条件.

驯顺的Deligne-Simpson问题

The tame Deligne-Simpson problem

本文证明了Crawley-Boevey对驯顺Deligne-Simpson问题猜想解的必要性，通过非阿贝尔Hodge对应和抛物权重变化将问题归结为仿射Dynkin图及其加顶点情形，并利用Kostov不存在性结果和简单表示的连通性分析完成证明。

说明弱Deligne-Simpson问题某些方面的例子

Examples illustrating some aspects of the weak Deligne-Simpson pro blem

我们考虑矩阵$A_j$的$(p+1)$ -元组的多样性（分别为. $M_j$）从给定的共轭类$c_j\subset gl(n,{\bf C})$（resp. $C_j\subset GL(n,{\bf C})$），使得$A_1+... +A_{p+1}=0$（分别$M_1... M_{p+1}=I$）。这个簇与弱{ \em Deligne-Simpson问题：给出了选择共轭类$c_j\subset gl(n,{\bf C})$的充要条件（分别为. $C_j\subset GL(n,{\bf C})$），使得存在具有矩阵$A_j\in c_j$的平凡中心化子的$(p+1)$ -元组（相应地，$M_j\in C_j$），其和等于0（分别其乘积等于$I$）。矩阵$A_j$（分别$M_j$）被解释为Fuchsian线性系统的矩阵-剩余（分别为作为正规线性系统的单值算子）在黎曼球面上。我们考虑的例子，这种品种的尺寸高于预期的，由于存在的$(p+1)$ -元组与非平凡的中心化;在一个例子中，两个维度之间的差异是O（n）。

关于Deligne-Simpson问题及其弱版本

On the Deligne-Simpson problem and its weak version

我们考虑{ \em Deligne-Simpson问题（DSP）（分别为.弱DSP）：给出了选择$p+1$共轭类$c_j\subset gl(n,{\bf C})$或$C_j\subset GL(n,{\bf C})$使得存在不可约$(p+1)$ -元组的充要条件。$(p+1)$ -具有平凡中心化器的元组）的和为零的矩阵$A_j\in c_j$或乘积为$I$的矩阵$M_j\in C_j$。}矩阵$A_j$（分别$M_j$）被解释为Fuchsian线性系统的矩阵-剩余（分别为作为正则线性系统的单值矩阵）的复时间微分方程。在本文中，我们给出了DSP的情况下，其中一个矩阵的特征值不同的情况下，可解的充分条件。

关于Deligne-Simpson问题的一些例子

Some examples related to the Deligne-Simpson problem

我们考虑矩阵$M_j$的$(p+1)$ -元组的多样性，从$GL(n,{\bf C})$的给定共轭类，使得$M_1... M_{p+1}=I$。这个品种是连接与Deligne-Simpson问题和矩阵$M_j$被解释为正规系统在黎曼球面上的单值算子。我们考虑在其他情况下，当尺寸的品种是高于预期的，由于存在的$(p+1)$ -元组与非平凡的中心。

The Deligne-Simpson problem for connections on $\mathbb{G}_m$ with a maximally ramified singularity

The classical additive Deligne-Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne-Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called…

关于\(\mathbb{P}^{1}\)上不规则\(G\)-联络的一个德利涅 - 辛普森问题

A Deligne-Simpson problem for irregular $G$-connections over $\mathbb{P}^{1}$

研究\(\mathbb{P}^{1}\)上不规则\(G\)-联络的德利涅 - 辛普森问题，给出代数和几何准则，用于解决经典群等斜问题及特定\(G\)的问题，对等上同调刚性联络分类并在多种类型中获新情况。

射影线上抛物丛的模堆叠、双曲表示和Deligne-Simpson问题

The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem

在“Hitchin的可积系统和Hecke本征层的量子化”中，Beilinson和Drinfeld介绍了光滑复等维叠加的“非常好”性质。证明了对于半单复群G，光滑复射影曲线X上G-丛的模叠加是“非常好的”，只要X的亏格g > 1。在投影线的情况下，当g = 0时，情况并非如此。然而，结果有时可以通过在标记点的集合处引入额外的抛物线结构并稍微修改“非常好”堆栈的定义来扩展到投影线。本文给出了复射影线上抛物型向量丛的模叠加非常好的一个充分条件。然后，我们使用这个属性来研究的Deligne-Simpson问题的解决方案的空间。

相对谱对应：抛物Higgs丛与Deligne--Simpson问题

Relative spectral correspondence for parabolic Higgs bundles and Deligne--Simpson problem

本文推广抛物Higgs丛的谱对应至相对情形，用曲面族上的线性系统刻画Hitchin像，证明OK条件是Deligne--Simpson问题可解的充分条件，并给出模空间非空性。

Linear differential equations on the Riemann sphere and representations of quivers

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties …

给定共轭类中不具有公共不变子空间且和为零的矩阵

On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero

我们确定这些k-元组的共轭类的矩阵，从它是可能的选择矩阵，没有共同的不变子空间和零。这是Deligne-Simpson问题的一个附加版本。我们推导出的结果，从我们早期的工作预投射代数和矩映射表示的箭图。我们的答案依赖于Kac-Moody李代数的根系。

Explicit constructions of connections on the projective line with a maximally ramified irregular singularity

The Deligne--Simpson problem is an existence problem for connections with specified local behavior. Almost all previous work on this problem has restricted attention to connections with regular or unramified singularities. Recently, the authors, together with Kulkarni and Matherne, formulated a version of the Deligne--Simpson problem where certain ramified singular points are allowed and solved it for the case of Coxeter connections, i.e., connections on the Riemann sphere with a maximally ramified singularity at zero and (possibly) an additional regular singular point at infinity. A certain m…

刚性表示的一些例子

Some examples of rigid representations

考虑Deligne-Simpson问题：{ \em给出了选择共轭类的充分必要条件$C_j\subset GL(n,{\bf C})$（分别为. $c_j\subset gl(n,{\bf C})$）使得存在矩阵$M_j\in C_j$的不可约$(p+1)$ -元组（分别为. $A_j\in c_j$）满足等式$M_1... M_{p+1}=I$（resp. $A_1+... +A_{p+1}=0$）}。矩阵$M_j$和$A_j$被解释为单值算子和黎曼球面上的Fuchsian系统的矩阵剩余。我们给出了矩阵$M_j$的这种$(p+1)$ -元组的存在性的新例子（分别是. $A_j$），它们是{ \em刚性}的，即，一旦类$C_j$（分别为$c_j$）是固定的。对于刚性表示，类$C_j$（分别为$c_j$）等于$2n^2-2$。

Some results on affine Deligne-Lusztig varieties

The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.

Moduli Problem of Hitchin Pairs over Deligne-Mumford Stack

We define the moduli problem of Hitchin pairs over Deligne-Mumford Stack and prove this moduli problem is represented by a separated and locally finitely presented algebraic space, which is considered as the moduli space of Hitchin pairs over Deligne-Mumford stack.

Affine Deligne-Lusztig varieties and folded galleries governed by chimneys

We characterize the nonemptiness and dimension problems for an affine Deligne-Lusztig variety $X_x(b)$ in the affine flag variety in terms of galleries that are positively folded with respect to a chimney. If the parabolic subgroup associated to the Newton point of b has rank 1, we then prove nonemptiness for a certain class of Iwahori-Weyl group elements x by explicitly constructing such galleries.

带标记有理曲线的Deligne-Mumford模空间的（上）同调

The (Co)homology of the Deligne-Mumford Moduli Spaces of Marked Rational Curves

本文整理了实带标记有理曲线的Deligne-Mumford模空间（上）同调的关键结果与未解决问题，涵盖拓扑与代数两个方向，并尝试用类operad语言刻画其同调结构。

曲线的 $P$-进 Simpson 对应中的提升独立性问题

Lift-independence problem in the $p$-adic Simpson correspondence for curves

研究曲线情形下 $p$-进 Simpson 对应中 Higgs 丛与 $v$-丛的等价性是否依赖于提升的选择，证明了小秩半稳定提升独立 Higgs 丛的 Higgs 场为零，并构造了非零 Higgs 场的半稳定提升独立 Higgs 丛。

关于Hilbert-Blumenthal模问题

On the Hilbert-Blumenthal moduli problem

本文比较了Rapoport，Deligne，Pappas和Kottwitz所考虑的Hilbert-Blumenthal Shimura簇的积分模型中的实乘法阿贝尔格式的不同条件.我们证明了Deligne/Pappas和Kottwitz研究的模型在Spec Z_（p）上是同构的。我们还检查了相关的本地模型，并证明它们是相等的。

On the Existence of Heterotic-String and Type-II-Superstring Field Theory Vertices

We consider two problems associated to the space of Riemann surfaces that are closely related to string theory. We first consider the problem of existence of heterotic-string and type-II-superstring field theory vertices in the product of spaces of bordered surfaces parameterizing left- and right-moving sectors of these theories. It turns out that this problem can be solved by proving the existence of a solution to the BV quantum master equation in moduli spaces of bordered spin-Riemann surfaces. We first prove that for arbitrary genus $g$, $n_{\text{NS}}$ Neveu-Schwarz boundary components, an…

$GL_n$ 的正 Coxeter 型基本轨迹

Basic Loci of Positive Coxeter Type for $GL_n$

本文在 $GL_n$ 情形下推广了 Görtz、He 和 Nie 关于基本仿射 Deligne-Lusztig 簇分解为 Coxeter 型经典 Deligne-Lusztig 簇之并的分层研究。

Drinfeld p-adic半平面的另一种描述

An alternative description of the Drinfeld p-adic half-plane

证明了Drinfeld p-adic半平面相对于非阿基米德局部域F的Deligne形式模型表示了极化O_F-模的模问题，其作用是整数环O_E在F的二次扩张E中的作用.通过建立与德林费尔德模问题的比较同构来进行证明。这种同构反映了二维维数分裂厄米空间C（E/F）的SL 2（F）和SU（C）（F）的偶然同构。