International Journal of Mathematics最新文献_第3页

Higgs bundles twisted by a vector bundle 被矢量束扭曲的希格斯束

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-05 DOI: 10.1142/s0129167x24410076

Guillermo Gallego, Oscar García-Prada, M. S. Narasimhan

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank $2$ vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin–Kobayashi correspondence for a generalization of Hitchin’s equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher-dimensional variety.

在本文中，我们考虑了光滑复投影曲线上希格斯束理论的广义化，其中希格斯场对曲线典型束的扭曲被秩 2 向量束所取代。我们定义了希钦映射，并给出了谱对应关系。我们还为希钦方程在这种情况下的泛化提出了希钦-小林对应关系。从某种意义上说，这一理论介于曲线上的希格斯束理论和高维综上的希格斯束理论之间。

引用次数: 0

Simplicity and tracial weights on non-unital reduced crossed products 非空还原交叉积的简约性和三项权重

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-05 DOI: 10.1142/s0129167x24500216

Yuhei Suzuki

We extend theorems of Breuillard–Kalantar–Kennedy–Ozawa on unital reduced crossed products to the non-unital case under mild assumptions. As a result simplicity of $C^{*}$ -algebras is stable under taking reduced crossed products over discrete $C^{*}$ -simple groups, and a similar result for uniqueness of tracial weight. Interestingly, our analysis on tracial weights involves von Neumann algebra theory.

Our generalizations have two applications. The first is to locally compact groups. We establish stability results of (non-discrete) $C^{*}$ -simplicity and the unique trace property under discrete group extensions. The second is to the twisted crossed product. Thanks to the Packer–Raeburn theorem, our results lead to (generalizations of) the results of Bryder–Kennedy by a different method.

我们在温和的假设条件下，将布雷亚德-卡兰塔-肯尼迪-奥泽关于非素数还原交叉积的定理扩展到非素数情况。结果是，在离散 C∗简单群上取还原交叉积时，C∗数组的简单性是稳定的，而三面重的唯一性也有类似的结果。有趣的是，我们对三面权的分析涉及冯-诺依曼代数理论。首先是局部紧凑群。我们建立了（非离散）C∗-简约性的稳定结果和离散群扩展下的唯一迹属性。其次是扭曲交叉积。得益于 Packer-Raeburn 定理，我们的结果以另一种方法引出了布赖德-肯尼迪结果的（概括）。

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引用次数: 0

Hitchin map on even very stable upward flows 甚至是非常稳定的上升流的希钦地图

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-04 DOI: 10.1142/s0129167x2441009x

Miguel González, Tamás Hausel

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the ${GL}_{n}$ case, we classify the type $(1, \dots, 1)$ examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, $4 n$ -spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.

我们定义了偶数非常稳定的希格斯束，并研究了限制于其向上流动的希钦映射。在 GLn 情况下，我们对类型 (1,...,1) 例子进行了分类，发现它们受偶数高度的根形成的根系统支配。我们讨论了实和四元格拉斯曼、4n 球和实 Cayley 平面的等变同调谱如何描述偶数向上流的希钦映射。这里所说的偶数上升流与准分裂内实形式的希格斯束模量空间中的上升流相同。后者由奥斯卡-加西亚-普拉达（Oscar García-Prada）及其合作者开创。

引用次数: 0

Framed parabolic sheaves on a trinion 三元组上的框架抛物面剪切

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-04 DOI: 10.1142/s0129167x24410027

Indranil Biswas, Jacques Hurtubise

We consider for structure groups $SU (n) \subset SL (n, ℂ)$ a densely defined toric structure on the moduli space of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces.

对于结构群 SU(n)⊂SL(n,ℂ)，我们考虑了三棱锥球面上有框抛物面剪切的模空间上的密集定义的环状结构，它退化为实际的环状结构。结合之前的退化结果，这些结果扩展到任意黎曼曲面的类似模空间。

引用次数: 0

Volume functionals on pseudoconvex hypersurfaces 伪凸超曲面上的体积函数

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-04 DOI: 10.1142/s0129167x24410052

Simon Donaldson, Fabian Lehmann

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface $M$ in a complex Calabi–Yau manifold (that is, a complex $n$ -manifold with a nowhere-vanishing holomorphic $n$ -form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in $R^{n}$ . We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces $M$ bounding compact domains $Ω \subset Z$ . That is, we study critical points of the volume functional $A (M)$ where the ordinary volume $V (Ω)$ is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere $S^{2 n - 1} \subset C^{n}$ does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in “most” directions but non-negative in directions corresponding to deformations of $S^{2 n - 1}$ by holomorphic diffeomorphisms. We are led to conjecture a “minimax” characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case

本文的重点是在复 Calabi-Yau 流形（即具有无处消失全形 n 形式的复 n 流形）中的伪凸超曲面 M 上定义的一种体积形式。我们首先定义这种体量形式，并指出它可以看作是 Rn 中凸超曲面上仿射不变体量形式的广义化。我们计算了第一个变化，这导致了仿射平均曲率的类似广义化。在第 2 节中，我们研究了以紧凑域 Ω⊂Z 为边界的伪凸超曲面 M 的约束变分问题。也就是说，我们研究的是普通体积 V(Ω) 固定的体积函数 A(M) 的临界点。临界点类似于恒定平均曲率子曲面。我们发现佐佐木-爱因斯坦超曲面满足条件，尤其是标准球 S2n-1⊂Cn 满足条件。本文的主要工作在第 3 节，我们计算了球面的第二次变化。我们发现它在 "大多数 "方向上都是负值，但在全形差分变形对应的 S2n-1 变形方向上却不是负值。我们由此猜想出球面的 "最小 "特征。我们还讨论了与仿射几何和凯勒-爱因斯坦几何的联系。我们研究这些问题的最初动机来自 n=3 的情况和我们之前论文 [S. Donaldson and F. Leinstein] 中研究的嵌入问题。Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208].这种情况有一些特殊之处。在第 4 节中，我们将回顾这一点，并从 M 上精确 3-forms 的交映结构和 M 的差分作用的矩映射的角度发展一些理论。

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引用次数: 0

Quantization of locally compact groups associated with essentially bijective 1-cocycles 与本质上双射的 1-Cocycles 相关联的局部紧凑群的量化

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-03 DOI: 10.1142/s0129167x24500277

Pierre Bieliavsky, Victor Gayral, Sergey Neshveyev, Lars Tuset

Given an extension $0 \to V \to G \to Q \to 1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$ -cocycle $η : Q \to \hat{V}$ , we define a dual unitary $2$ -cocycle on $G$ and show that the associated deformation of $Ĝ$ is a cocycle bicrossed product defined by a matched pair of subgroups of $Q ⋉ \hat{V}$ . We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of $G$ on $L^{2} (Q)$ and a unitary quantization map $L^{2} (G) \to HS (L^{2} (Q))$ of Kohn–Nirenberg type.

给定局部紧密群的扩展 0→V→G→Q→1，其中 V 是无性的，以及一个相容的本质上双射的 1 循环 η:Q→V̂，我们定义了 G 上的对偶单元 2 循环，并证明Ĝ 的相关变形是由 Q⋉V ̂ 的一对匹配子群定义的循环双交积。我们还讨论了从匹配对的 Kac 同调的角度对我们的构造的解释。我们的设置概括了 Etingof 和 Gelaki 对有限群的设置、Ben David 和 Ginosar 对其的扩展，以及我们早期对满足对偶轨道条件的局部紧凑群的研究。特别是，我们从杨-巴克斯特方程的每一个渐开非enerate集合理论解，或者更广义地说，从每一个支撑结构，都可以得到一个局部紧凑的量子群。在技术方面，新的关键点在于构建了 G 在 L2(Q) 上的不可还原投影表示和 Kohn-Nirenberg 类型的单元量子化映射 L2(G)→HS(L2(Q))。

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引用次数: 0

On codimension one holomorphic distributions on compact toric orbifolds 关于紧凑环状轨道上的一维全态分布

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-04-03 DOI: 10.1142/s0129167x24500241

Arnulfo Miguel Rodríguez Peña

The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we provide a classification for regular distributions on rational normal scrolls and weighted projective spaces. Additionally, under specific conditions, we prove that the singular set of a codimension one holomorphic foliation on a compact toric orbifold admits at least one irreducible component of codimension two, and we also present a Darboux–Jouanolou type integrability theorem for codimension one holomorphic foliations. Our results are exemplified through various illustrative examples.

我们确定了紧凑环状轨道上一般标度为一的全形分布的奇点数（以倍率计算）。因此，我们为有理正卷和加权投影空间上的正则分布提供了一个分类。此外，在特定条件下，我们证明了紧凑环状轨道上的一维全形叶状体的奇异集至少包含一个二维不可还原分量，我们还提出了一维全形叶状体的达尔布-朱阿诺卢型可整性定理。我们的结果将通过各种示例加以说明。

引用次数: 0

Endpoint estimates of variation and oscillation operators associated with Zλ functions 与 Zλ 函数相关的变化和振荡算子的端点估计值

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-03-28 DOI: 10.1142/s0129167x24500253

Yongming Wen, Yanyan Han, Xianming Hou

This paper obtains weak-type estimates, limiting weak-type behaviors for variation operators associated with $Z^{λ}$ functions. Besides, we give a new characterization of Hardy space via the boundedness of variation operators associated with $Z^{λ}$ functions.

本文获得了弱型估计，限制了与 Zλ 函数相关的变算子的弱型行为。此外，我们还通过与 Zλ 函数相关的变分算子的有界性给出了哈代空间的新特征。

引用次数: 0

Twistor space for local systems on an open curve 开阔曲线上局部系统的孪生空间

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-03-27 DOI: 10.1142/s0129167x24410131

Carlos T. Simpson

Let $X = \bar{X} - D$ be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank $2$ . We extend this construction to the case of any rank $r$ , although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.

假设 X=X¯-D 是一条光滑的准投影曲线。我们之前为秩为 2 的连接构造了一个具有 Hecke gauge groupoid 的 Deligne-Hitchin 模空间。我们将这一构造扩展到任意秩 r 的情况，但仍保留通性假设。与驯服谐波束相对应的优选截面的形式邻域受混合扭曲结构支配。

引用次数: 0

Implosion, contraction and Moore–Tachikawa 内爆、收缩和摩尔-立川

IF 0.6 4区数学 Q3 Mathematics

International Journal of Mathematics

Pub Date : 2024-03-20 DOI: 10.1142/s0129167x24410040

Andrew Dancer, Frances Kirwan, Johan Martens

We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type $A$ to a general reductive group, and interpret it in the context of the Moore–Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalized to the hyperkähler or complex symplectic situation.

我们考察了内爆构造，将其与高折射几何有关的某些方面从 A 型扩展到一般还原群，并在摩尔-立川范畴的背景下对其进行了解释。我们利用这些观点来讨论如何将交点几何中的内卷构造推广到超交点或复交点情形中。

引用次数: 0

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