Table Of ContentLecture Notes in Mathematics 5161
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David B. Massey
~L Cycles and
Hypersurface Singularities
r e g n~ i r p S
Author
David B. Massey
Department of Mathematics
Northeastern University
Boston, MA 02115, USA
Cataloging-in-Publication data applied for
Library oF Congress Cataloging-In-Publication Data
Massey, David B., 1959-
L~ cvcles and hypersur~ace singularities / David B. Massey.
p. cm. -- (Lecture notes In mathematics ; 1615)
Includes bibliographical references and Index.
ISBN 3-540-60395-6 (soFtcover)
1. StratiFied sets. 2. HypersurFaces. 3. Functions oF several
complex variables. 4. Singularities (Mathematics) I. Title.
I. Series: Lecture notes In mathematics (Sprlnger-Verag) : 1615.
QA3.L28 no. 1615
QA614.42
510 s--dc20
516.3'6 95-39592
CIP
Mathematics Su~ect Classification (1991): 32C18, 32B10, 32B15, 32C25
ISBN 3-540-60395-6 Springer-Verlag Berlin Heidelberg New York
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(cid:14)9 Springer-Verlag Berlin Heidelberg 1995
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Typesetting: Camera-ready TEX output by the author
SPIN: 10517520 46/3142-543210 - Printed on acid-free paper
For my father, Robert Brian Massey, on tile occasion of his 60th birthday.
PREFACE
In 1976, the result of L~ and Ramanujam L-R appeared. This result -
that. the constancy of the Milnor number in a one-parameter family of isolated
hypersurface singularities implies topological constancy - is an outstandingly
beautiful blend of algebraic geometry and topology.
In 1983, my dissertation advisor, Bill Pardon, gave me three problems to
choose from for my thesis: the first was to develop stratified Smith theory, the
second I cannot seem to remember, and the third was to generalize the result of
L~ and Ramanujam to families of hypersurfaces with one-dimensional singular
sets. The third problem was both my favorite and Bill's, and so the topic of my
dissertation was decided.
)2saM
While the results of my dissertation (see Masl and were mod-
erately interesting, there were several unsatisfying aspects of my work. The
numerical data that I required to be constant in a family seemed too strong and
did not seem to generalize easily to higher-dimensional singularities. Moreover,
in Val, Vannier gave a number of stronger results (using completely different
techniques).
In 1987, during my second year in a visiting position at the University
of Notre Dame, t "fixed" these problems - I developed the L~ cycles and L6
numbers of a hypersurface singularity of arbitrary dimension. The L~ numbers
are a generalization of the Milnor number and, by using them, I have been
able to generalize a number of results on isolated hypersurface singularities to
non-isolated hypersurface singularities. Some of these results appear in Mas2,
Mas3, Mas5, and M-S. However, I have now compiled a number of new
results and some interesting examples.
These notes summarize essentially all of the results that I know of concern-
ing the L~ numbers. The types of problems that I can attack with L~ cycles
and L~ numbers are problems concerning the Milnor fibration and problems
concerning how limiting tangent planes to level hypersurfaces approach a given
hypersurface. Some of the results here have appeared elsewhere, but I include the
proofs for completeness and because there have been some slight, but important,
improvements in many of the statements.
However, many of the results of this work have never appeared; new re-
sults include an application of a lemma of Lazarsfeld in Proposition 1.27, an
application of Siersma's result on isolated line singularities in Corollary 3.3, a
Pliicker formula for the L~ numbers in Corollary 4.6, the lexigraphic upper-
semicontinuity of the tuple of ~L numbers given in Corollary 4.14, the results
on hyperplane arrangements given in Chapter 5, the multi-parameter and non-
parametrized versions of results on Thom's a condition given in Chapter 6, all
the results on aligned singularities given in Chapter 7, and the new characteri-
zations of the L~ cycles given in Chapter 10.
In addition to giving these new results, a major purpose of these notes is
vIIf
to provide a large number of interesting examples. The examples include one in
which the L6 cycles move with the coordinate choice (2.3), the coordinate hyper-
planes in affine space (2.6), non-reduced plane curves (2.7), quasi-homogeneous
singularities (4.7 and 4.8), the specific example of the swallowtail singularity (4.9
and 4.10), and central hyperplane arrangements (5.2).
I believe a word or two is in order about how I chose the terminology Lf
cycles and L~ numbers. I originally was calling these objects invarianl cycles and
invariant numbers, but then I found that they were not nearly so "invariant" as
I first believed. At that point, I sat there not writing another word for fifteen
minutes because I could not. think of what else to call these mathematical devices.
Finally, I decided to name them after the man whose work had influenced me
far more than anyone else's had influenced me up to that point in nay career.
When I first showed L6 Dfing Tr~ng my work on the L6 cycles and L6 numbers,
it was with a great deal of trepidation. I informed Tr~ng that if he did not like
the work, I would certainly change the names of these cycles and numbers; I am
pleased to say that I have not had to alter my terminology.
As this work is a summary of my study of L6 cycles and L6 numbers from
1987 to the present, there are many people that I should thank:
Ambar Chowdhury for his encouragement and many helpful "conversations" at
The Commons; Mark Goresky, L6 Dfing Trs and Terry Gaffney for bringing
me to Northeastern University and for many, many invaluable discussions; a
special thanks to Terry for Theorem 10.17; Dirk Siersma for the bounds which
appear in the general L6-Iomdine formulas; Dan Cohen for his help with hy-
perplane arrangements and for leading me to the chain complex which appears
in Theorem 10.9; Dominic Welsh and Giinter Ziegler for pointing out the iden-
tity between L~ numbers and the M6bius function given in Theorem 5.6; Steve
Kleiman for conversations on the relationship between the +L cycles and the
work of Vogel and van Gastel, especially how it implies the Pliicker formula of
Corollary 4.6; Bob MacPherson for a number of conversations about the derived
category and perverse sheaves; Roberto Callejas-Bedregal for one long, bewil-
dering discussion about the L6 cycles, which led to Theorem 10.14; Alex Suciu
for the homology groups of the Milnor fibre of the swallowtail in Remark 4.10,
and for nmch encouragement over many cups of coffee; and Robert Gassier, Mike
Green, Alex Sueiu and, of course, L6 D~ng Tr~ng for making many helpful sug-
gestions and for spotting many typographical errors in the preliminary versions
of this manuscript.
Also, there are some institutions that I must thank:
the Mathematics departments of The University of Notre Dame and Northeast-
ern University, for having me as a visiting professor and treating me so well;
XJ
the National Science Foundation, for three years as a post-doctoral research fel-
low and then for summer support.; and The University of Paris VII, for two
one-month visits during which much was accomplished.
David B. Massey
Boston, MA
April 24, 1995
TABLE OF CONTENTS
Preface .................................................................... v
Table of Contents ........................................................ viii
Introduction ............................................................... 1
Chapter .1 Definitions and Basic Properties ................................ 8
Chapter 2. Elementary Examples ......................................... 31
Chapter 3. A Handle Decomposition of the Milnor Fibre .................. 37
Chapter 4. Generalized L~-Iomdine Formulas ............................. 42
Chapter 5. ~L Numbers and Hyperplane Arrangements ................... 16
Chapter 6. Thom's af Condition ......................................... 68
Chapter 7. Aligned Singularities .......................................... 75
Chapter 8. Suspending Singularities ...................................... 81
Chapter 9. Constancy of the Milnor Fibrations ........................... 86
Chapter 10. Other Characterizations of the ~L Cycles ..................... 92
Appendix. Privileged Neighborhoods and Lifting Milnor Fibrations ...... 105
INTRODUCTION
The L~ numbers and L6 cycles generalize the data given by the Milnor number
of an isolated hypersurface singularity. In this introduction, we wish to quickly
review why the Milnor number of an isolated hypersurface singularity is impor-
tant. We will then give some previously known general results on non-isolated
hypersurface singularities, and indicate the types of results that can be obtained
by the machinery contained in the rest of this book.
Let // be an open neighborhood of the origin in C n+l and let f : (//, 0) +--
(C, 0) be an analytic function. Then, the Milnor fibration Mi3, Lfi5, Ra of f
at the origin is an object of primary importance in the study of the local, ambient
topology of the hypersurface, V(f) := f-a(0), defined by f at the origin. Milnor
defined his fibration on a sphere of radius e; however, his Theorem 5.11 of Mi3
leads one to consider a more convenient, equivalent, fibration which lives inside
the open ball of radius e. Hence, throughout these notes, we will use the Milnor
fibration as defined below.
......................... ;~i
(cid:12)9 ~ i~
Figure 0.1. The Milnor Fibration inside a ball
o
For all e > 0, let B~ denote the open ball of radius e centered at the origin
in C n+l. For all /r > 0, let ~,D denote the closed disc centered at the origin in
C, and let 0D, denote its boundary, which is a circle of radius ./r Then, having
fixed an analytic function, f, there exists 0e > 0 such that, for all c such that
0 < e <~ e0, there exists r h > 0 such that, for all 7/such that 0 < /r <~ ~, the
o
restriction of f to a map ~B N f-l(0D~) ~ 0D, is a smooth, locally trivial
fibration whose diffeomorphism-type is independent of the choice of e and ./7
