Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinite towers of orbifolds uniformized by the complex 2-ball and some orbifolds over K3 surfaces uniformized by the 2-ball. Corresponding lattices are all arithmetic and commensurable. We also give, along with infinitely many reducible examples, an infinite series of irreducible curves along which the projective plane is uniformized by the product of two 1-balls.
For an orbifold X which is the quotient of a manifold Y by a finite group G we construct a noncommutative ring with an action of G such that the orbifold cohomology of X as defined in math.AG/0004129 by Chen and Ruan is the G invariant part. In the case thar Y is S^n for a surface S with trivial canonical class we prove that (a small modification of) the orbifold cohomology of X is naturally isomorphic to the cohomology ring of the Hilbert scheme of n points on S computed in math.AG/0012166 by Lehn and Sorger.
In this paper, we investigate the stability manifold of local models of orbifold quotients of elliptic curves. In particular, we describe a component of the stability manifold which maps as a covering space onto the universal unfolding space of the mirror singularity. The construction requires a detailed study of the McKay correspondence for $A_N$ surface singularities and a study of wall-crossing phenomena.
We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror $M$ of the orbifold ball quotient $(X,M)$ and by taking the quotient by an involution, we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees $1,2$ and $3$. We also exhibit an arrangement of f…
Considering quasismooth varieities as global $\CC^*$ quotients, we present a Riemann-Roch formula via general Riemann-Roch formula for quotient stacks. Furthermore, we give a parcing formula for Hilbert series associated to a polarized quasismooth projectively Gorenstein algebraic varieties with orbifold curves and dissident points, which is an extension of the result in \cite{BRZ}.
Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When $K$ is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian $K$-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary $\operatorname{SU}_2$-modules yield symplectic q…
We prove a result on the singularities of ball quotients $Γ\backslash\CC H^n$. More precisely, we show that a ball quotient has canonical singularities under certain restrictions on the dimension $n$ and the underlying lattice. We also extend this result to the toroidal compactification $(Γ\backslash\CC H^n)^*$.
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we conside…
We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne-Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show in Theorem 7.4 that the two ball-quotient constructions can be unified in a geometric way.
We study the Fano surface S of the Fermat cubic threefold. We prove that S is a degree 81 abelian cover of the degree 5 del Pezzo surface and that the complement of the union of 12 disjoint elliptic curves on S is a ball quotient. The lattice of this ball quotient is related to the Deligne-Mostow lattice number 1.
We describe periods of irreducible holomorphic manifolds of $K3^{[n]}$-type with a non-symplectic automorphism of prime order $p\geq 3$. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective, by introducing the notion of $K(T)$-generality.
We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric invariant theory. We illustrate this with the moduli spaces of smooth quartic curves and rational elliptic surfaces.
We define, for smooth projective orbifold pairs $(X,D)$ notions of `slope Rational connectedness', and of orbifold `slope Rational quotient' . These notions extend to this larger context the classical notions of rationally connected manifold and `rational quotient' (sometimes called `MRC fibration'). Our notions and proofs work entirely in characteristic zero, and are based on the consideration of foliations with minimal positive slope with respect to some suitable movable class. The existence of covering or connecting families of `orbifold rational curves' is indeed presently unknown in the o…
In this paper we prove a characterization of quotients of Abelian varieties by the actions of finite groups that are free in codimension-one via some vanishing conditions on the orbifold Chern classes. The characterization is given among a class of varieties with mild singularities that are more general than quotient singularities, namely among the class of klt varieties. Furthermore we show that over a projective klt variety, any semistable reflexive sheaf with vanishing orbifold Chern classes can be obtained as the invariant part of a locally-free sheaf on a finite Galois cover whose associa…
A three-dimensional orbifold $(Σ, γ_i, n_i)$, where $Σ$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with th…
We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. We furthermore introduce a notion of stability for Na…
We describe hypergeometric functions of Deligne-Mostow type for open subsets of the configuration space of six points on P^2, induced from those for seven points on P^1. The seven point ball quotient example DM(2^5,1^2) does not appear on Mostow's original list, but does appear on Thurston's corrected version. We show that DM(2^5,1^2) is a finite cover of the moduli space of cubic surfaces M_C endowed with the ball quotient structure G_C\B^4 of Allcock, Carlson, and Toledo. This answers a question of Allcock about the commensurability of G_C with the monodromy groups of Deligne-Mostow hypergeo…
The purpose of this note is to show that $2K$ of any smooth compact complex two ball quotient is very ample, except possibly for four pairs of fake projective planes of minimal type, where $K$ is the canonical line bundle. For the four pairs of fake projective planes, sections of $2K_M$ give an embedding of $M$ except possibly for at most two points on $M$.
We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.
We define hypergeometric functions using intersection homology valued in a local system. Topology is emphasized; analysis enters only once, via the Hodge decomposition. By a pull-back procedure we construct special subsets S_{pi}, derived from Hurwitz spaces, of Deligne-Mostow moduli spaces DM(n,mu). Certain DM(n,mu) are known to be ball quotients, uniformized by hypergeometric functions valued in a complex ball (i.e., complex hyperbolic space). We give sufficient conditions for S_{pi} to be a subball quotient. Analyzing the simplest examples in detail, we describe ball quotient structures att…
S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we show that this space essentially coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves. More specifically, we give an explicit description of this GIT quotient, and show that the birational map from this space to Kondo's space is resolved by the blow-up of a single point. This provides a modular interpretation of the points in the boundary of Kondo's space. Connections with the slope nine space in the Hassett-Keel program…
Let $(({\mathbb B} / Γ_1)', T(1))$ be á torsion free toroidal compactification with abelian minimal model $(A_1, D(1))$. An arbitrary principal isogeny $μ_a : A_2 \rightarrow A_1$, $a \in {\mathbb C}$ pulls-back $(A_1, D(1))$ to the abelian minimal model $(A_2, D(2))$ of a torsion free toroidal compactification $(({\mathbb B} / Γ_2)', D(2))$. The present work makes use of the isogeny pull-backs of abelian ball quotient models, towards the construction of infinite series of mutually non-birational co-abelian torsion free Galois covers $(({\mathbb B} / Γ_n)', T(n))$ of a ball quotient compactifi…
Let X' be the toroidal compactification of the quotient of the complex 2-ball by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is an abelian surface, birational to X'. The present work can be viewed as an illustration for the presence of a plenty of non-compact co-abelian torsion free toroidal compactifications. More precisely, it shows that all the admissible values for the volume of a torsion free quotient of the complex 2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers. The article provides three types of infinite series of finite unra…
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.
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 the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=Γ\backslash \B_n$, in particular vanishing and non--vanishing results. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system with the $L^2$-cohomology contributing to the lowest weight part.
We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We shall concern only the quotient singularity of hypersurface type. The abelian group $A_r(n)$ for $A$-type hypersurface quotient singularity of dimension $n$ is introduced. For $n=4$, the structure of Hilbert scheme of group orbits and crepant resolutions of $A_r(4)$-singularity are obtained. The flop procedure of 4-folds is explicitly constructed through the process.
The moduli space of cubic threefolds in CP4, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.
We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild…
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 extreme by constructing arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.
We study the problem of non-existence of totally geodesic complex ball quotients in the open Torelli locus in a moduli space of principally polarized Abelian varieties using analytic techniques.
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 classes of all presently known non-arithmetic ball quotients.
In this article we provide further evidence for the monstrous proposal of Daniel Allcock, by giving a plausible but still conjectural explanation for the deflation relation in the Coxeter group quotient of the orbifold fundamental group.
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 of $K_X + αD$ for $α\in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.
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 deformations of the curves away from the orbifold limit and the Fayet-Iliopoulos parameters in the corresponding quotient construction. We work out the orbifold limit of E_7, E_8, and some higher dimensional examples.
This paper exposes the fundamental role that the Drinfel'd double $\dkg$ of the group ring of a finite group $G$ and its twists $\dbkg$, $β\in Z^3(G,\uk)$ as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that $G$--Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of $\dkg$--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold $K$--theory of glob…
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 $e(S)$ is the (topological) Euler characteristic, and in the case of equality show that $S$ is a regular cover of a particular Deligne--Mostow orbifold. We conjecture that this inequality holds independent of arithmeticity, and note that work of Xiao makes progress on this conjecture…
We classify orbifolds obtained by taking the quotient of a three tori by abelian extensions of Z/n x Z/n automorphisms, where each torus has a multiplicative Z/n action (n=3,4 or 6). This 'completes' the classification of orbifolds of the above type initiated by Donagi and Faraggi (hep-th/0403272) and, Donagi and Wendland in the cases n=2.
Let $Γ\subset \mathbf{PU}(2,1)$ be a lattice which is not co-compact, of finite Bergman-covolume and acting freely on the open unit ball $\mathbf{B} \subset \mathbb{C}^2$. Then the compactification $X = \bar{Γ\setminus \mathbf{B}}$ is a smooth projective surface with an elliptic compactification divisor $D = X \setminus (Γ\setminus \mathbf{B})$. In this short note we discover a new class of unramified ball-quotients $X$. We consider ball-quotients $X$ with $kod(X) = h^1(X, \mathcal{O}_X) = 1$. We prove that all minimal surfaces with finite Mordell-Weil group in the class described become after…
In connection with our previous investigation about Siegel threefolds which admit a Calabi--Yau model, we consider ball quotients which belong to the unitary group $\U(1,3)$. In this paper we determine a very particular example of a Picard modular variety of general type. Really we determine the ring of modular forms. This algebra has 25 generators, 15 modular forms $B_i$ of weight one and ten modular forms $C_j$ of weight 2. Both will appear as Borcherds products. We determine the ideal of relations. The forms $C_i$ are cuspidal. Their squares define holomorphic differential forms on the non-…
Let $X$ be a quotient of a bounded domain in $\mathbb C^n$. Under suitable assumptions, we prove that every subvariety of $X$ not included in the branch locus of the quotient map is of log general type in some orbifold sense. This generalizes a recent result by Boucksom and Diverio, which treated the case of compact, étale quotients. Finally, in the case where $X$ is compact, we give a sufficient condition under which there exists a proper analytic subset of $X$ containing all entire curves and all subvarieties not of general type (meant this time in in the usual sense as opposed to the orbifo…
We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.
