The End Curve Theorem for Normal Complex Surface Singularities

We prove the “End Curve Theorem” which states that a normal surface singularity (X, o) with rational homology sphere link Σ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end-curve function” is an analytic function (X, o) → (C, 0) whose zero set intersects Σ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice-quotient singularity” (X, o) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in C, where t is the number of leaves in the resolution graph for (X, o), together with an explicit description of the covering transformation group action on this abelian cover through diagonal matrices acting on C. The end curve theorem immediately implies the previously known results: (X, o) is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005). We consider normal surface singularities whose links are rational homology spheres (QHS for short). The QHS condition is equivalent to the condition that the resolution graph Γ of a minimal good resolution be a rational tree, i.e., Γ is a tree and all exceptional curves are genus zero. Among singularities with QHS links, splice-quotient singularities are a broad generalization of weighted homogeneous singularities. We recall their definition briefly here and in more detail in Section 1. Full details can be found in [19]. Recall first that the topology of a normal complex surface singularity (X, o) is determined by and determines the minimal resolution graph Γ. Let t be the number of leaves of Γ. For i = 1, . . . t we associate the coordinate function xi of C to the i–th leaf. This leads to a natural action of the “discriminant group” D = H1(Σ) by diagonal matrices on C (see Section 1). Under certain (weak) conditions on Γ, called the “semigroup and congruence conditions” one associates an explicit set of t − 2 equations in the variables xi, which 2000 Mathematics Subject Classification. 32S50, 14B05, 57M25, 57N10.