New non-arithmetic ball quotients from elliptic curves on Abelian surfaces.
problem Constructing non-arithmetic ball quotients from Abelian surfaces.
method Branched covers of Abelian surface quotients by finite groups.
result Alternative construction of lattices from elliptic curves.
New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.
problem Creating non-arithmetic lattices in projective orthogonal groups.
method Using anti-holomorphic involutions on complex arithmetic ball quotients, gluing fixed loci along geodesic subspaces.
result Explicit calculation of the volume of constructed non-arithmetic orbifolds.
New construction reveals group relationships for Klein quartic.
problem Understanding non-arithmetic ball quotients and their groups.
method Algebro-geometric construction of ball quotients.
result Reveals connections between orbifold fundamental groups and Klein quartic automorphism group.
Study non-arithmetic orbifold ball quotients using Jacobian of Bolza curve.
problem Identify and study non-arithmetic orbifold ball quotients.
method Use Jacobian of Bolza curve and birational transformations.
result Obtain orbifold ball quotient surfaces with interesting configurations.
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…
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.
This paper classifies commensurability of Deligne-Mostow lattices.
problem Classifying commensurability among Deligne-Mostow lattices.
method Geometric and algebraic approaches to commensurability relations.
result 104 Deligne-Mostow lattices form 38 commensurability classes.
Study geodesics on ball quotients to find nonvanishing sections.
problem Finding nonvanishing holomorphic sections on ball quotients.
method Analyzing sequences of pluricanonical bundles associated to closed geodesics.
result Obtain asymptotics of holomorphic sections on ball quotients.
Study non-existence of complex ball quotients in Torelli locus.
problem Non-existence of totally geodesic complex ball quotients in Torelli locus.
method Analytic techniques.
result Analytic techniques used to study non-existence.
The paper explores ball quotient compactifications and their properties.
problem Smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres.
method Use totally geodesic punctured spheres to prove ampleness of KX+αD for α∈(41,1). result First examples of bielliptic ball quotient compactifications are produced.
The study calculates the volumes of 3-ball quotients and their orbifold Euler characteristics.
problem Calculating the volumes and orbifold Euler characteristics of 3-ball quotients.
method Explicit description and presentation of 3-ball quotients, deduction of orbifold Euler characteristics.
result Explicit values for the orbifold Euler characteristics of 3-ball quotients.
This paper classifies ball quotients of the complex projective plane.
problem Understanding the structure of the complex projective plane as a ball quotient.
method Analyzing the branch locus as a line arrangement and smooth normal-crossing curves.
result The orbifold structure of (P2,D) is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover. Study foliations from complex ball to another via harmonic maps.
problem Rigidity of complex ball quotients.
method Lattice-equivariant harmonic map of small rank.
result Rigidity of complex ball quotients proven.
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…
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
Study on curves on specific arithmetic quotients of hyperbolic 2-ball.
problem No complex curves of certain genus on these arithmetic quotients.
method Volume estimates and understanding special subvarieties.
result For large discriminants, no complex curves of fixed genus.
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
New non-arithmetic lattice found in PU(3,1)
problem Arithmeticity of Couwenberg-Heckman-Looijenga lattices
method Study of arithmeticity and non-arithmetic lattices in PU(n,1)
result Found a non-arithmetic lattice in PU(3,1) not commensurable to Deligne-Mostow lattice
New insights into a complex hyperbolic braid group quotient.
problem Understanding a complex hyperbolic braid group quotient.
method Analyzing the moduli space of 12-tuples in CP1 and identifying loops.
result Identifying loops in the 9-ball quotient corresponding to standard braid generators.
Hybrid subgroups found in non-arithmetic PU(2,1) lattices.
problem Exploring hybrid subgroups in non-arithmetic PU(2,1) lattices.
method Exploring hybrid subgroups of certain non-arithmetic lattices in PU(2,1). Showing that Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
result Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
Paper shows non-arithmetic surface with unique geometric property.
problem Non-arithmetic surfaces with unique geometric properties.
method Example of a non-arithmetic surface with marked length variety rigidity.
result Found a non-arithmetic surface with marked length variety rigidity.
Paper finds new ball quotients from curve products.
problem Finding rational deformations of surfaces.
method Using cocompact lattices in PU(2,1).
result Shows existence of new ball quotients for surfaces.
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. The paper examines Euclidean domains with nearly maximal Yamabe quotients.
problem Understanding domains with nearly maximal Yamabe quotients in Euclidean space.
method Analyzes the properties of domains in R3 with nearly maximal Yamabe quotients, proving conditions for equality and comparing to quasi-conformal maps. result Domains with nearly maximal Yamabe quotients are diffeomorphic to balls and are close to a ball in a metric space sense.
Classifies non-arithmetic orbifolds in specific hyperbolic spaces.
problem Classifying non-arithmetic affine invariant orbifolds in Hodd(2, 2) and H(3, 1).
method Classification through Veech surfaces and rigidity results.
result Classification of non-arithmetic rank one orbifolds.
New lattices are linked to higher hypergeometric functions.
problem Understanding non-arithmetic lattices in PU(2,1).
method Showed all known non-arithmetic lattices are monodromy groups of higher hypergeometric functions.
result Non-arithmetic lattices in PU(2,1) are linked to higher hypergeometric functions.
New complex hyperbolic lattices discovered from triangle groups.
problem Finding new non-arithmetic complex hyperbolic lattices.
method General procedure to produce fundamental domains for complex hyperbolic triangle groups.
result Some triangle groups yield new commensurability classes, increasing the count to 22.
In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our…
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
New symplectic 4-manifolds with non-negative signatures are constructed using complex surfaces and quotients.
problem Creating new symplectic 4-manifolds with non-negative signatures.
method Using complex surfaces, Cartwright-Steger surfaces, and Hirzebruch's line-arrangement surfaces, along with quotients.
result Irreducible symplectic and non-symplectic 4-manifolds homeomorphic but not diffeomorphic to (2n−1)CP2#(2n−1)CPˉ2 are constructed. Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of 38π2, i.e., they attain all possible volumes of c…
The paper confirms conjectures about Stein manifolds formed by quotients of the ball.
problem Characterizing Stein manifolds formed by quotients of the ball.
method Analyzing discrete subgroups of PU(n,1) and their quotients.
result The quotient of the ball by geometrically finite groups is Stein.
Smooth manifold structure on Möbius transformations of quaternionic ball identified.
problem Identifying the manifold structure of Möbius transformations of quaternionic unit ball.
method Realizing M(B) as a quotient of Sp(1,1) and using Lie group properties. result The manifold M(B) is diffeomorphic to R4imesS3. The paper proves nonvanishing cohomology for ball quotient fundamental groups.
problem Proving nonvanishing cohomology for ball quotient fundamental groups.
method Using arithmetic lattices and profinite completions, the paper constructs an open subgroup with nontrivial cohomology.
result The virtual cohomological dimension of the fundamental group is at least 2n. 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 show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-a…
The paper embeds non-arithmetic hyperbolic manifolds into higher-dimensional spaces.
problem Embedding non-arithmetic hyperbolic manifolds into higher-dimensional hyperbolic spaces.
method Using totally geodesic submanifolds and commensurability classes.
result Many non-arithmetic hyperbolic manifolds can be embedded geodesically.
We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riem…
Proves a limit on hyperplanes in complex manifolds.
problem Limiting the number of hyperplanes in complex manifolds.
method Effective density theorem for periodic orbits, Margulis functions, restricted projection theorem, equidistribution result.
result Proves a quantitative finiteness theorem for hyperplanes.
New 4D hyperbolic spaces found with minimal volume.
problem Finding minimal-volume hyperbolic 4-manifolds.
method Applying Gromov and Piatetski-Shapiro's technique to construct non-arithmetic manifolds.
result Proved existence of at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds, and constructed the smallest known non-arithmetic one.
Study of complex projective manifolds using arithmetic lattices.
problem Holomorphic convexity for toroidal compactifications of ball quotients.
method Show that Albanese mapping on an étale covering space generates jets on the interior.
result Shafarevich conjecture on holomorphic convexity satisfied in dimension 2 for arithmetic lattices.
Uniformizes varieties with log-canonical singularities using ball quotients.
problem Uniformizing complex projective varieties with log-canonical singularities.
method Criteria based on Miyaoka-Yau inequality and log-resolutions of singularities.
result Criteria for isomorphism to Baily-Borel-Mok compactifications.
Study hyperbolic 2-spheres with cone points, describing spaces for n=3.
problem Characterize the space of hyperbolic 2-spheres with cone points.
method Analyzing the space C(a0,a1,…,an) for n=3 and n=4. result Detailed description of spaces for n=3 and examples for n=4. We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative K-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $\ball^m$ be the unit ball in Cm,…
We produce a family of new, non arithmetic lattices in PU(2,1). All previously known examples were commensurable with lattices constructed by Picard, Mostow and Deligne-Mostow, and fell into 9 commensurability classes. Our groups produce 5 new distinct commensurability classes. Most of the techniques are completely gen…
Interior C2 estimates for sum Hessian quotient equations on Riemannian manifolds
problem Interior C2 estimates for sum Hessian quotient equations on Riemannian manifolds method Interior C2 estimates for sum Hessian quotient equations on Riemannian manifolds result Interior C2 estimates at the center of a geodesic ball We consider the analogue of Hurwitz curves, smooth projective curves C of genus g≥2 that realize equality in the Hurwitz bound ∣Aut(C)∣≤84(g−1), to smooth compact quotients S of the unit ball in C2. When S is arithmetic, we show that ∣Aut(S)∣≤288e(S), where $e(S…