Orbifold Fundamental Group Research Articles

The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts

We prove a cyclotomic generalization of Enriquez and Furusho's result stating that the scheme DMR× of double shuffle and regularization relations between multiple zeta values arising from Racinet's formalism is a torsor of isomorphisms between “Betti” and “de Rham” objects. After reviewing the “de Rham” objects arising from a crossed product interpretation of Racinet's formalism, we construct the “Betti” objects based on the orbifold fundamental group of the quotient (C×∖μN)/μN with μN being the group of Nth roots of unity for N≥1. We then show that DMR× is a subtorsor of stabilizer schemes relating “Betti” and “de Rham” coproducts.

Flat vector bundles and analytic torsion on orbifolds

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.

The local fundamental group of a Kawamata log terminal singularity is finite

We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.

Relative homological representations of framed mapping class groups

Let $\Sigma$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated "framed mapping class group" on the homology of $\Sigma$ relative to its boundary (respectively marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.

Generators for a complex hyperbolic braid group

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the "monstrous proposal", which posits a relationship between this braid group and the monster finite simple group.

Convex projective generalized dehn filling

In dimension $d = 4, 5, 6, 7$, we find the first examples of complete finite volume hyperbolic $d$-dimensional manifolds $M$ with cusps such that an infinite number of orbifolds $M_m$ obtained by generalized Dehn fillings on $M$ admit a properly convex real projective structure. The manifolds $M$ are covering of hyperbolic Coxeter orbifolds and the orbifold fundamental groups $\Gamma_m$ of $M_m$ are Gromov hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d−2}.

Non-arithmetic lattices and the Klein quartic

Abstract We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Barthel–Hirzebruch–Höfer and Couwenberg–Heckman–Looijenga.

On Poincaré duality for orbifolds

In this paper we address the relation between the orbifold fundamental group and the topology of the underlying space. In particular, under the assumption that π1orb(O)=π1(|O|), we prove Poincaré Duality for orbifolds of dimensions 4 and 5.

Cherednik and Hecke Algebras of Varieties with a Finite Group Action

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles

Let $(M,\sigma)$ be a compact Klein surface of genus $g \geq 2$ and let $E$ be a smooth Hermitian vector bundle on $M$. Let $\tau$ be a Real or Quaternionic structure on $E$ and denote respectively by $\mathcal{G}^{\tau}_{\mathbb{C}}$ and $\mathcal{G}^{\tau}_{E}$ the groups of complex linear and unitary automorphisms of $E$ that commute to $\tau$. In this paper, we study the action of $\mathcal{G}^{\tau}_{\mathbb{C}}$ on the space $\mathcal{A}^{\tau}_{E}$ of $\tau$-compatible unitary connections on $E$ and show that the closure of a semi-stable $\mathcal{G}^{\tau}_{\mathbb{C}}$-orbit contains a unique $\mathcal{G}^{\tau}_{E}$-orbit of projectively flat 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 $(M,\sigma)$.

Finite group actions on moduli spaces of vector bundles

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 orbifold fundamental groups.

ALBANESE VARIETIES OF CYCLIC COVERS OF THE PROJECTIVE PLANE AND ORBIFOLD PENCILS

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, that is, a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is nontrivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary.

A general theory of orbifolds, and essential 2-suborbifolds respecting graph products

We make an analogy of Culler–Morgan–Shalen theory for 3-orbifolds. Our main goal is to show that there exists a non-empty system of incompressible and not boundary parallel 2-suborbifolds respecting a graph product of the orbifold fundamental group.

Cherednik and Hecke Algebras of Varieties with a Finite Group Action

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

The complex Lorentzian Leech lattice and the bimonster (II)

LetDDbe the incidence graph of the projective plane overF3\mathbb {F}_3. The Artin group of the graphDDmaps onto the bimonster and a complex hyperbolic reflection groupΓ\Gammaacting on1313dimensional complex hyperbolic spaceYY. The generators of the Artin group are mapped to elements of order22(resp.33) in the bimonster (resp.Γ\Gamma). LetY∘⊆YY^{\circ } \subseteq Ybe the complement of the union of the mirrors ofΓ\Gamma. Daniel Allcock has conjectured that the orbifold fundamental group ofY∘/ΓY^{\circ }/\Gammasurjects onto the bimonster.In this article we study the reflection groupΓ\Gamma. Our main result shows that there is homomorphism from the Artin group ofDDto the orbifold fundamental group ofY∘/ΓY^{\circ }/\Gamma, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections inΓ\Gamma. This answers a question in Allcock’s article “A monstrous proposal” and takes a step towards the proof of Allcock’s conjecture. The finite groupPGL⁡(3,F3)⊆Aut(D)\operatorname {PGL}(3, \mathbb {F}_3) \subseteq \mathrm {Aut}(D)acts onYYand fixes a complex hyperbolic line pointwise. We show that the restriction ofΓ\Gamma-invariant meromorphic automorphic forms onYYto the complex hyperbolic line fixed byPGL⁡(3,F3)\operatorname {PGL}(3, \mathbb {F}_3)gives meromorphic modular forms of level1313.

Mapping class groups of trigonal loci

In this paper, we study the topology of the stack $$\mathcal {T}_g$$ of smooth trigonal curves of genus g over the complex field. We make use of a construction by the first named author and Vistoli, which describes $$\mathcal {T}_g$$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $$\mathcal {T}_g$$ , and of its substrata with prescribed Maroni invariant, and describe their relation with the mapping class group $$\mathcal {M}ap_g$$ of Riemann surfaces of genus g.

Anti-holomorphic involutions of the moduli spaces of Higgs bundles

We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas--Garc\'{\i}a-Prada--Hurtubise and Baraglia--Schaposnik.

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.

The fine structure of the moduli space of abelian differentials in genus 3

The moduli space of curves endowed with a nonzero abelian differential admits a natural stratification according to the configuration of its zeroes. We give a description of these strata for genus 3 in terms of root system data. For each non-open stratum we obtain a presentation of its orbifold fundamental group.

On cohomology of crystallographic groups with cyclic holonomy of split type

We disprove a conjecture stating that the integral cohomology of any n-dimensional crystallographic group Zn⋊Zm admits a decomposition:H⁎(Zn⋊Zm)≅⊕i+j=⁎Hi(Zm,Hj(Zn)) by providing a complete list of counterexamples up to dimension 6. This finishes the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi–Yau toroidal orbifolds. We also find a counterexample with odd order holonomy, m=9, in dimension 8.