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Zbl 0659.58002
Arnol'd, V.I.; Gusejn-Zade, S.M.; Varchenko, A.N.
(Arnold, V.I.; Gusein-Zade, S.M.; Porteous, H.)
Singularities of differentiable maps. Volume II: Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous. (English)
[B] Monographs in Mathematics, 83. Boston, MA etc.: Birkhäuser Verlag. viii, 492 p. DM 238.00 (1988). ISBN 0-8176-3185-2

The present volume is the second in a two-volume book presenting (the Russian approach to) the singularities of smooth maps. [Vol. I (1985; Zbl 0554.58001); see also the review of the Russian original in Zbl 0545.58001.] In fact this volume is essentially concerned with complex function singularities and contains three parts. \par The first one ($\simeq 170 pp.)$ is based on an excellent and widely quoted survey written by the second author in 1977 and describes the topology around an isolated complex hypersurface singularity. All the basic concepts in this area (e.g. local Picard-Lefschetz theory, vanishing cycles, distinguished basis, intersection form, monodromy operator) are carefully introduced and several interesting examples are offered. A lot of results are discussed here (with proofs varying much in completeness), but some important related results are not mentioned at all [e.g. the work of {\it W. Ebeling} on the arithmetic description of the monodromy groups of isolated complete intersection singularities in Invent. Math. 90, 653-668 (1987; Zbl 0633.32014) and we refer the reader to Ebeling's book, The monodromy groups of isolated singularities of complete intersections (Lect. Notes Math. 1293 (1987)) for further results and references]. \par The second part of the book ($\simeq 100 pp.)$ is devoted to an investigation of the relation between the asymptotic behaviour of the oscillatory integrals and the singularities of the phase functions in these integrals. This study has led the first author to the classification of the famous A-D-E simple singularities as early as 1972, given thus a strong impetus to the latter development of the subject. This part contains a lot of interesting information, sometimes valuable informal discussion of the basic results, but it is decidedly more of a survey than of a text-book from which a beginner might learn systematically. \par The third part ($\simeq 190 pp.)$ studies the integrals of holomorphic differential forms on the vanishing cycles. The behaviour of these integrals as the cycles turn around the singularity is clearly related to the monodromy operator of the singularity. And by the well-known work of the third author this behaviour reflects some other subtle invariants of the singularity (e.g. the mixed Hodge structure on the vanishing cohomology, the spectrum of the singularity). Here again most of the proofs are just informally discussed and the reader is referred to the original papers for details. \par The basic contributions of J. Steenbrink to this topic are quoted and a brief attempt is made to compare the two distinct approaches to the mixed Hodge structures on vanishing cohomology. However we feel that this point deserves a more detailed discussion, perhaps involving interesting connections with Algebraic Geometry, along the lines of the {\it J. Scherk} and {\it J. H. M. Steenbrink}'s paper [Math. Ann. 271, 641-665 (1985; Zbl 0618.14002)]. \par On the whole, the book collects together a huge amount of results and informal discussions on them (quite difficult to find elsewhere), but the presentation may cause difficulties for an unexperienced reader who would like to understand clearly (and fill in) all the details.
[A.Dimca]

MSC 2000:: *58-02 Research monographs (global analysis)
58C25 Differentiable maps on manifolds (global analysis)
58K99 None of the above, but in this section
58C35 Integration on manifolds
58A10 Differential forms
32Sxx Singularities
32A99 Holomorphic functions of several variables

Keywords: singularities of smooth maps; complex function singularities; hypersurface singularity; Picard-Lefschetz theory; vanishing cycles; intersection form; monodromy operator; asymptotic behaviour of the oscillatory integrals; phase functions; integrals of holomorphic differential forms

Citations: Zbl 0554.58001; Zbl 0545.58001; Zbl 0633.32014; Zbl 0618.14002

Cited in: Zbl pre06034389 Zbl 1132.41342 Zbl 1103.32015 Zbl 1041.58005 Zbl 1034.32013 Zbl 0908.42005 Zbl 0877.57015

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