This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X. We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyper-Kaehlerstructure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of X. We describe the moduli spaces and their geometry in terms of the Riemann period matrix of X.
This is the simplest case of the theory developed by Hitchin, Simpson and others. We emphasize its formal aspects that generalize to higher rank Higgs bundles over higher dimensional Kaehler manifolds.
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's rank-2 vector bundle on the projective plane, constructed from a line bundle on the double cover CP^1 \times CP^1 \to CP^2, is naturally a co-Higgs bundle, with the twisted endomorphism, or "Higgs field", also descending from the double cover. Allowing the branch conic to vary, we…
We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.
We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles
We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.
Using the Morse-theoretic methods introduced by Hitchin, we prove that the moduli space of $\SO_0(1,n)$-Higgs bundles when $n$ is odd has two connected components.
The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bundles. Topological and differential-geometric properties of the critical loci are addressed.
We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria …
We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered $λ$-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
I consider principal Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced by Bruzzo and Graña Otero. I prove that a principal Higgs bundle $\mathfrak{E}=(E,φ)$ is H-nflat is either stable or there exists a Higgs reduction of $\mathfrak{E}$ to a parabolic subgroup $P$ of $G$ such that the principal $L$-bundle $\mathfrak{E}_L$ obtained by extending the reduced Higgs bundle $\mathfrak{E}_P$ to the Levi factor $L$ is H-nflat and stable; and as consequence, $H^{*}(\mathfrak{E},\mathbb{R})$ is isomorphic to the cohomology ring of the associated graded object $\ma…
We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of ${\mathbb C}^*$-actions on semiprojective varieties, ${\mathbb C}^*$ characters of indices of ${\mathbb C}^*$-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier-Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscul…
We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Petersson-Weil Kähler form is computed. We also show that, under certain assumptions, a moduli space of Higgs bundles supports of natural hyper-Kähler structure.
A principal Higgs bundle $(P,ϕ)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $ϕ:X\to\text{Ad}P \otimes Ω^1_X$. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles. More precisely, given a faithful representation $ρ:G\to Sl(V)$ of $G$, we consider principal Higgs bundles as triples $(E,q,ϕ)$ where $E$ is a vector bundle with $\rk{E}=\dim V$ over the normalization $\xtilde$ of $X$, $q$ is a parabolic structure on $E$ and $ϕ:E\ab{}\to L$ is a morphism of bundles, …
Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach is simpler.
We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exceptional components of the maximal $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundle moduli space, which correspond to components solely consisted of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topologica…
We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the representation theoretic generalized Milnor-Wood inequality and the various different versions of Milnor-Wood inequalities for Higgs bundles that are known in the literature.
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.
We study the cameral and spectral data for the moduli space of polystable $\mathrm{SU}(p+1,p)$-Higgs bundles and deduce the latter from the former. As an application, we obtain that the Toledo invariant classifies the connected components of the regular fibers of the Hitchin map.
Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, García-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi-Yau moduli.
Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.
We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Riemann surface $X$. It is already known the gradient flow with initial data $(A_0,ϕ_0)$ converges to a critical point $(A_\infty, ϕ_\infty)$ of this functional. Using a modified Chern-Weil type inequality, we prove that the limiting twist Higgs bundle $(E, d_{A_\infty}", ϕ_\infty)$ is given by the graded twist Higgs bundle defined by the Harder-Narasimhan-Seshadri filtration of the initial twist Higgs bundle $(E,d_{A_0}",ϕ_0)$, generalizing Wilkin's results for u…
Let X be a smooth complex projective curve and S a finite subset of X. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian-Einstein connection if and only if it is polystable.
Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P^1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defined equation. For ranks r=2,3,4, we explicitly verify the conjectural Betti numbers emerging from the recent work of Chuang, Diaconescu, Pan, and Mozgovoy on the ADHM formula. We state the result for r=…
These are the lecture notes from my course in the January 2011 School on Moduli Spaces at the Newton Institute. I give an introduction to Higgs bundles and their application to the study of character varieties for surface group representations.
In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.
In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.
In this paper, using Donaldson's heat flow, we show that the semi-stability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian-Einstein structure on the Higgs bundle.
We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit towards the solution of the Hitchin equation. We relate the existence of balanced metrics to the Gieseker stability of the Higgs bundle.
This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.
We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds.
In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a compact Kähler manifold $(M, ω)$. We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorphic to the double dual of the graded Higgs sheaves associated to the Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle.
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.
In this paper, we consider the existence of approximate Hermitian-Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. By using the continuity method, we show that they are equivalent.
In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.
We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian.
Fix a $C^\infty$ principal $G$--bundle $E^0_G$ on a compact connected Riemann surface $X$, where $G$ is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang--Mills--Higgs functional on the cotangent bundle of the space of all smooth connections on $E^0_G$. We prove that this flow preserves the subset of Higgs $G$--bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs $G$--bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author…
In this paper, we study the existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds and discuss related consequence, specially on the applications in Higgs bundles, towards generalizing Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds setting.
In this paper we investigate the moduli space of parabolic Higgs bundles over a punctured Riemann surface with varying weights at the punctures. We show that the harmonic metric depends analytically on the weights and the stable Higgs bundle. This gives a Higgs bundle generalisation of a theorem of McOwen on the existence of hyperbolic cone metrics on a punctured surface within a given conformal class, and a generalisation of a theorem of Judge on the analytic parametrisation of these metrics.
We give a geometric characterisation of the topological invariants associated to SO(m,m+1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)-Higgs bundles.
The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.
For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $Σ$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $Σ$.
We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.
We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such a Higgs bundle is equivalent to a certain class of solutions to the affine Toda equations. We further explain this relationships in terms of lifts of harmonic maps.
We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kähler manifolds. We provide a simple and independent proof of Reznikov's result and show that our characteristic classes of Higgs bundles and the characteristic classes of flat vector bundles are compatible via the nonabelian Hodge theorem.
The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the $SL(r, \mathbb{C})$ character variety of a finitely presented group $Γ$ is homeomorphic to the moduli space of rank $r$ Higgs bundles over an admissible complex $X$ with $π_1(X) = Γ$. A key role is played by the theory of harmonic maps defined on singular domains.
