Table Of ContentNew Developments in Singularity Theory
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Series II: Mathematics, Physics and Chemistry -Vol. 21
New Developments
in Singularity Theory
edited by
D. Siersma
Department of Mathematics,
University Utrecht,
Utrecht, The Netherlands
C.T.C.Wall
Mathematical Sciences,
Liverpool, United Kingdom
and
V. Zakalyukin
Moscow Aviation Institute,
Moscow, Russian Republic
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Study Institute on
New Oevelopments in Singularity Theory
Cambridge, United Kingdom
31 July-11 August 2000
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-0-7923-6997-4 ISBN 978-94-010-0834-1 (eBook)
DOI 10.1007/978-94-010-0834-1
Printed on acid-free paper
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©2001 Springer Science+Business Media Oordrecht
Originally published by Kluwer Academic Publishers in 2001
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or by any information storage and retrieval system, without written permission from the
copyright owner.
Table of Contents
Preface vii
Part A: Singularities of real maps
Classifications in Singularity Theory and Their Applications 3
James William Bruce
Applications of Flag Contact Singularities 35
Vladimir Zakalyukin
On Stokes Sets 65
Yuliy Baryshnikov
Resolutions of discriminants and topology of their complements 87
Victor Vassiliev
Classifying Spaces of Singularities and Thom Polynomial.q 117
Maxim Kazarian
Singularities and Noncommutative Geometry 135
Jean-Paul Brasselet
Part B: Singular complex varietes
The Geometry of Families of Singular Curves 159
Gert-Martin Greuel and Christoph Lassen
On the preparation theorem for subanalytic functions 193
Adam Parusinski
Computing Hodge-theoretic invariants of singularities 217
Mathias Schulze and Joseph Steenbrink
Frobenius manifolds and variance of the spectral numbers 235
Claus Hertling
Monodromy and Hodge Theory of Regular Functions 257
Alexandru Dimca
vi
BiJurcations and topology oj meromorphic germs 279
Sabir Gusein-Zade, Ignacio Luengo and Alejandro Melle Hernandez
Un ilary reflection gT'O/IPS and aulomorphisms oj simple. hypersurJace
singularities 305
Victor V. Goryunov
Simple Singularities and Complex Reflections 329
Peter Siodowy
Part C: Singularities of holomorphic maps
Discriminants, vector fields and singular hypersurJaces 351
Andrew A. du Plessis and C. T. C. Wall
The theory oj integral dos'ure oj ideals and modules: Applications
and new developments 379
Terence Gaffney
(With an appendix by Steven Kleiman and Anders Thorup)
Nonlinear Sections oj Nonisolated Complete Intersections 405
James Damon
The vanishing Topology oj Non Isolated Singularities 447
Dirk Siersma
Preface
The primary purpose of the Advanced Study Institute was the intro
duction of new developments in singularity theory to a wide audience,
including many young mathematicians from all over the world. The
event was held at the Isaac Newton Institute in Cambridge, which
provided an excellent venue and a very pleasant atmosphere and sur
roundings; it attracted over a hundred participants.
Singularities arise naturally in a huge number of different areas of
mathematics and science. As a consequence, singularity theory lies at
the crossroads of paths connecting the most important areas of ap
plications of mathematics with some of its most abstract parts. For
example, it connects the investigation of optical caustics with simple
Lie algebras and the theory of regular polyhedra, while also relating
the wavefronts of hyperbolic PDE to knot theory, and the theory of
solid shape to commutative algebra.
The main goal in most problems of singularity theory is to under
stand the dependence of some objects of analysis, geometry, physics,
or some other science (functions, varieties, mappings, vector or ten
sor fields, differential equations, models and so on) on parameters.
For generic points in the parameter space, their exact values influ
ence only the quantitative aspects of the phenomena, their qualitative,
topological features remaining stable under small changes of parameter
values.
However for certain exceptional values of the parameters, the qual
itative, topological features of the phenomena may suddenly change
under a small variation of the parameter. This change is called a pere
stroika, bifurcation, metamorphosis, catastrophe etc in different branches
of science.
The meeting consisted of several courses consisting of a couple of
survey lectures, a large number of research lectures, and a problem
session. This volume consists of articles amplifying the survey lectures
given at the meeting. We can group them under three headings.
A: Singularities of real maps
The classification of singularities and applications is discussed by
Bruce, and plays a central role in the contribution of Zakalyukin on flag
contact singularities. The relation between combinatorics, Stokes sets
and bifurcation sets is studied in the paper of Baryshnikov. Interesting
developments on topological invariants dual to singular sets form the
theme of Vassiliev's work. He gives an outline of his programme for
vii
viii
studying discriminant spaces and their complements, with numerous
examples. The work of Kazarjan (reported by Vassiliev at the meeting)
uses a direct construction of a 'classifying space' for singularities to
develop a new method for calculating Thorn polynomials. Brasselet
writes about generalized index theorems on singular varieties, relating
them to non-commutative geometry.
B: Singular complex varieties
The geometry of families of plane curves, a classical subject with
many new results, is the topic of the article of Greuel, Lossen and
Shustin. Parusinski proves in his article a preparation theorem for
subanalytic functions. Interesting developments on the geometry of arc
spaces were presented by Denefl. This subject is related to questions
in Hodge theory and mirror symmetry. The computation of Hodge
theoretic invariants is the subject of the paper by Steen brink and
Schultze. The relation between Frobenius manifolds and singularities
was explained by Hertling. In his article Dimca combines the theory of
Hodge structures with the study of polynomials at infinity. These can
be regarded as giving examples of meromorphic functions; an account
of these, focussing on the zeta function, is given by Luengo, Melle and
Gusein-Zade. The relation between simple singularities and complex
reflection groups is developed in the articles of Goryunov and Slodowy.
c:
Singularities of holomorphic maps
Discriminants and vector fields on singular hypersurfaces form the
topic of the article by du Plessis and Wall. The transition from geomet
rical questions to algebraic ones relates this subject to the contribution
of Gaffney. He presents the theory of integral closure of ideals and
modules. Algebra also plays a role in Damon's work on nonlinear sec
tions of non-isolated complete intersections. Moreover, these results are
related to Sierma's article about the vanishing topology of non-isolated
singularities.
Dirk Siersma
Terry Wall
Volodya Zakalyukin
February 2001
1 this talk is not represented here; the speaker's transparencies are posted on
http://www.wis.kuleuven.ac.be/wis/algebra/NotesCambridge/default.htm
Part A: Singularities of real maps
Classifications in Singularity Theory and Their
Applications
James William Bruce
Mathematical Sciences, University of Liverpool
(jwbruce~liv.ac.uk)
1. Introduction
Classifying objects in mathematics is a fundamental activity. Each
branch has its own natural notion of equivalence and it is equally
natural to list the objects in question up to that equivalence. So,
for example, we have the notions of isomorphism for vector spaces
or for groups, diffeomorphism for manifolds, conjugacy for dynamical
systems.
In this article we consider classifications which arise in singular
ity theory. We start with some straightforward classifications in linear
algebra, and elementary algebraic geometry. We then move to the clas
sification of jets under the natural equivalence relations in singularity
theory, using the idea of a complete transversal. 'Ve also establish some
results on classification in infinite dimensional spaces, that is discuss
(formal) determinacy theorems.
Prerequisites We assume that the reader is familiar with the notion
of smooth manifold and mapping, the inverse function theorem, the
idea of a germ, some elementary facts concerning projective spaces,
projective curves and Lie groups.
There is a brief discussion of the history of the su bject in the final
section. I make no great claim to originality, but much of this material
emerged from joint work with Andrew du Plessis, Terry Wall and Neil
Kirk to whom I am grateful.
EXAMPLE 1.1. We start by considering some familiar objects in
linear algebra and their associated notions of equivalence.
(1) Two linear maps j,g : V --7 Ware equivalent if there are isomor
phisms ¢ : V --7 V and 1jJ : W --7 W with 1jJ 0 j 0 ¢-l = g.
(2) Two symmetric bilinear forms a, f3 on a vector space V are iso
morphic if there is an isomorphism ¢ : V --7 V such that a( ¢-, ¢-) =
f3(-,-).
(3) Two automorphisms j, 9 : V --7 V are are equivalent if there is an
isomorphism ¢ : V --7 V with ¢ 0 j 0 ¢-l = g.
3
D. Siersma el al. (eds.), New Developments in Singularity Theory, 3-33.
© 2001 Kluwer Academic Publishers.
