Higgs bundles for the Lorentz group

Metrics on semistable and numerically effective Higgs bundles

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 …

Good Wild Harmonic Bundles and Good Filtered Higgs Bundles

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.

Higgs reductions and numerically flat principal Higgs bundles

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…

Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

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…

栈曲线上希格斯丛模空间的泊松结构

Poisson Structures on Moduli Spaces of Higgs Bundles over Stacky Curves

本研究通过李代数胚在栈曲线的扭曲稳定希格斯丛模空间上构造泊松结构，给出新实例，并重点研究根栈上的抛物型希格斯丛模空间。

Geometry of moduli spaces of Higgs bundles

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 compactification of the moduli space of principal Higgs bundles over singular curves

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, …

Einstein-Hermitian connection on twisted Higgs 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.

Model Higgs bundles in exceptional components of the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character variety

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…

Higgs Bundles and Holomorphic Forms

A new class of Higgs bundles is introduced in a natural setting. Existence and nonexistence results for Higgs-Hermitian-Yang-Mills metrics are proved.

Bounded cohomology, Higgs bundles, and Milnor-Wood inequalities

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.

Spectral data for U(m,m)-Higgs bundles

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.

Higgs bundles for the non-compact dual of the unitary group

Using Morse-theoretic techniques, we show that the moduli space of U*(2n)-Higgs bundles over a compact Riemann surface is connected.

Vinberg 对与 Higgs 丛

Vinberg pairs and Higgs bundles

本文研究由半单复李代数循环阶化定义的 Vinberg 对，并考察其在 Higgs 丛理论中的作用。

通过向量丛扭曲的希格斯（Higgs）丛

Higgs bundles twisted by a vector bundle

本文推广光滑复射影曲线上的Higgs丛理论，用秩2向量丛替代典范丛扭曲希格斯场，定义了Hitchin映射、给出谱对应并阐述对应Hitchin方程的Hitchin-Kobayashi对应，填补了曲线与高维簇Higgs丛理论的中间空白。

Cameral data for SU(p+1,p)-Higgs bundles

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.

Toric co-Higgs bundles on toric varieties

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.

Higgs bundles and the real symplectic group

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.

Nilpotent Higgs bundles and the Hodge metric on the Calabi-Yau moduli

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.

Hermitian-Einstein connections on polystable parabolic principal Higgs bundles

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.

Convergence of Yang-Mills-Higgs flow for twist Higgs pairs on Riemann surfaces

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…

Hermitian-Einstein connections on polystable orthogonal and symplectic parabolic Higgs bundles

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 on P^1

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=…

Representations of surface groups and Higgs bundles

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.

Higgs bundles -- Recent applications

We present an overview of some recent applications of Higgs bundles and the Hitchin fibration.

Curvature restrictions on a manifold with a flat Higgs bundle

We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.

Parabolic Bundles and Parabolic Higgs bundles

This is a survey article about parabolic bundles and parabolic Higgs bundles.

Numerically flat Higgs vector bundles

After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.

Higgs向量丛的正性

Positivity of Higgs Vector Bundles

本文提出了一种改进的Higgs向量丛正性定义，解决了Higgs场为零时正性与向量丛正性不一致的问题，并通过Barton-Kleimann型准则刻画了Higgs正向量丛的性质。

$L^2$ representation of Simpson-Mochizuki's prolongation of Higgs bundles and the Kawamata-Viehweg vanishing theorem for semistable parabolic Higgs bundles

In this paper, we provide an $L^2$ fine resolution of the prolongation of a nilpotent harmonic bundle in the sense of Simpson-Mochizuki (an analytic analogue of the Kashiwara-Malgrange filtrations). This is the logarithmic analogue of Cattani-Kaplan-Schmid's and Kashiwara-Kawai's results on the $L^2$ interpretation of the intersection complex. As an application, we give an $L^2$-theoretic proof to the Nadel-Kawamata-Viehweg vanishing theorem with coefficients in a nilpotent Higgs bundle.

A Fourier transform for Higgs bundles

We define a Fourier-Mukai transform for Higgs bundles on smooth curves and study its properties. It is shown that the transform of a stable zero-degree Higgs bundle is an algebraic vector bundle on the cotangent bundle of the Jacobian of the curve. We show that this transformed bundle admits a natural extension to an algebraic vector bundle over a projective compactification of the base. The main result is that the original Higgs bundle can be reconstructed from this extension. We also compute certain invariants of the transformed bundle.

Semistable Higgs bundles of small ranks are strongly Higgs semistable

We give a proof of the conjecture that a semistable Higgs bundle is strongly Higgs semistable in the case of small ranks, based upon the fact that there exists a gr-semistable Griffiths-transverse filtration on a $\nabla$-invariant semistable vector bundle. The latter is a generalization of a result of Carlos Simpson.

Counting twisted Higgs bundles

We count invariants of the moduli spaces of twisted Higgs bundles on a smooth projective curve.

Parabolic principal Higgs bundles

In ``Ramified G-bundles as parabolic bundles'' (J. Ramanujan Math. Soc. (2003) Vol. 18) with Balaji and Nagaraj we introduced the ramified principal bundles. The aim here is to define the Higgs bundles in the ramified context.

关于由向量丛扭曲的 Higgs 丛的若干评注

Remarks on Higgs bundles twisted by a vector bundle

本文研究紧 Riemann 曲面上秩二向量丛扭曲的 Higgs 丛，通过 Hecke 变换刻画相伴线丛扭曲对，并建立其模空间与谱覆盖上秩一无挠 Higgs 层正像之间的谱对应。

Quantization of Hitchin's equations for Higgs Bundles I

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.

An Introduction to Higgs Bundles via Harmonic Maps

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.

Unipotent flat bundles and Higgs bundles over compact Kaehler manifolds

We characterize those unipotent representations of the fundamental group $π_1(X,x)$ of a compact Kaehler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of $π_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.

Hodge Theory and Higgs Bundle on Moduli Spaces of Smooth Varieties with Semi-Ample Canonical Line Bundles, Negativity and Hyperbolicity

This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.

Higgs bundles on Sasakian manifolds

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.

Counting Higgs bundles

We prove a closed formula counting semistable twisted Higgs bundles of fixed rank and degree over a smooth projective curve defined over a finite field. We also prove a formula for the Donaldson-Thomas invariants of the moduli spaces of twisted Higgs bundles over a curve.

Periodicity of Hitchin's uniformizing Higgs bundles

We link the periodicity of Hitchin's uniformizing Higgs bundle with the arithmetic geometry of its underlying curve. Some new relations are discovered. We also speculate on the whole class of periodic Higgs bundles.

Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles

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.

On Higgs bundles on Elliptic Surfaces

The aim of this paper is to establish an equivalence of certain categories of Higgs bundles on a non-isotrivial elliptic surface $π: X \rightarrow C$ with $χ(X) > 0$ and certain categories of Parabolic Higgs bundles on $C$

A note on co-Higgs bundles

We show that for any ample line bundle on a smooth complex projective variety with nonnegative Kodaira dimension, the semistability of co-Higgs bundles of implies the semistability of bundles. Then we investigate the criterion for surface $X$ to have $H^0(T_X)=H^0(S^2T_X)=0$, which implies that any co-Higgs structure of rank two is nilpotent.

卡拉比-丘流形上数值平坦希格斯丛与曲线半稳定希格斯丛的滤过

Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau manifolds

研究卡拉比-丘流形上的希格斯丛，证明数值平坦希格斯丛存在商为稳定局部自由H-nflat的Jordan-Hölder滤过，进而推出单连通卡拉比-丘流形上曲线半稳定希格斯丛判别式为零。

Stratifications on the Moduli Space of Higgs Bundles

The moduli space of Higgs bundles has two stratifications. The Bialynicki-Birula stratification comes from the action of the non-zero complex numbers by multiplication on the Higgs field, and the Shatz stratification arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs bundle. While these two stratification coincide in the case of rank two Higgs bundles, this is not the case in higher rank. In this paper we analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles.

Sasakian and parabolic Higgs bundles

Let $M$ be a quasi-regular compact connected Sasakian manifold, and let $N = M/S^1$ be the base projective variety. We establish an equivalence between the class of Sasakian $G$-Higgs bundles over $M$ and the class of parabolic (or equivalently, ramified) $G$-Higgs bundles over the base $N$.

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…

On Harder-Narasimhan reductions for Higgs principal bundles

The existence and uniqueness of H-N reduction for the Higgs principal bundles over nonsingular projective variety is shown. We also extend the notion of H-N reduction for (Γ, G)-bundles and ramified G-bundles over a smooth curve.

Spectral data for spin Higgs bundles

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…

Torelli theorem for the moduli space of parabolic Higgs bundles

In this article we extend the proof given by Biswas and Gomez of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.

Semistable Higgs bundles on Calabi-Yau manifolds

We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.

Topological invariants of parabolic $G$-Higgs bundles

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of…