通过半稳定曲线上的谱数据紧化秩二 Hitchin 系统

On automorphisms of semistable G-bundles with decorations

We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group $G$. For example, our result applies to the stack of semistable $G$-bundles, stacks of semistable Hitchin pairs, and stacks of semistable parabolic $G$-bundles. Similar arguments apply to Gieseker semistable $G$-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic $0$ every stack of semistable de…

稳定尖点曲线上亚纯希钦纤维化的模空间

The meromorphic Hitchin fibration over stable pointed curves: moduli spaces

本文在稳定尖点曲线叠上构造了半稳定亚纯希格斯丛模空间的万有部分紧化，建立了容许节点谱覆盖下纤维与紧化雅可比簇的 BNR 对应类比，并研究了留数受幂零共轭类约束时的平坦固有希钦态射。

Hyperpolygons and Hitchin systems

We study the hyperkähler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperkähler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these spaces for all ranks. Building on a natural analogy between hyperpolygons and parabolic Higgs bundles, we identify hyperpolygon spaces with certain degenerate Hitchin systems, and use this to establish their complete integrability, for ranks up to and including 3.

正交辛复形的恰当模空间

Proper moduli spaces of orthosymplectic complexes

本文为Bridgeland半稳定正交辛复形的模栈构造了恰当好模空间，并提出其可作为正交群和辛群主丛模空间的紧化候选。

光滑曲线上主丛模空间的自同构群

Automorphisms of moduli spaces of principal bundles over a smooth curve

本文研究特征零代数闭域上殆单群G在亏格≥4光滑射影曲线上半稳定G-丛模空间的自同构群，通过分析Hitchin纤维化奇异纤维得到结果，并附带刻画了Hitchin基中两类奇异除子的不可约分支。

Hitchin 模空间中的奇异拉格朗日子流形与共形极限

Singular Lagrangians in the Hitchin moduli space and conformal limits

本文在 SL(r, C)-Higgs 丛模空间中构造了通过多稳定不动点的拉格朗日子轨迹，并在特定条件下证明了其共形极限的存在性。

Higgs bundles over elliptic curves

In this paper we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the…

任意特征下约化群的亚纯 Hodge 模空间

Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic

本文在任意特征下，为约化群概型构造了配备有界极点 $t$-连接的 $G$-丛的亚纯 Hodge 模空间，证明了半稳定轨迹的光滑性，并定义了正特征下常态的 Hodge-Hitchin 态射。

半射影与非常稳定性在辛和正交抛物Higgs包模空间中的研究

Semiprojectivity and very stability in moduli of symplectic and orthogonal parabolic Higgs bundles

本文研究了辛和正交抛物Higgs包模空间的半射影性与非常稳定性，证明了稳定辛抛物包的非常稳定性条件，并讨论了其在正交情况下的应用。

曲线上$\mathrm{SL}_n$-希格斯丛模空间的交截上同调

On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

本文利用Ngô型支撑不等式研究曲线上任意次数SL_n-希格斯丛模空间的交截上同调，证明了奇异情形下的支撑定理及Hausel-Thaddeus拓扑镜像对称猜想的一个版本，并推广了Harder-Narasimhan定理。

射影曲面上带框架层的 Uhlenbeck-Donaldson 紧化

Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces

本文为射影曲面上斜率稳定带框架向量丛的模空间构造了 Uhlenbeck-Donaldson 型紧化，并证明其与 Gieseker 半稳定框架层模空间之间存在射影态射，从而可通过 Hitchin-Kobayashi 对应与带框架理想瞬子模空间比较。

Moduli of Higgs bundles over the five punctured sphere

We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $μ\in[0,1]^5$. The moduli space corresponding to the central weight $μ_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli…

The Kapustin-Witten equations and nonabelian Hodge theory

Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $λ$-connection …

代数曲线的线性稳定性与秩二 Clifford 指标及其应用

Linear stability and rank two Clifford indices of algebraic curves with applications

本文证明计算秩二 Clifford 指标的向量丛线性半稳定，给出线性稳定条件，并应用于相干系统，验证 Butler 猜想在特定范围内成立。

椭圆点之外 A 型群的 Dolbeault 几何 Langlands 对应

The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

本文证明了对 GL_r 及 Langlands 对偶对 SL_r/PGL_r 在 Hitchin 基的包含椭圆点的开集上的 Dolbeault 几何 Langlands 等价，该开集对应至多 A 型奇点的谱曲线。