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**Singularities of differentiable maps. Volume II: Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous.**
*(English)*
Zbl 0659.58002

Monographs in Mathematics, 83. Boston, MA etc.: Birkhäuser Verlag. viii, 492 p. DM 238.00 (1988).

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.

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 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].

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.

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.

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 J. Scherk and J. H. M. Steenbrink’s paper [Math. Ann. 271, 641-665 (1985; Zbl 0618.14002)].

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.

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 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].

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.

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.

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 J. Scherk and J. H. M. Steenbrink’s paper [Math. Ann. 271, 641-665 (1985; Zbl 0618.14002)].

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.

Reviewer: A.Dimca

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58C25 | Differentiable maps on manifolds |

58K99 | Theory of singularities and catastrophe theory |

58C35 | Integration on manifolds; measures on manifolds |

58A10 | Differential forms in global analysis |

32Sxx | Complex singularities |

32A99 | Holomorphic functions of several complex 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
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\textit{V. I. Arnol'd} et al., Singularities of differentiable maps. Volume II: Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous. Boston, MA etc.: Birkhäuser Verlag (1988; Zbl 0659.58002)