Bin Wang
In this paper, we talk about parahoric Hitchin systems over smooth projective curves with structure group a semisimple simply connected group. We describe the geometry of generic fibers of parahoric Hitchin fibrations using root stacks. We work over an algebraically closed field with a mild assumption of the characteristic. All of these can be treated as a generalization of GLn case in [SWW19]
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Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let $\mathrm{Bun}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^*\!\mathrm{Bun}_{\mathcal{G}}$. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that $\mathrm{Bun}_{\mathcal{G}}$ is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, thes…
We consider the local model of a Shimura variety of PEL type, with the unitary similitudes corresponding to a ramified quadratic extension of $\mathbb{Q}_p$ as defining group. We examine the cases where the level structure at $p$ is given by a parahoric that is the stabilizer of a selfdual periodic lattice chain and that is special in the sense of Bruhat--Tits theory. We prove that in these cases the special fiber of the local model is irreducible and generically reduced; consequently, the special fiber is reduced and is normal, Frobenius split, and with only rational singularities. In additio…
On the generic fibers and true base of parabolic $\mathrm{SO}_{2n}$-Hitchin systems
本文证实了抛物型 SO(2n)-Hitchin 系统的物理猜想，证明 Hitchin 映射通过同构于仿射空间的有限覆盖分解，并引入剩余幂零 Hitchin 系统分析一般纤维，揭示其自对偶性与特殊幂零轨道的联系。
In this paper we construct the Poincare line bundle for the stack of Higgs bundles on smooth projective curves and show that it induces a fully-faithful Fourier-Mukai transform on the category of quasi-coherent sheaves.
We analyze and completely describe the four cases when the Hitchin fibration on a $2$-dimensional moduli space of irregular Higgs bundles over $\mathbb{C}P^{1}$ has a single singular fiber. The case when the fiber at infinity is of type $I_0^*$ is further analyzed, and we give constructions of all the possible configurations of singular curves inelliptic fibrations having this type of singular fiber at infinity.
We set up a BNR correspondence for moduli spaces of Higgs bundles over a curve with a parabolic structure over any algebraically closed field. This leads to a concrete description of generic fibers of the associated strongly parabolic Hitchin map. We also show that the global nilpotent cone is equi-dimensional with half dimension of the total space. As a result, we prove the flatness and surjectivity of this map and the existence of very stable parabolic vector bundles.
Let $φ\colon X\rightarrow Y$ be a degree two Galois cover of smooth curves over a local field $F$ of odd characteristic. Assuming that $Y$ has good reduction, we describe a semi-stability criterion for the curve $X$, using the data of the branch locus of the covering $φ$. In the case that $X$ has semi-stable reduction, we describe the dual graph of the minimal regular model of $X$ over $F.$ We do this by adopting the notion of cluster picture defined for hyperelliptic curves for the case where $Y$ is not necessarily a rational curve. Using these results, we describe the variation of the p-adic…
Conformal blocks, parahoric torsors and Borel-Weil-Bott
通过Hecke对应原理，建立了torsor模空间上线束上同调的关系，证明了特征零下抛物群概形的猜想，并得到真空传播原理、中心荷独立性和射影平坦性等推论。
Irreducibility of Local Models
本文研究了带有抛物面水平结构的Shimura簇和$\bG$-Shtuka模空间的局部模型几何特殊纤维的不可约性，分类了局部模型不可约的情况，并证明了不同水平间映射的纤维同构于单个Schubert簇。
The Hitchin morphism for certain surfaces fibered over a curve
本文证明Chen-Ngô猜想对直纹曲面及非各向同性椭圆纤维化（的爆破）上的任何约化群成立，并进一步对经典群证明了限制在半调和Higgs丛模空间上的Hitchin态射的满射性。
We investigate the geometry of the special fiber of the integral model of a Shimura variety with parahoric level at a given prime place. To be more precise, we deal with the definition of central leaves in this situation, their local closedness, and the relationship between the folations for varying parahoric level. This is connected to the verification of axioms for integral models formulated by He and Rapoport.
Springer correspondence and mirror symmetries for parabolic Hitchin systems
本文通过构造幂零轨道闭包相关的一般Hitchin纤维并分析其与抛物型Hitchin系统一般纤维的关系，证明了B、C型抛物型Hitchin系统的SYZ与拓扑镜像对称，并给出Springer对偶和Lusztig典范商的新几何解释。
Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular. Our main result provides a fully faithful functor from a twist of the derived category of Pic$^τ$(X/B) to the derived category of X. Here Pic$^τ$(X/B) is the algebraic space classifying fiberwise numerically trivial line bundles. Next, we show t…
Laumon parahoric local models via quiver Grassmannians
本文利用箭图表示理论研究类型 A 中任意仿抛物子群的 Laumon 局部模型特殊纤维，刻画其不可约分支、自然分层和胞腔分解，并证明奇点消解与仿抛物子群间自然投影相容。
We investigate the geometry of the special fiber of the integral model of a Shimura variety with parahoric level at a given prime place. To be more precise, we deal with the EKOR stratification which interpolates between the Ekedahl-Oort and Kottwitz-Rapoport stratifications. In the Siegel case we give a geometric description by suitably generalizing the theory of $G$-zips of Moonen, Wedhorn, Pink and Ziegler to our context.
We study hyperkahler metrics and hyperholomorphic connections of Hitchin's moduli spaces after Gaiotto, Moore and Neitzke. Their construction via the twistor technique produces intricate wall crossing behaviors. For certain four dimensional Hitchin's moduli spaces local models and degeneration to local models near singular fibers of the Hitchin's fibration are understood.
In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\mathcal{M}^{ss}}$ is uniformized by a formal scheme $\mathcal{N}$ which is called Rapoport-Zink space. We show that the irreducible components of the induced reduced subscheme $\mathcal{N}_{red}$ of $\mathcal{N}$ are Deligne-Lusztig varieties and their intersection behavior is controlled by a certain Bruhat-Tits building. Also, we define special cycles in $\mathcal{N}$ and study their intersection multiplicities.
Microsheaves from Hitchin fibers via Floer theory
本文基于 Floer 理论证明光滑 Hitchin 纤维可决定全局幂零锥上的微层，且不同纤维导出正交微层，其自同构同构于纤维上同调，为几何 Langlands 对应的 Hecke 本征层提供了候选构造。
On parahoric $(\mathcal{G}, μ)$-displays
本文将仲 $(\mathcal{G}, \mu)$-展示推广到非 $p$-无挠基环并引入截断展示，研究其形变理论，从而把 Hodge 型整 Shimura 簇特殊纤维的 EKOR 分层实现为到截断展示代数栈的光滑态射纤维。
On Fourier-Mukai transforms of upward flows for Hitchin systems
本文利用Arinkin层和余维数估计，将Hausel和Hitchin关于Donagi-Pantev对偶在光滑Hitchin纤维开集上交织拉格朗日向上流与超全纯向量丛的结果，推广至整个Hitchin底上任意次数的Higgs丛。
Invariant vector bundles and Hitchin systems
本文研究亏格至少为2的光滑复射影曲线伽罗瓦覆盖上半稳定不变量向量丛的模空间，分类其连通分支，并确定相应希钦系统在光滑情形下的纤维结构。
Hitchin in [Duke Math. J. 54 (1), 91-114 (1987)] introduced a proper morphism from the moduli space of stable $G$-Higgs bundles ($G=\mathrm{GL}(n,\mathbb{C}),\mathrm{Sp}(2m,\mathbb{C})$ and $\mathrm{SO}(n,\mathbb{C})$) over a curve to a vector space of invariant polynomials and he described the generic fibers of that morphism. In this paper, we first describe the generic Hitchin fibers for the moduli space of stable parabolic projective symplectic/orthogonal Higgs bundles without fixing the determinant. We also describe the generic fibers when the determinant is trivial.
In this article we give an explicit description of the Hitchin fiber of SU(1,2) Higgs bundles $(L,F,γ,β)$ over a compact Riemann surface $X$ of genus $\ge 2$ with $q=γ\circβ$ having simple zeros and Toledo invariant $τ=2 \rm{deg} L$ satisfying $|\rm{deg} L|<g-1$. In particular we identify the data in an SU(1,2) Higgs bundle as a Hecke transformation $ι: F\to L^{-2}K\oplus LK$ at $Z(q)$. The Hitchin fiber is identified with a fiber bundle over $\text{Pic}^d X$ with unirational fiber, which is a GIT-quotient of a $\mathbb{C}^\times$-action on $\left(\mathbb{P}^1\right)^{4g-4}$. The base parametr…
In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and "twisted" spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spa…
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。
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。
Normed Fiber Functors
该研究借助Tannakian形式体系，将一般线性群Bruhat-Tits建筑的非阿基米德范数描述推广到非分歧约化群情形，并给出对应仿射群概形的模论刻画。
Let $S\to C$ be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on $H^*(S^{[n]},\mathbb{Q})$ associated with the natural map $S^{[n]}\to C^{(n)}$ implies that $S\to C$ is an elliptic fibration. The converse is also true when $S\to C$ is a Hitchin-type elliptic fibration.
Let $Σ$ be a smooth projective complex curve and $\mathfrak{g}$ a simple Lie algebra of type ${\sf ADE}$ with associated adjoint group $G$. For a fixed pair $(Σ, \mathfrak{g})$, we construct a family of quasi-projective Calabi-Yau threefolds parameterized by the base of the Hitchin integrable system associated to $(Σ,\mathfrak{g})$. Our main result establishes an isomorphism between the Calabi-Yau integrable system, whose fibers are the intermediate Jacobians of this family of Calabi-Yau threefolds, and the Hitchin system for $G$, whose fibers are Prym varieties of the corresponding spectral c…
Cyclic Spectral Data
本文研究特征零代数闭域上光滑簇的循环谱数据，证明非零循环谱数据的Hitchin纤维非空，并构造Hitchin态射截面及循环谱覆盖的平坦族。
The Bruhat-Tits stratification for basic unramified $GU(1,n-1)$ Rapoport-Zink spaces at arbitrary parahoric level
本文对签名 $(1,n-1)$ 的基本非分歧酉 Rapoport-Zink 空间在任意抛物子群水平下的约化特殊纤维进行分层，证明了闭分层的光滑性、不可约性并计算了维数，同时利用 Bruhat-Tits 指标描述了分层间的关联关系。
Let $X$ be a compact Riemann surface of genus $g \geq 2$, and let $D \subset X$ be a fixed finite subset. Let $\mathcal{M}(r,d,α)$ denote the moduli space of stable parabolic $G$-bundles (where $G$ is a complex orthogonal or symplectic group) of rank $r$, degree $d$ and weight type $α$ over $X$. Hitchin discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic curve is an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. In this paper we study the Hitchin fibers …
We show that the Hitchin integrable system for a simple complex Lie group $G$ is dual to the Hitchin system for the Langlands dual group $\lan{G}$. In particular, the general fiber of the connected component $\Higgs_0$ of the Hitchin system for $G$ is an abelian variety which is dual to the corresponding fiber of the connected component of the Hitchin system for $\lan{G}$. The non-neutral connected components $\Higgs_α$ form torsors over $\Higgs_0$. We show that their duals are gerbes over $\Higgs_0$ which are induced by the gerbe of $G$-Higgs bundles $\gHiggs$. More generally, we establish a …
Any irreducible Dynkin diagram $Δ$ is obtained from an irreducible Dynkin diagram $Δ_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $Δ$ and compact Riemann surface $Σ$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(Σ, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to t…
This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu's proof of the Langlands-Shelstad fundamental lemma. Ngô's approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of "characteristic polynomials". Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fi…
The aim of this paper is to give an explicit expression for Hitchin's connection in the case of rank 2 bundles with trivial determinant over curves of genus 2. We recall the definition of this connection (which arose in Quantum Field Theory) and characterize it for families of genus two curves. We then consider a particular family (over a configuration space), the corresponding vector bundles are (almost) trivial. The Heat equations which give the connection these bundles are related to the Lie algebra so(6). We compute some local monodromy representations. As a byproduct we obtain, for any re…
Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves
该研究针对曲面上的希格斯层模空间与希钦映射，证明了高维谱簇的Noether–Lefschetz型定理，给出一般希钦纤维非空的充要条件，并分析第二陈类变化对模空间几何的影响。
We propose a generalization of Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the affine Weyl group on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the affine Weyl group on the cohomology of parabolic Hitchin fibers, providing the first step towards a global Springer theory.
The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T-duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations…
Separation of variables for Hitchin systems with the structure group $SO(4)$, on genus two curves
本文针对亏格二曲线上结构群为 $SO(4)$ 和 $SL(2)$ 的 Hitchin 系统，利用 Jacobi 反演映射的类似物转移 Prymian 上的直线缠绕，实现了变量分离并求得轨迹。
The companion section for classical groups
本文利用 GL_n 的伴随矩阵构造为经典群和 G_2 构造 Chevalley 映射的典范截面，并借助谱覆盖上的典范张量，给出仿射 Springer 纤维与 Hitchin 纤维的显式格描述。
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.
In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup. The total space of Hitchin is obtained by taking the fiber product of the Hecke stack with the diagonal of the stack of G-bundles $Bun_{G}$; we prove a transversality statement between the intersection complex of the Hecke stack and the diagonal of $Bun_{G}$, over a sufficiently big open subset, in order to get local applications, such that the fundamental lemma for the spherical Hecke algebra. Alon…
Cellular pavings of fibers of convolution morphisms
本文证明，在任意域上的分裂群情形下，与仿射旗簇相关的卷积态射的所有纤维均可被仿射直线和去掉一点的仿射直线的乘积铺砌。
Families of Hitchin systems and N=2 theories
本研究围绕四维N=2理论关联的Hitchin系统族展开，分析其在Deligne-Mumford模空间上的全局行为，证明Hitchin基构成紧化模空间上的向量丛，还分类了幂零轨道与味对称子群的允许对。
A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus
本文在 Hitchin-小轨迹上，通过引入新的带连接周期层，建立了 $p$-进 Riemann--Hilbert 对应中层状等价，连接了可积连接与局部系统。
The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky-Kottwitz-MacPherson and Bezrukavnikov-Varshavsky. We first prove this result for the global Springer action, and…
This thesis, done under the supervision of Filippo Viviani, is devoted to the study of the spectral correspondence for $G$-Higgs pairs, in the case of $G=SL(r,\mathbb{C})$, $PGL(r, \mathbb{C})$, $Sp(2r,\mathbb{C})$, $GSp(2r,\mathbb{C})$, $PSp(2r,\mathbb{C})$, over any fiber. In the first chapter we introduce the compactified Jacobian stack parametrizing torsion-free rank-1 sheaves over a (possibly reducible, non reduced) projective curve and we describe the spectral correspondence for Higgs pairs in terms of the compactified Jacobian. In the second chapter, we study the notion of direct and in…
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。
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…
On tamely ramified $\mathcal G$-bundles on curves
本文证明光滑射影曲线上的仿射 Bruhat-Tits 群概形可表示为驯顺覆盖上约化群概形的不变量，并建立两类扭子及其模叠之间的联系，同时发展等变扭子的局部类型理论。
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