Ana Casimiro

Higgs bundles and Schottky representation

We relate Schottky representations to certain Lagrangian subspaces of the moduli space of Higgs G-bundles (G is a connected reductive algebraic group). It is a fundamental result in the theory of Higgs bundles, the so-called non-abelian Hodge theorem, that by considering the Hitchin equations for G-Higgs bundles, one obtains a homeomorphism between the Betti space, B, and the moduli space of semistable G-Higgs bundles over a Riemann surface X. By a remark of Baraglia-Schaposnik, when considering G-Higgs bundles over X with a real structure, one is naturally lead to representations into G of the fundamental group of a 3-manifold with boundary X. These are naturally related to Schottky representations, as we will present in this talk. Our approach via Schottky representations has one advantage: we obtain a simple argument that shows that the Baraglia-Schaposnik brane is indeed Lagrangian with respect to the natural complex structure of B (coming from the complex structure of G). More precisely, we obtain a simpler proof of the vanishing of the complex symplectic form on the strict Schottky locus. This joint work with Susana Ferreira (IPL) and Carlos Florentino (FCUL).

Camilla Felisetti

Intersection cohomology of the moduli space of Higgs bundles on a smooth projective curve

Let X be a smooth projective curve of genus g over ℂ. The character variety ℳ_B parametrizing conjugacy classes of representations from the fundamental group of X into SL(2,ℂ) is an affine irreducible singular projective variety. The Non-Abelian Hodge theorem states that there is a real analytic isomorphism between ℳ_B and the quasi-projective singular variety ℳ_Dol which parametrizes semistable Higgs bundles of rank 2 and degree 0 on X. During the seminar I will present a desingularization of these moduli spaces and I will compute the intersection cohomology of ℳ_Dol using the famous decomposition theorem by Beilinson, Bernstein, Deligne and Gabber. Moreover I will show that the mixed Hodge structure on the intersection co-homology is pure, showing evidence that an analogue of the P=W conjecture might hold for singular moduli spaces.

Carlos Florentino

Generating Functions for Hodge-Euler polynomials of GL(n,ℂ)-character varieties

Given a finitely generated group F and a complex reductive Lie group G, the G-character variety of F, X_FG=Hom(F,G)//G, is typically a singular algebraic variety whose geometric and topological (and sometimes arithmetic) properties can be studied via mixed Hodge structures (MHS). The most interesting cases are when F is the fundamental group of a Kähler manifold M, since then X_FG is homeomorphic to a space of G-Higgs bundles over M. Some special classes of character varieties have their MHS encoded in a polynomial generalization of the Euler-Poincaré characteristic: the Hodge-Euler polynomial, also called the E-polynomial. In this seminar, concentrating in the case of the general linear group G=GL(n,ℂ), we present a remarkable relation between the E-polynomials of X_FG and those of X^irr_FG, the locus of irreducible representations of F into G. We will also give an overview of known explicit computations of E-polynomials, as well as some conjectures and open problems.

Emilio Franco

Cartan branes on the Hitchin system

We study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) BBB-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup P, whose conjectural dual BAA-branes we identify. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of P. Secondly, when P is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the BBB-brane, and study the variation of the dual under this choice. We give evidence of both families of branes being dual under mirror symmetry via an ad-hoc Fourier-Mukai integral functor.

Peter gothen

SO(p,q)-Higgs bundles and higher Teichmüller components

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others, such as the Hitchin component in the moduli space of Higgs bundles, are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such "exotic" components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q>3). Time permitting, we will comment on possible generalizations. The talk will mainly be based on arXiv:1802.08093 which is joint work with Marta Aparicio-Arroyo, Steven Bradlow, Brian Collier, Oscar García-Prada and André Oliveira.

Viktoria Heu

The Riemann-Hilbert mapping in genus two

One possible formulation of the Riemann-Hilbert problem in higher genus is to ask which is the vector bundle underlying the holomorphic connection over a curve associated to a given monodromy representation. Since the monodromy is given in terms of the topological and not the complex structure of the curve, one may vary the latter and obtains, by the Riemann-Hilbert correspondence, an isomonodromic family of connections. In collaboration with F. Loray, we obtained the following result: In the moduli space of irreducible 𝖘𝔩₂ℂ-connections over genus two curves, the isomonodromic foliation is transversal to the locus of the trivial bundle and transversal to the locus of flat unstable bundles. In this talk, we will present some applications of this result and of its proof.

Norbert Hoffmann

Universal torsors over degenerating del Pezzo surfaces

Let S be a split one-parameter family of smooth del Pezzo surfaces degenerating to a del Pezzo surface with ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor (or principal G-bundle) over S whose restriction to the smooth fibres is the extension of structure group of the universal torsor under the Néron-Severi torus introduced and studied by Colliot-Thélène and Sansuc. The G-torsor is unique and infinitesimally rigid. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces.

It is joint work with Ulrich Derenthal.

Ángel Luis Muñoz-Castañeda

Compactification of the moduli space of stable principal G-bundles over a stable curve and beyond

Given a projective manifold X, a reductive group G and a faithful representation ρ:G→SL(V), A. Schmitt defined a singular principal G-bundle on it as a pair (ℱ,τ) given by a torsion free sheaf and a morphism of algebras τ: S^•(V⊗ℱ)^G→𝒪_X, and proved the existence of a compact moduli space for δ-(semi)stable singular principal G-bundles having the space of stable principal G-bundles as an open subscheme U. Bhosle proved the existence of a compact moduli space for δ-(semi)stable singular principal G-bundles over irreducible projective curves with at most nodes as singularities. An important feature of this moduli space is that, when the curve is smooth, it is isomorphic to the classical moduli space constructed by A. Ramanathan provided the rational parameter δ is large enough. When the base curve is reducible, we can generalize the definition of singular principal bundle by considering sheaves of depth one. In this talk, I will discuss the existence of a compact moduli space for δ-(semi)stable singular principal G-bundles over nodal (possibly reducible) projective curves and its behavior under variations of the base curve along ℳ_g.

Angela Ortega

Klein coverings of genus 2 curves

We consider étale 4 : 1 coverings of smooth genus 2 curves with the monodromy group the Klein group. Depending on the values of the Weil pairing restricted to the group defining the covering, we distinguish the isotropic and non-isotropic case. In this talk we will discuss the correspondence between the non-isotropic Klein coverings and the (1,4)-polarised abelian surface. As a consequence of this, one can show the existence of exactly four hyperelliptic curves in a general (1,4)-polarised abelian surface. We will also give several characterisations of the Klein coverings (isotropic and non-isotropic) leading to the result that the corresponding Prym maps are generically injective in both cases.

This is a joint work with Pawel Borówka.

Ana Peón-Nieto

Mirror symmetry on some non generic loci of the Hitchin system

Mirror symmetry for Hitchin systems has been proven to be realised by a relative Fourier-Mukai transform over the generic locus. In this talk I will explain how mirror symmetry operates on some natural hyperholomorphic branes on the Hitchin system, given by fixed points by tensorisation with a torsion line bundle. Generically, they are supported over the locus of singular integral spectral curves. Many aspects of their geometry are however more easily understood through branes supported on the most singular locus of a related Hitchin system. I will refer to this interplay during the exposition. This is joint work with E. Franco, P. Gothen and A. Oliveira.

Francesco Polizzi

Surface braid groups, finite Heisenberg coversand double Kodaira fibration

A Kodaira fibration is a smooth, connected holomorphic fibration f: S→B, where S is a compact complex surface and B is a compact complex curve, which is not isotrivial (this means that not all its fibres are biholomorphic to each other). Examples of such fibrations were originally constructed by Kodaira as a way to show that, unlike the topological Euler characteristic, the signature σ of a manifold is not multiplicative for fibre bundles. In fact, every Kodaira fibred surface S satisfies σ(S)>0, whereas σ(B) = σ(F) = 0, and so σ(S)≠σ(B)σ(F). On the other hand, in a classical work by Chern, Hirzebruch and Serre it is proved that the signature is multiplicative for differentiable fibre bundles in the case where the monodromy action of the fundamental group π₁(B) on the rational cohomology ring H^∗(F,ℚ) is trivial; thus, Kodaira fibrations provide examples of fibre bundles for which this action is non-trivial. In this talk, we show how to construct new examples of double Kodaira fibrations by using finite Heisenberg covers (i.e., Galois covers with Galois group isomorphic to a finite Heisenberg group) of a product Σ_b×Σ_b, where Σ_b is a smooth projective curve of genus b ≥ 2. Each cover is obtained by providing an explicit group epimorphism from the pure braid group P₂(Σ_b) to the corresponding Heisenberg group. In particular, we exhibit the first examples of surfaces that admit two distinct Kodaira fibrations with base genus 2, answering a stronger version of a problem from Kirby’s list in low-dimensional topology:

Theorem. There exists an oriented 4-manifold X of signature 144 that can be realized as a surface bundle over a surface of genus 2 with fibre genus 325 in two different ways.

This is joint work with A. Causin.

Martha Romero

On Galois group of factorized covers of curves

Let ψ:𝒴→𝒳, φ:𝒳→ℙ¹ be a sequence of covers of compact Riemann surfaces. In this work we study and completely characterize the Galois group G(φ◦ψ) of φ◦ψ under the following properties: φ is a simple cover of degree m and ψ is a Galois unramified cover of degree n with abelian Galois group of type (n₁,n₂,...n_S). We prove that G(φ◦ψ)≅(ℤ_n1×ℤ_n2×···×ℤ_ns)^m-1⋊S_m. Furthermore, we give a natural geometric generator system of G(φ◦ψ) obtained by studying the action on the compact Riemann surface Z associated to the Galois closure of φ◦ψ.

Florent Schaffhauser

Higher Teichmüller spaces for orbifolds

The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group of X into PGL(2,ℝ). It is a contractible space. For closed orientable surfaces, “Higher analogues” of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of π₁(X) that consist entirely of discrete and faithful representations. There are two known families of such spaces, namely Hitchin representations and maximal representations, and conjectures on how to find others. In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yield new examples of higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.

Tom Sutherland

Theta functions on moduli spaces of local systems

Moduli spaces of SL(2,ℂ)-local systems on punctured Riemann surfaces provided examples of log Calabi-Yau varieties, for which a generalisation of the classical theta functions has been developed by Gross and Siebert. We will see an explicit computation and a modular interpretation of these theta functions for the moduli space of local systems of the four-punctured sphere, which is the total space of a family of affine cubic surfaces. Further we will discuss the relationship with the complex geometry of a diffeomorphic family of rational elliptic surfaces with a singular fibre of type I₀^*, the complement of which is the corresponding moduli space of Higgs bundles.