Ngo Bao Chau
We propose a geometric interpretation of the theory of elliptic endoscopy, due to Langlands and Kottwitz, in terms of the Hitchin fibration. As applications, we prove a global analog of a purity conjecture, due to Goresky, Kottwitz and MacPherson. For unitary groups, this global purity statement has been used, in a joint work with G. Laumon, to prove the fundamental lemma over local fields of equal characteristics.
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In this paper, we explore the structure of the Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces.
The main tool in Ngô Bao Châu's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration. For GL(n) and a divisor of degree >2g-2, the theorem says that the relative cohomology is completely determined by its restriction to any dense open subset of the base of the Hitchin fibration. In this note we would like to present in this particular case, our extension of that theorem to the whole Hitchin fibration, including the global nilpotent cone.
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。
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…
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。
We describe certain Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp-Hitchin systems.
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 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.
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…
We prove that the direct image complex for the $D$-twisted $SL_n$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $GL_n$ is due to P.-H. Chaudouard and G. Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $δ$-regularity results for some auxiliary weak abelian fibrations.
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 $\…
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。
In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrictions of line bundles on the variety. As a consequence, we describe the group of sections of the Hitchin fibration for moduli spaces of Higgs bundles on curves.
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.
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…
In this paper, we study the cohomology of the truncated Hitchin fibration, which was introduced in a previous paper. We extend Ngô's main theorems on the cohomology of the elliptic part of the Hitchin fibration. As a consequence, we get a proof of Arthur's weighted fundamental lemma
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。
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。
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…
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。
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.
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…
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…
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 …
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。
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。
We generalize the topological recursion of Eynard-Orantin (2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle $T^*C$ of an arbitrary smooth base curve $C$. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal $\hbar$-deformation family of $D$-modules over an arbitrary projective algebraic curve $C$ of genus greater than $1$, from the geome…
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。
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.
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。
Let $X$ be a compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\cal A}^{1,1}$ be the set of connections on $E$ whose curvature lies in $Ω^{1,1}(E\times_{Ad} {\frak k})$, where ${\frak k}$ is the Lie algebra of $K$. Endow $\frak k$ with a nondegenerate biinvariant bilinear pairing. This allows to identify $\{\frak k}\simeq{\frak k}^*$. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $μ$ for the action of $K$ on $F$. Let ${\cal S}=Γ(E\times_K F)$. In this paper we study the equation $$ΛF_A+μ(Φ)=c$$ fo…
Algebraic cycles and Hitchin systems
本文通过结合傅里叶变换、邻近和消失循环技术以及Springer理论，证明了希奇顿系统的动机分解猜想、相对Hard Lefschetz对称性逆以及强perversity界，揭示了其在代数几何中的核心贡献。
We prove three new results about the global Springer action defined in \cite{GSI}. The first one determines the support of the perverse cohomology sheaves of the parabolic Hitchin complex, which serves as a technical tool for the next results. The second one (the Endoscopic Decomposition Theorem) links certain direct summands of the parabolic Hitchin complex of $G$ to the endoscopic groups of $G$. This result generalizes Ngô's geometric stabilization of the trace formula in \cite{NgoFL}. The third result links the stable parts of the parabolic Hitchin complexes for Langlands dual groups, and e…
Parabolic Multiplicative Affine Springer Fibers
该研究引入抛物型乘法仿射Springer纤维及其全局对应物抛物型乘法Hitchin纤维，通过相关纤维化证明前者等维并确定其维数，为相关代数几何研究提供新对象与结论。
Non-reduced components of global nilpotent cones
本文研究了全局nilpotent锥的非约分分量，通过群方案作用、GIT分层和变形到正常锥等方法，证明了在特定条件下，关联曲线的全局nilpotent锥、K3/阿贝尔/del Pezzo表面的模空间以及Beauville-Mukai系统纤维的homology类性质。
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…
In this talk we discuss the description of the moduli space of principal G-bundles on an elliptic fibration X-->S in terms of cameral covers and their distinguished Prym varieties. We emphasize the close relationship between this problem and the integrability of Hitchin's system and its generalizations. The discussion roughly parallels that of [D2], but additional examples are included and some important steps of the argument are illustrated. Some of the applications to heterotic/F-theory duality were described in the accompanying ICMP talk (hep-th/9802093).
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.
We present an overview of some recent applications of Higgs bundles and the Hitchin fibration.
Let $M\stackrelπ\arrow X$ be a principal elliptic fibration over a Kaehler base $X$. We assume that the Kaehler form on $X$ is lifted to an exact form on $M$ (such fibrations are called positive). Examples of these are regular Vaisman manifolds (in particular, the regular Hopf manifolds) and Calabi-Eckmann manifolds. Assume that $\dim M > 2$. Using the Kobayashi-Hitchin correspondence, we prove that all stable bundles on $M$ are flat on the fibers of the elliptic fibration. This is used to show that all stable vector bundles on $M$ take form $L\otimes π^* B_0$, where $B_0$ is a stable bundle o…
We define and study spectral data associated to U(m,m)-Higgs bundles through the Hitchin fibration. We give a new interpretation of the topological invariants involved, as well as a geometric description of the moduli space.
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