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.
We prove that, given the isomorphism class of the parabolic Deligne-Hitchin moduli space over a smooth projective curve, we can recover the isomorphism class of the curve and the parabolic points.
In this paper, we first prove the parabolic Beauvile-Narasimhan-Ramanan correspondence over an arbitrary field which generalizes the corresponding results over algebraically closed fields in [SWW22]. We use the correspondence and the p-adic integration methods developed by Groechenig- Wyss-Ziegler [GWZ20b] to prove the topological mirror symmetry for parabolic Hitchin systems on curves with arbitrary parabolic structures.
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.
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]
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 study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) $(BBB)$-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup $\mathrm{P}$, whose conjectural dual $(BAA)$-branes we describe. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of $\mathrm{P}$. Secondly, when $\mathrm{P}$ is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the $(BBB)$-brane, and study the va…
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…
In this article, we study the deformation theory of locally free sheaves and Hitchin pairs over a nodal curve. As a special case, the infinitesimal deformation of these objects gives the tangent space of the corresponding moduli spaces, which can be used to calculate the dimension of the corresponding moduli space. We show that the deformation of locally free sheaves and Hitchin pairs over a nodal curve is equivalent to the deformation of generalized parabolic bundles and generalized parabolic Hitchin pairs over the normalization of the nodal curve respectively.
We determine the image of the (strongly) parabolic Hitchin map for all parabolics in classical groups and $G_2$. Surprisingly, we find that the image is isomorphic to an affine space in all cases, except for certain "bad parabolics" in type $D$, where the image can be singular.
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.
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…
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
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.
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 …
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 construct an action of the graded double affine Hecke algebra (DAHA) on the parabolic Hitchin complex, extending the affine Weyl group action constructed in \cite{GSI}. In particular, we get representations of the degenerate DAHA on the cohomology of parabolic Hitchin fibers. We also generalize our construction to {\em parahoric} versions of Hitchin stacks, including the construction of 'tHooft operators as a special case. We then study the interaction of the DAHA action and the cap product action given by the Picard stack acting on the parabolic Hitchin stack.
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…
In this paper we study four families of moduli problems which give rise to two dimensional examples of the Hitchin map. Using a few Fourier-Mukai transforms on the corresponding spectral curves, we give isomorphisms between these moduli problems.
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.
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.
We will study the Hitchin's hamiltonian system for a modular stack of principal SL_2(C) bundle on a smooth projective curve which has a parabolic reduction at certain points. As an application we will obtain a generalization of the Beilinson-Drinfeld isomorphism, which is a quantization of the Hitchin's hamiltonian system.
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…
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…
We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can conclude that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasi projective variety.
Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, we obtain the quasiprojective version of the Corlette-Simpson correspondence between flat bundles and Higgs bundles.
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.
Let $X$ be a quasi-projective curve, compactified to $(Y,D)$ with $X=Y-D$. We construct a Deligne-Hitchin twistor space out of moduli spaces of framed $λ$-connections of rank $2$ over $Y$ with logarithmic singularities and quasi-parabolic structure along $D$. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on $X$ give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights $0,1,2$. The weight $2$ piece corresponds to the deformations of the KMS structure including parabolic weights and the residues …
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…
In this paper we study the relation between parabolic Higgs bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL($k,\Bbb R$). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hit…
In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of abelian varieties. More precisely, we consider the subset of the moduli space of Higgs bundles satisfying the analogous cohomological condition. We show that this set is Zariski closed and that if Sigma is the normalization of an irreducible component containing a stable point, then a connected component of a general fiber of the restriction of the Hitchin map …
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…
We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles, and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.
In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition o…
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…
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…
