Richard A. Wentworth, Michael Wolf
We show that for every nonelementary representation of a surface group into $SL(2,{\mathbb C})$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.
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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.
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.
In this paper, we prove a generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over a class of non-compact Gauduchon manifolds.
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.
Nahm Transform for Higgs bundles
我们构造了亏格至少为2的黎曼曲面上的Higgs丛的Nahm变换作为其对偶Jacobian的切丛的全空间上的超全纯联络。
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.
Harmonic Metrics for Higgs Bundles of Rank 3 in the Hitchin Section
本研究针对谱曲线为黎曼曲面2叶分歧覆叠的情形，探讨Hitchin截面中秩3希格斯丛的相容调和度量存在性，并给出了复平面$\u2102$与去心复平面$\u2102^*$上对应希格斯丛存在相容调和度量的条件。
Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields
研究紧Kähler流形上带非全纯Higgs场的Higgs丛，给出Donaldson泛函在对角Hermitian度量集合上取极小的充要条件，并证明在强假设下可解Hermitian-Einstein方程。
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.
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 some generalizations of the Hermitian-Einstein equation for diagonal harmonic metrics on cyclic Higgs bundles, including a generalization using subharmonic functions. When the coefficients are all smooth, we prove the existence, uniqueness, and convergence of the solution of their heat equations with Dirichlet boundary conditions. We also generalize two inequality estimates for solutions of the Hermitian-Einstein equation for cyclic Higgs bundles.
The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold $(X, ω)$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend this correspondence to the projectively flat bundles case. We prove that there is an equivalence of categories between the category of $ω$-semi-stable (poly-stable) Higgs bundles $(E, \overline{\partial}_{E}, ϕ)$ with $(2rc_{2}(E)-(r-1)c_{1}^{2}(E))\cdot [ω]^{n-2}=0 $ and the catego…
By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of a folded hyperkaehler 4-manifold and conjecture the existence of a family of such metrics parametrised by an infinite-dimensional analogue of Teichmueller space.
In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion $f$ associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of $f$. As an application, we give a complete picture for maximal $Sp(4,\mathbb{R})$-representations in the $2g-3$ Gothen components and the Hitchin components.
Regulated $L^2$ Norm on Certain Wild Higgs Bundles Over $\mathbb{CP}^1$
本文研究了复射影平面上的Higgs丛模空间上的受控L²范数，并证明了其极限与Fredrickson-Neitzke提出的核心纤维上的范数相关。
Lecture notes from the Third International School on Geometry and Physics at the Centre de Recerca Matematica in Barcelona, March 26--30, 2012.
On diagonal pluriharmonic metrics of $G$-Higgs bundles
本文研究紧致Kähler流形上全纯向量丛分解为线丛直和的Higgs丛，给出了对角度量满足Hermitian-Einstein方程的充要条件，并将其推广到$G$-Higgs丛，同时揭示了稳定性条件与该条件在环面作用下的联系。
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.
Extensions of Higgs Bundles
我们证明了希格斯包的扩展Hitchin-Kobayashi对应。所得结果推广了全纯丛扩张的已知结果。利用Simpson方法，我们构造了稳定对象的模空间.在附录中我们构造了Higgs丛的Bott-Chern形式
Co-Higgs bundles of Schwarzenberger type and the determinant morphism
研究Schwarzenberger型共Higgs丛的行列映射像，探讨其数学结构与几何性质。
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.
Metrics on semistable and numerically effective Higgs bundles
我们提供的概念的数值有效性和数值平坦性的Higgs向量丛紧Kähler流形的纤维度量。我们证明了满足这些条件的几个性质的捆绑，特别是我们表明，数值平坦的希格斯包有消失的陈类，他们承认过滤的concentients是稳定的平坦的希格斯包。我们比较这些定义与那些以前给出的情况下，投射品种。最后，我们研究了数值有效性与半稳定性之间的关系，建立了任意维射影流形上Higgs丛的半稳定性判据。
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…
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
在紧致Kähler流形上Higgs丛的模空间上构造了Peterson-Weil型Kähler形式.证明了这种形式的纤维积分公式，由此得出Peterson-Weil形式是射影流形上Higgs丛的模空间上的具有Quillen度量的行列式线丛的曲率.计算了Peterson-Weil Kähler形式的曲率。我们还证明了在一定的假设下，Higgs丛的模空间支持自然超Kähler结构.
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…
Higgs Bundles and Holomorphic Forms
在自然环境中引入了一类新的希格斯包。证明了Higgs-Hermitian-Yang-Mills度量的存在性和不存在性。
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.
Using Morse-theoretic techniques, we show that the moduli space of U*(2n)-Higgs bundles over a compact Riemann surface is connected.
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。
We present an overview of some recent applications of Higgs bundles and the Hitchin fibration.
Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that a…
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。
Higgs fields, bundle gerbes and string structures
我们使用丛gerbes和他们的连接和弯曲，以获得一个明确的公式德拉姆代表的字符串类的循环群丛。这与早期关于卡路里的工作有关。
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.
We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.
Let $M$ be a compact complex manifold of dimension at least three and $Π: M\rightarrow X$ a positive principal elliptic fibration, where $X$ is a compact Kähler orbifold. Fix a preferred Hermitian metric on $M$. In \cite{V}, the third author proved that every stable vector bundle on $M$ is of the form $L\otimes Π^*B_0$, where $B_0$ is a stable vector bundle on $X$, and $L$ is a holomorphic line bundle on $M$. Here we prove that every stable Higgs bundle on $M$ is of the form $(L\otimes Π^*B_0,Π^*Φ_X)$, where $(B_0, Φ_X)$ is a stable Higgs bundle on $X$ and $L$ is a holomorphic line bundle on $…
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.
We determine the asymptotic behavior in the limit of large Higgs fields of the sectional curvatures of the natural $L^2$ hyperkähler metric $G_{L^2}$ of the moduli space $\mathcal M$ of rank-$2$ Higgs bundles on a Riemann surface $Σ$ away from the discriminant locus. It is shown that their leading order part is given by a sum of Dirac type contributions on $Σ$, for which we find explicit expressions.
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=…
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