arXiv research

Study non-arithmetic orbifold ball quotients using Jacobian of Bolza curve.

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

This paper classifies ball quotients of the complex projective plane.

We give an explicit description of the 3-ball quotients constructed by Couwenberg-Heckman-Looijenga, and deduce the value of their orbifold Euler characteristics. For each lattice, we also give a presentation in terms of generators and relations.

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…

We consider the analogue of Hurwitz curves, smooth projective curves $C$ of genus $g \ge 2$ that realize equality in the Hurwitz bound $|\mathrm{Aut}(C)| \le 84 (g - 1)$, to smooth compact quotients $S$ of the unit ball in $\mathbb{C}^2$. When $S$ is arithmetic, we show that $|\mathrm{Aut}(S)| \le 288 e(S)$, where $e(S…

Paper finds new ball quotients from curve products.

Orbifold uniformization of complex algebraic variety via polystable parabolic Higgs bundle

A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the fundamental domain of any lattice contains a ball whose radius depends only on the group …

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 Bar…

Quotients of torus endomorphisms have parabolic orbifolds.

In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.

The paper studies orbifold splice quotients and log covers of surface pairs.

For a finitely generated discrete group $Γ$, the $Γ$-sectors of an orbifold $Q$ are a disjoint union of orbifolds corresponding to homomorphisms from $Γ$ into a groupoid presenting $Q$. Here, we show that the inertia orbifold and $k$-multi-sectors are special cases of the $Γ$-sectors, and that the $Γ$-sectors are orbif…

Develops new methods for isospectral orbifolds and regulator quotients.

We introduce the $Γ$-Euler-Satake characteristics of a general orbifold $Q$ presented by an orbifold groupoid $\mathcal{G}$, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as t…

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the A…

Characterizes orbifolds with upper curvature bounds as reflectofolds.

The paper examines gradient Ricci solitons on orbifolds and proves their rigidity properties.

Study non-existence of complex ball quotients in Torelli locus.

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for …

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

We construct a new family of compact orbifolds with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another known family, these examples classify all 4-dimensional orbifolds that are quaternion Kaehler quotients by a torus of real Grassmannians.

The paper classifies cusp types in hyperbolic knot complements.

Study of $D_2$ ALF manifolds via hyperkahler quotients of affine spaces.

We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of th…

We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural…

We study groups generated by three half-turns in the Lobachevsky $3$-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a m…

We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti…

This is the writeup of a lecture given at the May Wisconsin workshop on mathematical aspects of orbifold string theory. In the first part of this lecture, we review recent work on discrete torsion, and outline how it is currently understood in terms of the B field. In the second part of this lecture, we discuss the rel…

Linear bound on Betti numbers of negatively curved orbifolds.

The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.

We deal with a Lie group G acting by isometries on a Riemannian manifold M, such that the quotient M/G is an orbifold, or, equivalently, all slice representations are polar. We show that any smooth orbifold symmetric 2-tensor on M/G lifts to a smooth G-invariant symmetric 2-tensor on M. The proof relies on a fact about…

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness o…

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which als…

New geometric model for equivariant cohomotopy using bordism.

We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors. As a corollary, we complete the classification of commensurability…

We study the Bott-Chern cohomology of complex orbifolds obtained as quotient of a compact complex manifold by a finite group of biholomorphisms.

Study cohomology of ball quotients and their compactifications.

Study on curves on specific arithmetic quotients of hyperbolic 2-ball.

Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.

Generalizes mean-value inequality to orbifold setting.

The paper extends a fibration theorem to all orbifolds, proving an isomorphism conjecture.

The purpose of the article is to study a foliation associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex ball quotients.

New insights into a complex hyperbolic braid group quotient.

We obtain asymptotics of sequences of the holomorphic sections of the pluricanonical bundles on ball quotients associated to closed geodesics. A nonvanishing result follows.

We classify representations of compact connected Lie groups whose induced action on the unit sphere has an orbit space isometric to a Riemannian orbifold.

We develop a graphical representation of polynomial invariants of unitary gauge groups, and use it to find the algebraic curve corresponding to a hyperkahler quotient of a linear space. We apply this method to four dimensional ALE spaces, and for the A_k, D_k, and E_6 cases, derive the explicit relation between the def…