The residue calculus in several complex variables

Gordon, Gerald Leonard

doi:10.1007/bfb0068114

Cited by 4 publications

(6 citation statements)

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“…We would now like to recall some topological facts proven by the author in [5]. Suppose V0 = U xi ~ w, where the X i are submanifolds of complex codi- Given I~l=k, we set ct'=~u {j} forj~ and X~, is a submanifold of X~, hence we have the Thom-Gysin map Zk+l,j: Hp(X~,)--~ Hp+I (X~-X~,).…”

Section: 3mentioning

confidence: 98%

A geometric study of the monodromy of complex analytic surfaces

Gordon

1977

Invent Math

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In this article we will be interested in studying a complex analytic deformation W over the unit disk whose generic fibre 14, is a manifold of complex dimension n. In particular, we shall study the case n = 2 in detail although the techniques will go through in the more general situation. Furthermore in the n=2 case, the techniques should be applicable to studying the monodromy of schemes using etale cohomology.In Section 2 we shall make some topological reductions using resolution of singularities and semi-stable reduction to study the action of T,-I,, where T, is the monodromy, i.e., the action on homology induced from going once around the unit circle, and I, is induced from the identity. We show after a suitable reduction, in particular T, is unipotent, that the action of T,-I, is the same as a boundary operator, (2.4.1). We also show that to study T, -I,, one must study Ker g,, where g, : V~ ~ V o is the collapsing map, i.e., g is the composition of the inclusion of the generic fibre 1/1 into W and the retraction of W onto the singular fibre V o over the origin.In Section 3 we shall study the Ker g,, (3.4), for t1=2, and show that Kerg, c~Hz(V 0 is generated by 3 subspaces: G~, G2 and G3, (3.4). We conjecture that G1 | G2 | G3 c H2(VO, i.e., there are no relations among them.In Section 4, we show by purely topological results that G 1 =(T,-I,)G2, if we take closed support. This is also true for compact support if the proper transform has only one component, i.e., the original V o is irreducible, (4.4.2).However, if V 0 is not irreducible, it can be false that G1 c Image (T,-I,), (4.4.3).In Section 5 we study G 2. We are able to show that if all of the components of the proper transform of V 0 are quasi-Kahler, i.e., each is bianalytically homeomorphic to a compact Kahler manifold minus at most a proper subvariety, then G2 ~ Image (T, -I,), (5.3).In Section 6 we show using the results of Gordon [7] that if all the components of the proper transform are quasi-Kahler, then G3clmage T,-I,, when we take closed support, (6.2); and hence in the middle dimension the local invariant cycle problem, (2.2.1), is true for any analytic deformation of non-*

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Section: 3mentioning

confidence: 98%

A geometric study of the monodromy of complex analytic surfaces

Gordon

1977

Invent Math

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“…Then (2.4) actually shows that the Leray spectral sequence of j: W -VczW collapses at However, it is easy to give topological examples where the spectral sequence does not degenerate at the approprimate codimension. E.g., let S x and S 2 be two 2-dimensional spheres in iϋ 4 which intersect transversely in two points P and Q. Let T(S) be a regular tubular neighborhood of S = S, U S 2 in i?…”

Section: We Note That (23) Is a Topological And Not An Analytic Factmentioning

confidence: 99%

“…Let T(S) be a regular tubular neighborhood of S = S, U S 2 in i? 4 and form DT(S), the double of T(S), by glueing T(S) to itself along its boundary. Then DT(S) is a compact 4-dimemsional manifold and S is two submanifolds in general position of real codimension two.…”

Section: We Note That (23) Is a Topological And Not An Analytic Factmentioning

confidence: 99%

On the degeneracy of a spectral sequence associated to normal crossings

Gordon¹

1980

Pacific J. Math.

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“…Here I prove his conjecture that the filtration of a cohomology class ß of a triangulable space is the smallest integer q such that ß is represented by a cocycle whose support has dimension q (Theorem 8.8). (An incomplete proof of Zeeman's conjecture is given in [7].) In another paper [21] I will show that there is a simple relation between the Zeeman filtration and the Deligne weight filtration of the rational cohomology of a complex projective variety.…”

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confidence: 99%

“…The following result was motivated by the work of G. L. Gordon [7], [8]. Again fix a piecewise-linear structure on X.…”

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confidence: 99%

Zeeman’s filtration of homology

McCrory¹

1979

Trans. Amer. Math. Soc.

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Abstract.Geometric interpretations of Zeeman's filtrations of the homology and cohomology of a triangulable space are given, using an analysis of his spectral sequence for Poincaré duality.The failure of a space to satisfy Poincaré duality is reflected by a topologically invariant filtration of its homology groups. In some sense, the filtration of a homology class measures the "degree of freedom" of its cycles. This filtration was first studied by Zeeman, who defined it using a spectral sequence for Poincaré duality. (This spectral sequence was also discovered independently by Cartan and by Fáry.) Zeeman started a geometric investigation of the spectral sequence for spaces such as singular algebraic varieties, for which it reveals a wealth of information by relating their local and global homology [30]. I was introduced to Zeeman's spectral sequence by Sulhvan (cf. [25, p. 202] and [26]).In this paper I show that the Zeeman filtration of a homology class a in a triangulable space is equal to the largest integer q such that a is represented by a cycle in the complement of any closed subspace of dimension less than q (Theorem 8.3). If S is a stratification of the space X, then the homology class a of I has filtration > q if and only if a is represented by a cycle whose intersection with each stratum S of S has codimension > q in S (Theorem 8.4). This new geometric interpretation of the Zeeman filtration has been generalized recently by M. Goresky and R. MacPherson to define "intersection homology theories" for stratified spaces [9].Zeeman also defined a filtration of the cohomology of a space. Here I prove his conjecture that the filtration of a cohomology class ß of a triangulable space is the smallest integer q such that ß is represented by a cocycle whose support has dimension q (Theorem 8.8). (An incomplete proof of Zeeman's conjecture is given in [7].) In another paper [21] I will show that there is a simple relation between the Zeeman filtration and the Deligne weight filtration of the rational cohomology of a complex projective variety.

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The residue calculus in several complex variables

Cited by 4 publications

References 18 publications

A geometric study of the monodromy of complex analytic surfaces

A geometric study of the monodromy of complex analytic surfaces

On the degeneracy of a spectral sequence associated to normal crossings

Zeeman’s filtration of homology

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