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Florent Schaffhauser

Publications and source records attributed to Florent Schaffhauser.

At least 19 recordsLinked to original sources
arXiv2026

Connected components of real loci in moduli spaces of vector and Higgs bundles over a Klein surface

Let $X$ be a Riemann surface of genus $g \geqslant 2$ and let $\sigma : X \to X$ be an antiholomorphic involution on $X$. Let $\mathcal{N}(r,d)$ be the moduli space of semistable vector bundles of rank $r$ and degree $d$ on $X$, with the induced real structure. Using a gauge-theoretic approach, we determine the number of connected components of the real locus of $\mathcal{N}(r,d)$ for general $r$ and $d$. We show in particular that, when the base curve has real points, quaternionic vector bundles can exist for even rank and degree but that the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ is still equal to that of $\mathbb{R}\mathrm{Pic}_d$. In contrast, when the base curve has empty real locus and $r$ and $d$ are not coprime, the number of connected components of $\mathbb{R}\mathcal{N}(r,d)$ can be smaller than that of $\mathbb{R}\mathrm{Pic}_d$. We then generalize these results to real loci of moduli spaces of Higgs bundles and apply them to the study of the topology of certain $(A,A,A)$ and $(A,B,A)$ branes in the associated hyperk\"ahler quotient.

math.AG↗
arXiv2026

Hodge numbers of moduli of principal bundles on a curve

We prove an inversion theorem for recursive formulas satisfied by certain families of converging power series in two variables. These power series are indexed by the Harder-Narasimhan types of principal $G$-bundles of degree $d \in π_1 G$ on a smooth projective curve $X$, where $G$ is a connected complex reductive group. As an application, we obtain a closed formula for the Hodge-Poincaré series of moduli stacks of semistable principal $G$-bundles of degree $d$. We also compute the variation of Hodge structure of the moduli stack of all principal $G$-bundles over $X$, as a function of the period matrix of that curve.

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arXiv2026

Harder-Narasimhan filtrations of decorated vector bundles

A decorated vector bundle is a vector bundle equipped with a reduction of structure group to a complex reductive subgroup $G \subseteq \mathbf{GL}(r,\mathbb{C})$. Examples include symplectic and special-orthogonal vector bundles, as well as vector bundles with trivial determinants. In this expository paper, we provide direct constructions of Harder-Narasimhan filtrations of symplectic and special-orthogonal vector bundles, and use them to construct canonical reductions in the sense of Atiyah and Bott. We compare these canonical reductions to those constructed by Biswas and Holla. Lastly, we set up the obstruction theory necessary to define Harder-Narasimhan types of principal bundles, and stratify the moduli stack of principal $G$-bundles.

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arXiv2025

Flat torsors in complex geometry

These notes grew out of a mini-course given by the second-named author at Casa Matemática Oaxaca in the Fall of 2022. Their purpose is to provide an exposition, directed at graduate students, of the basic properties of complex analytic group bundles and torsors under them, including the flat case.

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arXiv2025

Real Bialynicki-Birula flows in moduli spaces of Higgs bundles

Let $X$ be a compact Riemann surface $X$ of genus $\geqslant 2$ and let $σ:X \to X$ be an anti-holomorphic involution. Using real and quaternionic systems of Hodge bundles, we study the topology of the real locus $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ of the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ on $X$, for the induced real structure $(E,ϕ) \to (σ^*(\overline{E}),σ^*(\overlineϕ))$. We show in particular that, when $\mathrm{gcd}(r,d)=1$, the number of connected components of $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ coincides with that of $\mathbb{R} \mathrm{Pic}_d(X)$, which is well-known.

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arXiv2023

Orbifolds and the modular curve

We provide an account of the construction of the moduli stack of elliptic curves as an analytic orbifold. While intimately linked to Thurston's point of view on the subject (discrete groups acting properly and effectively on differentiable manifolds), the construction of the modular orbi-curve and its universal family of elliptic curves ends up requiring a bit more technology, in order to allow for non-effective actions. The paper is entirely expository and makes no claims to originality: its main goal is to be self-contained enough in order to be useful to young researchers who are entering the field and are interested in the interactions between differential and algebraic geometry.

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arXiv2022

Maximality of moduli spaces of vector bundles on curves

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 numbers of the real locus to the Hodge numbers of the complex locus and showing that moduli spaces of vector bundles over a maximal curve actually satisfy a property which is stronger than maximality and that we call Hodge-expressivity. We also give a brief account on other varieties for which this property was already known.

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arXiv2020

Hitchin components for orbifolds

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 Higgs bundles, and the deformation theory of real projective structures on $3$-manifolds.

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arXiv2019

Symmetric differentials and the dimension of Hitchin components for orbi-curves

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 Hitchin component for orbifold groups to an application of the orbifold Riemann-Roch theorem.

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arXiv2019

Parabolic vector bundles on Klein surfaces

Given a discrete subgroup $Γ$ of finite co-volume of $\mathrm{PGL}(2,\mathbb{R})$, 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 $Σ$. We then prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are in bijective correspondence with equivalence classes of real and quaternionic unitary representations of $Γ$. Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.

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arXiv2019

Rational points of quiver moduli spaces

For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\bar{k}$-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 of this decomposition using quiver representations over division algebras, and we reinterpret this description using twisted quiver representations. We also see that moduli spaces of twisted quiver representations give different forms of the moduli space of quiver representations.

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arXiv2018

Group actions on quiver varieties and applications

We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkaehler quiver varieties, as fixed loci of such actions.

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arXiv2016

Vector bundles over a real elliptic curve

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 interpretation of its points. We also study, for arbitrary rank and degree, the moduli space of indecomposable vector bundles of rank r and degree d, and determine its isomorphism class as a real algebraic variety.

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arXiv2015

On the Narasimhan-Seshadri correspondence for Real and Quaternionic vector bundles

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 perspective to prove a version of the Narasimhan-Seshadri correspondence in this context: S-equivalence classes of semi-stable Real and Quaternionic vector bundes are in bijective correspondence with equivalence classes of certain appropriate representations of orbifold fundamental groups of Real Seifert manifolds over the Klein surface M.

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arXiv2015

Lectures on Klein surfaces and their fundamental group

The goal of these lectures is to give an introduction to the study of the fundamental group of a Klein surface. We start by reviewing the topological classification of Klein surfaces and by explaining the relation with real algebraic curves. Then we introduce the fundamental group of a Klein surface and present its main basic properties. Finally, we study the variety of unitary representations of this group and relate it to the representation variety of the topological fundamental group of the underlying Riemann surface.

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arXiv2015

Differential geometry of holomorphic vector bundles on a curve

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.

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arXiv2013

The Yang-Mills equations over Klein surfaces

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 point of view, these Lagrangian quotients are connected sets of real points inside a complex moduli variety endowed with a real structure; when the rank and the degree are coprime, they are in fact the connected components of the fixed-point set of the real structure. This presentation as a quotient enables us to generalize the methods of Atiyah and Bott to a setting with involutions, and compute the mod 2 Poincare polynomials of these moduli spaces in the coprime case. We also compute the mod 2 Poincare series of moduli stacks of all real and quaternionic vector bundles of a fixed topological type. As an application of our computations, we give new examples of maximal real algebraic varieties.

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