Singularity Theory: Proceedings of the European Singularities Conference, August 1996, Liverpool and Dedicated to C.T.C. Wall on the Occasion of His 60th Birthday 0521658888, 9780521658881

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LONDON MATHEMATICAL SOCIETY LECTURE NOIB SERIES Managing Editor: Professor N.J. Hitchin, Mathematics Institute, University of Oxford, 24-29 St Giles, Oxford OX! 3LB, United Kingdom The titles below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 46 59 66 86 87 88 89 90 92 93 96 97 98 99 100 104 105 !07 !09 113 114 116 119 121 128 130 131 132 133 134 137 138 139 140 141 144 146 148 149 150 151 152 153 155 156 158 159 160 161 162 163 164 166 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182

p-adic Analysis: a short course on recent work, N. KOBLITZ Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds II, M.J. FIELD Topological topics, l.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS Polytopes and symmetry, S.A. ROBERTSON Representations of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JA TEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Compactification of Siegel moduli schemes, C.-L. CHAI Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) Lectures on the asymptotic theory of ideals, D. REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y.-C. CHANG Representations of algebras, P.J. WEBB (ed) Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Descriptive set theory and the structure of sets of uniqueness, A.S . KECHRIS & A. LOUVEAU Model theory and modules, M. PREST Algebraic, extremal & metric combinatorics, M.-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UHL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR & A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) Introduction to uniform spaces, I.M. JAMES Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Helices and vector bundles, A.N. RUDAKOV et al Solitons, nonlinear evolution equations and inverse scattering, M. ABLOWITZ & P. CLARKSON Geometry of low-dimensional manifolds I, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) Classification theories of polarized varieties, T AKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & R.J. BASTON (eds) Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMA YER (eds) Groups St Andrews 1989 volume I, C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds) Lectures on block theory, BURKHARD KULSHAMMER Harmonic analysis and representation theory, A. FIGA-TALAMANCA & C. NEBBIA Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares, A.R. RAJWADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology I, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) Lower K- and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD et al Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory II, G.A. NIBLO & M.A. RO LLER (eds)

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183 184 185 186 187 188 189 190 191 192 194 195 I 96 197 198 199 200 201 202 203 204 205 207 208 209 210 211 212 214 215 216 217 218 220 22 I 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 257 260 26 1 263 264

Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, 0. MANZ & T.R. WOLF Complexity: knots, colourings and counting, DJ.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, D.J. BENSON Finite geometry and combinatorics, F. DE CLERCK et al Symplectic geometry, D. SALAMON (ed) Independent random variables and rearrangement invariant spaces, M. BRAVERMAN Arithmetic of blowup algebras, WOLMER VASCONCELOS Microlocal analysis for differential operators, A. GRJGJS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI et al The algebraic characterization of geometric 4-manifolds, J .A. HILLMAN Invariant potential theory in the unit ball of C 11 , MANFRED STOLL The Grothendieck theory of dessins d'enfant, L. SCHNEPS (ed) Singularities, JEAN-PAUL BRASSELET (ed) The technique of pseudodifferential operators, H.O. CORDES Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds) Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA & N. YUi Hilbert C*-modules, E.C. LANCE Groups 93 Galway/ St Andrews I, C.M. CAMPBELL et al (eds) Groups 93 Galway/ St Andrews II, C.M. CAMPBELL et al (eds) Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOW ALENKO et al Numbertheory 1992- 93, S. DAVID (ed) Stochastic partial differential equations, A. ETHERIDGE (ed) Quadratic forms with applications to algebraic geometry and topology, A. PFISTER Surveys in combinatorics, 1995, PETER ROWLINSON (ed) Algebraic set theory, A. JOYAL & I. MOERDIJK Harmonic approximation, S.J. GARDINER Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) Analytic semigroups and semilinear initial boundary value problems, KAZUAKI TAIRA Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) A mathematical introduction to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA, S. SCARLATTI Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Ergodic theory of zd actions, M. POLLICOTT & K. SCHMIDT (eds) Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds) The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS Finite fields and applications, S. COHEN & H. NIEDERREITER (eds) Introduction to subfactors, V. JONES & V.S. SUNDER Number theory 1993-94, S. DA YID (ed) The James forest, H. FETTER & B. GAMBOA DE BUEN Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREA YES et al Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) Clifford algebras and spinors, P. LOUNESTO Stable groups, FRANK 0. WAGNER Surveys in combinatorics, 1997, R.A. BAILEY (ed) Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) Model theory of groups and automorphism groups, D. EV ANS (ed) Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al p-Automorphisms of finite p-groups, E.I. KHUKHRO Analytic number theory, Y. MOTOHASHI (ed) Tame topology and o-minimal structures, LOU VAN DEN DRIES The atlas of finite groups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds) Characters and blocks of finite groups, G. NAVARRO Grtibner bases and applications, B. BUCHBERGER & F. WINKLER (eds) Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO, R. STOHR (eds) The q-Schur algebra, S. DONKIN Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) An introduction to noncommutative differential geometry and its physical applications, 2ed, J. MADORE Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al (eds) Groups St A ndrews 1997 in Bath, JI. C.M. CAMPBELL et al (eds) Singul ari ty theory, BILL BRUCE & DA YID MOND (eds) New trends in algebra ic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds)

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London Mathematical Society Lecture Note Series. 263

Singularity Theory Proceedings of the European Singularities Conference, Liverpool, August 1996. Dedicated to C.T.C. Wall on the occasion of his 60th birthday.

Edited by

Bill Bruce University of Liverpool DavidMond University of Warwick

UCAMBRIDG E

V

UNIVERSITY PRESS

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CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/978052 l 65888 l © Cambridge University Press 1999

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 A catalogue record for this publication is available from the British Library

ISBN 978-0-521-65888-l Paperback Transferred to digital printing 2010 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

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Contents Preface

vii

ix

Introduction Summaries of the Papers

xv

Complex Singularities Singularities Arising from Lattice Polytopes K. ALTMANN

1

Critical Points of Affine Multiforms on the Complements of Arrangements 25 J.N. DAMON

Strange Duality, Mirror Symmetry and the Leech Lattice

W.

55

EBELING

Geometry of Equisingular Families of Curves

79

G-M. GREUEL AND E. SHUSTIN

Arrangements, KZ Systems and Lie Algebra Homology

109

E.J.N. LOOIJENGA

The Signature of J(x, y)

+ zN

131

A. NEMETH!

Spectra of K-Unimodal Isolated Singularities of Complete Intersections 151 J.M. STEENBRINK

Dynkin Graphs, Gabrielov Graphs and Triangle Singularities

T.

163

URABE

Stratifications and Equisingularity Theory Differential Forms on Singular Varieties and Cyclic Homology

175

J.P. BRASSELET ANDY. LEGRAND

Continuous Controlled Vector fields

189

A.A. DU PLESSIS

Finiteness of Mather's Canonical Stratification

199

A .A. DU PLESSIS V

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vi

Contents

Trends in Equisingularity Theory T. J. GAFFNEY AND D. MASSEY Regularity at Infinity of Real and Complex Polynomial Functions

207 249

M. TIBAR

Global Singularity Theory A Bennequin Number Estimate for Transverse Knots V.V. GORYUNOV AND J.W. HILL Abelian Branched Covers of the Projective Plane

265 281

A. LIBGOBER

Elimination of Singularities: Thom Polynomials and Beyond 0. SAEKI AND K. SAKUMA

291

Singularities of Mappings An Introduction to the Image-Computing Spectral Sequence

305

K.A. HOUSTON

On the Classification and Geometry of Corank-1 Map-Germs from Three325 Space to Four-Space K.A. HOUSTON AND N .P. KIRK Multiplicities of Zero-Schemes in Quasihomogeneous Corank-1 Singular353 ities C" -. C" W.L. MARAR, J.A. MONTALDI AND M.A.S. RUAS

Butterflies and Umbilics of Stable Perturbations of Analytic Map-Germs 369 (C5 , 0) -. (C4, 0) T . FUKUI

Applications of Singularity Theory Singular Phenomena in Kinematics

P.S.

379

DONELAN AND C.G. GIBSON

Singularities of Developable Surfaces

403

G . ISHIKAWA

Singularities of Solutions of First Order Partial Differential Equations

419

S. lZUMIYA

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Preface Singularity Theory is a broad subject, involving a substantial number of mathematicians in most European countries. It was natural that a group should put together an application to the EU to set up a European Singularities Network (ESN) in 1993. It was equally natural that Terry Wall, of the University of Liverpool, should be selected to head up the bid and, subsequently, the organising committee. One of the activities funded by the ESN was an international review meeting. Given that Terry Wall was born in 1936, a number of his friends, colleagues, and ex-students (some in all three categories) decided that, to honour him on his 60th birthday, the meeting should bear his name. It was held from the 18th to the 24th of August 1996, at the University of Liverpool. It is a sign of Terry's continued vigour in the subject that we felt it best to give a Web Page address (see the end of this preface) for a list of his current publications, rather than print that list at the time of going to press. The meeting was attended by 88 mathematicians, 74 of them from 14 different countries outside the UK, and there were 61 talks. The festivities included a party hosted by Terry and his wife Sandra at their home, a multinational football (soccer) match, and a trip to the Lake District to experience a traditional English downpour. The papers presented here are a selection of those submitted to the Editors for inclusion in the proceedings of the meeting. Pressure of space has meant that a substantial number of other high quality submissions could not be included. In the introduction we have given a brief review of the subject and have attempted to set the scene for the various contributions in this collection. We would like to thank the EU for sponsoring the ESN, and consequently much of the expense associated to the meeting. We would also like to thank the London Mathematical Society for its financial help, which made it possible to include 10 mathematicians from the Moscow School of singularity theory in our invitations. We are grateful to Andrew Ranicki for coming and telling us of Terry Wall's exploits in his earlier life as a topologist, and to our t ea-ladies Emily and Joan, for serving up the refreshments at the conference with such style. Finally many thanks t o Wendy Orr for doing so much of the organisational work in her usual calm, friendly and efficient manner, and t o Neil Kirk for his enormous help with a great deal of the t echnical editing. vii

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Preface

Web Address The first URL below refers to the Web pages for The Department of Mathematical Sciences at The University of Liverpool, England. Follow the link to the Pure Mathematics Research Division (this can be reached directly via the second URL if you prefer) and then follow the link to staff and postgraduates. (We decided not to give a direct link as these have the annoying habit of changing over time. The addresses below should be fairly stable!) http://www.liv.ac.uk/Maths/ http://www.liv.ac.uk/PureMaths/

October 1998

Bill Bruce University of Liverpool

David Mond University of Warwick

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Introduction We start with a rapid survey, which we hope may be of some use t o nonexperts. Singularity theory is a broad subject with vague boundaries. It is concerned with the geometry and topology of spaces (and maps) defined by C 00 , polynomial or analytic equations, which for one reason or another fail to be smooth, (or submersions/immersions in the case of maps). It draws on many (most?) other areas of mathematics, and in turn has contributed to many areas both within and outside mathematics, in particular differential and algebraic geometry, knot theory, differential equations, bifurcation theory, Hamiltonian mechanics, optics, robotics and computer vision. It can be seen as a crossroads where a number of different subjects and projects meet . . In order to classify its current diverse productions, we centre our discussion around the contribution of five major figures: Whitney, Thom, Milnor, Mather and Arnold. The first of these to work on singularities was Whitney, who was led to study singularities in the process of proving his immersion theorem. An n-manifold M can be immersed in (2n - 1)-space, even though immersions are not dense in the space of all maps M - R.2n-l : singularities persist under small deformations. To remove them one needs a large deformation, and a good understanding of the singularities themselves. Whitney wanted a short list of the singularities that persist; besides finding the list for maps Mn - R.2n-l ([18]) he also did so for maps M 2 - R.2 ([19]) . The persistence of certain singularities under deformation leads to the idea of a stable map: one which is essentially unchanged if we deform it a little. Two maps M - N are right-left equivalent if one can be converted into the other by composing with diffeomorphisms of source and target; a smooth map is stable if its equivalence class is open, in some reasonable topology, in C 00 (M, N). The weaker notion of topological stability replaces diffeomorphisms by homeomorphisms in source and target. Whitney characterised the stable mappings Mn - R.2n-l and M 2 - R.2 , and in doing so introduced many of the central ideas and techniques in the subject. Thom was led to study singularities while considering the question of whether it is possible to represent homology classes in smooth manifolds by embedded submanifolds. With his transversality theorem ([16]) he gave the subject a push towards a kind of modern Platonism. Given smooth mani-

ix

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Introduction

folds M and N, the jet bundle Jk(M, N) is the space of Taylor polynomials of degree k of germs of smooth maps from M to N. It has a natural structure as a smooth bundle over M x N, using the coefficients in the Taylor polynomials as fibre coordinates. If f : M _, N is a smooth map, then by taking its k-th degree Taylor polynomial at each point, we obtain a map yk f : M _, Jk(M, N), the jet extension map. Thom saw the jet bundle as a version of the Platonic world of disembodied ideas, partitioned into attributes (the orbits of the various groups which act naturally on jets) as yet unattached to the objects (functions and mappings) which embody them. A map f : M _, N embodies an attribute W faithfully if its jet extension map jk f : M _, Jk(M, N) is transverse to W. Thom showed that for all M, N and W, most maps M _, N (actually a residual set of maps) have this property. This theorem put the "generic" into every subsequent theorem about generic behaviour of maps, in particular Mather's theorems that stable maps M _, N are dense in 0 00 ( M, N) if (dim M, dim N) are in his range of nice dimensions (Mather [8]- [13]), and that topologically stable maps are dense, in all dimensions [14]. It does the same for the applications of singularity theory to differential geometry, a field which is represented in this collection by the papers of Ishikawa and Donelan-Gibson. Thom also contributed the idea of a versal unfolding. Closely related to the idea of a slice to a group orbit, a versal unfolding of a map or mapgerm f is a finite-dimensional family f>.., >. E A which explores every nearby possibility of deformation, up to some specified notion of equivalence. Versal unfoldings of minimal dimension are unique up to isomorphism, and the structure of its miniversal unfolding provides one of the best ways of understanding a singularity. A germ f : (M, x) -, (N, y) has a versal unfolding with respect to a given equivalence relation if it has "finite codimension"; that is, if its orbit in the jet bundle Jk(M, N), with respect to this equivalence relation, has codimension which is eventually independent of k. The term "versal" is the intersection of "universal" and "transversal", and one of Thom's insights was that the singularities of members of families of functions or mappings are versally unfolded if the corresponding family of jet extension maps is transverse to their orbits (equivalence classes) in jet space. This insight, and Thom's Platonist leanings, led him to Catastrophe Theory. He identified and described the seven orbits of function singularities which can be met transversely in families of four or fewer parameters: these were his seven elementary catastrophes, which were meant to underly all abrupt changes (bifurcations) in generic four-parameter families of gradient dynamical systems. With the eye of faith some of his followers were able to see the elementary catastrophes in every field of science; without it, their critics objected to the invocation that this entailed of such invisibles as real variables and smooth potential functions in fields like politics and prison riot s, and to t he triteness of some of its conclusions. Nevertheless many of Thom 's ideas in bifurcation theory and gradient dynamical systems have provided the basis for lat er development , and the controversy surrounding

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xi

Catastrophe Theory should not mask the importance of his contribution to the subject. Besides proving Thom's conjectures about the genericity of stability, Mather made possible the systematic classification of germs of functions and maps. There are several standard equivalence relations, induced by the action of the various groups of diffeomorphisms on the space of map-germs. Mather showed that if a germ J : (M, x) - t (N, y) has finite codimension with respect to a given equivalence relation, then it is finitely determined with respect to that relation: there is an integer k such that any other germ having the same k-jet as f, is equivalent to it. Subsequently other authors, beginning with Jean Martinet, Terry Gaffney and Andrew du Plessis, improved on Mather's estimates for the determinacy degree (the smallest such k); see the excellent survey article of Wall [17]. Their work led to a number of papers providing lists of polynomial normal forms for equivalence classes of low codimension. Terry Wall is also a prodigious list-maker - see for example [105],[110],[129] in his list of publications. Indeed, the presence of a large taxonomy is one of the features of contemporary singularity theory, and significant theoretical advances have often come from the desire to understand features empirically observed in the lists. The paper of Houston and Kirk in this collection uses recently developed techniques from classification theory to obtain a new list, namely a list of right-left equivalence classes of singularities of maps from 3-space to 4-space. One of the most influential mathematicians working in singularity theory is V.I. Arnold, also a prodigious list-maker. In the early seventies, in connection with a project to extend the stationary phase method for the estimation of oscillatory integrals in quantum mechanics, Arnold produced extensive lists of singularities of functions ([1H4]) making spectacular advances in the techniques of classification, and in understanding the structure of the objects and the lists he found. He also introduced his notions of Lagrangian and Legendrian singularities, which systematise many of the mathematical ideas behind Catastrophe Theory. The reader is recommended Arnold's little book [5], both for a lively, personal, and probably overly harsh view of Catastrophe Theory and, more interestingly, for its exposition of the important contributions made by his school to a number of interesting geometrical problems, using techniques from singularity theory. One of Arnold's contributions was the crucial concept of simple singularity- one whose classification does not involve continuous invariants - which has proved extremely fruitful. One of the oddities of a subject with many lists is that the same lists may occur with different headings, and none more so than the short list of simple singularities determined by Arnold. The singularities in Arnold's list had already appeared as Kleinian singularities, and as the "singularities which do not affect adjunction" in a paper of Du Val in 1934. In [6], Durfee gives 15 different characterisations. Several of them , due to Brieskorn, come from t he theory of algebraic groups. Arnold also noticed a mysterious duality between some of his list s, which was subsequently in-

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Introduction

terpreted by a number of mathematicians and extended by Terry Wall and Wolfgang Ebeling. Ebeling's paper in this collection surveys this and further extensions of Arnold's strange duality, and relates it to mirror symmetry. Singularity theory typically focuses on local behaviour, on germs of spaces and maps. Isolated singular points of algebraic and analytic varieties are studied by considering only what goes on inside a conical neighbourhood, small enough to exclude the global topology and geometry of the variety. One then looks not only at the singular space, or map, but also at what goes on within this neighbourhood as the singularity is perturbed. Building on earlier work of Burau, Zariski, Mumford and Brieskorn, Milnor developed this approach in his 1968 book [15] (see also [7]), showing that if the zero set of a polynomial f : C' -+ C has an isolated singular point, then the intersection of a nearby non-singular level set of f with a small ball B, around the singular point is homotopically equivalent to the wedge of a finite number of n-dimensional spheres. The number of these spheres reflects the complexity of the singularity; it is known as the Milnor number, µ, and the intersection itself is the Milnor fibre. In fact the restriction off to B, defines a fibration, the Milnor fibration, over a small punctured disc around the critical value. A loop around zero in this disc lifts to a diffeomorphism of the Milnor fibre, inducing an automorphism of its homology. The determination of the monodromy automorphism of the singularities in Arnold's lists was one of the great achievements of his school. The study of the topology and algebraic geometry of isolated singular points of complex hypersurfaces remains the focus of a great deal of activity. It has developed a formidable technical armoury, crowned by Deligne's mixed Hodge theory, which places canonical filtrations on the cohomology of the Milnor fibre. The application of mixed Hodge theory to singularity theory was developed independently in the mid seventies by Joseph Steenbrink and Alexander Varchenko. Steen.brink's paper in these proceedings is concerned with his extension of this work to the construction of a mixed Hodge structure on the cohomology of the Milnor fibre of an isolated complete intersection singularity. The idea of looking at what goes on in the neighbourhood of a singular point generalises naturally to other contexts in singularity theory: one should look for the nearby stable object. For an unstable map-germ f, the analogue of the Milnor fibre is therefore a stable perturbation; for a non-generic hyperplane arrangement, one sees what happens when it is moved into general position. Both cases are treated in papers in this volume: Marar-MontaldiRuas and Fukui compute the number of stable singularities of given type appearing in stable perturbations of certain unstable map-germs, using commutative algebra techniques centred on the properties of Cohen-Macaulay rings, and Damon proves a conjecture of Varchenko on multi-valued functions on the complements of affine hyperplane arrangements, by viewing the arrangements as "singular Milnor fibres" (nearby objects, stable in an appropriate class of spaces) of certain singularities. Singularity theory is also concerned with the global behaviour of singu-

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Introduction

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lar spaces. Singular algebraic and analytic spaces can be stratified, that is, given a locally finite partition into manifolds. One of Whitney's most fruitful contributions was to give geometric conditions (which now bear his name) for the local triviality (product-like structure) of a stratification. The study of stratified spaces led to the development, by Goresky and MacPherson, of Intersection Cohomology, in which a version of Poincare duality for singular spaces is recovered, and to the introduction of the category of perverse sheaves. Whitney stratifications were also the key to Mather's proof of the topological stability theorem, and to work of many authors on conditions for topological triviality of families of functions and maps. Recent developments are surveyed here in the paper of Terry Gaffney and David Massey. Two papers on stratifications, by Andrew du Plessis, provide the first correct proofs of key results in the area of real stratification theory, and the paper of Brasselet and Legrand looks at differential forms on stratified spaces and describes a version of Connes's celebrated theorem in which the de Rham cohomology of a smooth manifold is computed from the cyclic homology of its algebra of smooth functions. Singular sets arise naturally as the images and discriminants of smooth mappings. The paper of Kevin Houston presented here describes a spectral sequence for computing the homology of such sets from the alternating homology of the multiple point sets of the map. One of the areas of greatest interest over the last few years has concerned the jacobian conjecture: that a polynomial map C" -+ C" with nowherevanishing (and therefore constant) jacobian determinant, is an isomorphism. Attempts by Le and Weber to prove this conjecture by looking at the behaviour of polynomials at infinity have led to a number of papers on this topic, represented here by the contribution of Mihai Tibar. We hope we have convinced you that singularity theory is indeed a broad subject. This brief introduction can only hope to cover a part of the large area of mathematics it encompasses. Indeed, the topics represented by the articles in this volume, some of which are mentioned above, give a good indication of the diversity of the subject. The following section provides short summaries of each paper, which we hope the reader will find useful.

References [l] V.I. Arnold, Remarks on the stationary phase method and Coxeter numbers, Russian Math. Surveys, 23:5 (1968), 19-48. [2] V.I. Arnold, Normal forms of functions in neighbourhoods of degenerate critical points, Russian Math. Surveys, 29:2 (1974), 19-48. [3] V.I. Arnold, Critical points of smooth functions and their normal forms, Russian Math. Surveys, 30:5 (1975), 1-75.

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[4] V.I. Arnold, Critical points of functions on a manifold with boundary, the simple Lie groups Bk, Ck and F 4 and singularities of evolutes, Russian Math. Surveys, 33:5 (1978), 99-116. [5] V .I. Arnold, Catastrophe theory, Springer-Verlag, New York, Heidelberg, and Berlin, 1986. [6] A. Durfee, Fifteen characterisations of rational double points and simple critical points, Enseign. Math., 25 (1979), 131-163. [7] E.J.N. Looijenga, Isolated singular points of complete intersections, Cambridge Univ. Press, London and New York, 1984. [8] J.N. Mather, Stability of C 00 -mappings. I: The division theorem, Annals of Math., 87, No. 1 (1968), 89-104. [9] J.N. Mather, Stability of C 00 -mappings. II: Infinitesimal stability implies stability, Annals of Math., 89, No. 2 (1969), 254-291. [10] J.N. Mather, Stability of C 00 -mappings. III: Finitely determined map germs, Publ. Math. J.H.E.S., 35 (1968), 127-156. [11] J.N. Mather, Stability of C 00 -mappings. IV: Classification of stable germs by their JR-algebras, Publ. Math. I.H.E.S., 37 (1969), 223-248. [12] J.N. Mather, Stability of C 00 -mappings. V: Transversality, Advances in Mathematics, 4, No. 3 (1970), 301-336. [13] J.N. Mather, Stability of C 00 -mappings. VI: The nice dimensions, Proceedings of the Liverpool Singularities Symposium I, Lecture Notes in Mathematics 192, Springer-Verlag, New York, 1971. [14] J.N. Mather, Notes on topological stability, Preprint Harvard University, 1970. [15] J.W. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Stud., vol 61, Princeton Univ. Press, Princeton, NJ, 1968. [16] R. Thom, Une lemme sur les applications differentiables, Bol. Soc. Mat. Mexicana, (1956), 59-71. [17] C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc, 13 (1981), 481-539. [18] H. Whitney, The singularities of a smooth n-manifold in (2n -1)-space, Ann. of Math., (2) 45 (1944), 247-293. [19] H. Whitney, On singularities of mappings of euclidean spaces, I. Mappings of the plane into the plane, Ann. of Math ., (2) 62 (1955), 374-410.

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Summaries of the Papers Complex Singularities Singularities Arising from Lattice Polytopes K. ALTMANN Torie geometry begins with a procedure for assigning to each integral polyhedral cone Cl in ]Rn an algebraic variety V.,., and with each inclusion of one cone in another a morphism of the corresponding varieties. The varieties obtained in this way are called toric varieties, and include affine space, projective space, and many other important algebraic varieties. Since its introduction twenty years ago, toric geometry has become an important tool for algebraic geometers; the combinatorics of lattice polytopes encodes important information about the algebraic geometry of the toric varieties they give rise to. Klaus Altmann's paper begins with a brief introduction to toric geometry and goes on to survey some of the links between the geometry of polytopes and the singularities of toric varieties, including their deformation theory. Critical Points of Affine Multiforms on the Complements of Arrangments J.N. DAMON A hypersurface (D, 0) in a complex manifold (X, 0) is a free divisor if the sheaf Der(log D) of vector fields on X tangent to Dis free as an Ox-module. Free divisors occur throughout singularity theory, as discriminants of stable maps and of versal deformations of singularities. If¢> : (Y, y) -+ (X, 0) is transverse to the free divisor D outside y then E = cp- 1 (D) is an almost free divisor, and if ef>t is a deformation of¢> which is everywhere transverse to D , then Et = ¢>11 (D) is a singular Milnor fibre of E. The freeness of D allows one to calculate the rank of the vanishing homology of Et as the length of a certain module defined in terms of¢> and Der(log D), and Damon has made spectacular use of this fact in a number of papers. Here he further develops these ideas to prove a strengthened version of a conjecture of Varchenko on critical points of multivalued functions defined on the complement of a hyperplane arrangement. Varchenko's conjecture originated in his work on the KZ equation described in this volume in the paper of Looijenga. Strang e Duality, Mirror Symmetry and the Leech Lattice

W. EBELING This paper surveys a series of curious dualities, originally concerning Arnold's xv

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list of 14 exceptional unimodal (or "triangle") singularities, and later extended by the author and Terry Wall to cover a wider class. The author describes interpretations of this strange duality, due to Pinkham, and, independently, to Dolgachev and Nikulin, and shows that these too can be extended to cover the Ebeling-Wall version. He also describes the connection, observed by M. Kobayashi, between Arnold's strange duality and mirror symmetry, and the relation found by K. Saito with the Leech lattice, famous as a hunting ground for sporadic simple groups. Geometry of Equisingular Families of Curves G.-M. GREUEL AND

E.

SHUSTIN

The simplest, and for many the most familiar singularities, are those presented by plane curves. Their local topology was first investigated in the late 1920's, and they have provided an excellent proving ground for theories and theorems ever since. This paper addresses three central problems concerning plane curve singularities. Suppose we are given a natural number d and a finite list of types of plane curve singularities. We can then ask: (1) Is there an (irreducible) curve of degreed in the complex projective plane CP2 whose singularities are precisely those in the given list? (2) Is the family of such curves smooth, and of the expected dimension? (3) Is this set connected? These questions have a long history, appearing in the works of Plucker, Severi, Segre and Zariski. They are part of a more general problem: that of moving from local information to global results. The authors have made spectacular recent progress, which is surveyed in this paper. They extend their discussion to curves on more general surfaces. Combined with recent ideas of Viro their results also have application in real algebraic geometry. Arrangements, KZ Systems and Lie Algebra Homology E.J.N. LOOIJENGA The Knizhnik-Zamalodchikov differential equation first appeared in theoretical physics in connection with models for quantum field theory. A rather difficult paper ofVarchenko and Schechtman established connections between this equation and the cohomology of local systems on the complements of hyperplane arrangements: under certain circumstances a complete set of solutions to the KZ equation is provided by generalised hypergeometric integrals associated to the hyperplane arangement. Looijenga's paper gives a largely self-contained account of the work of Varchenko and Schechtman and other related work, in the process providing considerable clarification of some of the main ideas. In particular it contains a clear and concise sheaftheoretic treatment of the cohomology of the complement of a hyperplane arrangement.

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Summaries of the Papers The Signature of f(x, y)

A.

xvii

+ zN

NEMETH!

If f : (C", 0) -----+ (C, 0) and g : (C"', 0) -----+ (C, 0) have isolated singularity then so does the sum f + g : (cm+n, 0) -----+ (C, 0). Classical theorems of Thom and Sebastiani describe the Milnor fibre F1+9 of f + g and its monodromy automorphism in terms of the Milnor fibres and monodromy automorphisms off and g. In case m = 2 and n = l, Ff+g is a 4-manifold, so that the intersection pairing on its middle dimensional homology is symmetric, and thus a quadratic form. One of the most important invariants of a simply connected 4-manifold is the signature of this quadratic form; it is often referred to simply as the signature of the manifold. In this paper Andras Nemethi collects together results, many of them his own, which compute the signature of Ff+g in terms of invariants off and g. The Spectra of K- Unimodal Isolated Singularities of Complete Intersections

J.M.

STEENBRINK

Steenbrink and W. Ebeling recently extended Steenbrink's earlier construction of a mixed Hodge structure (MHS) on the Milnor fibre of a hypersurface singularity: an MHS can now be defined on the Milnor fibre of any isolated complete intersection singularity (ICIS) (X,x). The construction involves embedding (X,x) as a hypersurface in an ICIS (X',x) of minimal Milnor number. In fact, as in the case of hypersurface singularities, the MHS itself is very hard to calculate explicitly, and the closest one can get to it is the Hodge numbers (the dimension of the graded pieces of the associated bigraded structure) and the spectrum of the singularity, which is defined in terms of the monodromy action on the MHS. The spectrum is a collection of rational numbers, symmetric about its midpoint. In a certain precise sense it is semi-continuous under deformation, and indeed constant under µ-constant deformation. This semi-continuity is, together with its calculability, the key to its importance: it can be used to show that certain adjacencies of singularities do not occur. In this paper Steenbrink calculates the spectrum for the unimodal ICIS's of embedding codimension at least 2, the calculation for hypersurface singularities having been made by Goryunov some years ago. The modality of a singularity is the number of continuous parameters involved in its classification; thus, "simple" = 0-modal. The unimodal hypersurface singularities were classified by Arnold in 1973, and the remainder of the list of unimodal ICIS 's was determined in 1983 by Terry Wall (paper 105 in the list of his publications). Dynkin Graphs, Gabrielov Graphs and Triangle Singularities T. URABE It is of interest to describe not only the topology of the Milnor fibre of a singularity (X, x), but also the constellations of singularities which can appear on singular fibres in a deformation of (X, x). In this paper the author gives a

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simple algorithm for determining which constellations of simple singularities can appear on the fibres of a deformation of a triangle singularity. The 14 triangle singularities are a class of unimodal hypersurface singularity; under the name "exceptional unimodal singularities" they are the subject of the strange duality reported on by Ebeling in his paper in this volume. The name "triangle singularities" arose because they can be realised as quotients of C2 by certain finite groups of automorphism, constructed from triangles in the hyperbolic plane. The algorithm is in terms of the combinatorics of Dynkin diagrams (weighted graphs which encode information about the intersection form on the Milnor fibre).

Stratifications and Equisingularity Theory Differential Forms on Singular Varieties and Cyclic Homology J.-P. BRASSELET ANDY. LEGRAND

Brasselet has worked for a number of years on the global topology of stratified sets. In collaboration with Goresky and MacPherson, the creators of intersection homology, he has proved a version of the de Rham theorem for simplicial spaces. Another special feature of manifolds is provided by the de Rham theorem which (roughly) asserts that the cohomology of the complex of differential forms and exterior derivatives yields (over the reals) a ring isomorphic to the singular cohomology ring. The introduction of intersection homology theory led naturally to a search for a result analagous to de Rham's for a Whitney stratified set. Clearly what is required is some extension of the notion of smooth differential forms to stratified spaces. This paper is inspired by the celebrated theorem of Alain Connes, which recovers the de Rham cohomology of a smooth manifold by means of the cyclic homology of the Frechet algebra of smooth functions on the manifold. Brasselet and Legrand prove a version of this result for Whitney stratified spaces, in the process introducing a new definition of differential form on a stratified space. The key is the definition of a Frechet algebra of functions on the regular part of the stratification. In this paper the authors associate to these objects a complex of differential forms and a Hochschild complex of forms on the regular part, with poles along the singular part. They then relate the de Rham cohomology of the first part, and the cyclic cohomology of the second, with the intersection homology of the stratified set. The next two papers are motivated by questions emerging from the proof of the topological stability theorem. Continuous Controlled Vector fields

A.A. DU PLESSIS Frequently smooth equivalence of map-germs is too fine ; it preserves too much detail, distinguishing phenomena one wished it did not. One then has to replace diffeomorphism by homeomorphism. Whereas diffeomorphisms

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are produced by integrating smooth vector fields, to produce the homeomorphisms needed to prove topological stability, for example, one has to relax the smoothness condition on the vector fields while retaining enough control to ensure that one can integrate and obtain continuous flows. This is done by stratifying the space and constructing the vector fields on the individual strata. To ensure that the flows on the strata fit together to yield a family of homeomorphisms, the stratification has to satisfy Whitney's regularity conditions, and the vector fields themselves must satisfy certain technical conditions (be "controlled" ). In this paper the author looks at the following important special case. Suppose we are given a stratified manifold and a projection to a space which is a submersion on strata. This is the situation which arises when trying to prove that a given stratification is a topological product. If the stratification is Whitney regular (or indeed satisfies a weaker condition known as C- regularity, due to K. Bekka) then one can lift a smooth vector field in the target to one in the source which is controlled, hence integrate it and obtain the required trivialisation. Here du Plessis shows that one can also ensure that the lifted field is actually continuous. Finiteness of Mather's Canonical Stratification A.A. DU PLESSIS One of the triumphs of singularity theory is Mather's theorem that topologically stable maps are dense. In a series of papers Mather showed how to construct a "canonical" Whitney-regular stratification of each jet space, with the property that a proper map whose jet-extension is transverse to the stratification is topologically stable. In particular such mappings are open and dense in the space of all proper mappings. Mather also claimed that this stratification has only finitely many connected strata. This is an important fact, since it implies that there are only finitely many topological types of topologically stable map-germs for any given source and target dimension. This paper gives the first complete proof of this fact. It also gives a new simpler construction of the canonical stratification, which is of substantial independent interest. Trends in Equisingularity Theory T. J. GAFFNEY AND D. MASSEY In many situations classification up to smooth equivalence yields continuous families of inequivalent types (or moduli). Finding coarser relations which yield finite classifications is the main aim of equisingularity theory. There are a number of different approaches: the purely topological, developed first by Le and Ramanujam, using Smale's h-cobordism theorem; the stratification-theoretic, which looks for invariants whose constancy should guarantee Whitney regularity, developed initially by Teissier; and a third approach using filtrations on the spaces of map-germs, due mainly to Damon. In this paper the authors survey work on the second approach, discussing ex-

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tensions (for which they are responsible) of the early work of Teissier, to cover the case of families of complete intersection singularities, and of non-isolated hypersurface singularities. Regularity at Infinity of Real and Complex Polynomial Functions

M. TIBAR Let f : Kn ---. K be a polynomial mapping, with K the real or complex numbers. Classical singularity theory gives us a good understanding of the local topology of the fibres of f, but the global topology still presents a challenge. A value a E K is called typical if the map f is a local fibration near a. There are only finitely many atypical values. Some are critical values, where the mapping fails to be a fibration for obvious reasons, and the local results alluded to above explain how the fibres off change near a. The other atypical points arise, roughly speaking, because of problems "at infinity". To obtain control of the mapping at infinity the author uses two regularity conditions: t- and p--regularity. The first hinges on a compactification of the map f, the second seeks a control function in the affine space which defines a codimension one foliation which can be used to control lifts of a constant (non-zero) vector field in K. The paper investigates the relationships between these triviality conditions and affine polar curves, over both JR and C.

Global Singularity Theory A Bennequin Number Estimate for Transverse Knots V.V. GORYUNOV AND J.W. HILL Vassiliev's approach to knot invariants, motivated by ideas from singularity theory, has proved astonishingly successful. The paper presented uses this approach to consider Legendrian knots in the standard contact 3-space JR3 • It is well known that any topological knot type in a contact 3-manifold has a Legendrian representative, that is, there is a curve of the same knot type whose tangent lines are contained at each point in the associated contact plane. When the contact structure is coorientable any Legendrian knot inherits a natural framing. A result of Bennequin shows that the self-linking (or Bennequin) numbers of all canonically framed Legendrian representatives of a fixed knot are bounded from one side. A new proof is presented here of the fact that these Bennequin numbers are bounded from below by the negative of the lowest degree of the framing variable in its HOMFLY polynomial. For knots with at most 8 double points in their diagrams these two numbers coincide. Elimination of Singularities: Thom Polynomials and Beyond

o. SAEKI AND K. SAKUMA The t heory of elimination of singularit ies is concerned wit h the following global problem: given a smooth map f : M ---. N, does there exist a smooth map g homot opic t o f which has no singularities of a prescribed type I:? The

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first, and most basic, obstruction to the existence of such a map g comes from the Thom polynomial, a polynomial in the Stiefel-Whitney classes of M and N which is dual to the homology class represented by the closure of the set of points of type I:, since this only depends on the homotopy type of f. This article discusses circumstances in which this is the only obstruction. Abelian Branched Covers of the Projective Plane

A. LIBGOBER It is a classical problem, first studied by Zariski and van Kampen, to understand how the type and relative position of the singularities of a plane projective curve are reflected in the fundamental group of its complement. In this paper the author outlines a relationship between the fundamental group of the complement of a reducible plane curve, and certain geometric invariants depending on the local type and configuration of the singularities on the curve.

Singularities of Mappings Multiplicities of Zero-Schemes in Quasihomogeneous Corank-1 Singularities

W.L.

MARAR, J.A. MONTALDI AND M.A.S. RuAS

Butterflies and Umbilics of Stable Perturbations of Analytic Map-Germs

T. FUKUI In the nice dimensions, every map-germ f of finite codimension can be perturbed so that it becomes stable. Since the bifurcation set is a proper analytic subset of the (smooth) base of a versal unfolding, any two stable perturbations are smoothly equivalent, and thus there is a well-defined stable perturbation ft off. Understanding it is an important problem, both for theoretical reasons, and for applications to other subjects. Besides the homology of its image, there are other integer invariants of importance, such as the number of nodes on a stable perturbation of a germ of a parametrised plane curve, or the number of critical points of a stable perturbation of a function-germ. These two papers give formulae for related 0-dimensional invariants of weighted homogenenous map-germs, in t erms of their weights and degrees. Both use the principal of conservation of multiplicity for Cohen-Macaulay modules. Fukui's paper considers map-germs f : (C5 , 0) - t (C4 , 0) . Here there are two local singularity types occuring at isolated points in a stable perturbation, namely A 4 and D 4 • Marar, Montaldi and Ruas look at corank-1 map-germs (C"-, 0) - t (C"- , 0) and compute all 0-dimensional invariants, including those involving multi-germs . An Introduction to the Imag e-Computing Spectral Sequence K.A. HOUSTON

The images of analytic maps of complex manifolds are generally highly singular, even when the maps are stable: parametrised plane curves have nodes,

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and surfaces mapped stably into 3-space have normal crossings and Whitney umbrellas. On the other hand, Sard's theorem assures us that the generic fibres of a map are smooth. Consequently, in order to understand the topology of a space which is the image of a finite map between smooth manifolds, one has to use methods which take account of this fact. A useful handle on such questions is provided by the multiple point spaces of the map: the closure of the set of k-tuples of pairwise distinct points with the same image. The symmetric group Sk acts naturally on the k-th multiple point space, permuting the points. This paper surveys work on a spectral sequence which relates the homology of the image to the alternating part of the homology of the multiple point spaces, and discusses applications to the computation of the homotopy type of generic images. On the Classification and Geometry of Carank-1 Map-Germs from ThreeSpace to Four-Space K.A. HOUSTON AND N.P. KIRK Classifying map-germs up to right-left equivalence is far from easy, and rather few lists of equivalence classes have been obtained. The tables in this paper summarise a new classification and also show geometrical information about the germs listed. The key geometrical invariant of a germ f : (en, 0) ---+ (en+I, 0) is its image Milnor number µr(f), the rank of the middle homology of the image of a stable perturbation. It is conjectured that the image Milnor number and the Ae-codimension of a map-germ satisfy the Milnor-Tjurina relation: µr(f) ~ Ae-codim(f) with equality if f is weighted homogeneous, in analogy with the relation between Milnor number and Tjurina number of an isolated complete intersection singularity. This has been proved only when n = 1 and n = 2. The examples here provide the first support for the conjecture outside the range of dimensions where it has been proved. The paper also provides a brief survey of current classification techniques, and of results on the topology of map-germs f : (en, 0) ---+ (O', 0) for n < p.

Applications of Singularity Theory Singularities of Developable Surfaces

G.

ISHIKAWA

Differential geometry, in particular the differential geometry of curves and surfaces in 3-space, has provided an exciting area of application for singularity theory. Initially it seems slightly surprising that singularity theory should have anything new to say about such a well-studied area. In fact its success here has been quite dramatic. It has taken place on three fronts: the understanding of higher order phenomena (classical differential geometry concerns itself largely with quadratic forms); the understanding of generic phenomena, via transversality and classification theorems (classical geometry illustrat es itself with a surprisingly small collection of often very special surfaces ); and the underst anding of the

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singular objects which arise naturally in connection with smooth embeddings (such as the developable surface of a curve, and the dual and focal sets of a surface). In this paper the author surveys results on developable surfaces, those whose Gaussian curvature vanishes identically. It is a classical result that there are three types of analytic developable surface: cones, cylinders, and tangent developables (the union of the lines tangent to a space curve). In a natural sense the most general and interesting of these are the tangent developables. They are highly singular along the curve itself, and classical differential geometers found these singularities a source of embarrassment. In singularity theory they are, however, an object of great interest. After a brief discussion of some of the classical ideas using projective duality, Ishikawa describes the smooth and topological classification of the singularities of these surfaces.

Singular Phenomena in Kinematics P.S. DONELAN AND C.G. GIBSON The study of the kinematics of rigid bodies received substantial impetus during the industrial revolution, and has more recently received new stimulus from robotics. Classical results such as the solution by Watt and Peaucellier of the problem of the conversion of circular to rectilinear motion, and Kemp's theorem that any plane algebraic curve can be traced by a mechanism, lie on a fascinating interface between geometry and engineering. The advances of kinematics in the 19th century can, in part, be traced to developments in the algebraic geometry of plane curves. The thesis set out here is that the generic phenomena of rigid body kinematics can be understood using singularity theory. The basic object under review can be described as follows. Suppose given a mechanism of some form, for simplicity in the plane, with r degrees of freedom. If we attach another copy of the plane to the mechanism we will obtain an r-parameter family of motions of the plane, and in particular for each point of the plane a map from an r-dimensional space to the plane. If r = 1 this gives a 2 parameter family of curves. Most are immersed with transverse intersections, but for certain special tracing points we must expect, for example, cusps. Such points will be of some engineering interest. Understanding, in a general context, the types of singularity that can occur, and the nature of the corresponding exceptional tracing points is part of the task set here. The paper gives a detailed survey of the results obtained to date in this area. Singularities of Solutions of First Order Partial Differential Equations

S.

IZUMIYA

Singularity theory has applications to any area whose fundamental tools involve the calculus, and there is now a substantial body of work on its applications to bifurcation problems in ordinary differential equations. This part of the subject now has an independent existence, rather separate from

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singularity theory itself. Izumiya's paper surveys some applications of singularity theory to partial differential equations. He confines himself to Hamilton-Jacobi equations and single conservation laws. Both equation types arise in a number of applications (eg. to the calculus of variations, differential games, and gas dynamics). Classically, solutions are obtained by the method of characteristics; initially such characteristics are smooth, but beyond a certain critical time they become multivalued. The classification of the associated singularities is discussed, as is the evolution of shock waves, in the two variable case.

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Singularities Arising from Lattice Polytopes Klaus Altmann

1

Introduction

(1.1) Assume that an affine variety Y C (Sw is defined by certain binomials a b z - z (a, b e JNW); for example take Y := [zn - xy] C (T3. Then, the ring of regular functions on Y equals the semigroup algebra @[S] with S obtained from Nw via identifying a and b. If, moreover, the semigroup S is easy to handle (for instance, if S is the set of lattice points in a rational, polyhedral, convex cone in some Mk), then one might hope that important features of the algebraic variety Y can be encoded with combinatorial data. This is, more or less, the main idea of the concept of (affine) toric varieties. A lot more has been done: if a bunch of polyhedral cones comes in a so-called fan, then the affine toric varieties associated to these cones glue together; if the fan arises from the inner normals of a lattice polytope, then we obtain a projective variety. These ideas have been developed over the last 20 years, and many textbooks are now available. For a detailed treatment we refer to [Da], [Fu], [Ke], or [Od]. For a short introduction to the subject without proofs see §2. (1.2) As just mentioned, lattice polyhedra are related to projective toric varieties. Hence it is no surprise that (affine) cones over projective varieties, in the algebro-geometric sense, arise from cones over lattice polytopes in the sense of convex geometry. Doing toric geometry, one has to deal with cones and their duals as well (cf. (2.1)). This implies that lattice polytopes have a second chance to induce a certain class of affine toric varieties; it was first observed by Ishida in [Ish], that this class consists exactly of the affine toric Gorenstein varieties. A more detailed explanation of both methods to construct singularities from lattice polytopes is given in (2.6) and (2.7). (1.3) The main purpose of the present paper is to give a survey of the deformation theory of toric singularities (or equivalently, of affine toric varieties) known so far. In the very beginning, deformation theory appeared as the investigation of how complex structures may vary on a fixed compact,

2

Altmann

smooth manifold (cf. [Kod]). In a similar manner, we may regard deformations of germs of analytic spaces. If Y — (Y, 0) is such a germ (often called "singularity", since smooth germs are not the interesting ones), we define the following functor: Defy ((T, 0)) :=

{isomorphism classes of flat g : Z —> T , together with ^ ( O ) -^ Y} .

Good references for facts about deformation theory of germs are Artin's Lecture notes [Art], the long introduction to Palamodov's paper [Pa], or Stevens' Habilitationsschrift [St 2]. In many cases, for instance for isolated singularities, there exists the so-called mini-versal deformation. By definition, it yields every possible deformation via specialization of parameters, i.e. via base change. The mini-versal deformation is, up to non-canonical isomorphism, uniquely determined and may be considered a source of numerical invariants of the original singularity. If Y is a complete intersection, then every perturbation of the defining equations yields a (flat) deformation of Y; in particular, its mini-versal deformation is a family over a smooth base space with well-known dimension. However, as soon as we leave this class of singularities, the structure of the versal family or even the base space will be more complicated. The first example showing that the situation is not always as boring as in the complete intersection case was given by Pinkham, cf. [Pi]. He studied the cone over the rational normal curve of degree four; here the versal base space consists of two irreducible components of dimension one and three, respectively. In the following we will see similar examples in higher dimensions; even non-reduced points will occur as versal base spaces. (1.4) What are the major issues we are concerned with? In the sequel we will discuss the following three: (1) Study the vector spaces T y of infinitesimal deformations and T y containing the obstructions for lifting deformations onto larger base spaces. There are different ways of defining them (cf. (3.1)-(3.3)); but in any case, they are as multigraded as the semigroup algebra (P[S] is. Hence, the problem is to spot the multidegrees R contributing to these vector spaces and to determine the dimensions of Ty(—R). (The minus sign just makes some of the formulas easier.) (2) Let us return to the trivial example of the i4n_i-singularity mentioned in the very beginning. The algebra A := (F[x, y, z]/(zn — xy) is Z2graded via deg x := [n, —1], deg y := [0,1], and deg z := [1,0]. The infinitesimal deformations equal T r = QJ[z\/{zn~l), and the one-parameter deformation assigned to the element zn~k e Ty equals (zn — xy) + k k the total space of this deformation t{k)zn-k^ Substituting T := z +t( \ is defined by a binomial again, by Tzn~k - xy.

Singularities Arising from Lattice Polytopes

3

The general goal is to look for so-called genuine deformations, i.e. for those which are no longer infinitesimal, but defined over parameter spaces which are reduced or even smooth. To be able to use the language of polyhedral cones, we would like to remain somehow in the category of toric varieties. It turns out to be a good idea to look for deformations having toric total spaces as just seen for An_i. (3) The best possible result is the description of the whole versal deformation. This might be done by listing equations, by providing information about its irreducible components, or by different methods. After a short introduction into the subject of toric varieties we will discuss the progress in each of these questions in a separate section. Emphasis will be put on both cones over projective varieties and toric Gorenstein singularities, i.e. on those toric varieties induced by lattice polytopes. Our survey does not contain any proofs; the claims are either standard, or easy exercises, or we refer to the original papers.

2

Convex geometry and toric varieties

(2.1) We are going to introduce briefly the notions we need from convex geometry. It should be considered a good opportunity to fix notation, on the one hand, and to get readers from algebraic geometry in the mood for cones and polytopes, on the other hand. References for the details can be found in the new book [Zi] or the appendix in [Od]. Convex cones: Throughout the paper we use the word cone for rational, convex, polyhedral cones. If TV, M are two mutually dual, free Abelian groups of finite rank, then a cone a C NM := N ®% M can be given either by its fundamental generators m

a = (a\ ...,am)

:= £ i R > 0 • o*

(a 1 ,... ,am e N)

1=1

or by finitely many inequalities O;j

= l,...,K}

( r \ . . . ,rK e M ) .

The elements a1 G N and r7 G M can be normalized by asking for primitive ones, i.e. which are not proper multiples. (I hope the reader will not be confused by abuse of notation: We use the symbol (...) for both the pairing between the mutually dual lattices N, M as well as for indicating the generators of cones; " N2 such that /( Ya.2 is an open embedding identifying the first variety with the open subset [xr ^ 0] C Yai. Moreover, every open embedding in our category arises that way. Definition: / / E is a fan in NJR (i.e. a finite collection of cones such that d,r € S always implies r D cr < T,O~, and such that E contains with every cone all of its faces), then the toric variety Y% is obtained by gluing together the affine pieces Ya (a G E^ along common faces ofE-cones. A map between toric varieties Y^i, Y^-i is given by a %-linear f : iV1 —> N2 such that for each G\ E E 1 there is some (j2 G E 2 meeting /(£, on the other; Minkowski addition translates into the tensor product. Hence, it seems quite natural that, for projective toric varieties, the vector space V(P) of (2.3) appears as V(P) = Pic Yp (g>^ M.

Altmann Examples: (1) The /^-dimensional fan spanned by the canonical basis vectors e 1 , . . . , ek of 2Zk and — e, with e := e1 -\ h efc, defines the projective space JPfc. (2) The subdivision of the smooth cone a = JR>0 = (e 1 ,... , efc) into the union of &i := ( e 1 , . . . , e\ . . . , efc, e) (i = 1 , . . . , k) describes the blowing up of the origin in the affine fc-space. (3) If a — ( a 1 , . . . , a m ) is an arbitrary cone (with apex), then the normalized blowing up of Ya in the closed orbit is given by the subdivision of a into the union of o~r := {a G o~ \ (a,r) < (a,E)} with r running through the Hilbert basis E of o~y D M. In general, every fan can be subdivided into a "smooth" fan. That means, every toric variety admits a toric desingularization. (2.6) Let P C (LJJ, L) be a lattice polytope. Embedding P via P C LM = LJR x {1} ^-> Ljfi x M =: MJH into the next dimension, we obtain rational, polyhedral cones 0 • P C M ^ and a := crvv C A/JR. Since o f f [ d P n M ] = e d > o # ° ( y P , £ ^ ) , the toric variety 1^- equals the affine cone over the projective variety (Yp, Cp). There is a distinguished point a* := (0,1) G a C iV^; the equation [a* = 1] recovers P from Yp is provided by the ^-linear map ?r : A^ —»> N/Z - a* = L]R . It sends proper faces of a isomorphically onto E-cones, meaning that Ya \ {0} is a (F*-bundle over YP. (2.7) Finally, we would like to explain Ishida's relation (cf. [Ish], Theorem 7.7.) between lattice polytopes and Gorenstein singularities: up to sign, the differential form UJ0 := ^ A • • • A ^ on the torus ((F*)k ^ Spec(T[M] = Yo C FCT does not depend on the special choice of coordinates. Multiples xr • LJO are holomorphic on y a if and only if r belongs to the interior of crv. Since Ya is normal, this means that we can describe the canonical module as LUY = ( ®r (T[z]s -> / -H. 0

and 0 -^ / -> (T[i] -> A -» 0.

Then, the easiest way to define the vector spaces Ty and Ty is

ly

. -

via

.C^-A-

with ^ ) ; = ( | i , . . .

and

These definitions seem to depend on the embedding of y , but they do not. In the above straightforward formulas, the relation of the vector spaces Ty to deformation theory becomes apparent. For instance, if £ 6 Hom^7^2, A), then the infinitesimal deformation represented by £ may be obtained oy replacing fj with the perturbed equation fj + £%(fj). (3.2) A more fancy way to obtain Ty and Ty is to consider them as the first and second Andre-Quillen cohomology groups, respectively. This cohomology

10

Altmann

theory is obtained from the cotangent complex which is defined for any GJalgebras; it is closely related to Hochschild and Harrison cohomology. A nice introduction into this subject may be found in [Lo], §3.5 - §4.5. To calculate the vector spaces Ty for affine toric varieties Ya, the Harrison cohomology approach was most successful. It yields the following results (cf. [A1S1]): The Ty admit an M-(multi) grading as does any other natural module over A — (F[av n M]; let us fix an element R G M. For any face r < a we define

K? := [av C\(R- intr v ) n M] \ {0} . Definition: Let K C M be an arbitrary subset of the lattice M. A function / : K —> (£ is called quasilinear if f(r) + f(s) = f(r + s) for any r and 5 with r,s,r + s e K. The vector space of quasilinear functions is denoted by Horn (K, (P). The sets Kf admit the following properties: (i) K* = (crv H M) \ {0}, and Kf := ^ = {r e K o | ( ^ , r ) < ( a \ E ) } with z = 1 , . . . , m are "thick strips" along the facets of crv. (ii) For r / 0 the equality if.f = fla^r ^ ^ nolds. Moreover, if a is a topdimensional cone, K^ = K0C\(R — intcr v ) is a (diamond shaped) finite set. (iii) The dependence of the sets K^ on r is a contravariant functor. This gives rise to the complex Horn (K?, d) with *

^,(T)

(0 < p < dima)

and the usual differentials. (iv) If r < a is a smooth face, then the injections Hom(span (r i^^, (E) ^^ Horn (i^^, (T) are also isomorphisms. Moreover,

span^f = p| span^iff, and the latter vector spaces equal span^if/ 1 = M#, (a*)-1, or 0 if (a\R) > 2, = 1, or < 0, respectively. Now we can express the Andre-Quillen cohomology groups Ty in terms of the sets * T h e o r e m : ([A1S1]) 1) Let a be an arbitrary rational, polyhedral cone with apex in 0. Then, for every R G M,

Singularities Arising from Lattice Polytopes

T^(-R) = iF(Hom" (iff, (P))

11

/or p = 0,1,2.

Moreover, for p — 0,1, £/ws vector space equals ^ ( ( s p a n ^ i f f )*). 25 Gorenstein in codimension two, then Ty(-R) = if 2 ((span^iff)*). 3) Let Ya he an isolated singularity. Then, the Andre-Quillen cohomology in degree —R equals

{

# p (Hom (iff, (P)) = i P ^ s p a n ^ i f f )*) for 0 < p < dim a - 1 # d i m f f (Horn" (iff ,(P))

/or p = dim a

j

for p>dima

+ l.

m

Here, the vector spaces HA (K\(L) are defined as the cohomology of the complex Cq(K; (P) := [ip : { ( r i , . . . , rq) e Kq \ J2vrv G i f } -> (T | 99 is shuffle invariant} with differential 6* : Cq~\K\ Q) -+ Cq(K; Q) 9-1

, •. •, rq)

:=

Remark: The general framework for the above theorem is the existence of a spectral sequence E{'q = ®dimT=pHAq(K*; (P) = » T ^ 9 " ^ - ^ ) together with the vanishing result HAq(K^; (E) — 0 for smooth faces r < cr and 2. (3.3) The results for Ty and Ty were previously obtained, under some additional assumptions and by different methods, in [Al 2]. The paper uses the formula Ty = E x t ^ r ^ ^ , A), which is true for normal rings A and p — 1,2. Moreover, [Al 2] contains the relation between the above combinatorial formulas for Ty and those mentioned in (3.1). For Ty it looks like the following: Recall from (2.4) that E C crv n M denotes the Hilbert basis. It provides a natural map TT : ZE —» M which is dual to iV —>• 2ZE mentioned in Example (2.4)(2). Then, the ideal I C QJ[z\ denning Ya is generated, as a (F-vector space, by the binomials z_a — z6 with a, b € 1NE and ?r(a) = TT(6). With E? := E O iff we denote by LQ{E*) the (T-vector space consisting of the linear relations among £ f -elements, i.e. La(E^) = (kerTr^) D (EE*. In particular, there is an exact sequence 0 -> Lfl?(£7f) -> ( P ^ -> span^Ef -> 0.

12

Altmann

Theorem: ([Al 2]) Let R 6 M. The above sequence for the corresponding complexes gives

//, moreover, ip : LQ{E) —» (T induces some element ofTY{-R), then the A-linear map I/I2 —* A sending z" — z,b onto (p(a — b) • £7r(a)--R yields the same element via the TY -formula in (3.1). (3.4) What do the above formulas mean for the simplest example, the twodimensional cyclic quotient singularities? We use the notation of Example (2.4)(3), i.e. a = ((1,0); (-q,n)) C M2. We develop ^ _ i n t o a (negative) continued fraction n

i—

(with av > 2).

~

Then E C a v n Z? is given as the set £ = {r 1 ,..., rw} C ([0,1], [w, g]) n with

# r t;-i + r ^+i

z= av • r u (v = 2 , . . . , Ti; - 1) (cf. [Ri] or [Od]).

Pinkham has already obtained that, if w > 4, dim TY = Y%=2 av — 2. The previous theorems should give the same result. Denoting a1 = (1,0) and

a2 = (—q,n), they mean

TY(-R) = (

y

There are four different cases for the multidegree R € M = Z2; we assume iy > 3, i.e. y is not smooth: (i) R = r2 (or analogously R = rw~l)\ We obtain E? = {r1} and Ef = {r 3 ,..., r^}, i.e. span^Ef = (a1)-1, s p a n ^ f = (F2 (or (a 2 ) 1 , if w = 3), and span^J?^ = 0. Hence TY(-R) ^ (T (or 0, if w = 3). (ii) i? = rv (3 < v < w; - 2): We obtain E? = {r\..., r^1} and E? = {rv+1,..., rw}, hence span^Ef = s p a n ^ f = (E2, span^E^ = 0, and the theorem yields TY(-R)^(F2. (hi) R = p • rv (2 < v < w — 1, 2 < p < av for -u; > 4; or v = 2 = w — 1, 2 < p < a2 for w = 3): We obtain E? = { r 1 , . . . , ^ } and E? = {rv,..., r w }, i.e. span^Ef = span^Ef = (T2 and span^-Bg = (D - R. The theorem yields TY(-R) ^ (T.

Singularities Arising from Lattice Poly topes

13

(iv) For the remaining Re M, either E? C E% or E? C E f or # ( £ f n f > 2. In all these cases the theorem yields T^(-R) = 0. Summing up, we get Pinkham's formula back. The vector space Ty may be obtained in a similar way. (3.5) We would like to have an interpretation for the vector spaces T^(—R) in terms of convex geometry. In fact, we do not know any for T£, but for T$ this was done in [Al 4]: To any R G M we assign the crosscut Q(R) := aC\[R = 1], cf. Example (2.2), and two ^-vector spaces V(R), W(R) defined as follows: V(R) := V(Q(R)) is the space of "generalized" (allowing negative dilatations) Minkowski summands of Q(R) introduced in (2.3), and \Y(R)

•= /f?#{Q(-^)" vert i ces

not

belonging to N}

The latter provides coordinates Si for each vertex a1 G Q(R) \ N, i.e. for each fundamental generator a1 £ o~ with (al,R) > 2. To each compact edge dtj = ataj we associate a set of equations G^- dealing with elements of V(R) © W(R). These sets are of one of the following three types: (0) Gij = 0, (1) Gij = {si — Sj = 0} provided both coordinates exist in W(R), set Gij = 0 otherwise, or (2) G^ = {Uj — Si = 0, tij — Sj = 0}, dropping equations that do not make sense. Restricting V(R) 0 W(R) to the (at most) three coordinates t^, s^ Sj, the actual choice of G^ is made such that these equations yield a subspace of dimension 1 + dimTAi aj\(—R)', since (a1,a?) is a two-dimensional cone, we do know dim TL oJ-\(—R) from the previous section. Anyway, if Y is smooth in codimension two, then Gij is always of type (2). Theorem: ([Al 4]) The infinitesimal deformations ofYa in degree —R are given by T^(-R) = {(£, s) e VW(R) 0 WW(R) | ( t s) fulfills all the equations G^} / o • Q(R) that preserve up to homothety the stars of non-lattice vertices ofQ(R).

14

Altmann

In the last three sections we would like to mention some applications (3.6) and examples. First, assume that Ya is smooth in codimension two. For the cone a C NJR this assumption means that any two-dimensional face is generated by two elements of N forming a part of a ^-basis of TV. Proposition: There are only finitely many R £ crvflM such that Ty(—R) ^ 0. If, moreover, the three-dimensional faces of a are simplicial (i.e. built from only three fundamental generators), then Ty(-R) — 0 for each R € intcrvnM. Degrees from — (int crv D M) somehow play the role of negative degrees when dealing with singularities with ((T)*-action. In contrast to the Gorenstein case below, there might be infinitesimal deformations in non-negative degrees, even for isolated singularities. In particular, it is not always true that cones which are smooth in codimension three provide rigid singularities: Example: Let a = (a, 6, £, y, z) be the (four-dimensional) cone over some double tetrahedron.

a = (i, 0,0,0) b = (0, 1,0,0) y = (0, 0,1,0) X = (1, 1,1,-1) z = (1, 1,1,1)

The two partial cones generated by a, 6, y, x and a, 6, y, z, respectively, are smooth. So are the proper faces of cr, i.e. Ya is an isolated singularity. Let R = [0,0,1,0]; then the compact part of Q(R) consists just of two edges containing the lattice-vertices x,y,z.

Q(R) =

In particular, dimTY{—R)

= 1.

Singularities Arising from Lattice Polytopes

15

(3.7) If Ya is the affine cone over a projective toric variety Yp (cf. (2.6)), then we may use the distinguished element a* e N to transform Mmultidegrees into ordinary integers. In particular, elements R e PC\M C M yield Ty(—R) C Ty{—1). For cones over smooth, projective surfaces, the reverse implication is also true: Proposition: Let P be a plane lattice polygon defining a smooth, projective surface. Except when P is isomorphic to the unit square (yielding Y = [xy — zw = 0] C (F3), we have (nrl \

.

m

7^1 (

u\

T^ (

1^

£ft

T1^ (

/?^

Moreover, it is possible to list the series of polygons P provided Ty (—1) / 0; they are at most hexagons such that the inner lattice points are contained in a line. These polygons are closely related to the so-called reflexive polygons occurring in [Bat]. (3.8) Finally, we consider (J-Gorenstein singularities Ya provided by some lattice polytope Q C MJR sitting in height g (cf. (2.7)). Since R* € M is a distinguished element, we may expect the homogeneous part Ty(—R*) to be special. Theorem: ([Al 1]) Let a = cone(Q) be a Q-Gorenstein cone which is at least three-dimensional. Assume that Ty is finite-dimensional (for instance, if Ya has an isolated singularity). Then, (1) Ya is rigid unless it is Gorenstein. (2) Let Ya be Gorenstein, i.e. g = 1; then T^ = T^(-R*) = VW(Q)/(F • 1. If moreover, Ya is smooth in codimension three, then Ya is rigid, too. If the restriction dim Ty < oo is dropped, the situation will be more complicated. Examples are the three-dimensional, non-isolated Gorenstein singularities; they are given by plane lattice polygons Q. Each Q-edge aiai^rl (with a m + 1 := a 1 ) of length n stands for a one-dimensional An-\-singularity inside Ya. Besides deformations with degree in — N • R*, edges of length n provide (n—1) infinite series of degrees Re M with dim Ty(—R) — 1. Since

these deformations are, for each edge separately, unobstructed. They lift the versal deformation of A n _i. Details may be found in [Al 4],

16

4

Altmann

^-parameter families

(4.1) In the previous chapter we have seen that infinitesimal deformations of Ya in degree — R are closely related to Minkowski summands of the crosscut Q(R) = a n [R = 1]. Now, we would like to describe how certain Minkowski decompositions of Q(R) give rise to genuine deformations living over smooth parameter spaces. We begin with negative degrees; let R e Z and pr^ 9 : N -> 2Zq induce regular functions g : X& —> (T and (^\ . . . , ^9) : X^ —>• (T9, respectively. If restricted on Y^, they coincide. T h e o r e m : ([Al 1]) The resulting map (g1 - g,..., and has Ya ^-> X& as special fiber.

gq - g) : X& -* (Eq is flat

(4.2) Having both a description of Ty and a recipe to construct certain deformations, one should ask for the Kodaira-Spencer map. Here it comes: Let R e a v D M and Q(R) = Qo + • • + Qq be a decomposition satisfying (i) and (ii) mentioned above. Denote by (dl)v the vertex of Qv induced from a* E Q(R), i.e. a* = (a% + . . . + (a%.

Singularities Arising from Lattice Polytopes

17

T h e o r e m : ([Al 4]) The Kodaira-Spencer map of the corresponding toric deformation Xp —+ dq is Q : l y ( - i J ) C VW(R) 0 WW(R)/(F • (1,1) sending ev onto the pair [Qv, sv] e V (R) ® W (R) (v = l,...,q) v 1

Jo | 1

with

belongs to the lattice N if the vertex (al)v of Qv belon if (dl)v is not a lattice point.

R e m a r k : Setting e° := - ( e 1 + • • • + e«), we obtain g(e°) = [Qo, s°] with 5° denned similar to s_v in the previous theorem. (4.3) If R £ crv fl M, then the previous construction fails. Moreover, in [Al 1] it was shown that deformations such that • the total space together with the embedding of the special fiber still belong to the toric category and • the Kodaira-Spencer map points injectively into some homogeneous

piece T^(-R) C T^ are of the above type, including R € crv D M. Nevertheless, in [Al 4] it has been shown that Minkowski decompositions of Q(R) fulfilling (i) and (ii) yield ^-parameter deformations in the general case, too. They are obtained from the previous construction for the adapted cone a' := a fi [R > 0] via blowing down. This causes the total space to lose its toric structure. (4.4)

Examples:

(4.4.1) Cones over Del Pezzo surfaces: Del Pezzo surfaces arising from IP2 by blowing up not more than three points are toric. The affine cones over them are Gorenstein. Hence they fit in both patterns (2.6) and (2.7), and the polytopes P C MJR and Q C NJR are mutually dual. Let Q be the hexagon from (2.2); it provides the cone over the Del Pezzo surface of degree 6. Identifying the hyperplanes [R = 0] and [R = 1], the two different Minkowski decompositions shown in (2.2) induce families with one and two parameters, respectively. The total spaces are the cones over IP1 x F 1 x IP1 and IP2 x IP2. (4.4.2) Pinkham's example: The two-dimensional cone a := ((1,0), (—1,4)) yields the cone over the rational normal curve of degree four; see Example (2.4)(3). Ty consists of two one-dimensional parts in the degrees —[1,1] and -[1,3], respectively, and of a two-dimensional part in degree —[1,2]. Considering the latter, we obtain the real interval [-1/2,1/2] C M1 as crosscut

18

Altmann

Q(R). According to our rules (i) and (ii), it allows two different Minkowski decompositions [-1/2,1/2] = [-1/2,0] + [0,1/2] = {1/2} + [-1,0] inside M1 = [R = 0] = [R = 1]. The associated one-parameter deformations yield two lines in the versal base space of Ya; they belong to the two different components of dimension one and three, respectively. (4.4.3) The An_\-singularity from (1.4)(2) is given by the cone a = ((1,0), (1,n)). The vector space TY is spread in the degrees — [fc, 0] with 2 < k < n, and for each k the previous method yields the perturbation (zn — xy) + ) On the other hand, it is even possible to get all these families at one blow: despite TY vanishing in this degree, we consider R* — [1,0]. The cross cut Q(R*) is an integer interval of length n. It splits into a Minkowski sum of n unit intervals causing an (n — l)-parameter family via the previous construction. However, since TY(—R*) — 0, the Kodaira-Spencer map vanishes. The reason is that we have not obtained the versal deformation of An-i, but an n!-fold cover of it.

5

Versal deformations

(5.1) The two-dimensional toric singularities, i.e. the two-dimensional cyclic quotient singularities from Example (2.4)(3), have been investigated in [Ri], [KoSh], [Arn], [Ch], and [St 1], Using Kollar/Shepherd-Barron's fundamental result about the one-to-one correspondence between so-called Presolutions and components of the versal base space, Christophersen and Stevens have been able to give a detailed description of these components: For a given two-dimensional cone a let a 2 , . . . , aw_i be the positive integers introduced in (3.4). Since the An_i-singularities behave a little bit differently, assume w > 4 for simplicity. We define K(a) as the finite set K(a)

:=

{(fc2,...,fc«/_i) € %w~2\ 0 < kv < av, and the continued fraction k2 — -^— is well defined and yields 0 } .

Theorem: ([Ch], [St 1]) The set K(a) parametrizes the irreducible components of the reduced, versal base space of Ya. Moreover, the component assigned to k_ is smooth and has dimension Y0 represent Minkowski summands Qt of M>0 • Q, we may define the tautological cone as C(Q) := {(a, t)\te

C(Q); a e Qt} C M2 x V(Q) C M2 x Mm.

C(Q) is, like C(Q), a rational, polyhedral cone. By definition, there is a natural projection TT : C(Q) —» C(Q) yielding a := cone(Q) as the pull back of M>o • 1 Q C(Q). We obtain the following fiber product diagram of rational polyhedral cones:

C(Q)

C(Q) -

M >o

20

Altmann

with A denoting the diagonal embedding. Applying the functor of building toric varieties from cones yields affine varieties ^c(Q) an( ^ &c(Q) fitting into the fiber product diagram

with closed embeddings in the vertical directions. The map SC(Q) •—> (Pm is finite, and, for the sake of simplicity, we will pretend in the following that it is a closed embedding, too. Denote by M C (Um j(£ • 1 the largest closed subscheme such that M := £~1(M) C (Fm is also contained in Sc(Q)Theorem (3.8) says that both M and T^ sit in the same space Wm/(T • L Theorem: ([Al 3]) Restricting the map Xd{Q) -> (Fm/(F • 1 onto M C @rn/(F -I, the outer frame of the above commutative diagram yields the versal deformation ofYa. (5.4) What does M look like? To answer this question, we define for each integer k > 1 the (vector-valued) polynomial

generating an ideal

Since YliL\d% = 0, this ideal may alternatively be generated by £ l 9 . . . , tm — ti) with k > 1, i.e. J originally comes from the subring QJ\bi tj | 1 < i,j < m].

Theorem: ([Al 3]) (1) M, and M, are the affine schemes associated to the ideals J C QJ[t\ and J O Q[ti - tj) C (S[ti - tj), respectively. (2) If Q is contained in two different strips defined by pairs of parallel lines of lattice-distance < k0 each, then the polynomials g^ with k > k0 are not necessary for generating J or J P\(F[ti — tj). Since gi(t) are the linear forms defining V(Q) C Mm, we see that M is not only contained in (Fm, but also in V^(Q). Example: Let Q be the hexagon from (2.2). Starting with dl := (0,0)(l, 0), the anticlockwise oriented edges equal ^ = (1,0); d2 = (l,l);

d3=(0,l);

Singularities Arising from Lattice Polytopes

21

Obviously, Q is contained in at least three different strips of thickness two. Hence, J is generated in degree < 2. We obtain

(5.5) Finally, we would like to mention the structure of the underlying reduced spaces of M and M. Let Q = Ro H f- Rq be a decomposition of Q into a Minkowski sum of q + 1 lattice polytopes. Then, denoting with [Ry] G C(Q) the point representing the summand i£y, the m-tuples [Ro],..., [i?g] G iRm have only 0 and 1 as entries and sum up to ( 1 , . . . , 1). In particular, the (q + l)-plane (V • [Ro] + . . . + (F • [J?J C (Tm (or its g-dimensional image via £) is contained in M (in M, respectively). It is not difficult to see that there are no other points, i.e. Aired and M,red equal the union of those flats corresponding to maximal Minkowski decompositions of Q into lattice polytopes. Example: Considering the hexagon again, the linear equations of M allow the substitution t := ti, Si := ti —13, s2 :— ^4 ~hi a n d s 3 := ti —1 4 . The two quadratic equations transform into si s3 = s2 s3 = 0. Since M = l~l(M), the equations do not contain t\ it is still a variable for M, but not for M. We see that M is the union of a line and a two-plane. The corresponding one- and two-parameter families are exactly those obtained in (4.4.1) corresponding to the Minkowski decompositions drawn in (2.2). (5.6) In the same manner as we did with the hexagon, we may investigate the lattice rectangle Q = conv{(0,0); (2,1); (2,2); (1,2)} corresponding to the cone over the Del Pezzo surface of degree eight. Here the versal base space equals the fat point

References [Al 1] Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tohoku Math. J. 47 (1995), 151-184. [Al 2] Altmann, K.: Obstructions in the deformation theory of toric singularities. E-print alg-geom/9405008; to appear in J. Pure Appl. Algebra. [Al 3] Altmann, K.: The versal deformation of an isolated, toric Gorenstein singularity. E-print alg-geom/9403004; to appear in Invent. Math. [Al 4] Altmann, K.: One-parameter families containing three-dimensional toric Gorenstein singularities. E-print alg-geom/9609006.

22

Altmann

[Al 5] Altmann, K.: P-Resolutions of cyclic quotients from the toric viewpoint. E-print alg-geom/9602018. [A1S1] Altmann, K.; Sletsj0e, A.B.: Andre-Quillen cohomology of monoid algebras. E-print alg-geom/9611014. [Am]

Arndt, J.: Verselle Deformationen zyklischer Quotientensingularitaten. Dissertation, Universitat Hamburg, 1988.

[Art]

Artin, Michael: Lectures on deformations of singularities. Bombay: Tata Institute of Fundamental Research, 1976.

[Bat]

Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geometry 3 (1994), 493-535.

[Ch]

Christophersen, J.A.: On the components and discriminant of the versal base space of cyclic quotient singularities. In: Singularity Theory and its Applications, Warwick 1989, Part I: Geometric Aspects of Singularities, pp. 81-92, Springer-Verlag Berlin Heidelberg, 1991 (LNM 1462).

[Da]

Danilov, V.I.: The geometry of toric varieties. Russian Math. Surveys 33/2 (1978), 97-154.

[Fu]

Fulton, W.: Introduction to toric varieties. Annals of Mathematics Studies 131. Princeton, New Jersey, 1993.

[Ish]

Ishida, M.-N.: Torus embeddings and dualizing complexes. Tohoku Math. Journ. 32, 111-146 (1980).

[Ke]

Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings I. Lecture Notes in Mathematics 339, Springer-Verlag, BerlinHeidelberg-New York, 1973.

[Kod]

Kodaira, K.: Complex manifolds and deformation of complex structures. Grundlehren der mathematischen Wissenschaften 283. Springer-Verlag 1986.

[KoSh] Kollar, J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91, 299-338 (1988). [Lo]

Loday, J.-L.: Cyclic homology. Grundlehren der mathematischen Wissenschaften 301, Springer-Verlag 1992.

[Od]

Oda, T.: Convex bodies and algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3/15), Springer-Verlag 1988.

[Pa]

Palamodov, V.P.: Deformations of complex spaces. Russian Math. Surveys 31 (1976), 129-197.

[Pi]

Pinkham, H.: Deformations of algebraic varieties with Gm-action. Asterisque 20 (1974), 1-131.

[Ri]

Riemenschneider, O.: Deformationen von Quotientensingularitaten (nach zyklischen Gruppen). Math. Ann. 209 (1974), 211-248.

Singularities Arising from Lattice Polytopes

23

[SI]

Sletsj0e, A.B.: Cohomology of monoid algebras. J. of Algebra 161 No. 1 (1993), 102-128.

[St 1]

Stevens, J.: On the versal deformation of cyclic quotient singularities. In: Singularity Theory and its Applications, Warwick 1989, Part I: Geometric Aspects of Singularities, pp. 302-319, Springer-Verlag Berlin Heidelberg, 1991 (LNM 1462).

[St 2]

Stevens, J.: Deformations of singularities. Habilitationsschrift, Universitat Hamburg, 1995.

[Sm]

Smilansky, Z.: Decomposability of polytopes and polyhedra. Geometriae Dedicata. 24 (1987), 29-49.

[Zi]

Ziegler, G.M.: Lectures on polytopes. Graduate Texts in Mathematics 152 (1995), Springer-Verlag.

Institut fur reine Mathematik Humboldt-Universitat zu Berlin Ziegelstr. ISA D-10099 Berlin Germany e-mail: [email protected]. de

Critical Points of Affine Multiforms on the Complements of Arrangements James Damon*

To Terry Wall on his sixtieth birthday.

Introduction We consider a n affine hyperplane arrangement 4 c P which is a finite union of hyperplanes Hi denned by affine linear forms ^ = 0. In his work on the classical asymptotics of the KZ equation [Va], Varchenko conjectures

For generic complex values A«, the function (multiform) defined as a product of powers of affine linear forms f = X\t%i has only nondegenerate critical points and the number of critical points equals the Euler characteristic of the complement of A. He proves this result when A is the complexification of a real arrangement, extending an earlier result of Aomoto [Ao]. In [OT1], Orlik and Terao give a proof of this conjecture. They do so by relating both of these numbers to the Euler characteristic of a certain complex of logarithmic differential forms. In this paper, we prove several results (Theorems 1 and 2) which also yield this conjecture as a corollary. These theorems give a topological proof of the conjecture and illuminate the direct relation between the Euler characteristic and the number of critical points. More generally, these results extend Varchenko's conjecture to nonlinear arrangements A on smooth complete intersections X C (Cn so that / is now a product of higher degree polynomials restricted to X. We express the relative Euler characteristic as the sum of Milnor numbers of the critical points of f\ X\A (allowing degenerate critical points). Moreover, we show this relative Euler characteristic equals a "relative singular Milnor number" of a "singular Milnor fibration"on the complete intersection X. These results are valid when the A^ take positive integer values. However, by applying an observation of Varchenko, and extending a lemma of OrlikTerao, we still obtain the full conjecture. Moreover, in the case A is an "almost free arrangement", we can explicitly give a formula for the number of critical points in terms of the exponents of A and the degrees of the defining equations of X. * Partially supported by a grant from the National Science Foundation

26

Damon

The basis for these results is the point of view used in [Dl] that a central arrangement (in which all of the hyperplanes Hi pass through 0) defines a (highly) nonisolated singularity at 0, and any affine arrangement is the "singular Milnor fiber" of a generic hyperplane section of such a central arrangement. Then, local singular properties of the central arrangement translate into local properties of the "singular Milnor fiber" of the section, which in turn yield global properties of the affine arrangement. This applies as well to nonlinear arrangements on complete intersections. In particular, a weak version of Varchenko's conjecture for the sum of the Milnor numbers in the case where all A^ = 1 already follows from the proof of proposition 5.2 of [Dl] using the methods of [DM, 4], To obtain the full strength of Varchenko's conjecture we allow a degeneracy in the section (which occurs when there are parallel flats in the affine arrangement), and then the singular Milnor fiber arises via translation of the section. In §1, we recall the basic properties of algebraic transversality to singular varieties and introduce a version for multiforms. We also recall relevant properties of singular Milnor fibers and numbers. In §2, we state the local version of the main theorem and deduce the global results, including Varchenko's conjecture, giving several consequences and examples in §3. In §4, we establish the basic local result that relates the Euler characteristic of a nonisolated singularity Z defined by / on the Milnor fiber of a complete intersection V with the sum of the Milnor numbers of / restricted to the Milnor fiber of V. This extends the line of argument of Siersma [Si] and Looijenga [Lo], which in turn, extend the original argument of Le [Le2], Then, in §5 we deduce the various results and their corollaries, including specific formulas for the number of critical points. This author is especially grateful to both Alexander Varchenko and Hiroaki Terao for valuable conversations regarding these questions.

1

Transversality and Singular Milnor Fibers

We consider in this section the local singularity theory involving algebraic transversality conditions needed to define and determine the singular Milnor fibrations and numbers. We begin by recalling from [DM], [Dl], and [D3] how transversality of singular varieties can be defined and how we can define singular Milnor fibers. Then we extend the transversality conditions to multiforms. Because later we shall specifically relate local and global situations, we will find it convenient to work with local constructions in (Cn+1 (and global constructions in (En). Algebraic and Geometric Transversality : Let (V, 0) C (Dn+1 be a germ of an analytic variety, and let I(V) denote the ideal of holomorphic germs vanishing on V. If # n+ i denotes the module of germs of holomorphic vector fields on (Cn+1 at 0, then, following Saito [Sa]

Critical Points of Affine Multiforms

27

we define Derlog(F) = {C € 6n+l : C p , 0, then Derlog(G) = {C € 0n+1 : (/), or just JJL(V) if the map germ / is clear from the context. In the case i denotes the inclusion of a sufficiently generic ^-dimensional plane, then fiv(i) is independent of i and is denoted by /Xfc(V), the k-th higher multiplicity. These multiplicities extend the higher multiplicities introduced by Teissier [Te] for hypersurfaces, and are closely related to those introduced by Le-Teissier [Le-T] for general singularities. The singular Milnor numbers and higher multiplicities can be explicitly computed in the case that V is a free divisor (which means, see [Sa], that Derlog(V) is a free 0 w -module, necessarily of rank pf), then V is called an almost free divisor and fiv'(f) c a n be computed as vv{f), the length of a determinantal module, see [DM, thms 5,6] and [Dl, §4]. This is further extended in [Dl,§7,8] to almost free complete intersections, which are the transverse intersections off 0 of almost free divisors.

Critical Points of Affine Multiforms

29

To see how singular Milnor fibers and numbers are relevant for Varchenko's conjecture, consider V an ICIS defined by V = G'~1(0) for homogeneous G1 : (C n+1 ,0 -> 07,0 with V transverse to (Cn = 0 sufficiently small, Vte z : V n ( C x { t } ) n Be is the Milnor fiber of V; and moreover, Vt,€ the "singular Milnor fiber" of V Pi A

Figure 1.

As explained above, Vi,£ f) A is homotopy equivalent to a bouquet of spheres of real dimension n — p — 1. Furthermore, it follows from [Dl] and [DM, §4] that the number of such spheres, which is the "singular Milnor number"of V C\ A, equals the sum of the Milnor numbers of Q\ Vt, e \A To extend these results to arbitrary affine arrangements (which may have parallel flats) we must allow a degeneracy in the condition on the section of V. In addition, we show that the preceding results remain valid globally on X = VC\(\ where 1 = (1,...,1)). Then, algebraic transversality of V to A off 0 actually implies a stronger result. For z G V\A, dQ(z)\TzV ^ 0 (see e.g. [DM,§3] or [Dl,lemma 2.10]). This is equivalent to (JO\TZV ^ 0 for any z E V\A. Moreover, ior z eV D A, HV is homogeneous then the Euler vector field e is tangent to V, and u(e) = (1 /Q)dQ(e) — 5. Thus, V is transverse to Q off 0 in the sense that UJ\TZV ^ 0 in a (punctured)

30

Damon

neighborhood of 0. This form of transversality has a natural extension to F (see also [OT1,§3]). For an ICIS V defined by G and a (nonlinear) arrangement A, we consider Derlog(*4., G) defined to be Derlog(.A) fiDerlog(G). If we sheafify this module we obtain an object, also denoted by Derlog(>A, G), which is still coherent. Definition 1.5 We say that V is transverse to the multiform F = off 0 if for each z G V\{0} in a punctured neighborhood of 0, there is ( G Derlog(./t, G) such that UJX{() ^ 0 at £. By Saito [Sa,§l], since C G Derlog(>4), it follows the map

CJA(C)

£ O^n+i 0-

We

define

evF : Derlog(^G) - Ov,o ^ 0 so that for |s| and \t\ sufficiently small, Vt Pi F~l(s) ( = X O g~l{t) D h~l{s)) is stratified transverse to S€, and that F\ Vt\A (= h\ X D ^~ 1 (t)\/i~ 1 (0) has only a finite number of singular points, all of which approach 0 as t approaches 0. First, by assumption, V = X Pi (Cn+1 is (algebraically) transverse to both A — ^~1(A/) and F in a punctured neighborhood of 0. We choose e0 > 0 so that Beo lies in this neighborhood. Step 1: Establishing the property of the critical points of F\ Vt\A As V is also algebraically transverse to F, for each z € V\{0} in a punctured neighborhood of 0, there is ( 6 Derlog(fcA, G) such that $*LU\(() ^ 0. As Derlog(^4, G) is a finitely generated O^n+i -module, we can find a finite number of Q € Derlog(fc4, G) so that {**O;A(C»)> ^ = 1,.. •, A 0 such that if |t| < 7, and 0 < ex < e < e0, then E Pi g~l{t) n 5 e = 0. Thus, there are only a finite number of singular points of h\ Vt\A (= E C\ g'1^)). These lie in B6l and they approach 0 as t approaches 0. Step 2: Stratified Transversality to Se We must show that there is a Milnor radius e > 0 so that for all \s\ and |*| sufficiently small, Vt D F~1(s) (= X Pi ^ -1 (^) ^ ^~ 1 ( 5 )) ^s stratified transverse to Se. By the parametrized transversality theorem and the algebraic criterion for transversality, it follows that (by possibly shrinking e0) we may assume that for 0 < e < e0, the following are (stratified) transverse to S£: V = A'Pi(/~1(0),

vnA= xng-\o) nft-^o), and A. Now V nAn S€ (or more generally V Pi A Pi (BCO\J3C1)) is compact, and the condition for algebraic transversality to closed analytic subsets is an open condition. Hence, it is enough to show that z e V D A P) Se has an

42

Damon

open neighborhood Uz such that X D g~1(t) D h"1(s) is stratified transverse to each Se at all points of Uz. We know that X, (Cn+1 = ^ ( O ) , and A x (C = / r ^ O ) are in algebraic general position off 0 in Beo. Moreover, X is an ICIS defined by Gi(z, t) — G(z) — (3{t), so it is h-holonomic, as is (Cn+1 for zn+2We also claim that A is h-holonomic for F. Given this, it follows that the geometric tangent spaces to X, (Cn+1, and A x (C (off the axis {0} x (C) are given by Tiog(Gi)(z), Tz(Cn+1, and Tlog(F)(z). These spaces are in general position and their intersection contains a vector transverse to TzSe (e = ||^||). By the coherence of the associated sheaves, it follows that for each z' € Uz, a neighborhood of z, these tangent spaces at z' remain in general position and there is still a vector in the intersection of these tangent spaces at zf which is still transverse to Tz>Se> (ef = ||z'||). Thus, there is 7 > 0 so that for |s|,|t| 0 so that if |t| < 7 then Vt = {z G Be : G^(z) = 0, z = 1 , . . . ,p, and GQ(z) = t} is the Milnor fiber of G, and Vt fl ^4 is the singular Milnor fiber of the complete intersection V Pi A (V = Vn for sufficiently small t. Then, by the (C*-action, the critical points of F\ X\A will be in a compact set G. By proposition 2.5 for \t\ sufficiently small, there is a homeomorphism il) from X to the Milnor fiber Vt = VD((Cn x {t})nBe which sends A = XdA to Vt H A and is the restriction of the (C*-action (and so holomorphic) on G. Now we can see that proposition 2.5 together with the local version of Theorem 1 implies the global version. Since ij)\C is a biholomorphic diffeo-

Critical Points of Affine Multiforms

43

morphism, it is sufficient to prove (5.1)

(the sum of the Milnor numbers of F\ VtnA)

= \x(Vt, VtC\A)\

and that this relative Euler characteristic is given by

(5.2)

\x(vu vtnA)\ = dim(C Hn~p(vt) + dim(C Hn-p-\vt

n A).

However, Vt is the Milnor fiber of V and Vt C\ A is the singular Milnor fiber of V D A. Hence, Hn-p(Vt)

= fi{V)

and

dim^ Hn-p-l{Vt

n A) = n{V D A).

Thus, both (5.1) and (5.2) will follow provided we may apply the local version of Theorem 1 to F, V, and A. However, the global conditions exactly ensure that this is so. • Proof of Theorem 2. By proposition 2.10, there is a Zariski open subset A(4, X) C (Cfc such that for A = (Ai,..., Xk) e A(A, X) and / = J] i\\ then f\ X\A has only nondegenerate critical points and the number is constant. Thus, to prove the theorem it is enough to determine the number for some

\eA(A,X). We use an observation of Varchenko that if IN denotes the positive integers then lNfc is not contained in a proper Zariski closed subset of (Cfc. (This can be seen by a naive argument involving induction on A:. If k = 1 then the only proper Zariski closed subsets of (C are finite. By induction, if lNfc C B C (Cfc for a Zariski closed B, then for X1 Ro and \t\ < 1,

(5.3)

(^,t,(^,tnA))-(^,A).

Second, the critical set of Ft for \t\ < 1 is compact (because there are only a finite number of branches to the curve of critical points as in theorem 1). Thus, we can increase J^o and find an s0 > 0 such that the critical points of Ft for \t\ < 1 lie in BRo, and the critical values lie in BSQ. Third, we claim Lemma 5.4 There is an si > s0, 0 < t0 < 1 arbitrarily small, and an Ri ^ Ro so that for any t with t0 < \t\ < 1 1. F~1(s)n ft" is transverse to SR,t (in (C^J for all \s\ < si and R>R\; 2. the differentials d\Ft\2 and drt do not point in opposite directions for anyzeBR\F-\Bai).

Critical Points of Affine Multiforms

45

Let si, t0 and Ri denote the values given by Lemma 5.4. For a specific t\ with tQ < \ti\ < 1, we can argue as in §4. First, by (2) of Lemma 5.4 and the argument of lemma 4.2, F^l(BSl) D BRlM is a strong deformation retract of BRlM. Furthermore, by the proof of Lemma 4.1, F^1^) nBRlM = AnBRlytl is n — 2-connected and F^1(BS1) D BRlytl is obtained from Ft71(0) D BRutl by attaching n-cells, with the number of cells equal to the number of critical points of Ftl on (C^\v4tj (all of which have Milnor number 1). Hence, as in the proof of theorem 2 and using (5.3),

= number of critical points of / on R, There are e, 6 > 0 so that F0~1(s) is (stratified) transverse to 5e?0 for all \s\ < 6. Using the (C*-action, we obtain for any si > 0 an Ri > Ro such that F0~1(s) is transverse to 5#,o for all \s\ < si and all R > i?i. We also suppose Ri is large enough using s2 in (1) of Lemma 5.5 and that H^'H > Ri. Thus, if A is transverse to (Cn at (z1,0), then there is a neighborhood U of (zf, 0) in (Cn+1 so that if (z,t) G U with F(z,t) = s and \\z\\ =R> R1 then F ' ^ s ) is transverse to 5#,0 at (z,t). If A is not transverse to (Cn at (z',0), then such a neighborhood exists by (1) of Lemma 5.5. Again, this contradicts (z',0) being in the closure of W2Thus, W2 H (C^ is bounded for each 0 < \t\ < 1, and hence for all such t by the (D*-action. Thus, given 0 < \t\ < 1, there is an R\ > RQ such that W2 D (C^ C ^jRl5t- Hence, (1) of Lemma 5.4 follows since to < \t\ implies s2\t\N > si. • Proo/ 0/ Lemma 5.6. Suppose for some 0 < \t'\ < 1, Af C\ (C^ is unbounded. Hence, for each A; there is a (wk,t') G JV with 5^ = \\wk\\ ^ fe. Then, by the H + -action, (zk,tk) = ( ^ • wfc, ^ • t') G Af and ||^|| = i? for all jfe. Also, tk —> 0. By the compactness of SR, there is a convergent sequence of (zk, h) € M n 5^ tfc with limit (2/, 0) G (closure of J\f) D 5^,0• Proof of Lemma 5.5. Let K be the smallest flat of A containing z' and contained in (Cn. We let I = {i : L^K) = 0} and / ' = { 1 , . . . ,n}\J. We may choose coordinates so that the line through z' is the 2:fc-axis and the Li for i G / only depend on ( ^ + i , . . . , z n + 1 ). By the minimality of if, all of the L{ for i E I' depend on zk, which we shall single out and denote by y. Furthermore, we may assume that y is chosen to equal 1 for z'. We let ui = Z ^ e / ^ T f a n d ^2 = J2ier ^i^-f s o t n a t ^A = ^1 + ^2Likewise, we let Fi = 1 1 ^ / ^ * ^ " anc ^ ^ 2 = I l i e / ' ^ T 1 ' s o w e m a y ^ a c t o r F — F\ - F2 and so F ^ = rfF^. Since Fi and uj\ don't involve y, both are constant along lines parallel to the y-axis.

Critical Points of Affine Multiforms

47

If y = u + iv, then ^ = -^ + ^ . We see

dFl)

=

|Fl|

( | F 2 |) =

dy

dy

=\F\2-g(y,z)

(5.7) where g(y, z) = $ ^ € / , ^ f

| F l | |F 2 |^ 2 (|)

dy for

L

% = aiV + 40*0 w i t n a* ^ 0 for z € /'. Then,

and (5.9) *€/'

Thus, in a neighborhood of z' = (1,0), Re(g(y, z) > 0 so by (5.8) and (5.9), d|F t | 2 and drt are both positive on ^ and so do not point in opposite directions. For (1), we may view Fx as a function on (Cn+1~fc. By (1.8), its restriction to slices z n + i = t is singular off F1~1(0) on a finite number of lines Ci in the complement of i ? f 1 (0) where the coefficients of dzk+i,..., dzn in uj\ vanish. On such a line £«, i C2\y\N2 for \y\ > R^> Ro). Here A^2 = deg(F2). Furthermore, we may assume on Z with \y\ > Ri, that \F\ is increasing with increasing \y\ (by (5.8) and (5.9)) and that lines parallel to the y-axis are transverse to SRJ- Let Z>R, resp. ZR, denote the subset of Z with \y\ > i?, resp. \y\ = R. Then, in Z>Rl and on Mi = (

From (5.10), if Cx = min{|q|}, then |F| > sx\t\™ on M{ D ZR provided (5.11)

^ ^ :

As |2; n+1 | = \t\ is bounded on Z, given s1 > 0, we may increase Ri so that (5.11) holds on Z>Rl. Let C = {z E (Cn+1 : for t = zn+1, \Ft\ < s^t}1*}. ChoosingZof theform.B'xCCxB", thenC' = Cn{B'x{y : \y\ = Ri}xdB") is compact and misses all the M{. Then \\coi\\ is nonvanishing on C and —>• oo as z —> 0 along 1R+ orbits of (Cn+1~fc on which LUI is not zero. Since \F\ increases with \y\ on Z>Rl, we conclude H^xH > C > 0 on C D Z>Rl. As CJX contains j - in its kernel, ker^x) will be transverse to all SR^ on C fl Z>RX. Since ||o;2|| —* 0 uniformly as |y| —> oo on Z>^17 by further increasing i^i, ker(cji + CJ2) will remain transverse to S ^ on CC\Z>R1, completing the proof. D

Damon

48

6

Proofs of Propositions 2.5 and 2.10

Proof of Proposition 2.5. Step 1: Constructing the Homeomorphism There are two steps for constructing the homeomorphism. By the (C*-action t - ( z i , . . . , zn+i) — i >• (t • zi,..., t • 2 n +i), w e obtain a biholomorphic diffeomorphism ^1 between I = V n ( f x { l } ) with V Pi ((Cn x {£}) such that ^ i ( l n i ) ( = -0i(ATl>l)) = VnAD{(Enx{t}). Thus, ^1 has all of the required properties. Second, we shall construct for \t\ sufficiently small, a homeomorphism i\)2 from VH((Cn x {£}) to the Milnor fiber Vt = VH((Cn x {t}nBe) which sends ,4Pi V D ((Cn x {t}) to AnVt and is the identity on Be, for 0 < e' < e, with e' arbitrarily close to e. Furthermore, given a compact subset C oi X, we may choose |t| small enough so that t • C C Bet. Then, ^>2 ° ^i will be the desired homeomorphism (see fig. 6).

Figure 6. To construct ^ 2 we shall show that there is a 6 > 0 so that if \t\ < 6, then ] / t - y n ( f x {t}) is transverse to An ((Cn x {*}), and both Vt and Vt 0 A are (stratified) transverse to Sr for r > 6. Hence, by the openness of transversality (to closed Whitney stratified sets) they are transverse for r > e;, where e' < e but sufficiently close. Then, we can stratify Vt by Vt\A and Vt fl W« where W{ ranges over the strata of a Whitney stratification of A. By the transversality condition, ||z||2|Vt is proper and has no critical points on any strata for | > e1. Thus, by stratification theory, there is a homeomorphism of pairs (v t \fl e i, (vt n A)\Be>) ~ (v t \s e i, y t n ^in 5,0 x [