Monodromy of the SL_2 Hitchin fibration
Laura P. Schaposnik
计算了SL（2，C）Hitchin系统模2上同调的单值作用，并给出了结果在半稳定SL（2，R）Higgs丛的模空间中的应用.
We calculate the monodromy action on the mod 2 cohomology for SL(2,C) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2,R) Higgs bundles.
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We describe certain Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp-Hitchin systems.
We study the monodromy of the Hitchin fibration for moduli spaces of parabolic G-Higgs bundles in the cases when G=SL(2,R), GL(2,R) and PGL(2,R) A calculation of the orbits of the monodromy with Z2-coefficients provides an exact count of the components of the moduli spaces for these groups.
On the monodromy of the Hitchin connection
对于任意亏格g > 1，我们给出了一类亏格g的光滑复射影曲线，使得水平l不同于1，2，4，8的广义SL（2）-θ函数层上Hitchin联络的单值表示的象包含一个无限级无限阶元.
We prove that the perverse Leray filtration for the Hitchin morphism is locally constant in families, thus providing some evidence towards the validity of the $P=W$ conjecture due to de Cataldo, Hausel and Migliorini in non Abelian Hodge theory.
In this paper we give a complete description of the Hitchin fibration on all 2-dimensional moduli spaces of rank 2 irregular Higgs bundles with two poles on the projective line. We describe the dependence of the singular fibers of the fibration on the eigenvalues of the Higgs fields, and describe the corresponding motivic wall-crossing phenomenon in the parameter space of parabolic weights.
We are interested in studying the variation of the Hitchin fibration in moduli spaces of parabolic Higgs bundles, under the action of a ramified covering. Given a degree two map $π$ : Y $\rightarrow$ X between compact Riemann surfaces, we may pull back a Higgs bundle from X to Y , the lifted Higgs bundle tends to have many apparent singularities, then we perform a suitable birational transformation in order to eliminate them. This correspondence preserves the Hitchin fibrations and then its restriction to a general fiber gives a map between Abelian varieties. The aim of this paper is to descri…
Hitchin fibrations are Ngô fibrations
本文研究光滑射影曲线上 $\mathcal{L}$-值 $G$-Higgs 丛的 Hitchin 纤维化，证明其具有弱 Abel 纤维化结构，并在 $\mathcal{L}$ 为典范丛的约化有效除子扭转时证明其上界与 $δ$-正则性。
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 compute the supports of the perverse cohomology sheaves of the Hitchin fibration for $GL_n$ over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every $n \geq 3$. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare th…
A support theorem for the Hitchin fibration: the case of $SL_n$
我们证明了$D$ -扭曲$SL_n$希钦纤维化的直接镜像复形是由它对椭圆轨迹的限制决定的，其中谱曲线是积分的。对于$GL_n$的类似结果是由于P. - H. Chaudouard和G.劳蒙在此过程中，我们证明了相对Prym群格式的Tate模是可极化的，并且证明了某些辅助弱交换纤维化的$δ$ -正则性结果.
The Hitchin Fibration for Symmetric Pairs
本研究引入并刻画对称空间希钦纤维化的正则商，针对特定GL2n空间给出谱覆盖的不变量论方法，研究拟分裂对的正则中心化子群概形及相关群概形的伽罗瓦描述，推导希格斯丛与对应希钦系统的若干基本结论。
We prove that the moduli spaces of twisted $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles on a smooth projective curve have the same (stringy) class in the Grothendieck ring of rational Chow motives. On the level of Hodge numbers this was conjectured by Hausel and Thaddeus, and recently proven by Groechenig, Ziegler and the second author. To adapt their argument, which relies on p-adic integration, we use a version of motivic integration with values in rational Chow motives and the geometry of Néron models to evaluate such integrals on Hitchin fibers.
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。
In this paper, we explore the structure of the Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces.
Rationality of moduli of elliptic fibrations with fixed monodromy
证明了椭圆K3曲面和椭圆有理曲面的模空间具有固定单值群的合理性结果。
Intersection of two quadrics: modular interpretation and Hitchin morphism
本文证明两个二次曲面光滑交的余切丛上的拉格朗日纤维化在赋予扭曲 Spin-丛模空间结构后即为希钦态射，从而推广了三维情形下的经典结果。
Un théorème du support pour la fibration de Hitchin
吴宝洲证明朗兰兹-谢尔斯塔德基本引理的主要工具是一个关于希钦纤维化椭圆部分的相对上同调的支持定理。对于GL（n）和度因子>2g-2，该定理表明相对上同调完全由其对希钦纤维化基底的任何稠密开子集的限制决定。在本文中，我们将在这种特殊情况下，我们的推广，该定理的整个希钦纤维化，包括整体幂零锥。
On bifurcation braid monodromy of elliptic fibrations
我们定义了一个单值同态不可约家庭的定期椭圆纤维化的映射类群的穿孔球。我们只计算椭圆纤维化的单值性，它不包含除I_1和I*_0以外的奇异纤维。存在极大群，它们可以是代数的单值群，也可以是代数的单值群。可微家庭的椭圆曲面，我们给出了一个代数准则的平等，我们可以检查适用的情况下，I_1纤维的数量是最多6。
A Kodaira fibration is a non-isotrivial fibration $f\colon S\rightarrow B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space $\mathcal{M}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of $\mathcal{M}_3$ parametrizing curves whose…
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…
The monodromy of compact Lagrangian fibrations
研究紧致拉格朗日纤维化的monodromy表示，证明在周期映射通常浸没时表示是不可约的，在纤维为isotrivial时恢复Kim-Laza-Martin的结果，证明其纤维是椭圆曲线幂的isogeneous。
Fibration de Hitchin et endoscopie
我们提出了一个几何解释的理论椭圆内窥镜，由于朗兰兹和Kottwitz，在希钦纤维化。作为应用，我们证明了Goresky，Kottwitz和MacPherson的一个纯度猜想的一个全局模拟.在与G. Laumon的方法证明了等特征的局部域上的基本引理。
Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations
在亏格为$g>1$的紧致黎曼曲面上，计算了$L$-扭曲$SL(n,\mathbb{C})$和$GL(n,\mathbb{C})$-Higgs丛模空间的Hitchin纤维化的单值性.我们要求线丛$L$要么是标准丛，要么满足$deg(L) > 2g-2$。单值群由与奇异谱曲线的消失环相关的皮卡德-莱夫谢茨变换生成。我们明确地构造这样的消失圈，并使用它来表明，$SL(n,\mathbb{C})$ monodromy群是一个{\em斜对称消失格}在杨森的意义。利用$\mathbb{Z}$上消失格的分类，我们完全确定了$SL(n,\mathbb{C})$和$GL(n,\mathbb{C})$ Hitchin纤维化的单值群的结构.作为一个应用，我们确定的图像的限制映射从上同调的模空间的希格斯丛的上同调的非奇异纤维的希钦纤维化。
The singular Hitchin fibration, cameral data, and representation theory
本文研究具有任意约化结构群的Higgs丛模栈上的Hitchin纤维化，利用Higgs场的中心化子分析其奇异轨迹，通过阿贝尔化纤维化分解Hitchin映射，并推广Donagi-Gaitsgory的相机数据描述纤维，最后应用于实群并揭示与李代数表示论的联系。
Mirror of Orbifold Singularities in the Hitchin Fibration: the case $(\text{SL}_n,\text{PGL}_n)$
研究 Hitchin 纤维化中轨道奇点的镜像，证明 $\ ext{PGL}_n$ 模空间奇点对应 $\ ext{SL}_n$ 可约纤维，并验证 Fourier-Mukai 变换满足分数 Hecke 特征性质。
Relative Fundamental Lemmas for Spherical Hecke Algebras and Multiplicative Hitchin Fibrations: the Jacquet--Rallis Case
本文通过乘法希钦纤维化方法，证明了局部函数域上球面海加以尔代数的雅各布森-拉里斯基本引理。
Mordell-Weil Groups of Linear Systems and the Hitchin Fibration
本文研究光滑射影簇上线性系的相对Picard格式的有理截。证明了若线性系无基点且非整因子的轨迹余维数至少为2，则相应Picard格式的所有有理截面都来自于线丛对簇的限制.作为一个结果，我们描述的希钦纤维化的模空间的希格斯包曲线的部分。
In this paper a monodromy invariant for isotropic classes on generalized Kummer type manifolds is constructed. This invariant is used to determine the polarization type of Lagrangian fibrations on such manifolds - a notion which was introduced in an earlier paper of the author. The result shows that the polarization type of a Lagrangian fibration of generalized Kummer type depends on the connected component of the moduli space.
Dihedral Monodromy and Xiao Fibrations
我们构造了三个新的纤维族$π: S \to B$，其中$S$是一个代数复曲面，$B$是一条曲线，它违反了肖关于一般纤维的相对不规则性和亏格的猜想. $π$的纤维是某些超椭圆曲线的三角循环覆盖，它们给出了$P^1$的具有二面角单值的覆盖。作为应用，我们还证明了Brill-Noether范围内大有效因子和nef有效因子的存在性。
We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a…
Multiplicative Hitchin fibrations and Langlands duality
本文识别了底空间等同且一般纤维为对偶Beilinson $1$-动机的成对（扭曲）乘性Hitchin纤维化，涵盖无扭曲与扭曲情形，结果与Elliott和Pestun的猜想一致。
Monodromy of rank 2 twisted Hitchin systems and real character varieties
我们介绍了一种新的方法来计算单值的Hitchin映射，并使用它来完全确定单值的$L$扭曲$G$ -希格斯丛的模空间，为集团$G = GL(2,\mathbb{C})$，$SL(2,\mathbb{C})$和$PSL(2,\mathbb{C})$。我们还确定了扭曲陈类的正规轨迹，这阻碍了存在的一部分的模空间的$L$ -扭曲希格斯丛秩$2$和度$deg(L)+1$。通过对具有$\mathbb{Z}_2$ -系数的单值作用的轨道计数，统一地得到了实群$G = GL(2,\mathbb{R})$，$SL(2,\mathbb{R})$，$PGL(2,\mathbb{R})$，$PSL(2,\mathbb{R})$的特征标簇的分支数，以及具有最大Toledo不变量的$Sp(4,\mathbb{R})$ -特征标簇的分支数.我们还使用我们的结果为$GL(2,\mathbb{R})$计算单值的$SO(2,2)$ Hitchin地图，并确定组件的$SO(2,2)$字符品种。
Multiplicative Hitchin Fibrations and the Fundamental Lemma
本文在特征超过 Coxeter 数两倍的条件下，运用乘法 Hitchin 纤维化，证明了形式幂级数环上约化群球面 Hecke 代数的标准内窥基本引理。
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 …
Monodromy of elliptic surfaces
研究了射影直线上椭圆纤维化的单值群。
In his proof of the fundamental lemma, Ngô established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant vanishing statement for torsors over loop groups $R((t))$ from a general formula for $\mathrm{Pic}(R((t)))$. In the build up to the product formula, we present general algebraization, approximation, and invariance under Henselian pairs results for torsors, give short new proofs for the Elkik approximation theorem and the Chevalley isomorphism $\…
Global geometrization of local smooth integral models in the Hitchin fibration for $\mathrm{GL}_n$: The Bass case
研究\(\mathrm{GL}_n\)（巴斯情形）局部轨道积分，先提出用亨赛尔环上概型平滑方法，本文在全局希钦纤维化框架内将其几何化，给出有限域上特定谱曲线紧致化雅可比\(\ell\)-adic上同调封闭公式。
Let $f:S\to B$ be a fibration of curves and let $f_\astω_{S/B}=\cU\oplus \cA$ be the second Fujita decomposition of $f.$ In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $\cU$. The main result states that their vanishing on a general fibre of $f$ implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we p…
La fibration de Hitchin-Frenkel-Ngo et son complexe d'intersection
在这篇文章中，我们构造了希钦纤维化的集团以下的计划概述的Frenkel-Ngo在SL_{2}的情况下。这种构造使用了Vinberg半群作为决定性的工具。Hitchin的全空间是通过取Hecke栈的纤维积与G-丛$Bun_{G}$的栈的对角线得到的;我们证明了Hecke栈的交复形与$Bun_{G}$的对角线之间在一个足够大的开子集上的横截性陈述，以得到局部应用，例如球面Hecke代数的基本引理。随着这个定理的证明，我们建立了关于局部域中单连通群的点的积分共轭类的一个结果。
Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
我们将Eynard-Orantin（2007）的拓扑递归推广到Hitchin纤维化的谱曲线族。拓扑递归中的谱曲线被定义为复平面曲线，用任意光滑基曲线$C$的余切丛$T^*C$中的类属曲线代替。然后，我们证明，这些光谱曲线是量化的，使用新的形式主义。更确切地说，我们构造的$D$ -模的形式$\hbar$ -变形家庭的正则生成元的任意射影代数曲线$C$的亏格大于$1$，从一个规定的家庭的光滑Hitchin谱曲线与基本群$π_1(C)$的$SL(2,\mathbb{C})$ -字符簇。我们表明，通过这些$\hbar$ -变形$D$ -模的WKB近似的半经典极限恢复的初始家庭的Hitchin谱曲线。
A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the "geometric" monodromy, valued in the mapping class group of the fiber.
On Hitchin's Connection
本文的目的是给出亏格为2的曲线上秩为2的平凡行列式丛的Hitchin联络的一个显式表达式。我们回顾了这种连接的定义（这是在量子场论中出现的），并描述了它的家庭亏格两条曲线。然后我们考虑一个特殊的族（在配置空间上），相应的向量丛是（几乎）平凡的。给出这些丛的联系的热方程与李代数so（6）有关。我们计算一些局部单值表示。作为一个副产品，我们得到，任何表示空间V的so（2g+2），一个平坦的连接的平凡丛纤维V上的配置空间，从而单值表示的纯辫子群。
Le lemme fondamental pondéré. II. Énoncés cohomologiques
在本文中，我们研究截断希钦纤维化的上同调，这是在以前的文件中介绍。我们推广了Ngô关于希钦纤维化椭圆部分的上同调的主要定理。从而得到了Arthur加权基本引理的一个证明
Kodaira's classification of singular fibers in elliptic fibrations and its translation into the language of monodromies and Lefschetz fibrations has been a boon to the study of 4-manifolds. In this article, we begin the work of translating between singular fibers of genus 2 families of algebraic curves and the positive Dehn twist factorizations of Lefschetz fibrations for a certain subset of the singularities described by Namikawa and Ueno in the 70s. We look at four families of hypersurface singularities in $\mathbb{C}^3$. Each hypersurface comes equipped with a fibration by genus 2 algebraic…
Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy
本文通过分支覆盖构造和辫子单值性方法，直接计算Xiao亏格二Lefschetz纤维化的几何单值性分解，得到类型$(4,3)$的有序正分解及七个因子的Artin坐标。
Towards a Global Springer Theory I: The affine Weyl group action
我们提出了一个广义的施普林格表示的上下文中的群体在一个全球性的功能领域。全球对应的格罗滕迪克同时解决是抛物希钦纤维化。本文构造了仿射外尔群在抛物Hitchin纤维化的直像复形上的作用。特别是，我们得到的仿射外尔群上同调的抛物Hitchin纤维的表示，提供了第一步走向全球的Springer理论。
A Hitchin-Kobayashi correspondence for Kaehler fibrations
设$X$是紧Kaehler流形，$E\to X$是主$K$丛，其中$K$是紧连通李群.设${\cal A}^{1,1}$是$E$上曲率位于$Ω^{1,1}(E\times_{Ad} {\frak k})$中的联络集，其中${\frak k}$是$K$的李代数。赋予$\frak k$一个非退化的双不变双线性对。这允许识别$\{\frak k}\simeq{\frak k}^*$。设$F$是Kaehler左$K$ -流形，且存在$K$作用于$F$的矩映射$μ$.设${\cal S}=Γ(E\times_K F)$。本文研究了方程$$ΛF_A+μ(Φ)=c$$对$A\in {\cal A}^{1,1}$和一个截面$Φ\in {\cal S}$，其中$c\in{\frak k}$是一个固定的中心元.研究了复规范群在${cal A}^{1,1}\times{\cal S}$上的作用轨道中包含方程的解，并在${cal A}^{1,1}\times{\cal S}$上定义了一个正泛函，它推广了Yang-Mills-Higgs泛函，其局部极小值与方程的解重合.
Le lemme fondamental pondéré I : constructions géométriques
这项工作是我们的加权基本引理证明的几何部分，它是Ngô Bao Châu的Langlands-Shelstad基本引理证明的扩展。Ngô的方法是基于对Hichin纤维化的椭圆部分的研究。这种纤维化的全空间是Hitchin丛的代数堆叠，其底空间是“特征多项式”的仿射空间。在椭圆集上，Hitchin纤维化是正常的，其纤维在有限域上的点数可用轨道积分表示。本文研究了大于椭圆集的开集上的Hitchin纤维化，即“一般正则半单集”上的Hitchin纤维化。纤维一般既不是有限型，也不是分离型的。通过与Arthur的截断类比，我们引入了$ξ$ -稳定Hitchin丛的子栈.我们证明了它是一个Deligne-Mumford叠加，在基域上是光滑的，在“特征多项式”的底空间上是适当的。此外，有限域上的$ξ$ -稳定纤维的点数可以表示为加权轨道积分之和，这出现在Arthur-Selberg迹公式中。
Topology of representation spaces of surface groups in PSL(2,R) with assigned boundary monodromy and nonzero Euler number
在本文中，我们完成了SL（2，R）中穿孔曲面的基本群的表示空间的拓扑描述，在穿孔和非零欧拉数处具有规定的行为，遵循Hitchin在非穿孔情形中所采用的策略，并利用平坦丛和Higgs丛在抛物情形中的Hitchin-Simpson对应.这推广了Boden-Yokogawa和Nasatyr-Steer以前的结果。本文件的相关部分旨在概述这一主题。
We give an expression for the pull back of the Hitchin connection from the moduli space of genus two curves to a ten-fold covering of a Teichmüller curve discovered by Veech. We then give an expression, in terms of iterated integrals, for the monodromy representation of this connection. As a corollary we obtain quantum representations of infinitely many pseudo-Anosov elements in the genus two mapping class group.
Families of lattice polarized K3 surfaces with monodromy
我们扩展的概念，晶格极化K3表面的家庭（不一定是简单连接）的基础上，在某种程度上，使控制行动的单值代数周期，并讨论使用这个新的理论在研究家庭的K3表面承认纤维辛自同构。然后，我们给出了应用这些想法的研究卡-丘三重承认晶格极化K3表面的纤维化。
We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.
Milnor fibrations of arrangements with trivial algebraic monodromy
本文研究具有平凡代数单值性的超平面配置 Milnor 纤维化，重点分析上同调跳跃轨迹与基本群下中心序列商，并构造出 Betti 数相同但基本群不同构的一对配置。
Faisceaux caractères sur les espaces de lacets d'algèbres de Lie
本文建立了Grothendieck-Springer仿射纤维化的基础结果，证明了仿射层的可构造性与perversity性质，赋予对称群并构造特化态射，同时证明了可应用于Hitchin纤维化的l-adic复形同伦结果。
The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5
本文通过平铺基空间和分支覆盖方法，确定了 Xiao 亏格二纤维化在次数 3、4、5 下的消失循环，并给出了新的正分解。
Splitting Algebraic Singular Fibrations via Perturbation of Branch Covers
本文针对亏格二纤维化中另外两类奇异纤维，改用非超椭圆多项式族重建曲线，通过分支覆盖扰动将其形变为 Lefschetz 纤维化，并恢复中心纤维紧致化行为及相应单值分解。
We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injecti…
The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
排列理论中的一个中心问题是确定作用在复超平面排列$\mathcal{A}$的Milnor纤维的同调群$H_q(F(\mathcal{A}),\mathbb{C})$上的代数单值的特征多项式$Δ_q$是否由相交格$L(\mathcal{A})$确定。在简单的组合条件下，我们表明，$Δ_1$的因子对应于一定的特征值的顺序一个权力的一个总理$p$的多重性是等于Aomoto-贝蒂数$β_p(\mathcal{A})$，这反过来又是从$L(\mathcal{A})$提取。当$\mathcal{A}$定义了一个只有二重点和三重点的投影线的排列时，这导致了代数单值性的组合公式。为了获得这些结果，我们涉及网的基础拟阵的$\mathcal{A}$共振品种的积极特征。利用网的模不变量，我们发现了拟阵的一个新的可实现性障碍（在$\mathbb{C}$上），我们估计了$\mathcal{A}$的第一个复共振簇中的必要分量的数目。我们的方法还揭示了一个相当意想不到的连接模块谐振与几何的${\rm SL}_2(\mathbb{C})$代表品种，这是由毛雷尔-嘉当方程。
Seiberg-Witten differentials on the Hitchin base
该研究利用李理论与cameral数据，显式刻画了Hitchin纤维化基的Zariski开子集上、权1 Hodge结构变分所带Seiberg-Witten微分的Gauss–Manin协变导数。
The Picard-Lefschetz formula and a conjecture of Kato, the case of Lefschetz fibrations
Kato的一个猜想是，投射簇半稳定退化的上同调单值算子可以用纤维积退化的正规交叉模型的特殊纤维上的代数圈来表示。我们证明了这一猜想在简单的情况下产生的Lefschetz纤维化的半稳定退化。
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