Narrow Results

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.