Narrow Results

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over $ℂ$). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

Let be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over equipped with a holomorphic symplectic form. Fix a projective structure on X. Using , we construct a quantization of a certain Zariski open dense subset of the symplectic variety .

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.

Given a closed, oriented surface X of genus g ≥ 2, and a semisimple Lie group G, let be the moduli space of reductive representations of π₁X in G. We determine the number of connected components of , for n ≥ 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in is homotopically equivalent to .

We calculate the monodromy action on the mod 2 cohomology for SL(2, ℂ) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2, ℝ)-Higgs bundles.

Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi–Pantev formula holds on maximal rank symplectic leaves of the G-generalized Hitchin system.

We investigate representations of Kähler groups $\mathrm{\Gamma }={\pi }_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If ${dim}_{ℂ}X\ge 2$, these appear if and only if $X$ is a ball quotient, and essentially reduce to the diagonal embedding $\mathrm{\Gamma }<\text{SU}(n,1)\to \text{SU}(nq,q)\hookrightarrow \text{SU}(p,q)$. If $X$ is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.

In this paper, we investigate the relationship between twisted and untwisted character varieties, via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi–Yau categories introduced by Kontsevich and Soibelman. In terms of Donaldson–Thomas theory, this relationship is completely understood via the calculations of Hausel and Villegas of the $E$ polynomials of twisted character varieties and untwisted character stacks. We present a conjectural lift of this relationship to the cohomological Hall algebra setting.

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.

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank $2$ vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin–Kobayashi correspondence for a generalization of Hitchin’s equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher-dimensional variety.

We prove that a blowing-up formula for the intersection cohomology of the moduli space of rank $2$ Higgs bundles over a curve with trivial determinant holds. As an application, we derive the Poincaré polynomial of the intersection cohomology of the moduli space under a technical assumption.

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.

There are two families of geometric structures associated to a surface, with both structures related to representations of the fundamental group of the surface into SL(2, ℂ). These are projective structures on the surface, and Higgs bundles for a given conformal structure of the surface. This note discusses the links between the two.

We study the $2k$-Hitchin’s equations introduced by Ward from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and $2k$-Hitchin’s equations, we review some elementary facts on complex geometry and Yang–Mills theory. Then, we study some properties of holomorphic vector bundles and Higgs bundles and we review the Hermite–Yang–Mills equations together with two functionals related to such equations. Using some geometric tools we show that, as far as Higgs bundles are concerned, $2k$-Hitchin’s equations are reduced to a set of two equations. Finally, we introduce a functional closely related to $2k$-Hitchin’s equations and we study some of its basic properties.

Let $X$ be a compact Riemann surface of genus $g\ge 2$ and let $D\subset X$ be a fixed finite subset. We considered the moduli spaces of parabolic Higgs bundles and of parabolic connections over $X$ with the parabolic structure over $D$. For generic weights, we showed that these two moduli spaces have equal Grothendieck motivic classes and their $E$-polynomials are the same. We also show that the Voevodsky and Chow motives of these two moduli spaces are also equal. We showed that the Grothendieck motivic classes and the $E$-polynomials of parabolic Higgs moduli and of parabolic Hodge moduli are closely related. Finally, we considered the moduli spaces with fixed determinants and showed that the above results also hold for the fixed determinant case.

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable connections, and of holomorphic integrable connections over a compact Kähler manifold. In addition, we describe the moduli space of Higgs bundles and how it relates to the representation variety. We attempt to avoid abstraction, but strive to present and clarify the unifying ideas underlying the theory.