Isolated Singularities of Algebraic Surfaces with C ∗ Action

Orlik, Peter; Wagreich, Philip

doi:10.2307/1970772

Cited by 166 publications

(124 citation statements)

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“…THEOREM 1. Let {{J£ ,Jf), ( It follows from Theorem 2 and an argument due to Orlik and Wagreich [13] (see also Arnold [0]) that any isolated singularity (V, 0) in C 3 defined by a weighted homogeneous polynomial has the same topological type as one of the following seven classes of singularities with the same weights as (V,0).…”

Section: {T)mentioning

confidence: 99%

“…Hence in order to prove Theorem A, we may assume that both (V, 0) and (W, 0) are one of the seven classes above. Now we need two important results due to Neumann [12] and Orlik-Wagreich [13] about the abstract topology of these singularities. Neumann's results say that the minimal resolutions of these singularities are determined by the fundamental groups.…”

Section: {T)mentioning

confidence: 99%

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Topological types and multiplicities of isolated quasi-homogeneous surface singularities

Yau¹

1988

Bull. Amer. Math. Soc.

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ABSTRACT. TWO germs of 2-dimensional isolated quasi-homogeneous hypersurface singularities have the same topological type if and only if they have the same characteristic polynomial and the same fundamental group for their links. In particular, multiplicity is an invariant of topological type, an affirmative answer to Zariski's question in this case.Let (V, 0) and (W, 0) be germs of isolated hypersurface singularities in C n+1 . We say that (V, 0) and (W,0) have the same topological type if there is a germ of a homeomorphism from (C n+1 , V,0) to (C n+1 , W, 0). In his retiring Presidential address to the American Mathematical Society in 1971, Zariski asked whether (V, 0) and {W,0) have the same multiplicity if they have the same topological type. He expected that topologists would be able to answer his question in relatively short order. However the question appears to be much harder than what Zariski thought. Even special cases of Zariski's problem have proved to be extremely difficult. Only recently Greuel [4] and O'Shea [14] proved independently that topological type constant families of isolated quasi-homogeneous singularities are equimultiple. For quasihomogeneous surface singularities, Laufer [5] explained the constant multiplicity for a topological type constant family of singularities from a different viewpoint. However it is not known whether two quasi-homogeneous singularities having the same topological type can be put into a topological type constant family. Let (V, 0) be a dimension two isolated hypersurface singularity. Lê and Teissier [8] observed that A'Campo's work [1] can often be used to give positive results towards Zariski's question. Let C(V,0) be the reduced tangent cone. Let PC(V, 0) denote the hypersurface in CP 2 over which C(V, 0) is a cone. Then, the work of A'Campo shows that the multiplicity of (V, 0) is determined by the topological type of (V, 0) in case the topological Euler number x(PC(^0)) is nonzero. The same arguments also show that, for isolated hypersurface two-dimensional singularities, the embedded topology and the multiplicity determine x(PC(^0)))-However, so far, by using A'Campo's result, one can only prove that a surface in C 3 having at 0 a singularity of multiplicity 2 cannot have the same topological type at 0 as another surface of multiplicity different from 2.

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Section: {T)mentioning

confidence: 99%

Section: {T)mentioning

confidence: 99%

Topological types and multiplicities of isolated quasi-homogeneous surface singularities

Yau¹

1988

Bull. Amer. Math. Soc.

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“…., zm) = 0}. This contributes to a project begun by Milnor [3] and continued by Milnor and Orlik [4], Orlik and Wagreich [9], Orlik [6], Orlik and Randell [8], and others, to compute invariants which will help to describe the topology of a hypersurface defined by a complex polynomial near an isolated singularity. The results of this paper can be described briefly.…”

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confidence: 90%

On the Monodromy at Isolated Singularities of Weighted Homogeneous Polynomials

Cooper¹

1982

Transactions of the American Mathematical Society

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Then AT is (am -3)-connected, Hm_2(K) is the direct sum of a free abelian group of rank q + (-l)m and a cyclic group of order d/q, and Hm_x(K) is free abelian of rank q + (-l)m. I am grateful to Peter Orlik for the survey article [7] which brought the problem to my attention and which has a good bibliography on this and related subjects.1. Preliminaries. Let /: C" -* C be a polynomial, and let V = f~x(0). If V has a singularity at z0, that is, all the partial derivatives of/vanish there, let St denote the (2m -l)-sphere in C" of radius e, centered at z0, and write K = V n Se. Then for all e sufficiently small, the portion of V inside Se is homeomorphic to the cone over K [3, p. 18], and so we are interested in the topology of K.

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“…The difficulty lies in computing the "multiplicity" of V¡ n VQ in Vl, i.e., the Euler class of the induced normal bundle. In the case when F is a surface, Orlik and Wagreich [20] Hs-2q+1{T{P¡, V)) by Alexander and Poincare duality. Then R^'^V) <* Zv, i.e., the constant sheaf whose stalk over each point of V is Z and whose stalk off of Fis zero, so that Ep'° * HP{W), Ep'2q-X * HP{V) and Ep>° = 0 for p > 1 and s + 0, 2ç/ -1.…”

Section: Alsomentioning

confidence: 99%

The residue calculus in several complex variables

Gordon¹

1975

Trans. Amer. Math. Soc.

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ABSTRACT. Let If be a complex manifold and V an analytic variety.Then homology classes in W -V which bound in V, called the geometric residues, are studied. In fact, a long exact sequence analogous to the Thom-Gysin sequence for nonsingular V is formed by a geometric construction. A geometric interpretation of the Leray spectral sequence of the inclusion of W -V C V is also given.If the complex codimension of V is one, then one shows that each cohomology class of W -V can be represented by a differential form of the type 9 A X + T) where X is the kernel associated to V and 61 V is the Poincare residue of this class.

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Isolated Singularities of Algebraic Surfaces with C ∗ Action

Cited by 166 publications

References 12 publications

Topological types and multiplicities of isolated quasi-homogeneous surface singularities

Topological types and multiplicities of isolated quasi-homogeneous surface singularities

On the Monodromy at Isolated Singularities of Weighted Homogeneous Polynomials

The residue calculus in several complex variables

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