Related papers: Universal abelian covers of surface singularities

C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive…

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational…

In this paper we classify Lagrangian spheres in $A_n$-surface singularities up to Hamiltonian isotopy. Combining with a result of A. Ritter, this yields a complete classification of exact Lagrangians in $A_n$-surface singularities.

We prove the "End Curve Theorem," which states that a normal surface singularity $(X,o)$ with rational homology sphere link $\Sigma$ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good…

The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety $X$ with a subgroup of finite rank is contained in a finite union of cosets contained in $X$. In this article, we prove a uniform…

We study Glois embeddings of K3 surfaces in the case where the Galois groups are abelian. We show several properties of K3 surfaces concerning the Galois embeddings. In particular, if the Galois group G is abelian, then G is isomorphic to…

We present the complete list of all singularity types on Gorenstein $\mathbb{Q}$-homology projective planes, i.e., normal projective surfaces of second Betti number one with at worst rational double points. The list consists of $58$…

We obtain a new important basic result on splice-quotient singularities in an elegant combinatorial-geometric way: every level of the divisorial filtration of the ring of functions is generated by monomials of the defining coordinate…

We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a…

An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and $p_g(S) = 2$. Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The…

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or…

Let G be a finite abelian subgroup of SL(n,C), and suppose there exists a toric crepant resolution phi: X -- > C^n/G. We prove that for each component E of the exceptional set of phi there exists an open subset U of X that contains E and is…

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical…

A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb…

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed…

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urz\'ua over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$…

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of…

We prove that the affine closure of the cotangent bundle of the basic affine space of a complex semisimple group has conical symplectic singularities, which confirms a conjecture of Ginzburg and Kazhdan. We also show that this variety is…