Stable Higgs Bundles Research Articles

Hitchin map on even very stable upward flows

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the [Formula: see text] case, we classify the type [Formula: see text] examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, [Formula: see text]-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.

Cohomology of the moduli of Higgs bundles on a curve via positive characteristic

For a curve of genus g and any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by the complex Lie group \mathrm{GSp}(2g), between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As a consequence, we prove two structural results concerning the cohomology of Higgs moduli which are predicted by the P = W Conjecture in Non-Abelian Hodge Theory: (1) Galois conjugation for character varieties preserves the perverse filtrations for the corresponding Higgs moduli spaces. (2) The restriction of the Hodge–Tate decomposition for a character variety to each piece of the perverse filtration for the corresponding Higgs moduli space also gives a decomposition. Our proof uses reduction to positive characteristic and relies on the non-abelian Hodge correspondence in characteristic p between Dolbeault and de Rham moduli spaces.

Mirror symmetry for Nahm branes

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.

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 cominuscule Higgs bundles.

Curvature Formulas Related to a Family of Stable Higgs Bundles

In this paper, we investigate the geometry of the base complex manifold of an effectively parametrized holomorphic family of stable Higgs bundles over a fixed compact K\"{a}hler manifold. The starting point of our study is Schumacher-Toma/Biswas-Schumacher's curvature formulas for Weil-Petersson-type metrics, in Sect. 2, we give some applications of their formulas on the geometric properties of the base manifold. In Sect. 3, we calculate the curvature on the higher direct image bundle, which recovers Biswas-Schumacher's curvature formula. In Sect. 4, we construct a smooth and strongly pseudo-convex complex Finsler metric for the base manifold, the corresponding holomorphic sectional curvature is calculated explicitly.

Analytically stable Higgs bundles on some non-Kähler manifolds

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kahler, we solve the Hermitian–Einstein equation on analytically stable Higgs bundles.

Stable Higgs bundles on ruled surfaces

Let $\pi : X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on $C$ under $\pi$ is a $L$-(semi)stable Higgs bundle. Conversely, if $(V,\theta)$ is a $L$-(semi)stable Higgs bundle on $X$ with $c_1(V)= \pi^*(\bf d \rm)$ for some divisor $\bf d \rm $ of degree $d$ on $C$ and $c_2(V)=0$, then there exists a (semi)stable Higgs bundle $(W,\psi)$ of degree $d$ on $C$ whose pullback under $\pi$ is isomorphic to $(V,\theta)$. As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on $X$ whenever $g(C)\geq 2$ and the base field is $\mathbb{C}$.

Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section $$\mathfrak{D}$$ . Let ( $$E_{\mathfrak{D}},\phi_{\mathfrak{D}}$$ ) be a stable Higgs bundle over $$\mathfrak{D}$$ , and (E, e) a Higgs bundle over V which is asymptotic to ( $$E_{\mathfrak{D}},\phi_{\mathfrak{D}}$$ ). In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang-Mills metric on (E,e).

Symplectic Geometry of a Moduli Space of Framed Higgs Bundles

Abstract Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure that is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold that is called here as ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${\mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.

Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.

Stable Higgs bundles over positive principal elliptic fibrations

Abstract Let M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 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 ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

Le nombre des systèmes locaux ℓ-adiques sur une courbe

Let X1 be a projective, smooth, and geometrically connected curve over Fq with q=pn elements where p is a prime number, and let X be its base change to an algebraic closure of Fq. The goal of this article is to announce a formula for the number of irreducible ℓ-adic local systems (ℓ≠p) with a fixed rank over X fixed by the Frobenius endomorphism. This number behaves like a Grothendieck–Lefschetz fixed point formula for a variety over Fq, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. We also sketch our method, which consists in passing to the automorphic side by Langlands correspondence, then calculating all the terms in Arthur's non-invariant trace formula and linking the geometric part of trace formula to the number of Fq-points of the moduli space of stable Higgs bundles.

Two-dimensional moduli spaces of rank 2 Higgs bundles overCP1with one irregular singular point

Abstract We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over C P 1 with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible parabolic structure such that the parabolic degree of the Higgs bundle is 0. Our method relies on elliptic fibrations of the rational elliptic surface, an equivalence of categories between irregular Higgs bundles and some sheaves on a ruled surface, and an analysis of stability conditions.

Analytic convergence of harmonic metrics for parabolic Higgs bundles

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.

MIXED HODGE STRUCTURES OF THE MODULI SPACES OF PARABOLIC CONNECTIONS

In this paper, we investigate the mixed Hodge structures of the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable parabolic Higgs bundles and the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$-stable regular singular parabolic connections. We show that the mixed Hodge polynomials are independent of the choice of generic eigenvalues and the mixed Hodge structures of these moduli spaces are pure. Moreover, by the Riemann–Hilbert correspondence, the Poincaré polynomials of character varieties are independent of the choice of generic eigenvalues.

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Let $(E,\overline{\partial}_E,\theta)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix the flat metric $h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$ $(t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,t\theta)$ such that $\det(h_t)=h_{\det(E)}$. We study the behaviour of $h_t$ when $t$ goes to $\infty$. First, we show that the Hitchin equation is asymptotically decoupled under the assumption that the Higgs field is generically regular semisimple. We apply it to the study of the so called Hitchin WKB-problem. Second, we study the convergence of the sequence $(E,\overline{\partial}_E,\theta,h_t)$ in the case where the rank of $E$ is two. We introduce a rule to determine the parabolic weights of a "limiting configuration", and we show the convergence of the sequence to the limiting configuration in an appropriate sense. The results can be appropriately generalized in the context of Higgs bundles with a Hermitian-Einstein metric on curves.

Indecomposable vector bundles and stable Higgs bundles over smooth projective curves

We prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve X dened over a nite eld is given by a polynomial (depending only on r;d and the genus g of X) in the Weil numbers of X. We provide a closed formula | expressed in terms of generating series- for this polynomial. We also show that the same polynomial computes the number of points of the moduli space of stable Higgs bundles of rank r and degree d over X. This entails a closed formula for the Poincar e polynomial of the moduli spaces of stable Higgs bundles over a compact Riemann surface, and hence also for the Poincar e polynomials of the twisted character varieties for the groups GL(r).

Bogomolov restriction theorem for Higgs bundles

Let ( E , θ ) be a stable Higgs bundle of rank r on a smooth complex projective surface X equipped with a polarization H. Let C ⊂ X be a smooth complete curve with [ C ] = n ⋅ H . If 2 n > R r ( 2 r c 2 ( E ) − ( r − 1 ) c 1 ( E ) 2 ) , where R = max { ( r s ) ( r − 1 s − 1 ) : 1 ⩽ s ⩽ r − 1 } , then we prove that the restriction of ( E , θ ) to C is a stable Higgs bundle. This is a Higgs bundle analog of Bogomolov's restriction theorem for stable vector bundles.

Stable Higgs bundles with trivial Chern classes. Several examples

We propose a new interpretation of Higgs bundles as vector bundles together with actions of the symmetric tangent tensor algebra. Via this interpretation, we construct new nontrivial examples of stable Higgs bundles with vanishing Chern classes.

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.