UNIVERSAL ABELIAN COVERS OF CERTAIN SURFACESINGULARITIESTOMOHIRO OKUMA

Every normal complex surface singularity with Q-homology sphere link has auniversal abelian cover. Neumann and Wahl conjectured 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”. We will discuss theuniversal abelian covers and an approach to the conjecture.
Let (X, o) be a normal complex surface singularity germ and Σ its link, i.e.,the boundary of a regular neighborhood of o ∈ X. We may assume that X is
homeomorphic to the cone over Σ. Let Γ denote the resolution graph of (X, o). Itis known that Γ and Σ determine each other ([1]). Assume that Σ is a Q-homology
sphere, or equivalently, that the exceptional set of a good resolution is a tree ofrational curves. Then G := H1 (Σ, Z) is ﬁnite. A morphism (Y, o) → (X, o) of germsof normal surface singularities is called a universal abelian covering if it induces anunramiﬁed Galois covering Y \ {o} → X \ {o} with covering transformation groupG. By our assumption, the universal abelian covering (Y, o) → (X, o) uniquely
exist; in fact, the link of Y is the universal abelian cover of Σ in the topologicalsense.
Neumann and Wahl introduced the splice diagram equations associated withΓ satisfying the “semigroup condition”([2], [4], [3]). Let Y denote the singularity deﬁned by the splice diagram equations obtained from Γ. They proved thatY is an isolated complete intersection surface singularity, and that if Γ also satisﬁes the “congruence condition”, then G acts on Y and the quotient Y /G is anormal surface singularity (it is called a splice-quotient singularity) with resolution
graph Γ, and the quotient morphism Y → Y /G is the universal abelian covering.
They conjectured that rational singularities and minimally elliptic singularities with
Q-homology sphere links are splice-quotient singularities. Our approach to the conjecture is as follows.
Let π : M → X be the minimal good resolution, and let A = i Ai be thedecompositionof the exceptional set A = π −1 (o) into irreducible components. Let
ZAi and AQ = AZ ⊗ Q. Let Āi ∈ AQ satisfy Āi · Aj = −δij . We denote
AZ =by ĀZ the subgroup of AQ generated by Āi ’s. Then H
1 (Σ, Z) is isomorphic to thegroup ĀZ /AZ . We can construct an OX -algebra A := g∈G π∗ OM (Dg ) such thatY = SpecanX A, where Dg are divisors on M , and if Dg is numerically equivalentto Cg ∈ ĀZ , then {Cg mod AZ |g ∈ G} = ĀZ /AZ ([6]). The algebra A and splice
diagram equations are connected by “monomial cycles”.
A component Ai is called an end-curve if (A − Ai ) · Ai ≤ 1. We denote by E(A)the set of end-curves. A connected component of A − Ai is called a branch of Ai .
A component Ai is called a node if (A − Ai ) · Ai ≥ 3.
1

2

TOMOHIRO OKUMA

Definition 1. Let D =ai Āi ∈ ĀZ , ai ≥ 0. If ai = 0 for/ E(A), then wemall Ai ∈call D a monomial cycle. For any monomial cycle D = i=1 ai Āi , we associate amonomialmx(D) :=xai i ∈ C[x1 , . . . , xm ].
i=1

The x induces an isomorphism between the semigroup of monomial cycles andthat of monomials of x1 , . . . , xm .
We consider the following three conditions.
Condition 2. For any branch C of any node Ai , there exists a monomial cycle D
such that D − Āi is an eﬀective integral cycle supported on C.
Condition 3. A is star-shaped, or for any branch C of any component Ai ∈/ E(A),the fundamental cycle ZC supported on C satisﬁes ZC · Ai = 1.
Let ni be a positive integer such that ni Āi ∈ AZ and suppose Cgi ≡ Āi(mod AZ ). Then, in H 0 (A), H 0 (OM (Dgi ))ni ⊂ H 0 (OX ).
Condition 4. For each end Ai ∈ E(A), there exists yi ∈ H 0 (OM (Dgi )) suchthat div(yini ) is of the form ni (Āi + H), where H has no component of A andA · H = Ai · H = 1.
Condition 2 is equivalent to the semigroup condition and the congruence condition ([3]). So under this condition, splice diagram equations are deﬁned and the“leading terms” are linear combinations of x(D)’s, where D’s are the monomial
cycles as in Condition 2.
Lemma 5. Condition 3 implies Condition 2. Condition 3 and 4 are satisfied if(X, o) is a rational or a minimally elliptic singularity.
Let S = C{x1 , . . . , xm } be the convergent power series ring.
Theorem 6 ([5]). Suppose that Condition 3 and 4 are satisfied. Then a homomorphism ψ : S → OY,o defined by ψ(xi ) = yi is surjective, and the kernel of ψ isgenerated by the splice diagram equations which are homogeneous with respect toG-grading.
Therefore, rational singularities and minimally elliptic singularities with Qhomology sphere links are splice-quotient singularities.
References
1. W. D. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–344.
2. W. D. Neumann and J. Wahl, Universal abelian covers of surface singularities, Trends insingularities, Trends Math., Birkhäuser, Basel, 2002, pp. 181–190.
3.
, Complete intersection singularities of splice type as universal abelian covers, Geom.
Topol. 9 (2005), 699–755.
4.
, Complex surface singularities with integral homology sphere links, Geom. Topol. 9(2005), 757–811.
5. T. Okuma, Universal abelian covers of certain surface singularities, arXiv:math.AG/0503733.
6.
, Universal abelian covers of rational surface singularities, J. London Math. Soc. (2)
70 (2004), 307–324.
Department of Mathematics, Gunma College of Technology, 580 Toriba, Maebashi,
Gunma 371, Japan
E-mail address: okuma@nat.gunma-ct.ac.jp