Moduli Space Of Higgs Bundles Research Articles

Spin(8,C)-Higgs Bundles and the Hitchin Integrable System

Let M(Spin(8,C)) be the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface X of genus g≥2. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of Spin(8,C) acts on M(Spin(8,C)), and those Higgs bundles that admit a reduction in the structure group to G2 are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M(G2)→M(Spin(8,C)). In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with M(Spin(8,C)). In particular, it is checked that the map M(G2)→M(Spin(8,C)) is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G2 and Spin(8,C)-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.

On the spectral variety for rank two Higgs bundles

AbstractIn this article, we study the Hitchin morphism over a smooth projective variety. The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which in general not surjective when the dimension of the variety is greater than one. Chen–Ngô introduced the spectral base, which is a closed subvariety of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen–Macaulayfications of the spectral varieties. For rank two Higgs bundles over a projective manifold, we explicitly construct a finite Cohen–Macaulayfication of the spectral variety as a double branched covering, thereby confirming Chen–Ngô's conjecture. Moreover, using this Cohen–Macaulayfication, we can construct the Hitchin section for rank two Higgs bundles, which allows us to study the rigidity problem of the character variety and also to explore a generalization of the Milnor–Wood‐type inequality.

Cohomological $\chi$-independence for Higgs bundles and Gopakumar–Vafa invariants

The aim of this paper is two-fold. Firstly, we prove Toda’s \chi -independence conjecture for Gopakumar–Vafa invariants of arbitrary local curves. Secondly, following Davison’s work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank r and Euler characteristic \chi which are not necessary coprime, and show that it does not depend on \chi . This result extends the Hausel–Thaddeus conjecture on the \chi -independence of E-polynomials proved by Mellit, Groechenig–Wyss–Ziegler and Yu in two ways: We obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.

Mirror symmetry and Hitchin system on Deligne–Mumford curves: Strominger–Yau–Zaslow duality

Abstract We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.

Narasimhan–Ramanan branes and wobbly Higgs bundles

Narasimhan–Ramanan branes, introduced by the authors in a previous paper, consist of a family of (BBB)-branes inside the moduli space of Higgs bundles, and a family of complex Lagrangian subvarieties. It was conjectured that these complex Lagrangian subvarieties support the (BAA)-branes that are mirror dual to the Narasimhan–Ramanan (BBB)-branes. In this paper, we show that the support of these branes intersects nontrivially the locus of wobbly Higgs bundles.

Non-Strict Plurisubharmonicity of Energy on Teichmüller Space

Abstract For an irreducible representation $\rho :\pi _{1}(\Sigma _{g})\to \textrm{GL}(n,\mathbb{C})$, there is an energy functional $\textrm{E}_{\rho }: {{\mathcal{T}}}_{g}\to \mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\textrm{GL}(n,\mathbb{C})/\textrm{U}(n)$. It follows from a result of Toledo that $\textrm{E}_{\rho }$ is plurisubharmonic, that is, its Levi form is positive semi-definite. We describe the kernel of this Levi form, and relate it to the $\mathbb{C}^{*}$ action on the moduli space of Higgs bundles. We also show that the points in $ {{\mathcal{T}}}_{g}$ where strict plurisubharmonicity fails (i.e., this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,\rho )$, the map $\textrm{E}_{\rho }$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

Arakelov–Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb {C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

On the Image of Hitchin Morphism for Algebraic Surfaces: The Case GLn

Abstract The Hitchin morphism is a map from the moduli space of Higgs bundles $\mathscr {M}_{X}$ to the Hitchin base $\mathscr {B}_{X}$, where $X$ is a smooth projective variety. When $X$ has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme $\mathscr {A}_{X}$ of $\mathscr {B}_{X}$, which is called the space of spectral data. They proved that the Hitchin morphism factors through $\mathscr {A}_{X}$ and conjectured that $\mathscr {A}_{X}$ is the image of the Hitchin morphism. We prove that when $X$ is a smooth projective surface, this conjecture is true for vector bundles. Moreover, we show that $\mathscr {A}_{X}$, for any dimension, is invariant under any proper birational morphism and apply the result to study $\mathscr {A}_{X}$ for ruled surfaces.

Relative log-symplectic structure on a semi-stable degeneration of moduli of Higgs bundles

In a recent paper [4], a semi-stable degeneration of moduli space of Higgs bundles on a curve has been constructed. In this paper, we show that there is a relative log-symplectic form on this degeneration, whose restriction to the generic fibre is the classical symplectic form discovered by Hitchin. We compute the Poisson ranks at every point and describe the symplectic foliation on the closed fibre. We also show that the closed fibre, which is a variety with normal crossing singularities, acquires a structure of an algebraically completely integrable system.

Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

AbstractWith , let be the G‐character variety of a given finitely presented group Γ, and let be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for E‐polynomials of and the one for , generalizing a formula of Mozgovoy–Reineke. The proof uses a natural stratification of coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald–Cheah for symmetric products; we also adapt it to the so‐called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the E‐polynomials, and Euler characteristics, of the irreducible stratum of ‐character varieties of some groups Γ, including surface groups, free groups, and torus knot groups, for low values of n.

A cohomological nonabelian Hodge Theorem in positive characteristic

We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. It is interesting that this last result is proved as follows: we prove a weaker version in positive characteristic; we lift and strengthen the weaker version to the result in characteristic zero; finally, we specialize the result to positive characteristic. The moduli spaces we work with admit certain natural morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology ring isomorphisms we find are filtered isomorphisms for the resulting perverse Leray filtrations.

Sheaves on non-reduced curves in a projective surface

Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack Mx (nC, χ) of pure sheaves supported at the non-reduced curve nC (n ≽ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism $${h_{L,\chi}}:{\cal M}_X^H\left({L,\chi} \right) \to \left| L \right|$$ sending each semistable 1-dimensional sheaf to its support has all its fibers of the same dimension for X Fano or with the trivial canonical line bundle and |L| contains integral curves.

An analytic approach to the quasi-projectivity of the moduli space of Higgs bundles

The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.

P=W conjectures for character varieties with symplectic resolution

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-etale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, or the intersection theory of some singular Lagrangian cycles. In particular, we study in detail a Dolbeault moduli space which is specialization of the singular irreducible holomorphic symplectic variety of type O'Grady 6.

Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

In this article, we show some domination results on the Hitchin fibration, mainly focusing on the $n$-Fuchsian fibers. More precisely, we show the energy density of associated harmonic map of an $n$-Fuchsian representation dominates the ones of all other representations in the same Hitchin fiber, which implies the domination of topological invariants: translation length spectrum and entropy. As applications of the energy density domination results, we obtain the existence and uniqueness of equivariant minimal (or maximal) surfaces in certain product Riemannian (or pseudo-Riemannian) manifold. Our proof is based on establishing an algebraic inequality generalizing a GIT theorem of Ness on the nilpotent orbits to general orbits.

Shatz and Bialynicki-Birula stratifications of the moduli space of Higgs bundles

Let $X$ be a compact Riemann surface of genus $g\geq 2$. In this paper we study how two memorable stratifications of the moduli space of Higgs bundles of rank $4$ over $X$, Shatz stratification and Bialynicki-Birula stratification, relate to each other and provide dimensions for the Harder-Narasimhan type of a semistable rank 4 Higgs bundle over $X$. In particular, we prove that the two stratifications do not coincide. Thus, we extend to rank 4 the work of Hausel [7], who proved that both stratifications coincide in rank 2, and of Gothen and Zúñiga-Rojas [6], who proved that such a thing no longer occurred in rank 3.

Parabolic Hitchin maps and their generic fibers

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.

Construction of the moduli space of Higgs bundles using analytic methods

Construction of the moduli space of Higgs bundles using analytic methods