Splice diagram singularities and the universal abelian cover of graph orbifolds

Given a rational homology 3-sphere M whose splice diagram \(\varGamma (M)\) satisfies the semigroup condition, Neumann and Wahl define a complete intersection surface singularity called a splice diagram singularity. Under an additional hypothesis on M called the congruence condition they show that the link of this singularity is the universal abelian cover of M. They ask if this still holds if the congruence condition fails. In this article we generalize the congruence condition to orientable graph orbifolds. We show that under a small additional hypothesis this orbifold congruence condition implies that the link of the splice diagram singularity is the universal abelian cover. By showing that any two-node splice diagram satisfying the semigroup condition is the splice diagram of an orbifold satisfying the orbifold congruence condition, we answer the question of Neumann and Wahl affirmatively for two-node diagrams. However, examples show this approach to their question no longer works for three nodes.     

Horospheres on abelian covers

We consider the strong stable foliation of the geodesic flow for a noncompact, connected abelian cover of a closed negatively curved manifold. We show that there exists proper leaves, and that non-proper leaves are dense.To Ricardo Mañé, in memoriam.     

Universal abelian covers of quotient-cusps

 The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a ℚ-Gorenstein singularity. Received: 3 February 2001 / Revised version: 8 March 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 14B05, 14J17, 32S25 This research was supported by grants from the Australian Research Council and the NSF (first author) and the the NSA (second author).     

Singular abelian covers of algebraic surfaces

In this paper we generalise F. Cataneses work on singular (/2)²-covers to arbitrary finite abelian covers of algebraic surfaces. We determine the contribution of singularities to the invariants , K ², q, p _g , P _n and the canonical sheaf. We use these computations to construct a surface of general type with birational canonical map ₁, p _g =4 and K ² =31. Mathematics Subject Classification (2000):14J29, 14J17, 14J25, 14E20     

Universal abelian covers of superisolated singularities

We give explicit examples of Gorenstein surface singularities with integral homology sphere link, which are not complete intersections. Their existence was shown by Luengo–Velasco, Melle–Hernández and Némethi, thereby providing counterexamples to the universal abelian covering conjecture of Neumann and Wahl (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hausdorff convergence and universal covers

We prove that if is the Gromov-Hausdorff limit of a sequence of compact manifolds, , with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then has a universal cover. We then show that, for sufficiently large, the fundamental group of has a surjective homeomorphism onto the group of deck transforms of . Finally, in the non-collapsed case where the have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the are only assumed to be compact length spaces with a uniform upper bound on diameter.

     

A note on finite abelian covers

We prove that any abelian cover over a smooth variety is defined by some cyclic equations. From the defining equations, we compute explicitly the normalization, branch locus, ramification indices, global invariants, and the resolution of singularities. As an application, we construct a new algebraic surface which is the quotient of ball.     

Flat covers in abelian and in non-abelian categories

A general existence theorem for flat covers in (e.g., quasi-abelian) locally finitely presented categories is obtained from an additive Ramsey type theorem. In the abelian case, it is shown that flat covers always exist. Applications to categories of separated presheaves or sheaves, localizations of Bousfield type, torsion-free classes of finite type, and categories of filtered objects or complexes, are given.     

Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -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 complete intersection singularity defined by a system of ``splice diagram equations'. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y₀, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G. Dedicated to Professor Jonathan Wahl on his sixtieth birthday. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Cellular covers of totally ordered abelian groups

Cellular covers of groups, and in particular, those of divisible abelian groups, were studied in [FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of groups, J. Pure Appl. Algebra 208, (2007), 61–76], [CHA-CHÓLSKI, W.—FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of divisible abelian groups. In: Contemp. Math. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 77–97], and continued in [FUCHS, L.—GÖBEL, R.: Cellular covers of abelian groups, Results Math. 53, (2009), 59–76] for abelian groups in general. In this note we are investigating cellular covers in the category of totally ordered abelian groups (called o-cellular covers; for definition see Section 2). Some results are similar to those on torsion-free abelian groups (unordered), while others are completely different. For instance, though kernels of o-cellular covers can not be non-zero divisible groups (Lemma 3.1), they may contain non-zero divisible subgroups (Example 3.2); however, the divisible part can not be much larger than the reduced part (Theorem 3.4). There are o-groups, even among the additive subgroups of the rationals, whose o-cellular covers form a proper class (Theorem 4.3).     

Minimal universal covers inE n

It is shown that any plane set of constant unit width contains a semi-circle of radius 1/2, and using this a minimal univeral plane cover is explicitly constructed. It is also shown that in an n-dimensional space with n>2 there are minimal universal covers of arbitrary large diameter. This paper was written while the author was a National Science Foundation Visiting Senior Fellow at the University of Washington, Seattle, Washington, U.S.A.     

Uniform universal covers of uniform spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces.     

The multiplicity of abelian covers of splice quotient singularities

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 resolution graph. In this paper we give a method to compute the multiplicity of an abelian covering of a splice quotient from its resolution graph and the Galois group.     

Compactifications of universal abelian threefolds with CM

We determine completions of semi-abelian degenerations of abelian threefolds with complex multiplication. This type of completion plays a crucial role in the determination of motivic decompositions of such degenerating families. To this end we generalize the notion of “relatively complete model” as introduced by Mumford, Faltings and Chai and explicitly compute it in the case at hand.     

Oriented tree diagram Lie algebras and their abelian ideals

We introduce oriented tree diagram Lie algebras which are generalized from Xu＇s both upward and downward tree diagram Lie algebras, and study certain numerical invariants of these algebras related to abelian ideals.     

Liftable homeomorphisms of rank two finite abelian branched covers

Archiv der Mathematik - We investigate branched regular finite abelian A-covers of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the...     

Bounded and L2 harmonic forms on universal covers

((Without abstract)) Submitted: April 1997, Revision: October 1997