Florent Schaffhauser

Universität Heidelberg · Institute of Mathematics

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic H...

We prove that moduli spaces of semistable vector bundles of coprime rank and degree over a non-singular real projective curve are maximal real algebraic varieties if and only if the base curve itself is maximal. This provides a new family of maximal varieties, with members of arbitrarily large dimension. We prove the result by comparing the Betti n...

For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\overline{k}$-valued points of moduli spaces of quiver representations over $k$; the fixed locus is the set of $k$-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of $k$. We provide a modular interpretation...

This note is based on a talk given at the 2019 ISAAC Congress in Aveiro, Portugal. We give an expository account of joint work with Daniele Alessandrini and Gye-Seon Lee on Hitchin components for orbifold groups (arXiv:1811.05366), recasting part of it in the language of analytic orbi-curves. This reduces the computation of the dimension of the Hit...

Given a discrete subgroup Γ of finite co-volume of PGL(2, R), we define and study parabolic vector bundles on the quotient Σ of the (extended) hyperbolic plane by Γ. If Γ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vector bundles on the orientation cover of degree two of Σ. We...

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of Higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball, and we compute its dimension explicitly. For example, the Hitchin component of the right-angled hyperbolic-poly...

Real and quaternionic parabolic vector bundles over a $n$-pointed Riemann surface equipped with an anti-holomorphic involution are studied.

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then use this invariant-theoretic...

These notes are based on a series of three 1-hour lectures given in 2012 at the CRM in Barcelona, as part of the event Master Class and Workshop on Representations of Surface Groups, itself a part of the research program Geometry and Quantization of Moduli Spaces. The goal of the lectures was to give an introduction to the general theory of Klein s...

We study two types of actions on moduli spaces of quiver representations over a field $k$ and we decompose their fixed loci using group cohomology. First, for a perfect field $k$, we study the action of the absolute Galois group of $k$ on the $\overline{k}$-valued points of this quiver moduli space; the fixed locus is the set of $k$-rational points...

These notes are based on a series of three 1-hour lectures given in 2012 at the CRM in Barcelona, as part of the event Master Class and Workshop on Representations of Surface Groups, itself a part of the research program Geometry and Quantization of Moduli Spaces. The goal of the lectures was to give an introduction to the general theory of Klein s...

We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of certain orbifold fundamental groups.

Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we determine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and degree d on the curve. When r and d are coprime, we describe the topology of the real locus and give a modular...

These notes are based on a series of five lectures given at the 2009 Villa de Leyva Summer School on "Geometric and Topological Methods for Quantum Field Theory”. The purpose of the lectures was to give an introduction to differential-geometric methods in the study of holomorphic vector bundles on a compact connected Riemann surface, as initiated i...

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projective curve. From the algebraic po...

We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semi-stable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a complex projective variety. We...

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this pap...

The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as Dostoglou and Salamon (Ann. of Math (2), 139(3), 581–640, 1994) and Salamon (Proceedings of the International congress of mathematicians, vol.1, 2 (Zürich, 1994), pp. 526–536. Birkhäuser, Basel, 1995): given a 3-manifold M with boundary ∂M...

In this note, we gather known applications of quasi-Hamiltonian geometry to the study of representations spaces of surface groups. We consider three apects of the geometry of representation spaces of surface groups: the symplectic structure that they carry, the number of connected components of representation spaces and the construction of Lagrangi...

In this paper, we characterize unitary representations of π:= π1(S2\{s1,... ,s1}) whose generators u1,..., u1 (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions u j = σjσj+1 with σ1+1 = σ1 Our main result is that such representations are exactly the elements of the fixed-point set of an anti-sym...

We show that the quotient associated to a quasi-Hamiltonian space has a symplectic structure even when 1 is not a regular value of the momentum map: it is a disjoint union of symplectic manifolds of possibly different dimensions, which generalizes a result of Alekseev, Malkin and Meinrenken. We illustrate this theorem with the example of representa...

In this note, we state and give the main ideas of the proof of a real convexity theorem for group-valued momentum maps. This result is a quasi-Hamiltonian analogue of the O'Shea-Sjamaar theorem in the usual Hamiltonian setting. We prove here that the image under the momentum map of the fixed-point set of a form-reversing involution defined on a qua...

The main result of this paper is a quasi-hamiltonian analogue of a special case of the O'Shea-Sjamaar convexity theorem for usual momentum maps. We denote by U a simply connected compact connected Lie group and we fix an involutive automorphism of maximal rank on this Lie group (such an automorphism always exists). We then denote by M a quasi-hamil...

Following the book Quantum groups by C. Kassel, we recall the construction of two (families of) representations if Artin’s braid group. The first representation is obtained analytically: it is the monodromy representation of the KZ connection. The second one is constructed algebraically, from the r-matrix of a certain enveloping algebra. The Kohno-...

In this note, we study the fixed-point set of an (anti-symplectic) in-volutionˆβvolutionˆ volutionˆβ induced on a quasi-hamiltonian quotient M//U = µ −1 ({1})/U by a (form-reversing) involution β defined on the quasi-hamiltonian space (M ω, µ : M → U). When the action of U on µ −1 ({1}) is free, we adapt the results of [2] to the quasi-hamiltonian...

Le principal résultat de la thèse est un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie. Ce théorème est appliqué à la construction de sous-variétés lagrangiennes dans les quotients quasi-hamiltoniens, en particulier dans les espaces de représentations de groupes de surfaces. La notion de représentation déco...

The purpose of this paper is to study the diagonal action of the unitary group U(n) on triples of Lagrangian subspaces of Cn. The notion of angle of Lagrangian subspaces is presented here, and we show how pairs (L1,L2) of Lagrangian subspaces are classified by the eigenvalues of the unitary map σL2∘σL1 obtained by composing the Lagrangian involutio...