Related papers: Multiple realizations of varieties as ball quotien…
We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are…
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…
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…
Let $\Gamma \subset \mathbf{PU}(2,1)$ 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{\Gamma \setminus…
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…
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…
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…
The moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient, as a Baily--Borel compactification of a ball quotient, and as a compactified $K$-moduli space. From all three…
We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of K\"ahler metrics satisfying certain nonpositive curvature properties on these compactifications.…
A general problem in complex cobordism theory is to find useful representatives for cobordism classes. One particularly convenient class of complex manifolds consists of smooth projective toric varieties. The bijective correspondence…
We classify the smallest finite volume complex hyperbolic surfaces with cusps which admit smooth toroidal compactifications and which are not birational to a bi-elliptic surface. Remarkably, there is only one such surface which appears to…
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…
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…
We prove a result on the singularities of ball quotients $\Gamma\backslash\CC H^n$. More precisely, we show that a ball quotient has canonical singularities under certain restrictions on the dimension $n$ and the underlying lattice. We also…
We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic…
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…
We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to…
The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism…
We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.
We define a class of surfaces corresponding to the ADE root lattices and construct compactifications of their moduli spaces as quotients of projective varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We exhibit modular…