Ball Quotients Research Articles

Complex Ball Quotients and New Symplectic $4$-manifolds with Nonnegative Signatures

Complex Ball Quotients and New Symplectic $4$-manifolds with Nonnegative Signatures

Erratum to "Special subvarieties of non-arithmetic ball quotients and Hodge theory"

Erratum to "Special subvarieties of non-arithmetic ball quotients and Hodge theory"

Uniruledness of some low-dimensional ball quotients

Uniruledness of some low-dimensional ball quotients

NON-ISOMORPHIC SMOOTH COMPACTIFICATIONS OF THE MODULI SPACE OF CUBIC SURFACES

Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient ${\mathcal {M}}^{\operatorname {GIT}}$ , as a Baily–Borel compactification of a ball quotient ${(\mathcal {B}_4/\Gamma )^*}$ , and as a compactified K-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup ${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$ , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ . The spaces ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact not the case. Indeed, we show the more refined statement that ${\mathcal {M}}^{\operatorname {K}}$ and ${\overline {\mathcal {B}_4/\Gamma }}$ are equivalent in the Grothendieck ring, but not K-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

A BALL QUOTIENT PARAMETRIZING TRIGONAL GENUS 4 CURVES

Abstract We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$ .

On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.

Projective Embeddings of Ball Quotients, Birational to a Bi-elliptic Surface

For a neat lattice $$\Gamma < SU(1,2)$$, whose quotient $${\mathbb B} / \Gamma$$ is birational to a bi-elliptic surface, we compute the dimensions of the cuspidal $$\Gamma$$-modular forms $$[ \Gamma,n]_o$$ and all modular forms $$[ \Gamma, n]$$ of weight $$n \geq 2. $$ The work provides a sufficient condition for a subspace $$V \subset [ \Gamma, n]$$ to determine a regular projective embedding of the Baily-Borel compactification $$\widehat{ {\mathbb B} / \Gamma}$$ and applies this criterion to a specific example.

Irregular cusps of ball quotients

AbstractWe study the branch divisors on the boundary of the canonical toroidal compactification of ball quotients. We show a criterion, the low slope cusp form trick, for proving that ball quotients are of general type. Moreover, we classify when irregular cusps exist in the case of the discriminant kernel and construct concrete examples for some arithmetic subgroups. As another direction of study, when a complex ball is embedded into a Hermitian symmetric domain of type IV, we determine when regular or irregular cusps map to regular or irregular cusps studied by Ma.

Special subvarieties of non-arithmetic ball quotients and Hodge theory

Let $\Gamma \subset \mathrm{PU}(1,n)$ be a lattice and $S_\Gamma$ be the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal complex totally geodesic subvarieties, then $\Gamma$ is arithmetic. We also prove an Ax--Schanuel Conjecture for $S_\Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_\Gamma$ inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unlikely intersections.

Residually Finite Lattices in PU(2,1)˜ and Fundamental Groups of Smooth Projective Surfaces

This paper studies residual finiteness of lattices in the universal cover of PU(2,1) and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in PU(2,1) or a finite covering of it. First, we prove that certain lattices in the universal cover of PU(2,1) are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in PU(2,1) to construct smooth projective surfaces that are not birationally equivalent to a smooth compact ball quotient but whose fundamental group is a torsion-free cocompact lattice in PU(2,1).

Degenerating Kähler–Einstein cones, locally symmetric cusps, and the Tian–Yau metric

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta\to 0$ of K\"ahler-Einstein metrics $\omega_\beta$ with a cone singularity of angle $2\pi \beta$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $\omega_\beta$ converges to the locally symmetric metric and further give asymptotics of $\omega_\beta$ when $X\setminus D$ is a ball quotient. Our second result deals with the case when $X$ is Fano and $D$ is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of $\omega_\beta$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,\omega_\beta)$ converges to an interval in the Gromov-Hausdorff sense.

Subvarieties of quotients of bounded symmetric domains

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each \(p \ge 1\), this criterion gives a precise condition under which the subvarieties \(V \subset X\) with \(\dim V \ge p\) are of general type, and X is p-measure hyperbolic. Then, we give several applications related to ball quotients, or to the Siegel moduli space of principally polarized abelian varieties. In the case of smooth compactifications of ball quotients, we obtain a conditional upper bound on the dimension on the exceptional locus, assuming the Green–Griffiths–Lang conjecture holds true. As another example of application, we give effective lower bounds for levels l so that the moduli spaces of genus g curves with l-level structures are of general type.

Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

Abstract In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

Symmetric differentials on complex hyperbolic manifolds with cusps

Let $(X,D)$ be a logarithmic pair, and let $h$ be a smooth metric on $T_{X \setminus D}$. We give a simple criterion on the curvature of $h$ for the bigness of $\Omega_X(\operatorname{log} D)$ or $\Omega_X$. As an application, we obtain a metric proof of the bigness of $\Omega_X(\operatorname{log} D)$ on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of $\Omega_X$ in the more specific case of the ball. We obtain effective ramification orders for a cover $X^\prime \to X$, étale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that $\Omega_{X^\prime}$ is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the ball quotient we consider.

On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics

Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $\Gamma $ is trivial, that is, when the ball quotient $\mathbb{B}^n/\Gamma $ is the unit ball ${\mathbb{B}}^n$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric.

Jet differentials on toroidal compactifications of ball quotients

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.

Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily–Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.

Cusp and b_1 growth for ball quotients and maps onto Z with finitely generated kernel

Let $M = \mathbb{B}^2 / \Gamma$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

Open Torelli locus and complex ball quotients

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.

Arithmetic occult period maps

Several natural complex configuration spaces admit surprising uniformizations as arithmetic ball quotients, by identifying each parametrized object with the periods of some auxiliary object. In each case, the theory of canonical models of Shimura varieties gives the ball quotient the structure of a variety over the ring of integers of a cyclotomic field. We show that the (transcendentally-defined) period map actually respects these algebraic structures, and thus that occult period maps are arithmetic. As an intermediate tool, we develop an arithmetic theory of lattice-polarized K3 surfaces.