Hitchin Hamiltonians in genus 2

SL_2 Hitchin可积系统亏格为2的自对偶性

Self-duality of the SL_2 Hitchin integrable system at genus two

我们重新讨论了Hitchin可积系统，其相空间是亏格为2的光滑复曲线上的全纯$SL_2$ -丛的模空间$N$的丛余切。$N$可以等同于三维二阶theta函数的射影空间。我们证明了$T^*N$上的Hitchin系统具有显著的对称性：它在位置和动量的交换下是不变的。这一性质允许完成的工作范Geemen-Previato的基础上，经典的结果几何的库默尔四次曲面，指定的明确形式的哈密尔顿的希钦系统。由此产生的可积系统类似于经典的Neumann系统，也是自对偶的。它的量子化产生了一个交换家庭的微分算子的二阶作用于齐次多项式在四个复变量。正如van Geemen-de Jong最近所证明的，这些算子实现了SU（2）群和亏格2曲线的Knizhnik-Zamolodchikov-Bernard-Hitchin联络。

Very stable bundles and properness of the Hitchin map

Let $X$ be a smooth complex projective curve of genus $g\geq 2$ and let $K$ be its canonical bundle. In this note we show that a stable vector bundle $E$ on $X$ is very stable, i.e. $E$ has no non-zero nilpotent Higgs field, if and only if the restriction of the Hitchin map to the vector space of Higgs fields $H^0(X, \mathrm{End}(E) \otimes K)$ is a proper map.

低通用性的Hitchin系统

Hitchin Systems at Low Genera

本文给出了Hitchin可积系统和Knizhnik-Zamolodchikov-Bernard联络在亏格0、1和2上的最简单的情形。特别地，利用它与经典Neumann可积系统的关系，构造了SL（2）亏格为2的Hitchin系统的作用角变量.

不变量向量丛与希钦系统

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 connection on the Veech curve

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.

希钦哈密顿系统的量子化与Beilinson-Drinfeld同构

A quantization of the Hitchin hamiltonian system and the Beilinson-Drinfeld isomorphism

本文研究光滑射影曲线上主SL_2（C）丛的模叠加的Hitchin哈密顿系统，该曲线在某些点上具有抛物约化。作为应用，我们将得到一个推广的Beilinson-Drinfeld同构，这是一个量化的希钦的哈密顿系统。

两个二次曲面的交：模解释与希钦态射

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。

Hitchin fibrations, abelian surfaces, and the P=W conjecture

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…

论希钦的联系

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。

Hitchin 纤维化是 Ngô 纤维化

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。

一些希钦哈密顿量的等单点变形解释

An Interpretation of Some Hitchin Hamiltonians In Terms of Isomonodromic Deformation

本文研究了紧致Riemann曲面上具有非正则奇点联络的框架主丛的模空间。这些空间已建成Boalch通过一个无限维辛减少。证明了由Atiyah-Bott形式导出的辛结构与由超上同调给出的辛结构一致。本文的主要结果适应工作的Krichever和Hurtubise给出一个解释的一些希钦哈密尔顿作为产生哈密尔顿向量场的模空间的不规则连接所产生的差异的isomonodromic流定义在两种不同的方式。这依赖于实现开放集的模空间的丛所产生的Hecke修改的固定丛。

On the Automorphisms of a Rank One Deligne-Hitchin Moduli Space

Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element o…

奇怪的二元性和希钦/WZW连接

Strange duality and the Hitchin/WZW connection

对于正亏格的紧致黎曼曲面X，在秩为r和水平为k的丛的模上的某个θ丛的截面空间允许一个自然映射到秩为k和水平为r的相似截面空间（的对偶）（奇怪的对偶同构）。当X在一个族中变化时，同构的两边都带有射影联络。我们证明了这个映射是（射影）平坦的。

Hecke曲线与Hitchin判别式

Hecke curves and Hitchin discriminant

设$C$是复数域上亏格为$g\geq 4$的光滑射影曲线，${\cal SU}^s_C(r,d)$是秩为$r$且具有固定行列式次数的稳定向量丛的模空间$d$ .在投影余切空间中，在${\cal SU}^s_C(r,d)$的一般点$E$处，存在一个由具有奇异谱曲线的余切向量组成的特殊超曲面${\cal S}_E$.在$E$上的射影切空间中，存在一个由${\cal SU}^s_C(r,d)$到$E$中Hecke曲线的切向量组成的特殊子簇${\cal C}_E$.我们的主要结果证明了超曲面${\cal S}_E$和簇${\cal C}_E$是对偶的.作为这个对偶关系的应用，我们证明了任何满射态射${\cal SU}^s_C(r,d) \to {\cal SU}^s_{C'}(r,d)$，其中$C'$是亏格$g$的另一条曲线，都是双正则的.对于${\cal SU}^s_C(r,d)$，这证实了一般的期望，即皮卡德数1的法诺簇，除了投射空间，没有非平凡的自态射，并且皮卡德数1的法诺簇之间的态射是罕见的。对偶关系也给出了非交换Torelli定理和Kouvidakis-Pantev关于${\cal SU}^s_C(r,d)$的自同构的结果的简单证明。

Construction of the Poincare sheaf for higher genus curves

Let $X$ be a smooth projective curve of genus $g$ and $L$ be a degree $l$ line bundle on $X$ with $l\geq 2g-1$. Denote the stack of rank two Higgs bundles on $X$ with value in $L$ by $\mathcal{H}iggs$ and the semistable part by $\mathcal{H}iggs_{ss}$. Let $H$ be the Hitchin base. In this paper we will construct the Poincare sheaf $\mathcal{P}$ on $\mathcal{H}iggs\times_{H}\mathcal{H}iggs_{ss}$ which is a maximal Cohen-Macaulay sheaf and flat over $\mathcal{H}iggs_{ss}$. In particular this includes the locus of nonreduced spectral curves.

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 fibration on moduli of symplectic and orthogonal parabolic Higgs bundles

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 …

Hitchin-Mochizuki morphism, Opers and Frobenius-destabilized vector bundles over curves

Let X be a smooth projective curve of genus g \textgreater{}1 defined over an algebraically closed field k of characteristic p \textgreater{}0. For p sufficiently large (explicitly given in terms of r,g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E …

Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen…

Flat rank 2 vector bundles on genus 2 curves

We study the moduli space of trace-free irreducible rank 2 holomorphic connections over a complex projective curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus $2$ case, connections as above are invariant under the hyperelliptic involution : they descend as rank $2$ logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical a…

论希钦制度

On the Hitchin System

Hitchin系统是曲线上半稳定向量丛的模空间的余切空间上的完全可积Hamilton系统。我们考虑秩二向量丛具有平凡行列式的情形。这样一个丛$E$在曲线的雅可比矩阵中定义了一个除数$D_E$，对于$D_E$的任何光滑点，我们定义了一个余切向量（希格斯场）。希格斯场上的希钦映射由除数$D_E$上的高斯映射确定。我们将结果应用到$g=2$的情况下，并显示如何希钦的系统是相关的经典线几何$\PP^3$。

任意约化群的Hitchin系统阿贝尔化的初等方法

An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups

研究亏格大于1的紧致Riemann曲面C上稳定主G-丛的模空间，其中G是约化代数群。我们明确地构造了一个映射F，它是从Hitchin映射的类属纤维到与C的一个适当的Galois覆盖相关联的广义Prym簇的映射。映射F有有限纤维。在G= PG 1（2）的情况下，可以检验F的类属纤维是关于Z/2 Z的2d-2个拷贝的乘积的主齐性空间，其中d是C上的标准丛的次数。然而，如果G的Dynkin图不包含类型$B_{n}$ n>0的分量，或者当换位子群（G，G）是单连通的，则映射F是单射的。

亏格2中的扭量Hecke特征层

Twistor Hecke eigensheaves in genus 2

本文遵循已有策略，在亏格2的光滑射影曲线上构造了沿wobbly轨迹带奇点的平坦联络，验证了相关$D$-模为Hecke特征层，并基于非阿贝尔Hodge对应与Hitchin纤维揭示了局部系统与谱簇的联系。

全纯丛和Hitchin系的模空间的二重陪集构造

Double coset construction of moduli space of holomorphic bundles and Hitchin systems

给出了黎曼曲线上全纯向量丛的模空间的一种不同于标准环群构造的双陪集空间的描述。我们的方法是基于全纯丛的等价定义，基于转移映射或一阶微分算子。利用这种方法，我们提出了两个独立的推导的希钦可积系统。我们定义了一个“超自由”的楼上系统，并从该系统出发，通过三步哈密尔顿约化得到了Hitchin系统。一个特别注意的是被给予肖特基参数化曲线。

Spectral Data for SU(1,2) Higgs Bundles

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…

Generalized Kähler metrics from Hamiltonian deformations

We give a new characterization of generalized Kähler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin's partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for arbitrary generalized Kähler structures. That is, we show that any generalized Kähler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin's deformation. Finally, we indicate a close r…

A Brief Survey of Higgs Bundles

Considering a compact Riemann surface of genus greater than two, a Higgs~bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin's work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as "Higgs fields" beacuse in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson us…

$\text{SL}(n)$ 群的希钦（Hitchin）纤维计数

Comptage de fibrés de Hitchin pour le groupe $\mathrm{SL}(n)$

该研究在有限域光滑射影曲线框架下，给出了$\boldsymbol{\text{SL}(n)}$群的稳定$D$-扭曲Hitchin丛数目的循环覆叠表达公式，并结合已有工作推导了对应模空间的多类不变量闭式公式，核心工具为自守诱导基本引理与相对上同调支撑定理。

Hitchin可积系统、谱曲线的变形和KP型方程

Hitchin integrable systems, deformations of spectral curves, and KP-type equations

本文确定了希钦纤维化中出现的一个有效谱曲线族。利用这个族，我们将代数曲线上稳定Higgs丛的模空间嵌入到Sato Grassmannian中。我们证明了定义在Higgs模空间上的自然代数完全可积的哈密顿系统Hitchin可积系统与KP方程是一致的。证明了这些模空间上的Serre对偶对应于作用在Grassmannian上的伪微分算子的形式伴随。从这一事实出发，我们用KP方程的约化来确定Sp（2 m）-Higgs丛的模空间上的Hitchin可积系统。我们还证明了通过取GL-Higgs丛的模空间上的李代数作用的辛商，构造了Sp（2 m）-Higgs模空间的对偶阿贝尔纤维化（SYZ镜像对偶）.

量子上同调、Hitchin系统与Fourier-Mukai比较

Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison

本文证明稳定Higgs丛模空间上除子的等变量量子积由Steinberg对应给出，并证明Fourier-Mukai变换与量子积相容，建立了量子上同调与Hitchin系统的联系。

代数曲线上的向量丛与Lax方程

Vector bundles and Lax equations on algebraic curves

建立了任意紧致Riemann曲面上带谱参数的零曲率方程的Hamilton理论。结果表明，该方程可以看作是一个无限维场推广的希钦系统的交换流。提出了椭圆Calogero-Moser系统的场模拟。基于代数曲线上稳定全纯向量丛的Tyurin参数，得到了Hitchin系统的一个显式参数化。

Optimum weight chamber examples of moduli spaces of stable parabolic bundles in genus 0

We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of the complexity of the actions of the corresponding groups of bundle automorphisms on stable parabolic structures. For the given choices of parabolic weights, $\mathscr{N}$ consists entirely of isomorphism classes of strictly stable parabolic bundles whose underlying Birkhoff-Grot…

与Kowalewski顶有关的亏格2曲线的塔

A tower of genus 2 curves related to the Kowalewski top

已知亏格2的几条曲线，使得Kowalewski顶的运动方程在其雅可比行列式上线性化。人们可以期望从超越方法通过解决运动方程的θ函数，他们的雅可比是同构的。本文重点讨论两个这样的曲线：Kowalewski的和Bobenko-Reyman-Semenov-Tian-Shansky的，后者所产生的解决方案的问题的谱曲线的方法。一个等距建立在这些曲线的雅可比矩阵之间的纯代数手段，使用Richelot变换的亏格2曲线。它表明，这是尊重哈密顿流。这两条曲线被构造成一个具有同构雅可比行列式的亏格为2的曲线的无限塔。

秩2 Higgs丛模空间的上同调环中的关系

Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles

黎曼曲面上秩为2且度为1的稳定丛的模空间具有由所谓的泛类生成的有理上同调。巴拉诺夫斯基、金-纽斯特德、西伯特-田和扎吉尔的工作提供了这些类之间的一套完整的关系，用属中的递归来表示。本文在Hitchin和Simpson的意义下，对Higgs丛的非紧模空间也做了同样的工作。这里有比稳定丛更多的独立关系，但在某种意义上，答案更简单，因为公式是完全显式的，而不是递归的。Kirwan关于全纯圆作用的等变上同调的结果是非常重要的。第一部分和第二部分一起描述了秩为2的希格斯丛空间的上同调环，其详细程度与已知的稳定丛基本相同。