UNIVERSAL ABELIAN COVERS OF COMPLEX SURFACESINGULARITIESJONATHAN WAHL

We are interested in questions studied recently in joint work with Walter Neumann, relating the geometry to the topology of germs of complex normal (henceisolated) surface singularities (Y, 0). For instance, one can consider isolated hypersurface singularities{f (x, y, z) = 0} ⊂ C3 ,an instructive example of which are the Brieskorn singularities
V (p, q, r) = {(x, y, z) ∈ C3 | xp + y q + z r = 0}.
Intersecting one of these surfaces with a small 5-sphere about the singular point
gives the link Σ of the singularity. In this situation, it is a compact oriented 3manifold, and the cone over the pair (S 5 , Σ) describes the local topology of thesingular point in C3 . One is in a similar situation for any normal surface singularity, except that the link now sits in a sphere of larger dimension; so we do notconsider “knot invariants”, which govern the embedding in spheres. (Recall thatfor irreducible plane curve singularities, the “link” itself is just a circle, and theinteresting part is the knotting in S 3 .) It is well-known that many geometric features of the singularity are not encoded in its link; for instance, the Brieskorn links
Σ(2, 7, 14) and Σ(3, 4, 12) are diffeomorphic, but have many different invariants(e.g., multiplicity and Milnor number).
A good resolution (Ỹ , E) → (Y, 0) is one for which the exceptional configuration consists of smooth projective curves intersecting transversally. The link can
be viewed as the boundary of a tubular neighborhood of E in Ỹ , and a deep theorem of W. Neumann shows that conversely the link determines the exceptional
configuration (with some well-known exceptions).
Now, E is a tree of rational curves if and only if the first betti number of Σ is 0,i.e. is a QHS or rational homology sphere. This is a common situation, e.g. for aBrieskorn with p prime to qr. A QHS link has finite first homology group, so it hasa (topological) universal abelian cover which is finite over it. This covering can berealized on the level of germs of singularities via a finite Galois map (X, 0) → (Y, 0),unramified off the singular points. We call (X, 0) the universal abelian cover, orUAC, of our original singularity (of course, we also are including the group and its
action in the definition). A purely algebraic definition of this cover may be given
as well.
The main point of this discussion is that in many important cases, the UAC of(Y, 0) is an explicitly given complete intersection intersection singularity, of so-called“splice type” , whose equations (up to higher order terms) and group action can bewritten down just from the topology of Σ. In other words, the topology of the linkof (Y, 0) produces nice equations not for the singularity itself, but for its universal
abelian cover. Such singularities we call “splice quotients”; they are based on a new1

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and combinatorially interesting class of singularities, called “complete intersectionsof splice type,” generalizing Brieskorn complete intersections. Our main theorem
states that if a link satisfies two fairly mild topological properties (the “semigroupand congruence conditions”), then it is the link of a splice quotient.
Our talk will explain all the constructions and conditions, motivating the result
first by looking at the rational double points and the equations produced by FelixKlein. We describe the first general result in this direction, due to W. Neumann
in the 1980’s, describing weighted homogeneous singularities with QHS link; their
UAC’s are Brieskorn complete intersections. Very recently, T. Okuma has proved
the important result that every rational surface singularity satisfies the splice andsemigroup condition, and is in fact always a splice quotient. Finally, we will describea Conjecture (which hopefully soon will be a Theorem) which explains precisely
when a given singularity is a splice quotient.
Department of Mathematics, The University of North Carolina, Chapel Hill, NC
27599-3250
E-mail address: jmwahl@email.unc.edu