Singularities of complex surfaces with solvable local fundamental group

Abstract. Any graph-manifold can be obtained by plumbing according to some plumbing graph I\ A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity.In this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3-manifolds M(TX) and A/fTj) obtained by plumbing according to two graphs Tx and T2 are homeomorphic (3-manifolds are oriented 3-manifolds throughout this paper, and homeomorphisms of 3-manifolds are orientation preserving). We then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one-parameter families of complex curves. These results are described below.Since the class of 3-manifolds obtainable by plumbing is precisely the class of graph-manifolds, which were classified, with minor exceptions, by Waldhausen [24], a calculus for plumbing is in some sense implicit in Waldhausen's work. Moreover, it has been known for some time that the calculus can be put in a form like the one given here, but the details have never appeared in the literature. A related calculus for plumbing trees has been worked by Bonahon and Siebenmann [1] in order to classify their "algebraic knots".We describe the calculus in greater generality than is needed for the present applications, since this involves minimal extra work, and the calculus is needed elsewhere ([4], [14], and [15]). In particular, in an appendix we describe two generalizations of it.The calculus consists of a collection of moves one can do to a plumbing graph T without altering the plumbed manifold M(T). To see these moves are sufficient, we describe how they can be used to reduce any graph to a normal form which is

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“…The next theorem answers a question of Wagreich [23]. It is practically, but not quite, a corollary of Theorem 3.…”

Section: Corollarymentioning

confidence: 74%

A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves

Neumann¹

1981

Transactions of the American Mathematical Society

132

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“…In particular, Tjurina's proof [11] that all rational double and triple points are taut implies that all rational double and triple embeddings of curves are formally ¿-rigid, while the nontaut examples of Brieskorn [3] and Wagreich [12] are non-¿-rigid.…”

Section: Versal Objects For [7j X ]mentioning

confidence: 99%

“…Introduction. Much progress has been made recently in the classification of normal singularities of complex analytic surfaces by considering their resolutions (see [3], [7], [11], [12], [13]). The present paper investigates deformations of formally embedded schemes with the aim of eventually using these objects in the algebraic category to classify singularities of dimension two and higher.…”

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confidence: 99%

Deformations of formal embeddings of schemes

Halperin¹

1976

Trans. Amer. Math. Soc.

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ABSTRACT. A family of isolated singularities of k-varieties will be here called equisingular if it can be simultaneously resolved to a family of hypersurfaces embedded in nonsingular spaces which induce only locally trivial deformations of pairs of schemes over local artin Ac-algebras. The functor of locally trivial deformations of the formal embedding of an exceptional set has a versal object in the sense of Schlessinger. When the exceptional set Xq is a collection of nonsingular curves meeting normally in a nonsingular surface X, the moduli correspond to Laufer's moduli of thick curves. When X is a nonsingular scheme of finite type over an algebraically closed field k and X0 is a reduced closed subscheme of X, every deformation of (X, Xq) to k [e] such that the deformation of X0 is locally trivial, is in fact a locally trivial deformation of pairs.

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“…Let /? be a singularity of a normal two-dimensional analytic space V. In [27] Wagreich introduced a definition for p to be weakly elliptic. Weakly elliptic singularities have occurred naturally in papers by Grauert [5], Hirzebruch [9], Laufer [19], Orlik and Wagreich [22], [23], Wagreich [28].…”

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confidence: 99%

“…Since TV/ is two dimensional and not compact, Wagreich [27] defined the singularity /? to be elliptic if x(^) > 0 f°r aL cycles D > 0 and x(^) = 0 for some cycles F > 0.…”

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confidence: 99%

Normal two-dimensional elliptic singularities

Yau¹

1979

Trans. Amer. Math. Soc.

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Abstract. Given a weighted dual graph such that the canonical cycle K' exists, is there a singularity corresponding to the given weighted dual graph and which has Gorenstein structure? This is one of the important problems in normal surface singularities. In this paper, we give a necessary and sufficient condition for the existence of Gorenstein structures for weakly elliptic singularities. A necessary and sufficient condition for the existence of maximally elliptic structure is also given. Hence, the above question is answered affirmatively for a special kind of singularities. We also develop a theory for those elliptic Gorenstein singularities with geometric genus equal to three.

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Singularities of complex surfaces with solvable local fundamental group

Cited by 48 publications

References 10 publications

A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves

A Calculus for Plumbing Applied to the Topology of Complex Surface Singularities and Degenerating Complex Curves

Deformations of formal embeddings of schemes

Normal two-dimensional elliptic singularities

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