The residue calculus in several complex variables

Gordon, Gerald Leonard

doi:10.1090/s0002-9947-1975-0430297-3

Trans. Amer. Math. Soc.

1975

DOI: 10.1090/s0002-9947-1975-0430297-3

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The residue calculus in several complex variables

Gerald Leonard Gordon¹

Abstract: 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 … Show more

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Cited by 2 publications

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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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A geometric study of the monodromy of complex analytic surfaces

Gordon

1977

Invent Math

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Stationary Points at Infinity for Analytic Combinatorics

2021

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The residue calculus in several complex variables

Cited by 2 publications

References 16 publications

A geometric study of the monodromy of complex analytic surfaces

A geometric study of the monodromy of complex analytic surfaces

Stationary Points at Infinity for Analytic Combinatorics

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