arXiv research

Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.

The paper proves properties of quantum representations and their Toledo invariants.

Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: …

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli …

To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface ov…

We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additi…

The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\C$-Fuchsian curves in complex hyperbolic spaces.

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley corre…

Study on surface group representations in PU(2,1) leading to convex-cocompact examples.

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existe…

Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$ are proven to be Hitchin.

We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.

Paper unifies three invariants for flat bundles over surfaces with boundary.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality…

Develops orbibundle theory for complex hyperbolic geometry.

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 study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…

This paper defines maximal measurable cocycles for surface groups into Hermitian Lie groups and studies their algebraic hulls.

In this paper, we give a parameterization of the SU(2,1) representation space of the Brieskorn homology spheres using the trace coordinates. As applications, we give an example which shows that the orbifold Toledo invariant in \cite{krebs} does not distinguish the connected components of the PU(2,1) representation spac…

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL(2,K) representation variety, where K is the field of real or complex numbers, answering a question of W. Goldman. We also prove the existence of faithful representations into PU…

This paper generalizes a topological invariant to cyclic Higgs bundles.

Unified method to calculate Gromov norm for Kähler classes of bounded symmetric domains.

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal …

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …

We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigati…

We study representations of lattices of PU(m,1) into PU(n,1). We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m-space to complex hyperbolic n-space. This allows us to give a differential geometric proof of rigidity …

We prove the Carlson-Toledo conjecture (for all Kahler groups which do not have property T of Kazhdan).We deduce a conjecture of Goldman-Donaldson for all 3-manifold groups which are rich in the sense of [Re2].

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by…

In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topo…

This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $Σ$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…

Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs b…

This research connects Higgs bundles to projective structures via conformal limits.

Using the methods of the previous paper [ABG], we show that the Teichmuller space T of all closed Riemann surfaces is fibred twice over the Teichmuller space H of hyperelliptic ones. Both fibre bundles π_1,π_2:T->H are real algebraic (rational). They define an embedding T->HxH. In addition, we indicate slight modificat…

Carlson and Toledo conjectured that any infinite fundamental group $Γ$ of a compact Kähler manifold satisfies $H^2(Γ,\R)\not =0$. We assume that $Γ$ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure ($\C$-VHS) on the Kähler manif…

We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially use the concept of a simple earthquake which is a particular case of a Fenchel-Nie…

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of Kähler groups on infinite dimensional re…

This paper is about cohomology of mapping class groups from the perspective of arithmetic groups. For a closed surface $S$ of genus $g$, the mapping class group $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrow Sp(2g, Z)$, under which the stable cohomology of $Sp(2g,Z)$ pulls back to algebra generated…

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regula…

This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined wi…

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exce…

The paper constructs a new metric on Kähler manifolds.

Study shows infinite kernels in topological monodromy for curve families.

Given a group $G$ and a class of manifolds $\CC$ (e.g. symplectic, contact, Kähler etc), it is an old problem to find a manifold $M_G \in \CC$ whose fundamental group is $G$. This article refines it: for a group $G$ and a positive integer $r$ find $M_G \in \CC$ such that $π_1(M_G)=G$ and $π_i(M_G)=0$ for $1<i<r$. We th…

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existenc…

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.