Related papers: Geometry of splice-quotient singularities

We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of…

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…

We prove a formula for the geometric genus of splice-quotient singularities (in the sense of Neumann and Wahl). This formula enables us to compute the invariant from the resolution graph; in fact, it reduces the computation to that for…

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has…

The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to…

Every normal complex surface singularity with $\mathbb Q$-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a…

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this…

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…

We prove a Slice Theorem around closed leaves in a singular Riemannian foliation, and we use it to study the $C^\infty$-algebra of smooth basic functions, generalizing to the inhomogeneous setting a number of results by G.~Schwarz. In…

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions,…

From a resolution graph with certain conditions, Neumann and Wahl constructed an equisingular family of surface singularities called splice quotients. For this class some fundamental analytic invariants have been computed from their…

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They…

This expository talk is an expanded version of a lecture at G.-M. Greuel's 60th Birthday Conference in Kaiserslautern in October, 2004. We survey recent work of Neumann-Wahl and others on the relation between topology and geometry of normal…

Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are…

A three-dimensional orbifold $(\Sigma, \gamma_i, n_i)$, where $\Sigma$ 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…

We construct one Yang-Mills measure on a compact surface for each isomorphism class of principal bundles over this surface. For this, we define a new discrete gauge theory which is essentially a covering of the usual one. We prove that the…

We show a Chern-Weil type statement and a Hilbert-Samuel formula for a large class of singular plurisubharmonic metrics on a line bundle over a smooth projective complex variety. For this we use the theory of b-divisors and the so-called…

Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and…

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral…