Real and Complex Singularities: Proceedings of the Australian-Japanese Workshop, University of Sydney, Australia, 5-8 September 2005

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Real and Complex Singularities Proceedings of the Australian-Japanese Workshop

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Real and Complex Singularities Proceedings of the Australian-Japanese Workshop University of Sydney, Australia 5-8 September 2005

Editors

Laurenti u Paunesc u University of Sydney, Australia

Adam Harris University of New England, Australia

Toshizumi Fukui Saitama University,lapan

Satoshi Koike Hyogo University of Teacher Education, Japan

N E W JERSEY

*

LONDON

\:

World Scientific

SINGAPORE

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BElJlNG

-

SHANGHAI

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HONG KONG

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TAIPEI

CHENNAI

Published by World Scientific Publishing Co. F'te. Ltd. 5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 U K ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

REAL AND COMPLEX SINGULARITIES Proceedings of the Australian-JapaneseWorkshop (with CD-ROM) Copyright Q 2007 by World Scientific Pub!ishing Co. F'te. Ltd All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying,recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-551-8 ISBN-10 981-270-551-1

Printed in Singapore by World Scientific Printers ( S ) Pte Ltd

V

PREFACE The First Australian-Japanese Workshop on Real and Complex Singularities (JARCS SYDNEY 2005) was held at the University of Sydney, Australia, during the period 5-8 September, 2005. There were 35 participants, mostly from Japan and Australia. The present volume contains mainly the texts of most of the invited talks delivered at the workshop. The Australian- Japanese cooperation in singularity theory has quite a long history, starting in the late 70s, with two main protagonists, TzeeChar Kuo in Australia and Takuo Fukuda in Japan. Since then we have witnessed a fruitful and intensive collaboration in the field of singularities. This volume contains expository articles on several topics in singularity theory, and research papers in real and complex singularities, topology of stable maps, algebraic geometry, differential geometry and dynamical systems. The text is enhanced by beautiful figures and illustrations, many of which appear fully-coloured in the attached CD-ROM. All the articles in this volume have been carefully refereed. We thank the contributors and the referees for their cooperation. We acknowledge the support by the Australian Mathematical Sciences Institute (AMSI) and the School of Mathematics and Statistics of Sydney University. The publication of this volume was supported by the Grantin-Aid in Science 15340017 of Japan Society for the Promotion of Science (JSPS). L. Paunescu 31 October 2006

vi

LIST OF PARTICIPANTS Jiro Adachi (Hokkaido University, Japan) j-adachiQmath.sci.hokudai.ac.jp

Joshua Boyd (University of New England, Australia) JBoyd3Qpobox.une.edu.au

Norman Dancer (University of Sydney, Australia) E.DancerQmaths.usyd.edu.au

Peter Donelan (Victoria University of Wellington, New Zealand) Peter.Done1anQmcs.vuw.ac.nz

Michael Eastwood (University of Adelaide, Australia) meastwooQmaths.ade1aide.edu.au

Takuo Fukuda (Nihon University, Japan) fukudaQmath.chs.nihon-u.ac.jp

Toshizumi Fukui (Saitama University, Japan) tfukuiQrimath.saitama-u.ac.jp

Adam Harris (University of New England, Australia) adamhQturing.une.edu.au

Jonathan Hillman (University of Sydney, Australia) jonhQmaths.usyd.edu.au

Goo Ishikawa (Hokkaido University, Japan) ishikawaQmath.sci.hokudai.ac.jp

Shuzo Izumi (Kinki University, Japan) [email protected]

Shyuichi Izumiya (Hokkaido University, Japan) izumiyaQmath.sci.hokudai.ac.jp

Satoshi Koike (Hyogo University of Teacher Education, Japan) koikeQhyogo-u.ac..jp

Tzee-Char Kuo (University of Sydney, Australia) tkuo0359Qmail.usyd.edu.au

Gus Lehrer (University of Sydney, Australia) guslQmaths.usyd.edu.au

John Mack (University of Sydney, Australia)

List of participants

[email protected]

Jun-ichi Miyachi (Tokyo Gakugei University, Japan) miyachiou-gakugei.ac.jp

Kimio Miyajima (Kagoshima University, Japan) [email protected]

Daniel Murfet (Australian National University, Australia) Daniel.Murfet0maths.anu.edu.au

Yayoi Nakamura (Kinki University, Japan) [email protected]

Amnon Neeman (Australian National University, Australia) [email protected]

Paul Norbury (University of Melbourne, Australia) [email protected]

Mariko Ohsumi (Nihon University, Japan) [email protected]

Pasarescu Ovidiu Florin (Tokyo University, Japan) [email protected]

Laurentiu Paunescu (University of Sydney, Australia) [email protected]

Miles Reid (University of Warwick, England) [email protected]

Nuno Romao (University of Adelaide, Australia) [email protected]

Kazuhiro Sakuma (Kinki University, Japan) [email protected]

Masatomo Takahashi (Hokkaido University, Japan) [email protected]

Kiyoshi Takeuchi (Tsukuba University, Japan) [email protected]

Nobuko Takeuchi (Tokyo Gakugei University, Japan) nobukoau-gakugei.ac.jp

Jim Ward (University of Sydney, Australia) [email protected]

Takahiro Yamamoto (Hokkaido University, Japan)

.

[email protected] hokudai .ac .jp

Akira Yasuhara (Tokyo Gakugei University, Japan) yasuharaau-gakugei.ac.jp

Ruibin Zhang (University of Sydney, Australia) [email protected]

vii


ix

ORGANIZING COMMITTEE of the first Australian-Japanese Workshop on Real and Complex Sangularities Laurentiu Paunescu (Chairman) Adam Harris Toshizumi Fukui Satoshi Koike

University of Sydney, Australia - University of New England, Australia - Saitama University, Japan - Hyogo University of Teacher Education, Japan -


xi

CONTENTS Preface

V

List of Participants

vi

Organizing Committee

ix

Integral curves for contact and Engel structures J. Adachi On the realisation of a map of certain class as a desingularization map K. Bekka, T. Fukui and S. Koike Hermitian pairings and isolated singularities J. Hillman Zariski’s moduli problem for plane branches and the classification of Legendre curve singularities G. Ishikawa Introduction to algebraic theory of multivariate interpolation S. Izumi Fundamental properties of germs of analytic mappings of analytic sets and related topics S. Izumi

1

33

46

56

85

109

Singularity theory of smooth mappings and its applications:

A survey for non-specialists S. Izumiya

124

xii

Contents

Birational geometry and homological mirror symmetry

176

L. Katzarkov Desingularization and equisingularity at undergraduate level

207

T.C. Kuo and L. Paunescu General self-similarity: An overview

232

T. Leinster Generalized Plucker-Teissier-Kleiman formulas for varieties with arbitrary dual defect

248

Y. Matsui and K. Takeuchi Derived Picard groups and automorphism groups of derived categories 271

J. Miyachi Analytic approach to deformation of normal isolated singularities

279

K. Miyajima An infinite version of homological mirror symmetry

290

A . Neeman Stable reduction and topological invariants of complex polynomials

299

P. Norbu y Singularities appearing in a stable perturbation of a map-germ

323

M. Ohsumi Existence problem for fold maps

342

K. Sakuma On completely integrable first order ordinary differential equations

388

M. Takahashi Cyclides as surfaces which contain many circles

419

N . Takeuchi Euler number formulas in terms of singular fibers of stable maps

T. Yamamoto

427

Contents xiii

Author Index

459


1

Integral curves for contact and Engel structures Jiro Adachi

Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan. E-mail: [email protected]. hokudai. ac.jp We study integral curves in 3-dimensional contact space and Engel space. Comparing the proofs of the classification of topologically trivial Legendrian knots in the standard contact bspace and the classification of integral circles in the standard Engel space, we discuss the differences and similarities of them.

Keywords: Engel structure, Horizontal curve, Legendrian knot.

1. Introduction Submanifolds in manifolds with some geometric structures are often interesting objects for geometry, for example, Legendrian submanifolds in contact manifolds and Lagrangian submanifolds in symplectic manifolds. In this article, we study integral submanifolds in contact 3-manifolds and Engel manifolds. We discuss some relations between topologically trivial Legendrian knots in tight contact 3-manifolds, that is studied in [l],and embedded integral circles in the standard Engel space, that is studied in [2]. Contact structures and Engel structures are interesting objects for differential topology. A contact structure is a distribution of corank 1 on an odd-dimensional manifold which is maximally non-integrable. An Engel structure is a distribution of rank 2 on a 4-dimensional manifold which is maximally non-integrable (see Section 2 for precise definition). It is known that all contact structures on manifolds of the same dimension are locally equivalent (Darboux). Engel structures also enjoy the same important property as contact structures. All Engel structures are locally equivalent (Engel). Then, for contact and Engel structures, global study is important. These days, there are a lot of break through on 3-dimensional contact topology ( [3], [4]).However, as far as the author knows, global studies of Engel manifolds are not so many ( [5], [6],.. . , etc). Recently, sufficient condition for the existence of an Engel structure is obtained by Vogel [7]:

2

J. Adachi

There exists an Engel structure on a 4-dimensional manifold if the manifold is parallelizable. Then, Engel manifolds must be going to be studied as an object for global differential topology. We study the following objects in this article. An embedded circle in a 3-dimensional contact manifold which is everywhere tangent to the contact structure is called a Legendrian knot. An embedded circle in the standard Engel space which is everywhere tangent to the Engel structure is called a horizontal loop. Horizontal loops might be one of key objects to Engel topology like Legendrian knots to contact topology. Legendrian knots take important roles in constructions and classifications of contact manifolds. Then, horizontal loops should be good tools for constructions and classifications of Engel structures. Engel structures and contact structures on 3-dimensional manifolds are so closely related that mutual contributions between Engel topology and 3-dimensional contact topology are expected. Now, we state the classification results after some basic notions needed to the statements (see Subsection 2.1, 2.3 for precise definitions). First, we introduce some notions concerning 3-dimensional contact topology. Contact structures on 3-manifolds are divided into two conflicting classes, tight and overtwisted. For an overtwisted contact structure, there exists a so called overtwisted disk which is an embedded disk tangent to the contact structure at any boundary point. Two Legendrian knots yoyo,71:S1 4 ( M ,5) in a contact 3-manifold ( M ,5) are said to be Legendrian isotopic if they are connected by a path of Legendrian knots. For Legendrian knots, there are two classical invariants called ThurstonBennequin’s invariant and rotation number. The classification of topologically trivial Legendrian knots in a tight contact 3-manifold is obtained by Eliashberg and F’raser in [l]: Theorem 1.1. Let y0,yl: S1 4 ( M , [ ) be topologically trivial Legendrian knots in a tight contact 3-manifold ( M , ( ) . These Legendrian knots yo and y1 are Legendrian isotopic i f and only i f they have the same Thurston-Bennequin’s invariant and rotation number: t b (yo) = t b (71) (70)= r (71).

According to Theorem 1.1,topologically trivial Legendrian knots in a tight contact manifold are classified as follows. The possible pair (r (y) ,t b (7)) of a rotation number and Thurston-Bennequin’s invariant for a Legendrian knot y is restricted. It is known that for a topologically trivial Legendrian knot, the pair (r (y) ,t b (7)) takes its value in {m,-1ml - 2k - 1) I k 2 0 , m : integers} (see [l]).Further, for each value in the set, an example of

Integral curves for contact and Engel structures

3

topologically trivial Legendrian knot is constructed. An image of a Legendrian submanifold by projection along Legendre fibers is called a front. A front of a Legendrian knot is an immersed circle with cusps of type (2,3) generically. It is known that such a curve is lifted to a Legendrian knot.

Remark 1.2. For any pair (m,-lml - 2k - 1) of integers with k >. 0, there exists a topologically trivial Legendrian knot y whose rotation number r (y) is m and Thurston-Bennequin’s invariant t b (y) is -1ml - 2k - 1. In fact, we have the following catalog L,,t of fronts of Legendrian knots (see Figure 1). By lifting an L,,t-type front vertically, we obtain a Legendrian knot z,,t with Thurston-Bennequin’s invariant t b = -(2t+s+l), and rotation number r = fs.Thus, we conclude that topologically trivial

(z,,t)

(z,,,)

Fig. 1. Catalog of fronts

Legendrian knots in a tight contact 3-manifold are classified by ThurstonBennequin’s invariant and rotation number, and there corresponds a class to any pair (m,-1ml - 2k - 1) of integers with k 2 0. The standard Engel structure E on B4 is defined as

E = { d y - z d x = 0 , d z - w d x = 0}, where ( x ,y , z , w ) are coordinates of R4.Two horizontal loops -yo, y1: S1+. (R4,E) in the standard Engel space (B4,E) are said to be horizontally homotopic if they are connected by a path of horizontal loops. We classify horizontal loops in (B4, E ) under this equivalence relation. An invariant under horizontal homotopy of horizontal loops is defined as follows. A rotation number of a horizontal loop y : S1-+ (R4, E ) is defined as a degree of y with respect to a trivialization of the Engel structure E. Let r ( y ) denote it. The following are proved in [2].

Theorem 1.3. Let yo, 71:S1 +. (R4, E ) be horizontal loops in the standard Engel space. These loops yo and y1 are horizontally homotopic if and only i f their rotation numbers coincide: T (yo) = r (71).

4 J . Adacha

According to Theorem 1.3, we can classify horizontal loops in the following Remark 1.4. We claim that there actually is a horizontal loop with rotation number k for any integer k E Z.In order to describe horizontal loops, we use terms of fronts (see Subsection 2.2). The image of a horizontal loop in the standard Engel space by the projection 7r: (z, y, z , w) I+ (z, y) is called a front of the loop. Generically, it is a non-vertical immersed curve with cusps of type (2,5). Conversely, we can lift such a plane curve to a horizontal loop in the standard Engel space.

Remark 1.4. For any integer n E Z,there is a horizontal loop y whose rotation number is the given integer n: ~(y)= n. In fact, we have the following catalog F k of fronts of horizontal loops (see Figure 2). By lifting an Fk-type front vertically, we obtain a horizontal loop F k with rotation number T ( F k ) = k . Thus, we conclude that horizontal loops in the standard

Fig. 2.

Catalog of fronts

Engel space are classified by the rotation number, and that for any integer, there corresponds a class. We can observe the results in this article from a view point of homotopy principle (h-principle for short). One of the most famous examples of h-principle is Whitney's theorem on immersions of a circle into a plane ( [8], [9]): regular homotopy classes of immersed circles are classified by their rotation numbers. The rotation number for horizontal curves is also the main tool for the classificationsin this article, especially in Theorem 1.3. And, the rotation number for horizontal curves is defined by taking a certain projection to a plane. Then, by taking such a projection, we obtain a version of Whitney's theorem with some conditions on some area (see Subsection 6.1). In other words, we find a new relation which satisfies hprinciple. On the other hand, in order to show h-principle there is a strong

Integral curves f o r contact and Engel structures

5

machinery, Gromov’s method (see [lo], [9]). A relation, homotopy among Legendrian immersions, similar to the relation in this article is studied by that method in [lo]. The results in this article should also be proved by that method. The result in this article can be generalized to higher dimensional cases. The notion of Engel structure is a special case of the notion of Goursat structure. The Goursat structure on a 3-manifold is a contact structure. However, the Darboux-type theorem, or local triviality, does not hold in higher dimensional cases. We extend Theorem 1.3 to the higher dimensional cases (see Subsection 6.2). The author would thank the organizers of Australian-Japanese Workshop on Real and Complex Singularities for the perfect organization and the hospitality. 2. Preliminaries

We introduce some basic miscellaneous notions in this section. First, we define some basic notions on contact structures. And then, we introduce some notions on Engel structures as extensions of those on contact structures. 2.1. Basic definitions

2.1.1. Contact structures A contact structure on a 3-manifold is a completely non-integrable tangent plane field. The complete non-integrability implies that this tangent plane field C is defined, at least locally, by a 1-form a which satisfies the inequality CY A da # 0 everywhere. In terms of Lie brackets, the complete non-integrability is described as follows. The plane field C on a 3manifold M can be regarded as a locally free sheaf of vector fields. Let [C,C] denote the sheaf of vector fields generated by all Lie brackets [ X ,Y ] of vector fields X , Y which are cross-sections of C c T M as a bundle. Let C h ( C ) be the Cauchy characteristic subdistribution of C , that is, Ch(C) := { X E C I [ X , C ] c C}. A hyperplane field C is a contact structure if it satisfies C [C,C ] = T M , Ch (C) = ( 0 ) . The standard contact structure on R3 is a hyperplane field Co on R3 defined by a 1-form QO := dy-zdx as its kernel, where x , y, z are coordinates of R3. It is known that any contact structure on a 3-dimensional manifold is locally equivalent to this structure (Darboux). A contact structure on a 3-manifold traces on an embedded surface a 1-dimensional foliation with singularities. Let F be an embedded surface

+

6 J. Adachi

in a contact 3-manifold ( M , C ) . The contact structure C traces on F a 1dimensional singular foliation. Such a foliation is called the characteristic foliation on F with respect to C , and FC denotes it. When C = { a = 0) and F are oriented, FC is oriented by the vector field X on F which satisfies XAvolF = i*a, where v o l ~is a volume form of F , and i : F L) M is the inclusion mapping. Generically, the characteristic foliation F c has a finite number of singular points where F is tangent to C. A singular point is called positive or negative depending on whether the orientations of F and C coincide at the point or not. Generically, the index of the vector field which defines the characteristic foliation locally at a singular point is fl.We call a singular point elliptic if its index is +1, and hyperbolic if it is -1. Because of the non-integrability of contact structures, the characteristic foliation Fc has topologically the focus type singularity at elliptic points, and FC has the saddle type singularity at hyperbolic points. These singularities are called simple singularities. Contact structures on 3-manifolds are classified into two conflicting classes. A contact structure C on M is said to be overtwisted if there exists an embedded disk D c ( M ,C ) the characteristic foliation Dc on which has no limit cycle. A contact structure which is not overtwisted is called tight. A Legendrian knot is an embedded circle in a 3-dimensional contact manifold which is everywhere tangent to the contact structure. Two Legendrian knots yo, 71:S1 + ( M , C ) are said to be Legendrian isotopic if there exists a smooth mapping H : S1 x I + ( M , C ) which satisfies that, setting H t ( s ) := H ( s ,t ) , HO = yo and H1 = 71,and that H t : S1 -+ ( M ,C ) is a Legendrian knot for any t E I = [0,1]. There are two classical invariant under Legendrian isotopy for Legendrian knots, Thurston-Bennequin's invariant and the rotation number. Let y be an oriented Legendrian knot homologous to zero in a contact 3manifold ( M ,C), and F c M an embedded surface with boundary 8 F = y which represents a relative homology class p E H2(M,7). Pushing y slightly along a vector field transverse to C , we obtain a new knot y'. The intersection number of y' and F is called Thurston-Bennequin's invariant of y with respect to p. Let t b (y,P) denote it. Let X be a vector field tangent to y which agrees with the orientation of y. The degree of X with respect to a trivialization of the bundle C ( Fdepends only on ,B E H z ( M , y ) . It is called the rotation number of y with respect to p. r (y, p) denotes it. When the choice of ,L? E H z ( M , y ) is clear, then we omit 0.In particular, we study the rotation number of a Legendrian knot in the standard contact R3 in Subsection 2.3. Studying Legendrian knots is one of important issues in

Integral curves for contact and Engel stmctures

7

contact topology (see [ll],[12], [l],[13] for example). 2.1.2. Engel structures An Engel structure is a maximally non-integrable distribution of rank two on a 4-dimensional manifold. Generally, it is defined as follows. Let M be a 4-dimensional manifold, and D a distribution, or a subbundle of the tangent bundle T M , of rank 2. We can regard D as a locally free sheaf of vector fields on M . Let [D,D] denote the sheaf of vector fields generated by all Lie brackets [ X ,Y] of vector fields X , Y which are cross-sections of D. Set D2 := D+[D,D ] ,and D3 := D2+[D2,D2].Then, an Engel structure on M is defined as a distribution D c T M of rank 2 which satisfies the following conditions: rank D: = 3,

rank D i = 4

(1)

at any point p E M . A certain Engel manifold is constructed from a 3-dimensional contact manifold. Let E be a contact structure on a 3-dimensional manifold N . By taking fibrewise projectivization of the contact structure E , we obtain a new 4-dimensional manifold PE = U z E ~ P ( E zOn ) . the 4-dimensional manifold PE, an Engel structure D ( E ) is defined as D ( E ) , := (d7r)-'l, where 7r: PE -+ N is the canonical projection, q = ( p , l ) E PE is a point, and 1 c TpN is a line (see [5]). Such a procedure is called a Cartan prolongation (see ~ 4 1 [51, , PI). An Engel structure has a characteristic direction. Let D be an Engel structure on a 4-dimensional manifold M . From this Engel structure D, a line field is defined as follows: L ( D ) := { X E D 1 [ X ,D2]c D2}.The line field L ( D ) is called the Engel line field. It is known that a contact structure is induced from an even-contact structure D2 on an embedded 3-dimensional manifold N c M which is transverse to the Engel line field L ( D ) . The contact structure is obtained as D2 n T N . Such a procedure is called a deprolongation (see [5], [14]). In this article, we work in the standard Engel space (R4, E ) , that is, an ordinary 4-dimensional space R4 endowed with the standard Engel structure. The standard Engel structure on R4 is defined as a kernel of the following pair w1, w2 of 1-forms: ~1 = d y - Zdx,

where ( x , y , z , w ) E

~2 = d z

-

wdx,

(2)

R4 are coordinates. Let E denote the standard Engel

8

J . Adachi

structure on R4:

E

:= ( ~ 1= 0 , w2 = O } = Span

We call the 4-dimensional space (R4, E ) endowed with the standard Engel structure the standard Engel space. It is clear that the standard Engel structure E actually satisfies the condition (1) of the definition. In this case, the Engel line fields is L ( E ) = Span {a/aw}. With respect to the standard Engel structure on R4, the induced contact structure on R3 c R4, the (5, y , 2)-space, is the standard contact structure C = {w1 = d y - z d x = O } . A horizontal cume I' c M in a Engel manifold ( M , D ) is a curve which is tangent to the Engel structure D everywhere: Tpr c D, at any p E r. Horizontal loops in the standard Engel space (R4,E) are dealt with in this article. A horizontal loop in ( I%',) is an embedding y: S1 + (R4,E) of an oriented circle S1 into (W4,E)which satisfies the . other words, y satisfies the conditions condition y * ( T p S 1 ) c ( E ) y ( p lIn y*w1 = 0 and y*w2 = 0 , where w1, w2 are 1-forms defined as equations (2). For horizontal loops, we introduce an equivalence relation. Two horizontal loops 7 0 , y1: S1 -+ (R4, E ) are said to be horizontally homotopic if there exists a smooth mapping H : S1x I + (R4, E ) which satisfies that, setting Ht(s) := H ( s , t ) , HO = 70and H I = 71,and that H t : S1-+ ( R 4 , E )is a horizontal loop for any t E I = [0,1]. We introduce an invariant under horizontal homotopy for horizontal loops, in a similar way to the rotation number for Legendrian knots, in Subsection 2.3. 2.2. Vertical projections and fronts

We introduce the vertical projection of the standard Engel space to the standard contact 3-space, and the vertical projection of the standard contact 3-space to a 2-dimensional plane. The image of a Legendrian curve by the second vertical projection is called a front of the Legendrian curve. The image of a horizontal curve in the standard Engel space by the first vertical projection is a Legendrian immersion with singularities. The image of a horizontal curve in the standard Engel space by the double projections is also called a front of the horizontal curve. Fronts of Legendrian knots and those of horizontal loops are important tools to prove Theorem 1.1 and Theorem 1.3 respectively. Further, we introduce the vertical lift of a certain plane curve to a Legendrian curve and, further, to a horizontal curve in the standard Engel space.


9

The vertical projections are defined as follows. Let (z, y, z , w) be coordinates of the standard Engel space (R4,E) as above. Set Q3 := {w = 0 ) 2 R3, and Q 2 := {w = 0 , z = 0 ) 2 R2. Let T I : R4 + Q 3 , (2, y , z , w) H (z,y , z ) , and 7r2: Q 3 + Q 2 , (2, y, z ) H (z,y) be the canonical projections. We call them the first and the second vertical projections. We should remark that the first projection is along the direction of the Engel line field, and the second one is along the Legendrian fiber with respect to the induced standard contact structure C = {w1 = dy - zdz = 0 ) on Q 3 R3 (see Subsection 2.1). Let I?: S1 -+ (R3,C) be a Legendrian knot in the standard contact 3-space. Then, 7r2 o r : S1 -+ Q2 R2 is an immersion with singularities. At a point where r is tangent to the z-direction, the projection 7r2 o r has a singular point. The image of 7r2 o I?: S1 + R2 is called a wave front, or front for short, of r. Generically, the singular points are cusps of type (2,3), where 7r2 o r is locally diffeomorphic to a curve t H (t2,t3)at t = 0. Fronts of Legendrian knots are important tools to study Legendrian knots. We take further projections from the upper side to study horizontal loops. Let y : S1-+ (R4, E ) be a horizontal loop in the standard Engel space. Then, 7r1 o y : S1 -+ (It3,C)is a Legendrian immersion with singularities. At a point where y is tangent to the w-direction, the projection 7r1 o y has a singular point. Such Legendrian curves with singularities are studied by Ishikawa [15] and ZhitomirskiY [16].We call the image of further projection 7rp o 7r1 o y : S1 + Q Z % R2 a front of y. It is a wave front of a Legendrian knot 7r1 o y. The mapping 7r2 o 7r1 o y is an immersion with singular points. The singular points are cusps of type (2,5), that is, points where 7r2 o 7r1 o y is locally diffeomorphic to a curve t H ( t 2 , t 5 )at t = 0. Contrary to the vertical projection, we can reconstruct a horizontal curve in the standard Engel space and a Legendrian knot from certain plane curves. We call these procedures vertical lifts. They are based on the prolongation procedure explained in Subsection 2.1. First, we take vertical lifts to Legendrian knots. Let f : S1 R2 Z Q 2 , f(s) = (fl(s),fi(s)), be an immersion with cusps of type (2,3). Assume that the plane curve f is nonvertical, in other words, it satisfies dfl(s)/ ds # 0 at regular points and d2f1(s)/ds2 # 0 at cusp points. We remark that the vertical projection of a Legendrian knot is actually nonvertical. Then, we can lift the given plane curve f(s) = (f1 (s), f2(s)) to a Legendrian curve f : S1 + R3 Q 3 , 7(s) = (fl(s>,fi(s),f3(s)), in R3 with the standard contact structure C = {dy - zdz = 0) whose vertical projection is the given f ( t ) .Its z-coordinate f 3 ( s ) is obtained as a slope of the curve f(s).

-

10 J . Adachi

Precisely, it is written down as follows:

It is clear, from the definition of the standard contact structure, that the obtained curve f(s) is Legendrian. Even at a cusp point, the curve is lifted to a smooth curve (see Example 2.1). When the plane curve f : S1 -+ R2 has no self-tangency, it is lifted to an embedding, that is, a Legendrian knot. Next, we observe vertical lifts to horizontal loops in the standard Engel space. Let f : S1 -, R2 2 Q 2 , f(s) = (fl(s),fi(s)), be an immersion with cusps of type (2,5). Assume that the same condition as above. Then, by the procedure as above, we obtain a Legendrian immersion with singularities. As cusp points of f are of type (2,5), we still have singularities (see Example 2.1). Furthermore, we can lift the obtained Legendrian curve f(s) = (fl(s),f 2 ( s ) , f3(s)) in (R3 Q3,C) to a horizontal loop f(s) = (fi(s),fi(s), f3(s), f4(s)) in the standard Engel space (R4, E ) whose vertical projection is the given plane curve f(s). Its w-coordinate f4(s) is obtained similarly to the above. It is obtained as a slope of a plane curve induced from f(s) by the projection pr to the (x,z)-plane. Note that the projected plane curve prof: S1 + R2 is a nonvertical immersion with cusps of type (2,3), namely locally equivalent to a curve t H ( t 2 ,t3). It is because of the assumption that f(s) is an immersion with cusps of type (2,5) (see Example 2.1). Then, the w-coordinate of f ( s ) is obtained as follows:

It is clear that the resulting curve f(s) is horizontal with respect to the standard Engel structure E = {dy - zdx = 0, dz - wdx = 0). We should remark that, contrary to the Legendrian case, self-tangencies of order one are allowed for the plane curve f. Such points are lifted to different points with different w-coordinates.

Example 2.1. We observe these lifts at a cusp point. Suppose that a nonvertical cusp of type (2,5) on the (x, y)-plane R2 E QZ is parameterized as g ( t ) = ( t 2 , t 5 )at t = 0. Then, it is lifted to a space curve g ( t ) := (t2,t5,5t3/2) in R3 2 Q 3 which is Legendrian for the standard contact


11

structure C = {dy - zdx = 0). The projection pr og(t) = ( t 2 ,5t3/2) to the (x,z)-plane has a cusp of type (2,3) at t = 0, where pr: (x, y,z) H (x, z ) is a canonical projection. Then, g ( t ) is lifted to a horizontal smooth curve g ( t ) := ( t 2 ,t5,5t3/2, 15t/4) in the standard Engel space (R4, E).

Example 2.2. As a global circle, an immersed circle on the (z, y)-plane with cusps of type (2,5) is lifted as in Figure 3.

Fig. 3. projections and lifts.

We can calculate the Thurston-Bennequin invariant of a Legendrian knot, from its front. A vector field a/ay is transverse to the standard contact structure C = {dy - zdx = 0). We obtain the front of Legendrian knot slightly perturbed from the given one along a / a y by perturbing the front of the given Legendrian knot along the y-direction. It is easy to check that the contribution to linking number is +1 at a crossing and at a right-looking (or left-looking) cusp in the front. As a result, we conclude the ThurstonBennequin invariant of the Legendrian knot obtained from the L,,t-type front (see Figure 1 in Remark 1.2) is t b (L,,t) = 2t s 1. Note that the sign is opposite from that in [12], [l].It is because our standard contact structure is a negative one, that is, w1 A dwl = -dx A dy A dz.

+ +

2.3. Horizontal projections and rotation number

We introduce the horizontal projection of the standard Engel space to a 2-dimensional plane. We define the rotation number of a horizontal loop in the standard Engel space from the horizontal projection of the loop. Similarly to vertical lifts, we can lift a certain immersed circle in a plane to a Legendrian loop in the standard contact space, and to a horizontal loop in the standard Engel space. Further, we count the rotation number

12

3. Adacha

of a horizontal loop obtained from a front of type Fk (see Figure 2 ) by the vertical lifts. Remark 1.4 is verified here. The horizontal projection is defined as follows. Let (x,y, z , w) be coordinates of the standard Engel space (R4,E). Set P3 := {y = 0) E R3, and P2 := {y = 0 , z = 0) R2. Let p l : R4 4 P3, (z, y, z , w)H (2,z , w), and p2 : P3 + P2, (x,z , w)H (x,w)be the canonical projections. We call them the first and the second horizontal projections. We define an invariant, the rotation number, of a horizontal loop in the standard Engel space. It is similar to the rotation number for Legendrian knots (see [ll]). Let y : S1 + (W4,E)be a horizontal loop. The projected curve p2 0 p l o y: S1 4 P2 W 2 is an immersed plane curve because the standard Engel structure is transverse to y and z-axes. Then, we can calculate the degree of the immersed oriented curve pzoploy : S1 -+ P2 E R2 with respect to x and w-axes, in other words, with respect to a trivialization of the Engel structure E . We call it the rotation nzlmber of y. Let T (y) denote it. This definition is independent of the choice of the trivialization by a similar reason to the case of Legendrian knots (see [ll]).It is because Hl(y) is null in H1(R4). Then, the rotation number is invariant under diffeomorphisms preserving Engel structure, and horizontal homotopies of horizontal loops. Example 2.3. Let us consider a horizontal loop obtained in Example 2.2. The rotation number of the curve is 1. We can count by using the picture in Figure 3 on the (x,w)-plane. We can count rotation numbers of horizontal loops obtained from fronts of type Fk (see Figure 2 ) . According to Example 2.2, a cusp of a front corresponds to an immersed curve with one twist with the reversed orientation. Thus, in order to calculate the rotation number of a horizontal curve from a front of type Fk, we have only to count the degree of an immersed curve in Figure 4.As a result, we obtain that the rotation number of a horizontal

Fig. 4. How to count rotation numbers

loop obtained from a front of type Fk is k.


13

We can lift a certain immersed closed plane curve to an immersed horizontal loop in the standard Engel space. We call this procedure a horizontal lift. Let g: S1+ P2 2 R2,g(s) = (gl(s),g4(s)), be an immersed closed curve in the (x,w)-plane. Suppose that the algebraic area bounded by the immersed curve is zero:

The condition guarantees that the lifted curve is closed. We remark that horizontal projections of horizontal loops in (R4, E ) and horizontal projections of Legendrian closed curves satisfy this condition. Then, the given immersed plane curve g(s) = (gl(s),g4(s)) satisfying the condition (5) above is lifted to a Legendrian immersed circle g(s) = (g1(s),g~(s),g4(s)) with singular points in the (2, z , w)-space P3 E R3 with the standard contact structure C’ = (w2 = dz - wdx = 0). The horizontal projection of g(s) is the original g(s). The z-coordinate g3(s) of g(s) is obtained as follows:

In order to lift the Legendrian curve g(s) to an immersed horizontal loop in (R4, E ) , we need an additional condition. To make it sure that the lifted curve is actually closed, we need a similar condition to equation (5). Suppose that the algebraic area bounded by the projection of g(s) to the (2, 2)-plane is zero:

1

Proas’)

93 (S)*dgl(s) = 0,

(6)

where pr : (x,z , w) H (x,z ) is a canonical projection. Then, the Legendrian curve g(s) = (91(s),g3(s), g4(s)) in P3 &+ R3 can be lifted to an immersed E ) whose vertical horizontal loop J ( s ) = (gI(s), g2(s), g3(s), g4(5)) in (R4, projection is the given plane curve g(s). The y-coordinate g2(s) of G(s) is obtained as follows: g2(s) := lSg3(U)’dg1(U).

Example 2.4. If an immersed curve as the top of Figure 3 is given, we can lift the curve from the top to the bottom of Figure 3.

We will consider this horizontal lift in Section 6.

14 J. Adachi

3. Reidemeister moves for contact and Engel structures

The Reidemeister moves for contact and Engel structures are introduced in this section. In other words, we determine finite number of moves of a front which can be lifted to Legendrian isotopies in the standard contact 3-space or horizontal homotopies in the standard Engel space. 3.1. Reidemeister moves f o r contact structures

The Reidemeister moves for Legendrian knots are given in the following theorem. It is proved in [17]. Theorem 3.1 (Legendrian Reidemeister moves, [17]).Let ro,rl: S1--+ (R3,C ) be Legendrian knots in the standard contact 3-space. These knots are Legendrian isotopic if and only if the front 7rz o (Sl) is moved to the other front 7rz o (5'') via a finite sequence of the following moves:

I1

I11 Note that cusps in the list above are of type (2,3). 3.2. Reidemeister moves f o r Engel structures

The Reidemeister moves for horizontal loops in the standard Engel space is given in the following theorem. It is proved in [2]. Theorem 3.2 (Engel Reidemeister moves). Let yo,71: S1-+ (R4, E) be horizontal loops in the standard Engel space. These loops are horizontally homotopic if and only if the front 7rz o 7r1 o yo (Sl) is moved to the other front ?rz o 7r1 o y1 (5'') via a finite sequence of the following moves: 0

an isotopy of R2 preserving the curve non-vertical,

Integral curves for contact and Engel structures 15

-

11-(1)

11- (2)

I11 Note that cusps in the list above are of type (2,5). Moves I and II(2) never occur in the Legendrian Reidemeister moves, and that move I in Theorem 3.1 never occurs in the Engel case. We should remark that move 11-(1)’ below (see Figure 5) can be obtained by a combination of the moves 11-(1) and 11-(2) in Theorem 3.2 (see Figure 6).

Fig. 5.

11-(1)’

Fig. 6. combination of 11-(1) and IL(2)

Then, the move 11-(1)’ is not essential in the list of moves in Theorem 3.2. 4. Classification of trivial Legendrian knots

In this section, a rough sketch of the proof of the classification of topologically trivial Legendrian knots, namely Theorem 1.1, is given. The arguments in this section are mainly due to Eliashberg and Fraser [l].

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4.1. Characteristic foliations on a spanning disk

We observe characteristic foliations on a disk which spans the given topologically trivial Legendrian knot. Let r: S1-+ ( M ,C), or its image L := I'(S1), be a topologically trivial Legendrian knot in a tight contact 3-manifold ( M , C ) . Since L is topologically trivial there exists a disk D c ( M , C ) which spans L. We observe characteristic foliation D c by perturbing the disk D. The goal of this subsection is to arrange the characteristic foliation to a certain form. The form we are going to obtain is defined as follows. The characteristic foliation Dc is said to be in elliptic form if it satisfies the following conditions (see Figure 7 for example): 0 0 0

0

the signs of singular points on the boundary L = d D is alternating, all singular points on the interior intD are elliptic, each singular point on the boundary L = d D is connected to singular points in the interior intD by leaves of D c , except connections with two singular points on L = d D next to it, each elliptic point in the interior intD is connected to at least two hyperbolic points on the boundary L = dD.

Fig. 7. elliptic form

It is proved in [l]that any topologically trivial Legendrian knot L in a tight contact manifold ( M ,C) has a spanning disk D c ( M ,C) characteristic foliation on which is in elliptic form. A rough sketch of the proof is introduced in this subsection. First, we should introduce an important tool to deal with characteristic foliations. The following Elimination Lemma is proved by Giroux and improved by Fuchs (see [HI, [19]).

Integml cumes for contact and Engel structures

17

Lemma 4.1 (Elimination Lemma). Let F be an embedded surface in a contact 3-manifold ( M ,C ) , and p , q E F elliptic and hyperbolic points of the characteristic foliation FC with the same sign. If there is a trajectory y of FC joining p and q, then there exists a Co-small isotopy ht : F -+( M ,C ) , t E [0,1], which has the following properties: ht is the identity on y and outside a neighborhood U of y, ho = id, the characteristic foliation FC o n F := hl ( F ) has no singular point

inF;nU. (see Figure 8)

Fig. 8. Elimination of singularities

Now, a rough story of the proof of the existence of a spanning disk with a characteristic foliation in elliptic form is introduced as follows. First, take a spanning disk D of the given Legendrian knot so that the characteristic foliation DC on D has a certain form near the boundary d D = L. The characteristic foliation DC is said to be in normal absorbing form along L = d D if signs of singular points on L is alternating and the positive ones are hyperbolic and the negative ones are elliptic. We can put the disk D so that the characteristic foliation Dc is in normal absorbing form along L by twisting D around L = d D keeping L fixed. Then, we apply the Elimination Lemma 4.1. Since the characteristic foliation Dc on the spanning disk D is in normal absorbing form along L c d D, there exists a transverse knot L' on D near L (see Figure 9). Then, we can apply arguments in [12] for transverse knots. According to [12], we can eliminate all negative elliptic and positive hyperbolic singular points inside by the Elimination Lemma 4.1. Such a reduced foliation on D is said to be reduced with normal absorbing f o r m o n the boundary. We should remark that, according to [12], there appears a Legendrian tree on D whose vertices are positive elliptic points and whose edges are separatrices of negative hyperbolic points. Tightness of the contact structure is important for the argument in this paragraph.

18 J. Adachi

Fig. 9.

Normal absorbing form

It is proved in [l]that the spanning disk D of L whose characteristic foliation Dc is in reduced form with normal absorbing boundary is perturbed to have a characteristic foliation in elliptic form. An important trick is the converse of the Elimination Lemma occurring at a boundary singular point. It is carefully observed in [20] (elliptic-hyperbolic conversion). By this trick, we can exchange an elliptic (hyperbolic) point on the boundary with a hyperbolic (elliptic) point with the same sign and another elliptic (hyperbolic) point with the same sign (see Figure 10). Then, we can eliminate hyperbolic

Fig. 10. elliptic-hyperbolic conversion

points inside and some other extra things by the Elimination Lemma 4.1. Thus, we obtain a characteristic foliation in elliptic form. We should remark that there appears a Legendrian tree on the perturbed D with a characteristic foliation Dc in elliptic form. In this case, the vertices are positive and negative elliptic points and the edges are leaves of DC connecting elliptic points directly with no singular point on them. We observe this tree in the next subsection. 4.2. Deformation of corresponding fronts

In this subsection, we deal with fronts which correspond to the obtained characteristic foliation on the spanning disk of the Legendrian knot in elliptic form. Characteristic foliation in elliptic form is not enough to discuss the classification. So, we use the arguments on fronts of Legendrian knots.


19

The goal of this subsection is to reduce the front corresponding to the characteristic foliation in elliptic form to one of the fronts in the catalog (see Figure 1). A tree with twigs is induced from a characteristic foliation in elliptic form. Let D be a spanning disk of a Legendrian knot L characteristic foliation on which is in elliptic form. Then, we can take a Legendrian tree T whose vertices are interior elliptic points of DC and whose edges are leaves of DC connecting interior elliptic points directly. Note that there are at least two separatrices of hyperbolic points on the boundary meeting at each interior elliptic point. We consider them twigs of the tree. Let us embed twigged trees into the (z,y)-plane so that the slope of edges are sufficiently horizontal and two twigs at each vertex are vertical. And, give each vertex the same sign as the corresponding elliptic point (see Figure 11 and Figure 7).

I

0

Fig. 11. twigged tree

A front is induced from a twigged tree embedded in R2 as follows. First, we should be careful about the twisting of the spanning disk D. Then, it is useful to change the twigged tree a little. Reverse the relative positions of edges and twigs to the right of each positive vertex and to the left of each negative vertex (see Figure 11 and Figure 15 for example). Now, we apply the following procedure to obtain a front from a twigged tree. (1) To each end vertex, we give the following (Figure 12) part of front according to the sign of the point. At a positive or negative vertex, fronts go upward or downward respectively: (2) To each non-end vertex, with exactly two twigs, we give a crossing of fronts. At a positive or negative vertex, fronts go upward or downward respectively, like operation (1) above (see Figure 13):

(3) To each extra twigs, we give a zig-zag of fronts. The zig-zag goes downward for a twig at a positive vertex and goes upward for a twig at a negative vertex (see Figure 14):

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J. Adachi

Fig. 12. fronts corresponding to the ends

Fig. 13. fronts corresponding to non-end vertices

Fig. 14.

fronts corresponding to twigs

(4) Connecting them suitably, we obtain a front for the twigged tree (see Figure 15, 7, and 11):

Fig. 15. example of corresponding front

Thus, we obtain a front from a twigged tree. Note that the correspondence is one t o one. Also, we should remark that we can construct a Legendrian knot by the Legendrian lift, which has a spanning disk D with the same twigged tree as the given one on it. Now, we handle fronts instead of characteristic foliations. We can sim-


21

plify the obtained front to one of the front in the list given in Remark 1.2 (see Figure 1). Basically, we make the given tree linear by the Legendrian Reidemeister moves (Theorem 3.1). The procedure is discussed in [12] (see Figure 16, for example).

Fig. 16. linearization of a tree

The problem is how to deal with zig-zags which appear in the third procedure of constructing fronts, or which correspond to twigs. It is proved in [l]that we can move a zig-zag to anywhere we want. Then, applying the argument above, we obtain a front corresponding to linear tree with zig-zags on one end. A cancelation of different type of zig-zags is discussed in [l].Then, as a result, we obtain a front in the catalogue in Figure 1. The proof of Theorem 1.1 is completed in [l]by proving the following lemma:

Lemma 4.2. Let L, L‘ be Legendrian knots bounding disks D , D‘ respectively, which have the dilgreomorphic characteristic foliation in elliptic form. Then, L and L’ are Legendrian isotopic. The rough story of the proof is the following. By taking slightly different spanning disks (so called exceptional spanning surfaces) from what we discussed, the Legendrian knots L and L’ shrink arbitrary close to the diffeomorphic skeletons which consist of the trees on the spanning disks and elliptic points on the boundary of the disks. It is proved those skeletons with germs of surface coincide by an ambient contact isotopy.

5. Classification of horizontal loops in the standard Engel space Theorem 1.3 is treated in this section. It is clear that rotation numbers of two horizontally homotopic loops are the same. We show that any horizontal loop is horizontally homotopic to one of horizontal loops of type Fk,k E Z.

22

J. Adachi

The proof in [2] is organized as follows. First, we reduce the problem to a problem of Legendrian knots with singularities. Then, we can apply arguments of Eliashberg and Fraser [l]for Legendrian knots. By applying their arguments, the given loop is homotoped horizontally to a horizontal loop with a front of a certain type (Subsection 5.1). Next, such a front is moved to that of Fk-type by Engel Reidemeister moves (Subsection 5.2).

5.1. Legendrian knots w i t h singularities Horizontal loops in the standard Engel space (R4, E ) can be identified with Legendrian knots with certain singularities in the standard contact space (R3, C). Let y : S1 -+ (R4, E ) be a horizontal loop. Then, 7r1 o y : S1 + (R3,C) is a Legendrian knot with some singularities. In other words, a space curve 7r1 o y is obtained as a single vertical lift of the front 7r2 o 7r1 o y (see Subsection 2.2). The Legendrian curve 7r1 o y : S1-+ (R3, C) is not an immersion at points corresponding to cusps of the front 7r2 o 7r1 o y,and has self-intersection at points corresponding to self-tangency of the front. If we care about these points, we can apply some arguments on Legendrian knots. That is, homotopy among such Legendrian knots can be lifted to horizontal homotopy. First of all, we remark that a self-intersection of a Legendrian knot corresponding to a horizontal loop is canceled by horizontal homotopy of the horizontal loop. As we mentioned above, a self-intersection of the corresponding Legendrian knot is a self-tangency of the front of the horizontal loop. In terms of fronts, a self-tangency of a front appears as that of regular curves, a regular curve and a cusp, or cusps. The last one is canceled easily. The former two are canceled by Engel-Reidemeister moves 11-(2) and 11-(1)' (see Subsection 3.2). An important fact is that any horizontal loop is horizontally homotopic to a horizontal loop corresponding to a topologically trivial Legendrian knot with singularities. It is known, in knot theory, that applying X-moves, we can make a knot topologically trivial (see [21]). An X-move is an operation on knot diagram, projection of knots to planes, defined as in Figure 17. We obtain the X-move for Legendrian knots, which is obtained by Engel Reidemeister moves I, 11-(1), and 11-(2) (see Figure 18). This implies that any horizontal loop is horizontally homotopic to a horizontal loop whose corresponding Legendrian knot is topologically trivial. Then, considering topologically trivial Legendrian knots with singularities is sufficient to show Theorem 1.3. We can apply arguments due to Eliashberg and Fkaser in [l]for topo-


23

Fig. 17. X-move

Fig. 18. X-move for Legendrian knots

logically trivial Legendrian knots. Roughly speaking, the argument due to Eliashberg and Fkaser [l]consists of two parts. Manipulations of characteristic foliations on a disk spanning the given topologically trivial Legendrian knot, and deformations of fronts. As for the manipulations of characteristic foliations, we can apply almost the same arguments to topologically trivial Legendrian knots induced from horizontal loops. On the other hand, arguments on deformations of fronts for horizontal loops are different from those for Legendrian knots. Reidemeister moves for fronts of horizontal curves are different from those for fronts of Legendrian curves (see Theorem 3.2, Theorem 3.1). We manipulate characteristic foliations on a disk D c R3 spanning the given topologically trivial Legendrian knot 7r1 o y : S1--+ (R3, C) with singularities which is induced from a horizontal loop y : S1 4 (R4, E ) . Such a curve has singular points where projections to the (2,y)-plane are (2,5)cusps and projections to (2,z)-plane are (2,3)-cusps generically. We take a spanning disk D so that the characteristic foliation Dc has elliptic singular points (see Figure 19) at singular points of 7r1 o y. We can take such

Fig. 19. elliptic corner

a disk

D as follows. First, we take surfaces around singular points of the

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J . Adacha

curve 7r1 o y so that the curve is leaves of the characteristic foliation on the surfaces, and that the singular points are elliptic corners. We extend such small surfaces to an annulus along the curve 7r1 o y so that it can be extended to a disk. And then, we extend the annulus to a disk. Thus, we obtain a disk D spanning D with elliptic corners as above. An elliptic corner can be changed to a smooth elliptic point by a Co-small perturbation of D keeping D c homeomorphic, according to [19].Applying such perturbations to all singular points, we obtain a topologically trivial smooth Legendrian knot I?: S1 ---f (R3, C) with a spanning disk 0, characteristic foliation & on which is homeomorphic to Dc. Then, we apply the argument in [I]to which is Co-close to 7r1 o y. For the Legendrian knot I',there corresponds a front. Note that the given Legendrian knot 7r1 o y with singularities is induced from the horizontal loop y. Then, the corresponding front must have the same number of cusps as the number of singular points of 7r1 o y. The number of cusps of the front corresponding to the smooth curve r might be less than the number of singular points of 7r10 y because of the Legendrian Reidemeister moves I. In such a case, we create pairs of singular points of D c on I? = d D corresponding to the Legendre Reidemeister moves I above by the converse operation of the Elimination Lemma. Then, the characteristic foliation on D corresponds to a front with the same number of cusps as the number of singular points of 7rloy. Perturbing the disk D we arrange the characteristic foliation DC so that the points on = d D corresponding to singular points of 7r1 oy correspond to cusps of the obtained front. If such a point is a boundary elliptic point of D c , then we can return I? to 7r1 o y around the point. If such a point is a boundary hyperbolic point, we make it elliptic by applying the elliptic-hyperbolic conversion (see Figure 20). Then, we can return I? to 7r1 o y. We should remark that a convex (resp.

r

r

Fig. 20.

cusp for boundary hyperbolic point

concave) elliptic corner for the initial spanning disk D might be changed to a concave (resp. convex) elliptic corner for the resulting disk D. In this case, we have just retaken the spanning disk and never changed the given Legendrian knot 7r1 o y. Consequently, we obtain a disk 8 with singularity h


25

spanning 7r1 o y the characteristic foliation on which is in elliptic-like form. In other words, the characteristic foliation & is in elliptic form almost everywhere with extra pairs of elliptic and hyperbolic points of the same sign (see Figure 8) on the boundary which correspond to pairs of cusps in the Legendrian Reidemeister move I, and with extra hyperbolic elliptic conversions (see Figure 20). Then, the front we obtained is the front induced from a characteristic foliation in elliptic form with some pair of cusps in the figure of the Legendrian Reidemeister move I whose cusps are of type (275). We continue the arguments in [l]on deformations of fronts. They conclude that any topologically trivial Legendrian knot in (R3, C) is isotopic to a Legendrian knot whose front is of type L,,t (Remark 1.2, see Figure 1). In their arguments, they apply Legendrian Reidemeister moves (Theorem 3.1). By one of those moves, a pair of cusps are canceled. In our case, such cancellations are impossible by the Engel Reidemeister moves (Theorem 3.2). As a result ,we obtain a front of type L,,t with some pair of cusps as in move I of the Legendrian Reidemeister move (Theorem 3.1). We should remark that such a pair of cusps does not affect any other Engel Reidemeister moves. In fact, a regular curve can move through the pair of cusps freely (See Figure 21).

Fig. 21.

cusps do not affect

5.2. Deformations of fronts

In this subsection, we deform a front of type L,,t with certain pairs of cusps to a front of type Fk by Engel Reidemeister moves. Our goal is a front of type Fk. Before describing the required moves, we observe important properties of the pairs of cusps in Legendrian Reidemeister move I. As we checked above, a regular curve can move through the pair of cusps freely. Further, a pair of cusps can move through a cusp. It is realized by canceling and

26

J. Adachi

creating a zig-zag by Engel Reidemeister move I (see Figure 22). Last of

Fig. 22. cusps move through a cusp

all, two consecutive pairs of cusps are canceled if they are on different side of a curve each other (see Figure 23).

Fig. 23.

cancellation of cusps

Now, we describe the move from a front of type L,,t with certain pair of cusps to a front of type 4 . First, we reduce a front of type L,,t to that of type LO,^. As is mentioned above, those pairs of cusps do not affect to Engel Reidemeister moves. Therefore, we can apply Engel Reidemeister moves directly to a front of type L,,t ignoring those pairs of cusps. We can cancel zig-zags on the right hand side of an L,,t-type front by the Engel Reidemeister move I, and make two consecutive twists one by move 11-(2). Then, a front of type L,,t is reduced to that of type Lo,o,that is, a flying saucer or lips. Now, we deform such a front to a front of type Fk. We can cancel one pair of cusps inside of Loptype front and one pair of cusps outside of L0,otype front together by the move in Figure 23. Thus, we obtain a front of type LO,Owhich has pairs of cusps only inside or outside. In the first case, we obtain a front of type F k by pulling one cusp of each pair to the outside as in Figure 22 by the Engel Reidemeister moves 11-(1), (2). In the second case, we consider in two steps. If there is only one pair of cusps outside of an Lo,o-type front, we obtain an Fo-type front by the move in Figure 22. If there are k 1 > 1 pairs of cusps outside, we obtain an Fo-type front with k pairs of cusps. Applying the move in Figure 22 to the other side, we

+


27

obtain a -Lo,o-type front with k pairs of cusps inside and one pair of cusps outside, where -Lo,o is a type of fronts which is the same as LO,Obut with the reversed orientation. Then, canceling a pair of pairs of cusps, we obtain a front of type -Lo,o with k - 1 pairs of cusps inside. We should remark that the orientation is reversed by the operation above (see Figure 24). In

Fig. 24.

cusps outside

the same way as in the first case, we obtain a front of type Fk. Last of all, in order to finish the proof of Theorem 1.3, we need an argument similar to Lemma 4.2. The important operation is to make a Legendrian knot with singular points induced from a horizontal loop shrink to the skeleton on the spanning disk in Subsection 5.1. Since we obtained a disk with a characteristic foliation in elliptic-like form, we can deform the spanning disk so that it is an so called exceptional spanning surface almost every where. The differences are pais of elliptic and hyperbolic points and additional hyperbolic points (see Subsection 5.1). Note that we obtain such a surface by small deformation of an exceptional spanning surface. We can apply the same argument to the elliptic corners as the argument for boundary smooth elliptic points. For each Legendrian knots without additional pairs of hyperbolic and elliptic points and hyperbolic points shrinking to the skeleton, we can make additional points. Then, the given Legendrian knot shrinks sufficiently close to the skeleton. Thus, Theorem 1.3 has been proved. 6. Observations

We observe some related issues and some generalization. 6.1. Variants of Whitney’s theorem

We obtain a variant of the Whitey theorem concerning regular homotopy classes of plane curves as a corollary of Theorem 1.3.

J. Adacha

28

We, first, review the original Whitney theorem (see [S]). Let

fo, f l : S 1 -+ R2 be immersions. These two immersions are said to be regularly homotopic if there exists a mapping H : S1 x I -+ R2 which satisfies the following conditions: 0 0

H ( s ,0) = fo(s), H ( s , 1) = fl(s) for any s E S1, setting ft(s) := H ( s ,t ) ,fs : S1-+ R2 is an immersion for any t E I .

A rotation number is defined as the degree of an immersion of an oriented circle to a plane, which is an invariant for regular homotopies of immersed circles. Let r ( f ) denote the rotation number of an immersion f : S 1 -+ R2. The following theorem is known as Whitney's theorem.

Theorem 6.1 (Whitney). Let fo, f l : S1 -+ R2 be immersions. They are regularly homotopic if and only if they have the same rotation number: r ( f 0 )= r ( f 1 ) .

A version of the Whitney theorem is obtained as a corollary of Theorem 1.3. In Theorem 1.3, we dealt with horizontal homotopies of embeddings of circles horizontal to the Engel structure. Let us recall that we obtain immersions of S1 to R2 by taking projections of horizontal loops (see Subsection 2.3). Then, by taking horizontal projections, we obtain a regular homotopy from a horizontal homotopy. We should remark that such an immersion f := p2 o p l o y: S1 -+ R2, obtained from some horizontal loop y, are characterized in terms of volumes as follows (see Subsection 2.3). An immersed closed curve g : S1 -+ R2, g(s) = (gl(s),g4(s)), is a horizontal projection of some Legendrian immersed knot in the standard contact 3-space if and only if it satisfies the following condition (see equations (5)): (1)

/2T

g4(s)dg1(s)= 0.

0

Furthermore, g is an immersed loop in the standard Engel space if and only if it satisfies the following condition additionally (see equation (6)): (2)

12T (I'

)

g4(U)dgl(u) d g l ( s ) = 0.

Condition (1)implies that the algebraic area bounded by the curve g is zero. Condition (2) implies that the algebraic area bounded by the plane curve g(s) := (gl(s),gg(s)), where g3(s) := J,"g4('1L)dgl(u), is zero. As a result, we obtain the following as a corollary of Theorem 1.3. We should remark

Integral curves for contact and Engel structures 29

that the statements of Theorem 1.3 is valid even for immersed horizontal loops.

Theorem 6.2. Let fo, f l : S1 --+ R2, fi(s) = (fj(s),ff(s)) , i = 0,1, be immersions satisfying conditions (1) and (2). They are homotopic among immersed circles satisfying the conditions (1) and (2) above if and only if they have the same rotation number: r ( f 0 )= r ( f 1 ) . 6.2. General Goursat structure case

We can generalize the arguments of this article from the arguments on contact and Engel structures to those on general Goursat structures. First, we review the definition of a Goursat structure. Let M be a manifold, and D c TM a distribution of rank 2. We defined distributions D2, D3 in Subsection 2.1. Furthermore, we define distributions DZ, i = 4,5,6,. . . inductively as Di := Di-l [Die', Di-l]. The distribution D is called a Goursat structure on M if it satisfies the following conditions: (1) there exists a positive integer s E Z for which D" = TM, (2) rankD;+l=rankD;+l,i= 1,2, ..., s - 1 , a t a n y p o i n t p ~ M.AGoursat structure on 3-dimensional manifold is a contact structure. And an Engel structure is considered as a Goursat structure on a 4-dimensional manifold. The standard Goursat structure DOon Rn is given as

+

Do = ( d 2 2 - 2 3 d X l = 0 ,

d 2 3 - ~ 4 d 2 1= 0,. . .

, dzn-1- ~ n d =~0)i

An important property of Goursat structure is that there are singular points where Goursat structures are not locally equivalent to the standard one if the dimension of the manifold is greater than 4. Classification problem of the germs of Goursat structures is still open (see [22]). Similarly to the case for an Engel structure, we can define a horizontal loop and some related notions. An invariant, the rotation number, under horizontal homotopies is also defined in a similar way to the Engel case. Rn-', . . . ,pn-2 : R3 --+ Horizontal projections p l : Rn -+ Rn-ll p2 : R"-' R2, are defined as canonical projections pi: (21,xi+l,2 i + 2 , . . . ,xn) H ( z I , I c ~ +.,z,), ~ , . . i = 1 , 2 , . .. , T I - 2. The rotation number for a horizontal loop y : S1 4 (R", DO) is defined as a degree of an immersion p,-2 o pn-3 o . . . o p l o y: S1 --+ R2. Let r ( y ) denote it. Vertical projections 7r1: R" -+ IWn-l, 7r2 : Rn-' 4 RnV2,. . . , 7rn-2 : R3 4 R2, are defined as canonical projections 7ri: (21, 2 2 , . . . ,xn-i, xn-i+1) H ( 2 1 , 2 2 , . . . ,xn-i),

-

30

J. Adachz

i = 1 , 2 , . . . , n - 2. The image rn-2

oy: S1 -+ B2 by these vertical projections of a horizontal loop y : S1 --+ (Rn, DO)is called a front of y. Generically, it is an immersion with cusps of type ( 2 , 2 n - 3 ) . o r n - 3 o ...07r1

We can generalize some results in this article for Engel structures to the case of Goursat structures. We remark that ~ " - 4 0 . . .07rloy: S1 --+ (R4, E ) is a horizontal loop for the standard Engel structure with singularities, and that 7rn-3 o ?rn-4 o . . . o 7r1 o y: S1 --+ (R3, C ) is a Legendrian knot for the standard contact structure with singularities. Then, we can apply the same arguments concerning fronts as the Engel case since we can lift a front to a horizontal curve in (Rn, DO)by taking further Cartan prolongations (see Subsection 2.1). As a corollary of Theorem 1.3, we obtain the following:

Corollary 6.3. Let y 0 , y l : S1 (Rn,Do) be horizontal loops in the standard n-dimension.al Goursat space, n 2 4. These loops yo and y1 are horizontally homotopic if and only if their rotation numbers coincide: ?- ( 7 0 ) = (71) -+

We can also apply similar arguments concerning horizontal projections. The conditions under which an immersed circle g ( s ) = ( g l ( s ) , g n ( s ) ) ,s E S1 [0,27r]/-, into B2 can be lifted to an immersed horizontal loop in (R", DO)is described as follows. We define functions gi (s),i = 2,3, . . . ,n- 1, inductively as follows:

gn-l(s) :=

/'"

gn(U)dgl(u),

I" 0

gi(s) :=

gi+l(u)dgl(u), i = n - 2 , n - 3 , . . . , 2 .

Then the required condition is

gi(27r) = 0,

i = 1,2, . . . ,n - 1,n.

(7)

Then, as a corollary of Theorem 1.3, we obtain the following:

Corollary 6.4. Let fo, f1: S1 -+ R2,fi(s) = (f/(s),fF(s)) , i = 0 , 1 , be immersions satisfying condition (7), n 2 4. They are homotopic among immersed circles satisfying the conditions (7) above if and only i f they have the same rotation number: r(fo) = r(f1). We remark the case when n = 4 is the case related to horizontal immersed circles in the Engel space.

Integral curves for contact and Engel stmctures 31

Acknowledgements This work is supported by Grants-in-Aid for Young Scientists (B), No. 17740027, The Ministry of Education, Culture, Sports, Science and Technology, Japan, and 21st Century COE Program “Mathematics of Nonlinear Structures via Singularities” Hokkaido University, and 21st Century COE Program “Topological Science and Technology” Hokkaido University.

References 1. Ya. Eliashberg, M. Fraser, Classification of topologically trivial Legendrian knots, CRM Proc. Lecture Notes, 15,Amer. Math. SOC.,Providence, RI, 1998, 17-51. 2. J. Adachi, Classification of horizontal loops in the standard Engel space, (preprint). 3. E. Giroux, GEomt!trie de contact: de la dimension trois vers les dimensions sup&rieures, Proceedings of the International Congress of Mathematicians, Vol. I1 (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 405-414. 4. K. Honda, T h e topology and geometry of contact structures in dimension three, to appear in the ICM 2006 proceedings. 5. R. Montgomery, Engel deformations and contact structures, Northern California Symplectic Geometry Seminar, Amer. Math. SOC.Transl. Ser. 2, 196, 1999, 103-117. 6. J. Adachi, Engel structures with trivial characteristic foliations, Algebr. Geom. Topol. 2 (2002), 239-255. 7. T. Vogel, Existence of Engel structures, thesis, Ludwig-MaximiliansUniversitat Munchen (2004). 8. H. Whitney, O n regular closed curves in the plane, Compositio Math. 4 (1937), 276-284. 9. M. Adachi, Embeddings and immersions, Translations of Mathematical Monographs 124,American Mathematical Society, Providence, RI, 1993. 10. M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 9, Springer-Verlag, Berlin, 1986. 11. D. Bennequin, Entrelacements et Equations de P f a f f , Asthrisque, 107-108 (1983), 87-161. 12. Ya. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, 1993, 171-193. 13. J. Etnyre, Legendrian and Transversal Knots, to appear in the Handbook of Knot Theory. 14. R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, 18,Springer-Verlag, New York, 1991. 15. G. Ishikawa, Classifying singular Legendre curues by contactomorphisms, J. Geom. Phys. 52 (2004), 113-126.

32

J. Adachi

16. M. Zhitomirskii, Germs of integral curves in contact 3-space, plane and space curwes, Isaac Newton Inst., Preprint NI00043-SGT, 2000. 17. J. Swiqtkowski, O n the isotopy of Legendrian knots, Ann. Global Anal. Geom. 10 (1992), 195-207. 18. E. Giroux, Convezite' e n topologie de contact, Comment. Math. Helv. 66 (1991), no. 4, 637-677. 19. Ya. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165-192. 20. M. F'raser, Classifying Legendrian knots in tight contact 3-manifolds, thesis, Stanford University, 1994. 21. A. Kawauchi, A survey of knot theory, Birkhauser Verlag, Basel, 1996. 22. R. Montgomery, M. Zhitomirskii, Geometric approach to Goursat flags, Ann. Inst. H. PoincarB Anal. Non LinBaire 18 (2001), 459-493.

33

On the realisation of a map of certain class as a desingularization map Karim Bekka Institut de Recherche Mathematique de Rennes, Uniuersite' de Rennes 1 Campus Beaulieu, 35042 Rennes cedex, France E-mail: karim. bekka0uniu-rennesl. fr

Toshizumi Fukui Department of Mathematics, Saitama University 255 Shimo-Okubo, Sakura-ku, Saitama 338-8570, Japan E-mail: tfukuiQrimath.saitama-u. ac.jp

Satoshi Koike Department of Mathematics, Hyogo University of Teacher Education 942-1 Shimokume, Kato, Hyogo 673-14S4, Japan E-mail: [email protected]

In this article we discuss some representation problem, related to the equisingularity problem for real algebraic singularities. Given a polynomial mapping f : Wn -+ WP with f(0) = 0, we show that f is represented by a blowing-up and, in the case n 2 p , is represented by a desingularisation map for some real algebraic variety. We further treat these properties also in the polynomially bounded o-minimal structure. Keywords: polynomial mapping, minimality, polynomially bounded.

Nash

mapping,

desingularisation,

o-

1. Introduction

Given a polynomial family of polynomial mappings (more generally, a Nash family of Nash mappings) with a semialgebraic parameter space, the family of real algebraic varieties (resp. Nash varieties) defined by their zeros admits a Nash trivial simultaneous resolution after taking a finite subdivision of the parameter space [11,14].Then it is natural to ask the following question: Given a Nash trivial simultaneous resolution for a family of algebraic varieties (or Nash varieties) with a semialgebraic parameter space

34

K . Bekka, T. FZlkui €4 S. Koike

J , is there a thin semialgebraic subset K of J such that the Nash trivialisation upstairs induces a semialgebraic triviality downstairs of the family of algebraic varieties over any connected component of J - K? The answer is not always yes. In [17],I. Nakai constructed a family of polynomial 0 ) + (R2, 0 ) ) in which even local topological moduli mappings: {fa : (R3, appear. The second author [lo] constructed a family of 4-dimensional real algebraic varieties { Va}in R6 and its Nash trivial simultaneous resolution, such that the restriction of the resolution mapping to the intersection of the strict transform of the zero-set and the exceptional divisor at some point is the Nakai family. Generalising the construction, the first and third authors proved that any polynomial mapping can be represented by the restriction of a blowing-up with smooth centre to the intersection of the strict transform of the zero-set and the exceptional divisor at some point. We have announced it in the survey [15] on Blow-Nash triviality. Let f : R" + RP be a mapping of some class, say A, such that f(0)= 0. Here we consider the polynomial class, the Ck Nash class and the Ck definable class in the polynomially bounded o-minimal category for k = 1,2,. . . ,00, w . We say that f is represented by a blowing-up, if there are a variety V of class A in R", m > n, a blowing-up n : M -+ Rm with centre C c V of class A and a diffeomorphism of class A, a : Rn --t V' n E n U where U c M , V' is the strict transform of V by n and E is the exceptional set such that nlv,nEnu o a = f with RP = RPx {O}"-P c R". We further say that f is represented by a desingular isation map, if the same assumptions as above are satisfied for a desingularisation mapping of some variety, namely, there are a variety V of class A in R", m > n, a desingularisation n : M + R" of V in R" of class A and a diffeomorphism of class d,a : R" + V' n E n U such that nlv,nEnuo u = f . In this paper, we discuss the problem whether a map of class A can be represented by a blowingup or more strongly by a desingularisation map. We have already shown the theorem of representation by a blowing-up for polynomial mappings. In the next section, we give a slightly different proof of it and improve the result to the theorem of representation by a desingularisation map in case n 2 p . In $3, we prove the theorem of representation by a blowing-up for Ck,k < 00, definable mappings in the polynomially bounded o-minimal category. The same result holds also for Ck,k < 00, Nash mappings. We describe it in $4 and make some remarks on the theorem of representation by a desingularisation map for C"" Nash mappings in case n 2 p .

On the realisation of a map of certain class 35

2. Polynomial mappings

As stated in the introduction, we have proved the theorem of representation by a blowing-up for real polynomial mappings. Here we give a proof of it in a slightly different form from [15]so that the arguments work also for complex polynomials.

Theorem 2.1 ( [15]Assertion 7.3). Let f = ( f l , . . . ,fp) : (Rn,O) + (RP, 0 ) be a polynomial mapping such that fj is not identically zero for some 1 5 j 5 p, and let m be a n arbitrary positive integer with m 2 n+p+l. Then there are an algebraic variety V in R" and a blowing-up 7r : M -+R" with a n ( m - n - 1)-dimensional centre such that 7rlv,nE at some point Q E M is f. Proof. Let a

+ +

where Q = ( ~ 1.., . ,an) and x" = 2"' . . . x Q m . Set Jal = a1 an, and let dj be the degree of f', 1 5 j 5 p . Define gj : (Rn+l,0) + (R,O), 15jIP,bY

Q

Then each gj is a homogeneous polynomial of degree d j . Let Y = ( Y I , * * 7 Ym-n-1) E ~ m - n - l. Define the variety

in R"

j E { I , .. . ,p) and g j ( t , x ) - yjtdj = 0 , j E { p + l , ..., m - n - 1 ) Y j = 0,

+

Then V is an algebraic set of dimension greater than or equal to max{n 1 , p ) (since V contains apdimensional subset ( 2 1 = . . . = xn = t = yp+l = . . . = yn = 0)). We can notice that V - { t = 0 ) is smooth and of dimension n 1 (since the rank of the jacobian of the system of equations defining V is maximal at these points). Consider the blowing-up 7r : M Rm with smooth centre C = { ( y , t , s ) E R" ; x1 = . * * = x, = t = 0 ) . Using a suitable chart U , we can express 7r as follows:

+

-

36

K. Bekka, T. Fukui €4 S. Koike

In U , the exceptional divisor E is {u E U ; urn-" = 0) and the strict transform V' of V is given by: V'=

{u E U ; uj=fj(urn-n+l,

...,u rn), j~

{ l l . . . , ~ }

and

j~ { p + l , ..., m - n - l }

u j = 0,

+

1.

It is a smooth submanifold of U of dimension n 1. Since E is transverse to V ' , V ' n E is a smooth submanifold of dimension n of U and can be identified with the ( U ~ - ~ + I ,... ,urn)-space. Then in U , : V' n E -+ RP x {O}rn-P is given by: 7rlV'"E

YI = f i ( ~ r n - n + l , . . . , ~ r n ) ,. . * , ~p = f p ( ~ r n - - n + l , . * . , ~ m ) . Thus we can regard 7rlv,nE at 0 E U as f. In other words, we can construct 0 a diffeomorphism o : R" -+ V' n E n U such that 7rlv,nEnu 0 (T = f. We make some remarks on the results above.

Remark 2.2. In the proof of Theorem 2.1, while V' is smooth in U , it may have singularities a t 00 outside U . Assume that: dimV

= n+

1 and d im C 5 n.

(*I

Then using the desingularisation theorem of Hironaka [12], we can desingularise V' without touching a neighbourhood of 0 E V' n E n U . Namely, we can obtain the statement of Theorem 2.1 for a desingularisation 7r of V (not just a blowing-up 7r of Rm).

+

In case dimV > n 1, V' in U will disappear as the strict transform of V after desingularisation of V. In what follows, we consider some special cases where conditions (*) are fulfilled. In case n = 1, the zero-set f - ' ( O ) is already desingularised. Therefore we assume n 2 2. Let n 2 p and m = n + p + 1 in Theorem 2.1. Then we have d im C = m - n - 1 = p 5 n.

Claim: If p 5 2, then dim V

=n

+ 1.

Proof. In fact, as we have already noticed, V={(y,t,z) EWn+p+l; gj(t,z)-yjtdj = o , j E { 1 , " ' ) p } } is of codimension p outside the hyperplane {t = 0). On the other hand, V n {t = 0) is defined by the system gj(o,2)=

c

Jal=dj

ag)z"=O,

j € { l, . . . , p } .

O n the realisation of a map of certain class 37

Since at least one of these polynomials is not identically zero, the codimension of V n { t = 0 ) in { t = 0) it is at least 1, which means it is at least of codimension 2 in Rm. Then we have n 1 2 dim V 2 max{m - 2, n 1). Now if p 5 2 and m = n + p + 1, then we have

+

+

+ p - 1, n + 1) = n + 1.

n + 1 5 dim V 5 max{n Remark 2.3.

i) The F'ukui observation for the Nakai family in [lo] is a special case where n = 3, p = 2 and m = 6. ii) If p 2 3, the dimension of V can be bigger than n 1. In fact, if for example fj(z) = z1 for each j E ( 1 , . . ., p } , then we have dim V = m - 2 = n + p - 1 > n + 1.

+

In the following, we show the theorem of representation by a desingularisation map, modifying V in order to satisfy conditions (*) . Theorem 2.4. Let f = ( f l , . . . , f p ) : (Rn,O) 4 (Rp,O),n 2 p , be a polynomial mapping such that f j $ 0 for some 1 2 j 5 p . T h e n there are a n algebraic variety W in Rm, m = n p 1, and a desingularisation ii : M 4 Rm of W in Rm such that iilwlnEat some point Q E M i s f.

+ +

Proof. In the proof of Theorem 2.1, let

x = {(y, t ,z) E R"

; g j ( t , z) - y j t d j = 0, j E { 1 , . . .,p } } ,

and let Y = X - { t = 0). Then Y is an algebraically constructible set in R" of dimension n 1. We denote by ? the algebraic closure of Y . Set W = r U C. Then W is an algebraic variety of dimension n 1 in R" such that C c W . Note that dim C = p 5 n < n 1 = dim W. Similarly to the above, we consider the blowing-up with centre C,7r : M 4 R". Then, in U the exceptional divisor E is = 0) and we can regard ~ l at 0 E U as f. We obtain the desingularisation of W as in Remark 2.2.

+

+

+

Remark 2.5. The corresponding result to Theorem 2.4 holds also for complex polynomial mappings f = (fi, . ,fp) : (en, 0) -+ (CP,0 ) , n 2 p . a

e

3. o-minimal mappings

In this section we establish a version of Theorem 2.1 in the o-minimal category. We will assume the reader familiar with the basic facts about

~

38

K . Bekka, T. fikui €5 S. Koike

o-minimal structure. The standard references are L. Van den Driss [8], L. Van den Driss and C. Miller [9] and M. Coste [6]. Let us first recall the definition of an o-minimal structure extending the field (R, .).

+,

Definition 3.1. Let S = U n ~ ~ S where n, for each n E N, S, is a family of subsets of R". We say that the collection S is an o-minimal structure on (R, +, .) if: 1) each S, is a boolean algebra 2) if A E S, and B E S, then A x B E Sn+m 3) let A E S,+, and 7r : Rnfm -+ R" be the projection on the first n coordinates, then 7r(A)E S,. 4) all algebraic subsets of R" are in S, 5) the elements of S1 are the finite unions of points and intervals.

A subset A of R" which belongs to S, is called a definable set in S. A map f : A 4 Rm is definable in S if its graph is a definable subset of Rn x R" in S, if in addition, it is Ck for some k E N,we call it a C k definable map in S. Let S be an o-minimal structure on (R,+,.). We recall from [9] the following notation:

Notation 3.2. Let p be a natural number. Let @; denote the set of all odd, strictly increasing bijections cp : R + R CP definable in S and pflat at 0 (that is cp(')(O) = 0 for 2 = 0 , . . . , p ) .

-

We quote also from this paper the following lemma (Lemma C.7. page 523):

Lemma 3.3. Let f : A x R* R be a definable function in S , A c R". Then, for any p E W, there exists cp E such that 1imt-o p ( t )f (z, t ) = 0 for each x E A .

A Ckversion of this lemma is given in the following: Lemma 3.4. Let f : U x R* -+ R be a Ck definable function in S , and let U be an open subset of Rn. Then, for any p E N,there exists cp E @; such that the function

is a

C k definable function.

O n the realisation of a map of certain class 39

Proof. Let h : U x R* + R denote any function from the collection of partial derivatives:

{Dijc,t,f ;

Q:

=

(w,.. . ,a,,

a,+l) E

Nn+' and IQI I k }

such that limt.+o O(t)h(x,t ) = 0 for By Lemma 3.3, there exists 0 E 0 each x E U . Then cp := 02k+1 satisfies the needs.

-

+,

Definition 3.5. A structure S on the field (R, .) is polynomially bounded if for any function f : R R definable in S, there exists N E N such that

If(t)lI tN for all sufficiently large t.

Lemma 3.6. If S is polynomially bounded, then f o r any cp E exists d E N and real number E > 0 such that:

a:,

there

Icp(t)l L td f o r any t E ( - 6 ,

E).

1 Proof. We take 8(s) := - where s =

+. Since 0 is

definable in a

cp(?>

polynomially bounded structure, there exists d E N and M E N such that lO(s)l I sd for Is1 > M . Therefore Icp(t)l L td for any t E ( - E , E ) with 0

E = M L '

Using the lemmas above, we can show the theorem of representation by a blowing-up for C k , Ic < 00, definable functions in a polynomially bounded o-minimal structure.

-

Theorem 3.7. Let S be a polynomially bounded o-minimal structure o n the field (R, .). Let f = (f1, . . . ,f p ) : (Rn,O) (RP, 0 ) be a C k definable mapping which i s not identically zero. T h e n there exist a C k definable variety V in Rm, m 2 n p 1 , and a blowing-up IT : M Rm with a n ( m - n - 1)-dimensional smooth definable centre C such that at some point Q E M is f.

+,

+ +

-

Proof. Since the components f j ( 1 5 j 5 p ) of f are Ck definable, it follows from lemmas 3.4 and 3.6 that for each j E ( 1 , . . ., p } there exists a positive integer dj such that the germ of function gj : (EXn+', 0 ) -+ (R, 0 ) defined by g j ( x , t ) := t d j f ( T ) is of class C k .Then each gj is a homogeneous C k definable function of degree dj and for all t , gj (0,t ) = 0.

40

K . Bekka, T . Fukui €4 S. Koike

Let y = (yl,. . .

E Rm--n-l . Define the variety V in R" by

gj(x,t) - y j t d j = 0, j E (1,. . . , p } and j~ { p + l , ...,m - n - l } Yj = 0,

+

Then V is a definable set of dimension greater than or equal to max{n 1,p}. We can also notice that V - {t = 0) is smooth and of dimension n+ 1, as above. Consider the blowing-up 7r : M R" with centre

c = {(y,t,x) E R"

-

; x1 = .. * = z, = t = 0).

The centre C is a smooth (m - n - 1)-dimensional definable manifold. Then the remainder of the proof follows in a similar way to the proof of Theorem 2.1. 0

+

Remark 3.8. In case n 2 p , we can choose V to be of dimension n 1 (= dimV'). In fact, it is enough to take instead of V, the definable set of dimension n 1, .rr(cl(V' - E ) ) U C,here cl means the topological closure (it is the same as the o-minimal closure).

+

Recently, M. Shiota has established a desingularisation theorem in the 0minimal category ( [22]),using "hole-blow ups" introduced by himself. Applying this theorem at infinity as in the proof of Theorem 2.4, we may have a kind of theorem of representation by a desingularisation map in the polynomially bounded @minimal category. Considering only polynomially bounded &minimal structure on the field (R, .) is not restrictive. Let us give some examples.

+,

Example 3.9. The smallest o-minimal structure on (R, +, .) is the structure of all semi-algebraic sets in R", n E N. Example 3.10. Ran is the structure generated by all restricted analytic functions i.e. functions f : Rn + R , n E N,such that f is a restriction of an analytic function on [-1, 11" and identically 0 out off [-1, 11". This structure consists of the "global subanalytic sets" i.e. subanalytic sets of P"(R) when we identify R" with an open set of P"(R),n E N. In particular, it contains the compact subanalytic sets. Example 3.11. is the structure generated by all restricted analytic functions and the power functions Pr(x) :=

xT if x 2 0, 0 if x < 0.

O n the realisation of a m a p of certain class

41

where r E R.

Example 3.12. (The quasianalytic Denjoy-Carleman structure) Let M = ( M o ,M I ,. . . ) be a sequence of real numbers with 1 5 MO 5 MI 5 ... and B = [al,b l ] x . .. x [a,, b,] with ai < bi for i = 1 , . . . ,n.We let C g ( M )be the collection of all functions f : B -+ R for which there exist an open neighbourhood U of B , a CM function g : U + R and a constant A > 0 (all depending on f) such that f = g l B and

where ) a ):= a1

+ .. . + an. We call @%(Ad) the Denjoy-Carleman class

on B associated to M . (If Mi = i! for all i 2 0, then C i ( M ) is the class of all real-valued functions on B that can be extended analytically to an open neighbourhood of B.) The sequence M is called logarithmically convex, if M: 5 Mi-lMi+l for all i > 0, and strongly logarithmically convex if the sequence ( M i / i ! )is logarithmically convex. The class @ i ( M )is called quasianalytic if for any f E @",M) and any z E B , the Taylor series of f at z uniquely determines f among all functions in @",M). It is well known (see [20] ) that C g ( M )is quasianalytic if and only if

&

In general, the classes @ i ( M )will not be closed under differentiation, however, the classes @ B ( M ):= U ~ o @ ~ ( M (are j ) always ) closed, where M ( j ) := ( M j ,Mj+l,. . .). For each n E N and f E @ [ - l , l ~ n ( Mwe ) , define Rn -+ R by := f(z)if 5 E [-1, lIn and := 0 otherwise. Let IWc(M) be the expansion of the real field by all ffor f E @1-1,lln and n E N. In [19], J.-P. Rolin, P. Speissegger and A. J. Wilkie have proved the following result: If M is strongly logarithmically convex and satisfies ( l ) ,then the structure R ~ ( Mis) an o-minimal polynomially bounded structure. It is also worth mentioning that in this framework, E. Bierstone and P.D. Milman have established a desingularisation theorem (see [5]).

7(z)

7:

F(z)

Remark 3.13. In [21] M. Shiota introduced the notion of X-sets which satisfy the following axioms: let X be the family of subsets of Rn,n E N, such that:

42

K. Bekka, T. Fukui d 5'. Koike

1) every algebraic set is in X 2) if X I c R" and X2 c R" are elements of X, then X I n X2, X 1 - X 2 and X I x X2 are elements of X 3) if X C R" is an element of X and p : R" R" is a linear map such that the restriction of p to is proper, then p ( X ) is an element of X 4) each element of X,X in R is locally a finite union of points and open intervals.

x

--f

An element of X is called a X-set and a mapping f : X + Y is a X-map if X , Y and the graph of f are X-sets. For example, semi-algebraic sets, subanalytic sets, o-minimal sets generated by finitely analytic functions and exponential functions are X-sets. For "polynomially bounded" X-sets the same result holds as Theorem 3.7. Here the definition of polynomially bounded is the transcription of the definition in the o-minimal category, replacing definable function by X-function. 4. Nash mappings

Let r = 1 , 2 , . . ,00, w . A semialgebraic set M C R" is called a C' Nash manifold, if it is a C' (regular) submanifold of R". Let M c R" and N c R" be C' Nash manifolds. A C k mapping f : M N , k 5 r , is called a Ck Nash mapping, if the graph of f is semialgebraic in Rm x R". Note that a C" Nash manifold is a C" Nash manifold, and a C" Nash mapping is a C" Nash mapping (B. Malgrange [IS]). We call a C" Nash manifold and a C" Nash mapping a Nash manifold and a Nash mapping, respectively. Since any C kNash mapping belongs to any polynomially bounded structure, using a similar argument to the proof of Theorem 3.7, we can show the following result in the CkNash case, k E N. +

--f

-

Theorem 4.1. Let f = ( f l , . . .,f p ) : (R", 0) (RP, 0 ) be a Ck Nash mapping which is not identically zero. Then there exist a Ck Nash variety V in R", m 2 n p + 1, and a blowing-up 7r : M R" with an ( m - n - 1)-dimensional Nash manifold C as centre such that 7 r 1 ~ at , ~ some point Q E M is f .

+

-

For a semialgebraic function f : R" -+ R, there is a nontrivial polynomial P E R [ X 1 , .. . ,X,, Y ] such that P ( X ,f ( X ) ) = 0 for all X E R". The minimal degree of a polynomial P , as above, is called the complexity of f. R. Ramanakoraisina (see [IS])shows that the set of Nash functions in n

~

On the realisation of a map of certain class 43

variables having complexity I d forms a semialgebraic family of functions parametrised by a semialgebraic set of dimension less than (n+:+l). As a consequence of this, he shows that there exists an integer M(n,d) such that i f f : Rn -+ R is a C k Nash function and k 2 M(n,d)then the function f is actually a Nash function, i.e. analytic. We may expect in the Nash case the same type of result as Theorem 4.1 i.e. iff is a Nash function, then we can take for the variety V a Nash variety (i.e. analytic). If so, the theorem of representation by a desingularisation map for Nash mappings follows similarly to the polynomial case, since the desingularisation theorem for the Nash varieties has been established by H. Hironaka [12,13] and E. Bierstone - P.D. Milman [2-41. But our proof and more precisely the “homogenisation process” we use is unfortunately not applicable; because in this case the complexity is increasing with the differentiability k. The following will convince the reader:

Example 4.2. Let f : R X

dm* Then f ( i >= d tkff(3 = ,/-

cL k -A

is not

t

X

JR be the Nash function defined by f(x) = For any k E N,the function gk(t,X)=

r n ’

c”,therefore there exists no k E N such that g k

is a Nash fkction.

Nevertheless, we can transfer the results from the polynomial case (i.e. Theorems 2.1 and 2.4) to the Nash case using the following Artin-Mazur description of a Nash mapping (see [l]and [7]):

Theorem 4.3 (Artin-Mazur Theorem). Let f : M JRP be a Nash mapping, where M C RN is a Nash manifold. Then there exist an algebraic variety V c Rk and a Nash embedding ,B : M t Reg(V) and a polynomial mapping g : Rk ---f JRP such that f = g o 0, where Reg(V ) is the smooth part of v. ---f

Let f : Rn t RP be a Nash map. Using the Artin-Mazur Theorem and Theorem 2.1 we obtain the following commutative diagram

44 K. Bekka, T. h k u a d S. Koike

Here, the polynomial mapping g : Rk 4 RP and the Nash embedding ,O are given by the Artin-Mazur Theorem, and V‘, E , U ,d = T~V,,-,E,-,U and the isomorphism u are the result of the application of Theorem 2.1 to g.

Acknowledgements This article was written up while the first author was visiting Hyogo. He would like to thank Hyogo University of Teacher Education for its support and hospitality. This research was partially supported by a Grant-in-Aid for Scientific Research (No. 15540071) of the Ministry of Education, Science and Culture of Japan.

References 1. M. Artin and B. Mazur, O n periodic points, Ann. Math. 81 (1965), 82-99. 2. E. Bierstone and P.D. Milman, Uniformization of analytic spaces, Journal of Amer. Math. SOC.2 (1989), 801-836. 3. E. Bierstone and P.D. Milman, A simple constructive proof of canonical resolution of singularities, Effective Methods in Algebraic Geometry, Progress in Math. 94, Birkhauser, 1991, pp. 11-30. 4. E. Bierstone and P.D. Milman, Canonical desingularisation in characteristic zero by blowing u p the m a x i m u m strata of a local invariant, Invent. math. 128 (1997), 207-302. 5. E. Bierstone and P.D. Milman, Resolution of singularities in DenjoyCarleman classes, Selecta Math., New Series 10 (2004), 1-28. 6. M. Coste, An introduction to o-minimal geometry, Instituti editoriali e poligrafici internazionali, Pisa 2000. (http://name .math.univ-rennesl .fr/michel. coste/polyens/OMIN.pdf) 7. M. Coste, J. M. Ruiz and M. Shiota, Global problems o n Nash functions, Rev. Mat. Complut. 17 (2004), 83-115. 8. L. Van den Dries, Tame topology and o-minimal structures, London Math. SOC.Lecture Notes series 248,Cambridge University Press, 1998. 9. L. Van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), 497-540

On the realisation of a map of certazn class 45 10. T. Fukui, Introduction to toric modifications with an application to real singularities, RIMS Kokyuroku 1122 (2000), 96-114. 11. T. Fukui, S. Koike and M. Shiota, Modified Nash triviality of a family of zerosets of real polynomial mappings, Ann. 1nst.Fourier 48 (1998), 1395-1440. 12. H. Hironaka, Resolution of singularities of an algebraic variety ouer a field of characteristic zero: I, 11, Ann. of Math. 79 (1964), 109-302. 13. H. Hironaka, Idealistic exponents of singularity, Algebraic Geometry, J.J. Sylvester Sympos., Johns Hopkins Univ., Baltimore 1976, pp. 52-125. 14. S. Koike, Nash trivial simultaneous resolution for a family of zero-sets of Nash mappings, Math. Zeitschrift 234 (2000), 313-338. 15. S. Koike, Finiteness theorems on Blow-Nash triviality for real algebraic singularities, Banach Center Publications 65 (2004), 135-149. 16. B. Malgrange, Ideals of diflerentiable functions, Oxford Univ. Press, 1966. 17. I. Nakai, O n topological types of polynomial mappings, Topology 23 (1984), 45-66. 18. R. Ramanakoraisina, Complexite' des fonctions de Nash, Comm. Algebra 17, 1395-1406 (1989). 19. J.-P. Rolin, P. Speissegger and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, Journal of Amer. Math. SOC. 16 (2003), 751-777. 20. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987. 21. M. Shiota, Geometry of Subanalytic and Semi-algebraic Sets, Progress in Math. 150,Birkhaiiser, Boston, 1997. 22. M. Shiota, Compactification of manifolds definable in o-minimal structure, preprint .

46

Hermitian pairings and isolated singularities Jonathan A. Hillman School of Mathematics and Statistics University of Sydney, NSW 8006 Australia E-mail: jonhOmaths.usyd.edu.au unuw.maths.usyd. edu. au We suggest a reformulation of the problem of determining algebraically the Blanchfield pairing (Seifert form) of the link of an isolated singularity, in terms of seeking an hermitian isomorphism of the local Gaul3-Manin connection with its dual in the category of such connections. Keywords: Blanchfield pairing, hermitian pairing, isolated singularity, knot, link, meromorphic connection.

1. Introduction Let f : (V,0) -+ (C, 0) be an analytic function defined on an open neighbourhood of 0 in Cn+' and with an isolated singularity at 0. The link L ( f ) = f-l(O) n S?+' is an ( n - 1)-connected (2n - 1)-manifold, and the complement Szn+' - f - l ( O ) is fibred over S1with ( n - 1)-connected fibre F , for E sufficiently small [20]. The isotopy type of the embedding of L ( f )in SZn+l is determined by the intersection pairing on H n ( F ;Z)together with the action of the monodromy [7,14].In particular, if L ( f ) is n-connected it is homeomorphic to S2n-1,and the pair (Szn+', L ( f ) )is a (2n - 1)-knot. (The enhanced intersection pairing is then equivalent to the Seifert form of the knot). The homology with complex coefficients and the characteristic polynomial of the monodromy may be obtained in terms of a connection on a relative de Rham module [4]. The purpose of this note is to reconsider the question LLhow may we best derive the enhanced intersection pairing from f?" in the light of expectations deriving from knot theory. Barlet used asymptotic expansions of integrals of wedge products of differential forms along f-'(s) for s near 0 in C to construct a pairing which is equivalent to

Hermitian pairings and isolated singularities 47

the complexified intersection pairing (with monodromy action) away from the summand on which the monodromy has eigenvalue 1 [2].In the classical case of plane curve singularities ( n = 1) Eisenbud and Neumann gave an iterative proceedure to construct the complexified Seifert form, in terms of Puiseux data [8]. However there remains the problem of finding a direct, purely algebraic construction in terms of local commutative algebra. For simplicity we shall concentrate on the case of plane curve singularities. In $2 we sketch how Poincar6 duality gives rise to the Blanchfield pairing for an odd-dimensional knot, which is the enhanced intersection form in the fibred case. In $3 we define meromorphic connection and state without proof some of the basic properties of the category of such connections. The key example is the local Gaufl-Manin connection associated to an isolated singularity, which we recall in $4.In the final section we suggest a reformulation of the basic problem in terms of seeking a natural “geometric” self-dual isomorphism from the local Gad-Manin connection to its dual. The interaction between knot theory and complex analytic geometry has been long and fruitful. (See [9] for early references, [5,8,12,17]and [20] for expositions, and [22] and [23] for some more recent work on plane curve singularities.) The formulation of the question in $4 is the only novelty in this note. 2. Hermitian pairings in knot theory

Let K : S1-+S3 be a knot with a tubular neighbourhood N % S1 x D 2 . The exterior of K is X = X ( K ) = S3 - in tN and the knot group is T = T K = r l ( X ( K ) ) .Let d be the commutator subgroup of T . The standard orientations of euclidean spaces R” (and the “outward normal last” convention for orienting S”-l = dDn c Rn)determine a preferred conjugacy class of meridians m for K (corresponding to loops { 1) x S1on dN = ax) and hence a generator for the abelianization 7rab = T / T ’ = H1(X; Z) 2 2. (Note that we shall use Z to denote the ring of integers, and 2 to denote the infinite cyclic group.) We may choose the homeomorphism N S1 x D2 so that S1 x { 1) c d N = d X is a longitude t for K , bounding a surface F in X and representing a generator of Ker(: H l ( d X ;Z) 4 H 1 ( X ;Z)). We may construct the infinite cyclic covering space X’ corresponding to 7r’ by splittingXalongF. (IfY=X-Fx(-S,d) thenL?Y=F-US1x[-l,l]UF+, and we set X’ = Y x Z / -, where ( f , n )E F+ x { n ) is identified with (f,n 1) E F- x { n l}.)In particular, if X fibres over S1 with fibre F then X‘ % F x R.

+

+

48 J . Hillman

Let A = Z [ t , t - l ] be the integral group ring of zab = 2 and let B ( K ) = H l ( X ; A ) be the homology with local coefficients A in degree 1. Then B ( K ) = H1(X’; Z) with the (left) A-module structure determined by the action of the covering transformation t corresponding to the meridians. Since X is homotopy equivalent to a finite 2-complex, x ( X ) = 0 and A is noetherian B ( K ) is a finitely generated A-module with a square presentation matrix. It follows easily from the Wang sequence of the fibration X 4 S 1 = K ( 2 ,1) corresponding to the abelianization : T + 2 that (t-l)B(K) = B ( K ) .Hence B ( K )is a torsion A-module andp.d.AB(K) = 1 (if B ( K ) # 0). Since ax‘ ”= S1 x Iw the module H l ( d X ;A) is generated by a lift of l , which bounds a copy of F in X’. Therefore B ( K ) E H l ( X ,d X ; A). Poincarb-Lefschetz duality and the Universal Coefficient spectral sequence give isomorphisms

H 1 ( X ,d X ;A)

g

H 2 ( X ;A)

Exti(B(K), A).

(The modules under the overlines are naturally right A-modules, and the overline denotes the conjugate left module structure, given by t.h = ht-’ for all h E H 2 ( X ;A) or Exti(B(K), A).) The Ext group is in turn isomorphic to Homh(B(K),Q(t)/A). Let D : B ( K ) --t HomA(B(K),Q(t)/A) be the composite of these isomorphisms. In particular, B ( K ) is Ztorsion free. (This also follows easily from the results in the previous paragraph). Let b K ( a , P ) = D ( P ) ( a )for all a,/? E B ( K ) .Then bK is A-linear in a and conjugate linear in P, and is nonsingular (since D is an isomorphism). It can be shown that bK is hermitian, i.e., that b K ( a , P ) = b K ( P , a ) for all a l p E B ( K ) . This pairing is the Blanchfield pairing for K . (See [ll] and [16].) If K is a fibred knot with fibre F then B ( K ) = H1(F;Z), with A-module structure determined by the monodromy, and bK is then equivalent to the intersection pairing on H l ( F ;Z) together with the isometric action of the monodromy [26]. For other knots the Blanchfield pairing may be computed in terms of the Seifert pairing associated to a choice of spanning surface for the knot. (The above construction is intrinsic, in that it does not depend on such a choice). If we use instead coefficients in a field IF we may identify the Blanchfield pairing with the cup product pairing of Hq(X’;P) with itself g F [21]. into H 2 , ( X f ,ax’;IF) In higher dimensions, if K is a (2q - 1)-knot and T is the Ztorsion submodule of H,(X; A) there is an analogous Blanchfield pairing on B,(K) = H,(X; A ) / T . (The submodule T is 0 if K is “simple”, i.e., if X’ is (q - 1)-


connected). This is a nonsingular (-l)q+l-hermitian pairing, and it is a complete invariant for simple (2q - 1)-knots. The Witt class of b~ in the appropriate Witt group is a complete invariant of knot concordance for all odd n > 1 [15]. If we consider instead an r-component link L : rS1 4 S3 each component determines a conjugacy class of meridians, and T~~ 2'. There is a preferred epimorphism : T -+ 2 , sending each meridian to the positive generator 1 E 2. The corresponding infinite cyclic covering is called the total linking number covering. It is no longer true that Hl(X(L);A) is always a torsion A-module or that Hl(X(L); A) Hl(X(L), aX(L); A). However these modules become isomorphic after localizing with respect to the multiplicative system {(t - l)n I n > 0). We then obtain a nonsingular hermitian pairing on the localization of the A-torsion submodule of Hl(X(L);A). Moreover Hl(X(L);A) is a torsion module if the reduced Alexander polynomial is nonzero, which is always the case for the links of plane algebraic curve singularities. (There is also an analogous pairing associated to the maximal abelian covering, but this may be of less interest in the present context [ll]).

=

3. Meromorphic connections

The material in this section is largely taken from [18],to which we refer for further details. (See also [3] and [6].) Let S = C { { s } } be the ring of germs of holomorphic functions at 0 E C , with field of fractions ff = S[s-l], and let 8, be the derivation given by a,(f)= $ for all f E $3. Let 2)= S[as]and D[S-'] = ff[&] be the rings of differential operators Cia," with coefficientsin S and .R, respectively. The latter rings are left and right noetherian, and have no nontrivial 2-sided ideals. A meromorphic connection over ff is a pair ( M a , ~where ) M is a finitely generated ff-vector space and a~ : M -+ M is a C-linear function such ) a8(f)m f a ~ ( mfor ) all f E ff and m E M .Thus M that a ~ ( f m = is a left D[s-']-module, with a,m = a ~ ( m )Conversely . every left D[s-l]module of finite dimension over K determines a meromorphic connection. M )( M ' , a ~ t is) a homomorphism g : M -+ A morphism g : ( M , ~ + M' of ff-vector spaces such that aMlg = gaM. Such a homomorphism is clearly D[s-l]-linear, and the category C of meromorphic connections over .R is equivalent to the category of left D[~-~]-modules of finite length. In particular, it is an abelian category. The nonunits in the endomorphism ring of an indecomposable module of finite length form a 2-sided ideal

+

50

J . Hallman

[l].Therefore the Krull-Schmidt Theorem holds in C: every meromorphic connection over ff has an essentially unique decomposition as a direct sum of indecomposable connections. Moreover Endc ( M ,&) is artinian and hence radically complete. Meromorphic connections have the striking property encapsulated in the classical “Cyclic Vector Lemma”. There is an m E M and a Ic E N such that M is generated by {m,d ~ m ,. .., Equivalently, M 2 D [ s - ’ ] / D [ s - ~ ] for P some P # 0. (See Corollary 4.2.8 or Proposition 4.3.3 of Chapter 1 of [HI.) A meromorphic connection ( M ,a ~over ) ff is regular if there is an Slattice L < M such that s ~ M ( L5)L.The category of regular connections is a full additive subcategory Greg. The indecomposable regular meromorphic connections are all of the form M ( a , n ) = (An,&,,), where a E C and n 2 1, and &,,(ei) = s-l(aei ei+l) for i < n and aa,,(e,> = s-lae,. Two such elementary connections M ( a , n ) and M ( P , p ) are isomorphic if and only if a - P E Z and n = p . We may extend the notion of meromorphic connection by defining a meromorphic connection over to be a pair ( M ,a M ) consisting of a finitely generated S-module and a @-linear function 8~ : M 4 ff 8 s M such that a ~ ( f m = ) $m f a ~ ( mfor ) all f E S and m E M . The category CS of meromorphic connections over S is an additive category in which idempotents split, and there is an obvious localization functor from Cs to

ak-’m}.

+

s

+

c.

4. The Gauss-Manin connection

Brieskorn has shown that the cohomology of the Milnor fibre H n ( F ;C) may be identified with the kernel of the topological Gad-Manin connection over a deleted neighbourhood of 0 in C, and that the characteristic polynomial of the monodromy may (in principle) be computed in terms of an algebraically defined local Gad-Manin connection on a relative de Rham cohomology module [4]. In this section we shall define the latter connection in the plane curve case. (See [12] for an elementary account of plane curve singularities). Let R = C{{x,y } } be the ring of germs of holomorphic functions at 0 E C2 and let f E R be the germ of an isolated singularity. Then the ideal (fz,f y ) contains a power of the maximal ideal. In particular, fz and f y have no common factor in R , f k E (fz,f,) for Ic large and p(f) = dim@R / ( f z ,fy) is finite. Let RP be the module of germs of holomorphic pforms at 0 in C2. Then Ro = R, R1 = Rdx @ Rdy, R2 = R1 A 0’ = Rdx A d y and RP = 0

H e n i t i a n pairings and isolated singularities 51

for p > 2. The cochain complex R* determined by the exterior derivatives d : RP + R P f l gives a resolution 0 -+ C + R 4 R1 --+ R2 -+ 0. Let f* : S = C { { s } } -+ R be the ring homomorphism sending s to f , defined by f * ( g ) = g o f , and let R; = W / d f A Rp-l. The exterior derivative on RP induces S-linear differentials d f : 025 + R": (via f*) and so we obtain a S-cochain complex 02;.Let H; = H 1 ( R ; ) , H' = R + / d R = R 1 / d R + Rdf and H" = R2/df A d R . Then H;, H' and HI' are S-modules, H; 5 H' and HI' E' Cok(b), where 6 : R -+ R is the S-linear derivation given by 6(g) = g z fy - g J X , for all g E R. Since f x and f y are relatively prime wedge product with df induces a monomorphism IE : H' -+ HI', with cokernel 0; E R / ( f z ,fy). Moreover d induces a C-linear bijection d' : H' -+ HI', with d ' ( H j ) = &(W'). (For if dr] = df A dg then d(r] g d f ) = 0 and so r] gdf = dr for some T E R, since R* is exact above degree 0.) This induces an S-linear isomorphism H ' / H ; E' Cok(rc) R;. The S-module H; supports a natural meromorphic connection, defined as follows. If r] E R1 represents a class in H j then dr] = df A $ = IE([$]), for some 1-form $. The class of $ in H' is well-defined, and d ( f k $ ) = fkd$ kf"'df A $ is in df A 0' = ( f Z , f y ) d xA dy, for k large. Hence fk$ E H;, and the function defined by V([r]]) = s - ~@ [ f k $ ] gives a well defined meromorphic connection over S. The pair (H;, V )is the local Gauj3Manin connection associated to f. Brieskorn used the coherence theorem of Grauert to show that the Smodules H , HI and H" are finitely generated and of rank p(f), and that ( H j ,V ) is regular [4].Moreover H" is torsion free as an S-module [25]. Hence so are H j and HI. If we identify H j and H' with submodules of f f @ s H jwe have V([r]]) = ~ - ' d ' ( [ r ] ] )and , so V maps H; bijectively onto HI. (The 2-variable weighted homogeneous case is treated by direct calculation in [El.) The apparent contradiction of the injectivity of the local Gaui3-Manin connection as just defined with the claim that the cohomology of F may be identified with the kernel of the topological Gad-Manin connection may be resolved by interpreting the equation sV(r]) = 0 as a linear system of lSt order ODES and extending coefficients from S to a larger ring, to include all possible solutions to such a linear system. Fix a basis (211,. . . , u p } for H; over S, and write s V ( v i ) = C j ~ ~ a i j v j , with coefficients sij E S. Let X I , . . . ,xp E S. Then

+

+

+

52

J. Hallman

and so the equation sV(v) = 0 corresponds to the linear system

over R. Choose E 0. Then 3 is a D[s-']-module with respect to the derivation a, = e-2?rizd dz and composition with the exponential function induces an inclusion of D[~-~]-modules e* : R + ,?. The space of solutions may then be identified with Homn[s-l]( ( H j ,V), g ) , under the correspondance between meromorphic connections and D[~-~]-rnodules of finite length given in 52. It is a consequence of the algebraic nature of V that the monodromy is quasiunipotent (i.e., the eigenvalues of A are roots of unity) and the solutions to this linear system are in Z[X], where 3 = R[sk;n > 11 is the algebraic closure of R and the derivation is extended so that $(sk) = ks"' for all rational exponents Ic and &(A) = s-'. (Thus X corresponds to log(s).) The solution space and monodromy together give H 1 ( F ;C) (the cohomology of the Milnor fibre) as a @A-module.See [4].

+

+

-

-

7

2


5. Duality pairings Quebbemann, Scharlau and Schulte have codified the notion of hermitian pairing in an additive category C with an involution *. Let E = f l . Then an e-hermitian pairing on an object C of C is defined to be an isomorphism cp : C --f C* such that cp* = ecp. This categorical point of view is particularly useful if the Krull-Schmidt Theorem holds for the objects of C and the endomorphism rings of objects are radically complete. For then the analysis of the decompositions of hermitian pairings into orthogonal direct sums may be reduced to corresponding questions for pairings over (skew) fields [24]. For example, let PA be the additive category of finitely generated torsion A-modules of projective dimension at most 1. If M is such a A-module let M* = HomA(M,Q(t)/A). Then M** is canonically isomorphic to M , and * extends to a functorial involution of PA. If K is a classical knot then b~ is a +1-hermitian pairing on B ( K ) in the sense of [24]. If instead we let PFAbe the category of torsion modules over PA = F[t,t-'], where F is a field, the Krull-Schmidt Theorem and radical completion hold. The Krull-Schmidt Theorem and radical completion also hold if C is the category of triples (A, B , E ) , where A and B are finite A-modules and E is an exact sequence 0 + A/2A -+ B 4 2A -+ 0, with (A, B , E)* = (A*,B*,E * )where N* = Exti(N, A), or if C is the category of finite stable CW-complexes CnY with H*(Y;Q)= 0 and with duality given by SpanierWhitehead duality. The latter two cases have been used to study the factorization of certain nontrivial classes of even-dimensional knots [10,13]. The utility of this approach in knot theory suggests reformulating our original question in similar terms. Since the cohomology of the Milnor fibre (as a CA-module) is given by solutions of the linear system determined by the local Gau&Manin connection we seek an hermitian self-duality of ( H j ,V) (or of (A 18sH j , V)) which induces the cup product pairing on H q F ;C). The category C has a natural duality given by ( M , ~ M )=* ( M * , a & ) , where M* = HomA(M,A) and a&(f)(m) = - f ( a ~ ( m ) ) for all f E M* and m E M . If ( M , ~ Mis) regular then so is ( M , ~ M ) * . In terms of D[~-~]-rnodulesof finite length, this duality is given by ( M ,a ~ ) =* Ext&-l]((M, a ~ D[s-']), ), where the overline indicates the conjugate left D[s-']-module structure, given by a.n = -na for all n E Ext&[,-ll((M, a ~ D[s-']). ), The equivalence of these duality functors follows easily from the Cyclic Vector Lemma. Let Sol(M, a ~ =)HomDr,-ll((M, a ~ g ) ),. The local duality theorem

54

J. Hillman

for sheaves of Dv-modules on V = 5, in [19] suggests t h a t there should be a natural isomorphism of Sol((M,&-)*) and Hom@(Sol(M,a ~ C). ) , Thus a self-duality cp : ( M , ~ M4) ( M , ~ M )such * that cp* = cp should induce a hermitian pairing Sol(cp) o n t h e @-vector space S o l ( M , a ~ )Our . basic question becomes “is there a natural isomorphism cp : (R8 s H j , V) -+ (R8 s Hf, V)* such t h a t cp* = cp, and which determines t h e cup product pairing o n H 1 ( F ;C)?”

References 1. F.W.Anderson and K.R.F’uller, Rings and Categories ofModules, Graduate Texts in Mathematics 13, Springer Verlag, Berlin-Heidelberg-New York (1974). 2. D.Barlet, Forme hermitienne canonique sur la cohomologie de la fibre de Milnor d’une hypersurface 8. singularit6 isolbe, Invent. Math. 81 (1985), 115153. 3. A.Bore1, P.P.Grive1, B.Kaup, A.Haefliger, B.Malgrange and F.Ehlers, Algebraic 2)-Modules, Perspectives in Mathematics, Academic Press (1987). 4. E.Brieskorn, Die monodromie der isolierten singularitaten von hyperflachen, Manus. Math. 2 (1970), 103-161. 5. A.Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer Verlag, Berlin-Heidelberg-New York (1992). 6. A.Dimca, Sheaves in Topology, Universitext, Springer Verlag, BerlinHeidelberg-New York (2004). 7. A.Durfee, Fibred knots and algebraic singularities, Topology 13 (1974), 4759. 8. D.Eisenbud and W.D.Neumann, Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Math. Study 110, Princeton University Press, Princeton (1985). 9. M.Epple, Die Entstehung der Knotentheor ie, Vieweg Verlag, BraunschweigWiesbaden (1999). 10. J.A.Hillman, Finite simple even-dimensional knots, J. London Math. SOC.34 (1986), 369-374. 11. J.A.Hillman, Algebraic Invariants of Links, Series on Knots and Everything vol. 32, World Scientific, Singapore (2002). 12. J.A.Hillman, Singularities of plane algebraic curves, Expos. Math. 23 (2005), 233-254. 13. J.A.Hillman and C.Kearton, Stable isometry structures and the factorization of Q-acyclic stable knots, J. London Math. SOC.39 (1989), 175-182. 14. M.Kato, A classification of simple spinnable structures on a 1-connected Alexander manifold, J.Math. SOC.Japan 26 (1974), 454-463. 15. C.Kearton, Blanchfield duality and simple knots, Trans. Amer. Math. SOC. 202 (1975), 141-160. 16. J.P.Levine, Knot modules. I, Trans. Amer. Math. SOC.229 (1977), 1-50.

Hermitian pairings and isolated singularities

55

17. E.Looijenga, Isolated Singular Points on Complete Intersections, London Math. SOC.Lecture Notes 77, Cambridge University Press, Cambridge (1984). 18. P.Maisonobe and CSabbah, V-Modules Cohkrents et Holonomes, Hermann et Cie, Paris (1993). 19. Z.Mebkhout, Thborhmes de bidualitb locale pour les V-modules holonomes, Arkiv Mat. 20 (1982), 111-124. 20. J.W.Milnor, Singularities of Complex Hypersurfaces, Annals of Math. Study 61, Princeton University Press, Princeton (1968). 21. J.W.Milnor, Infinite cyclic coverings, in Topology of Manifolds (edited by J.G.Hocking), Prindle, Weber and Schmidt, Boston (1968), 115-133. 22. L.Narvaez-Macarro, Cycles Bvanescents et faisceaux pervers: cas des courbes planes irrkductibles, Compositio Math. 65 (1988), 321-347. 23. L.Narvaez-Macarro, Cycles bvanescents et faisceaux pervers 11: cas des courbes planes rbductibles, in Singularities (edited by J.-P. Brasselet), London Math. SOC.Lecture Notes 201, Cambridge University Press, Cambridge (1994). 24. Quebbemann, H.-G., Scharlau, W. and Schulte, M. Quadratic and hermitian forms in additive categories, J. Algebra 59 (1979), 264-289. 25. Sebastiani, M. Preuve d’une conjecture de Brieskorn, Manus. Math. 2 (1970), 301-308. 26. HSTrotter, Knot modules and Seifert matrices, in Knot Theory - Proceedings, Plans sur Bex, Switzerland 1977 (edited by J.-C.Hausmann), Lecture Notes in Mathematics 685, Springer Verlag, Berlin-Heidelberg-New York (1978), 291-299.

56

Zariski’s moduli problem for plane branches and the classification of Legendre curve singularities Go-o Ishikawa Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan. E-mail :ishikawaOrnath.sci. hokudai.ac.jp In this survey article we show several recent results on the local classification of irreducible plane curve singularities, plane branches, based on Oscar Zariski’s lectures [17], and that of Legendre curve singularities. We compare exact classifications, the classification of plane branches and the classifications of Legendre curves, in the cases of simple and uni-modal singularities, and, as a consequence, we observe remarkable difference and mysterious similarity of these classification results. Also the comparison is achieved for a bi-modal case of characteristic (6,7) which Zariski treated. Keywords: plane curve singularity, Puiseux characteristic, contactomorphism.

1. Introduction. In his lecture note [17], Oscar Zariski posed the following problem on the local classification of plane branches, as the local and singular counterpart of the moduli problem of Riemann surfaces:

Problem 1.1 (Zariski’s problem). Classify plane branches of a given equi-singular class by local bi-holomorphic diffeomorphisms. We work in the complex analytic category and treat parametric holomorphic curve-germs ( C ,0) -+ (C2,0), the normalisations of irreducible components of plane curve germs, or simply, plane branches, in ( C 2 , 0 ) . Note that the moduli problem of Riemann surfaces deals with the classification of complex analytic curves with a fixed topological type. Two plane branches are equi-singular if and only if they are homeomorphic. To clarify our point of view, we recall three equivalence relations under which we are going to consider the classification problems: -+ ( C 2 , 0 )are called difleomorphic (resp. Plane branches f , g : (C,O) homeomorphic, symplectomorphic) if there exists a pair of bi-holomorphic

Zariski's moduli problem for plane branches

57

diffeomorphism-germs D : (C,O) 4 (C,O) and T : (Cz,O) 4 (C2,0) (resp. homeomorphism-germs D and T , a hi-holomorphic diffeomorphism-germ D and a hi-holomorphic symplectomorphism-germ T ) satisfying T o f = g 0 D. Then we designate f -diffeo g (resp. f -home0 g , f NSymp 9 ) . Two plane branches f and g are homeomorphic if they are diffeomorphic, and they are diffeomorphic if they are symplectomorphic. Then we introduce the differential moduli space, for a plane branch f : ( G O ) (C2,0), -+

Mdiff(f)

:= ( 9 :

(c,0 )

-+

(c2, 0) I g -home0 f}/ -diffeo,

the set of diffeomorphism equivalence classes of plane branches which are homeomorphic to the given f. Then Zariski's problem can be formulated also as follows: Problem 1.2 (Zariski's problem reformulated). Investigate the structure of the differential moduli space f o r a given plane branch f : (C,O) 4 (C2,0). Set f ( t ) = ( z ( t ) , p ( t ):)(C,O) (Cz,O).The order o f f is the minimal degree of the leading terms of z(t) and p ( t ) a t 0. If ordf = m, then f is symplectomorphic to (t", CE,, a k t ' ) ) . Set --f

P1

:= min{k

I ak # O,m/Yk}.

Then f is symplectomorphic to

Moreover set

el := G C D ( P l , m ) , Pz := min{k I ak

# 0 , e l lk}, e2 := G C D ( P 2 , e l ) .

In general, inductively, we set Pr+1

:= min{k

I ak # 0, er 1k

} ~e r f l ..-- GCD(Pr+1,e r ) .

I f f has an injective representative, namely, f is a normalisation of its image, then there exists s such that e , = 1. Then ( m , P l , P z , . . . ,Ps) is called the Puiseux characteristic, or simply the characteristic of f [15]. Throughout this paper we assume f is a normalization of its image. L6, Lejeune, Zariski independently showed that f and g are homeomorphic if and only if their Puiseux characteristics are equal: m ( f ) = m ( g ) , P 1 ( f )= P l ( f ) , P Z ( f ) = P z ( f ) , . . . . [9,10,16,171.

58

G. Ishzkawa

Moreover a sequence of positive integers (m,PI, P2, . . . , P,) Puiseux characteristic for a plane branch if

is the

m < Pi < P2 < . . . < P,, m X PI, el = GCD(P1, m), el XP2, e2 = GCD(P2,el), e2 X P 3 , . . . , e, = GCD(P,, e,-i) = 1. Thus we are led to consider the moduli space Mdiff(m,P1,P2,. . . ,PS) as the quotient space {g : (c,0 ) + (C2,0) I g has characteristic (m,pl,P2,. . . ,&)}/

“diffeo

.

Then we have

PI, 02, - ..,P s ) characteristic (m,Pi, P2,. . . ,Ps). Mdiff(f) = Mdiff(m,

if f is of In this article we describe the topology of the differential moduli space for a class of plane branches. Note that the space of plane branches 0(1,2) = {g : (C,O) -+ (C2,0)} is regarded as a topological space by the C” topology; the weakest topology such that the projections from 0 ( 1 , 2 ) to the jet spaces J k(1,2) (k = 1,2,3, . . . ) are continuous. The product of the group of diffeomorphisms on (C, 0 ) and the group of symplectomorphisms (resp. diffeomorphisms) on (C2,0 ) acts on 0(1,2). Contrarily the group of homeomorphisms does not act on 0(1,2). However we have seen that the decomposition of 0(1,2) by homeomorphism equivalence is rather simply understood. In [17], Zariski himself gave the classification for plane branches with Puiseux characteristic (2,21 f l),(3,31 l),(3,31 2), (4,5),(4,6,21 51,(5,6) or (677). In this survey we re-approach Zariski’s classification problem, taking care of three things; modality, symplectomorphism equivalence and Legendrisation. We first compare two classifications; the diffeomorphism classification and the symplectomorphism classification for simple (0-modal) and uni-modal plane branches. Note that Zariski did not mention modality nor the symplectomorphism equivalence in [17], However actually he classified almost all cases of simple singularities and some of uni-modal singularities. Moreover for the diffeomorphism classification, incidentally he used the “symplectic normal form”, which we will give later. Note that the notion of modality depends on the mappings space we consider. We treat the classification problem of plane branches by modality as holomorphic curve germs (C, 0 ) -+ (C2,0 ) . Then we study the classification of simple (0-modal) and uni-modal branches by symplectomorphism equivalence and then by diffeomorphism equivalence.

+

+

+

Zariski’s moduli problem for plane branches

59

A plane branch f is called simple if a finite number of diffeomorphism classes form a neighborhood of f in the space of plane branches ( C ,0) + (CZ, 0). A plane branch f is called uni-modal if a finite number of diffeomorphism classes and one-parameter families of diffeomorphism classes form a neighborhood of f . Similarly we can define the notion of r-modality for r = 0,1,2, . . . . Since we assume the plane branch f is a normalization, f is finitely determined [14], so we can define its modality in a finite jet space. As for symplectomorphism equivalence, we define the symplectic moduli space Msymp(f)

= Msymp(m,

PI, Pz,

* * 7

Ps)

bY { g : ( C ,0) -+ (C’, 0) I g has characteristic (m,PI, Pz, .

. .,,&)}/ Nsymp,

if f has characteristic ( m ,PI, pz, . . . ,/Is). There is a natural surjection

n : M s y m p ( m , PI, Pz, -

* * 7

Ps)

+

Mdiff(m,

PI, Pz,. * .

7

Ps)

defined by mapping a symplectomorphism equivalence class to its diffeomorphism equivalence class. The mapping II can be considered as a “singular fibration” in a sense. The fibre of II over f is given by Msymp/diff(f)

= (9 : ( C ,0)

-+

(C’, 0) I g is diffeomorphic to

f}/ wsyrnp.

Then, in terms of a naive language, the moduli space M d i f f ( f ) is of dimension 0 if f is simple, and of dimension 1 if f is uni-modal. For each f in 0(1,2) = {g : (C,O) --f (Cz,O)},we can find a finite dimensional deformation of f which gives a mini-transverse section to the orbit through f in 0 ( 1 , 2 ) for the group of symplectomorphisms on (C’, 0). The mini-transverse section turns out to give the symplectic normal form in our case, and the symplectic moduli space Msymp(f) is a finite quotient of an aEne space C” for some s depending on the homeomorphism class of f [7]. Therefore we see that Msymp(f) is a Hausdorff space. Also the differential moduli space &iff( f ) is a quotient space of C ” .However M d i f f ( f ) is far from being Hausdorff. Bruce and Gaffney [2] gave the diffeomorphism classification of simple plane branches over C . In our paper [6], we gave the symplectomorphism classification of simple plane branches over R. In our paper [7], we completed the diffeomorphism classification and the symplectomorphism classification of simple and uni-modal plane branches over C .

60

G. Ishikawa

In $2, $3 and 54, we review the symplectomorphism classification and the diffeomorphism classification of simple and unimodal singularities from [6,7]. See Tables 1, 2, 3 and 4. The symplectomorphism equivalence is finer than the diffeomorphism equivalence except only for the characteristics (2,2! l),(3,4) and (3,5). In $5, we associate a plane branch f ( t ) = ( z ( t ) , p ( t )with ) a Legendre -+ ( c 3 , 0 ) , f ( t )= ( z ( t ) , p ( t ) , y ( t in ) ) $5. A Legencurve-germ : (c,o) dre curve is an integral curve for the complex contact form (I: = d y - pdz on the (x,p,y)-space C3. We call the Legendrisation of f. Since f is a normalisation, is also a normalisation (namely, f has an injective representative). Conversely any Legendre curve F : ( C ,0 ) -+ ( C 3 ,0 ) which is a normalisation projects to a plane branch. Two Legendre curve-germs and are called contactomorphic if they are transformed by a bi-holomorphic contactomorphism on ( C 3 ,0) preserving the Pfaff system { a = 0) C T C 3 , up to a re-parametrisation ( C ,0) -+ ( C ,0 ) . Then we designate f w c o n t g. The group of contactomorphisms on (C3,0)acts on the space on the space of Legendre curve-germs. Thus we have the decomposition of the space of plane branches by contactomorphism equivalence via Legendrisation. Then we observe both a difference and a similarity of the contactomorphism classification and the diffeomorphism classificationof plane branches. Zhitomirskii [18]showed that, for simple plane branches, two classifications coincide (Table 5). We observe they become different if we proceed to classify uni-modal singularities (Table 6). Here we summarise the known facts and the facts we show in this paper:

+

f

f

f

-

f

-

Proposition 1.3. Let f and g be plane branches. (0) Iff Nsymp S, then f Ndiffeo 9. Iff -diffeo 9 , then f Nhomeo 9. (0’) That f Ndiffeo g does not imply that f Nsymp 9. That f Nhomeo g does not imply that f Ndiffeo g .

-

Ncont ii (Amol’d [I]). does not imply that f (1 ’) That f Ncont does not imply that f NCont g .

(1) 8 f

Nsymp

9 , then f

-

(2) F N c o n t

5 if

(3) That f

Ndifleo

-

and only if

(3’) That fNcont

N~~~~

fNdiff

g . That f

5 (Zhitomirskii [lS]). -

g does not imply that f does not imply that f

Ncont

-g .

Ndiffeo g .

In 56 Proposition 6.5, we observe (3) and (3’).

Nhomeo

g

Zariski’s moduli problem for plane brunches 61

Zariski never mentioned Legendre curves in his lecture note [17). A diffeomorphism on the (2,p)-plane does not induce a contactomorphism on (x,p,y)-space and vice versa. Therefore, there would be no reason to expect any relation, without performing exact classifications, between the classification by diffeomorphisms of plane branches and the classification by contactomorphism of Legendre curves. Nevertheless we have moreover the following:

Theorem 1.4. For a simple or a uni-modal plane branch f and its Legendrisation we have that the contact modality of is equal to the differential f) is homeomormodality of f. Moreover the contact moduli space Mcont( phic t o diflerential moduli space Mdiff(f).

7,

7

For the notion of the contact modality and the contact moduli space of plane branches, see $6. The homeomorphism in Theorem 1.4 between the differential moduli space and the contact moduli space is induced actually from a homeomorphism on Cs,the parameter space of the symplectic normal form, which maps the decomposition by diffeomorphism equivalence to the decomposition by contactomorphism equivalence. In $5 and 6 we explain the results stated above. In his lecture note [17] Zariski did give the diffeomorphism classification of plane branches of characteristic (6,7). The plane branches of characteristic (6,7) is not simple nor uni-modal, but bi-modal. So it does not appear in the list of Tables 3 and 4 in $4. In 97, we follow Zariski’s classification by means of our method, and moreover we give the contactomorphism classification of Legendrisation of plane branches of characteristic (6,7). As a result of exact classification results, we have

Theorem 1.5. If f is a plane branch of characteristic (6,7), then f and its Legendrisation are both bi-modal, and the contact moduli space Mcont(6,7) is homeomorphic to the differential moduli space Mdiff(6,7). We have defined three equivalence relations on the space 0 ( 1 , 2 ) of plane branches; the homeomorphism equivalence, the diffeomorphism equivalence and the symplectomorphism equivalence. Moreover we can define the contactomorphism equivalence via Legendrisations of plane branches ($5).We can illustrate the relations of these equivalence relations:

62

G. Ishikawa

symplectomorphic

+

===+

Uff

diffeomorphic

homeomorphic

#ff contactomorphic

By Proposition 1.3, we see the interrelations between any two of four equivalence relations except for the homeomorphism equivalence and the contactomorphism equivalence. Moreover we easily observe there are homeomorphic plane branches which are not contactomorphic in Table 5. In 95, $6, we classify Legendre curves for each equi-singular class, namely, for each homeomorphism equivalence class. However, after the exact classification results obtained in this paper, we observe the following:

Proposition 1.6. For simple or uni-modal plane branches, the contactomorphism equivalence implies the homeomorphism equivalence. In $8, we pose open questions. The subjects which we treat are very basic and therefore they are related t o a rather wide area, from the topological theory of algebraic knots [3] t o the theory of differential systems [5,12]. Moreover, as we realise in this paper, still there would be many things to be found and there would be missing theories in the basic study of singularities. The classification results in this paper are obtained, among others, in the joint work with S. Janeczko [6,7],in the papers [4,18]and in the recent joint work with P. Mormul[8]. In the exposition of this paper we omit the detailed proofs of results given in those papers; instead we try to give concrete examples of the classification and show the incidence of the classification results. In this paper all mappings are assumed to be holomorphic unless otherwise stated. 2. How to find symplectic normal forms. In this section, we give an outline of the process to find symplectic normal forms. Detailed proofs are given in our papers [6,7]. We use the infinitesimal method by mean of vector fields: Let fx(t) be a holomorphic family of plane branches. We fuc a A. Then as an infinitesimal deformation of the plane branch fx : ( C ,0) + (C', 0) we consider a holomorphic map-germ v : ( C ,0) + TC' to the tangent bundle of C2 covering

Zariski's moduli problem for plane branches 63

fx via the natural projection IT : T C 2 + C 2 .We call such a u a vector field over fx. Then, for a given u,we consider the equation = (fx)*E

+ (fx)*r],

on a pair of holomorphic vector fields E over ( C ,0) and r] over ( C 2 0) , with the conditions [ ( O ) = O , v ( O , O ) = 0. Here (fx)* : (TC,T&) + T C 2 is the differential mapping of fx and (fx)*r] = r](fx), the pull-back by fx. We call it Mather's equation of fx for u. (cf. Mather [ll]). It expresses the infinitesimal condition that the infinitesimal deformation by u is recovered by diffeomorphism equivalences. Note that Mather's equation is a nonhomogeneous linear equation over C . We call u soluble if Mather's equation for u has a solution (E, r ] ) satisfying the required conditions. Actually we always require that the solution ([, r ] ) depends on X holomorphically. Note that, if u is soluble, then cu, ( c E C ) is also soluble. Let f x ( t ) be of the form M

f A ( t ) = (tm,

bk(X)tk).

k=Pi

Then we consider in particular the vector field u along fx of the form u = tk o f) or, writing simply, u = (0, t k ) ,the infinitesimal deformation

(&

of fx in the term of t k . Then we call u = tk

(&

(3

o f)

soluble up to higher

+

order terms if (tk+ $ ( t ) ) o f) = (0, tk $ ( t ) ) is soluble for some $(t) with ord($) 2 k 1. If r] can be taken to be a Hamiltonian vector field, which generates a family of symplectomorphisms by integration, via the holomorphic symplectic form dp A d x on C 2 ,then u is called symplectically soluble. For a given characteristic (m,01,. . . ,Pg),we suppose f x ( t ) is a family of plane branches of Puiseux characteristic (m,01,. . . , Pg). Then we have

+

Lemma 2.1. (Janeczko-Ishikawa [7]) (1) There exists a natural number N depending only o n the characteristic such that, f o r any n 2 N and for any $(t) with ord($) 2 n, there exists a holomorphic function h x ( x , p ) on (C', 0 ) depending o n X holomorphically and satisfying $(t) = h(fx(t))and ord(h) 2 2 . (2) A n y vector field u = (O,p(t)) along fx is symplectically soluble i f ord(p) 2 N - m, for the number N in (1).

64

G. Zshikawa

Using Lemma 2.1, we can reduce fx to a family of polynomial mapgerms up to symplectomorphism equivalence. A family f x ( t ) , X E C", is called a symplectic normal form for the characteristic (m,P I , . . . ,Pg) if any plane branch of characteristic (m,,&,. . . ,&) is symplectomorphic to f x ( t ) for some X E C" and those X E Cs for which fc are symplectomorphic to a given plane branch form just a discrete subset in C". If there exists a symplectic normal form, then we have a surjective mapping C" 4 Msymp(m, PI,. . .,Pg) with discrete fibres. A plane branch f : (C',O) .+ ( C 2 , 0 )induces a C-algebra homomorphism f * : 0 2 + 01 defined by the composition f * ( h )= h o f for h E 0 2 . Here On, as usual, denotes the C-algebra of holomorphic function-germs on (Cn,0 ) . The dimension over C of the quotient space 0 1 1f * 0 2 is a wellknown invariant, &invariant of the plane branch; the number of double points in a small perturbation of f. Also we can extract from 0 1 1f * 0 2 information on symplectomorphism classes which homeomorphic to f. By using the infinitesimal method, we have the following algebraic criterion to find symplectic normal forms in the cases we study in this paper: Proposition 2.2. (Janeczko-Ishikawa (71) Consider a characteristic (m,PI) with relatively prime m, (resp. (4,6,2C 5 ) , (4,10,2& 9), e = i , 2 , 3 ,... ). (1) Let f be a plane branch of a characteristic as above. Then the quotient vector space 011f * 0 2 has a common monomial basis which depends only o n the characteristic. (2) I f O l / f * 0 2 has a monomial basis

+

t,

t2,..

.,

tm-1,

tm+l, . * ' , f l f " , .

+

. . , trs+m

where r1 +m, . . . ,r,+m are all exponents greater than m+Pl(j = 1 , . . . ,s ) . Then f is symplectomorphic t o

f x ( t ) = (t",

tP1

+ Xltr' + X 2 t Q + . + A $ - . ) ,

for some X = ( X I , . . . ,A,) E C". (3) I n (2), if the characteristic is

f x ( t ) = (t",

to1

(m,P1),then

the family

+ Xlt" + XztQ + * . . + A S P ) ,

X = ( X I , . . . ,A,) E C" is a symplectomorphism normal f o r m for the characteristic (m,PI). (4) I n (2), if the characteristic is (4,6,2e 5 ) , then s = C 1 and r1 = 7,rz = 9 ,..., = 2C+3,re = 2 e + 5 , r C + l = 2C+7. Among the

+

+


family

f c ( t )= (t", t6 + clt7 + c2t9 + . . . + ~

~

- + ~~

t ~+~~ t~ ++~~ ~t ~~ ~ ++

the subfamily

f x ( t )= (tm,t6 + ~

~+ Xt~ P~+ ~~) ,

+

~

X = ( A 1 , X z ) E C2,X1# 0, is a symplectic normal form for the characteristic ( 4 , 6 , 2 t 5). (5) I n (2), if the characteristic is (4,10,2! 9), then s = C 4 and r1 = 11,rz = 13,r3 = 15,...,re-1 = 21+7,re = 2C+g,re+l = 21+ 11, re+2 = 2C 13, re+3 = 2C+ 17, re+4 = 21 21. Among the family fc(t) = (t4 ,t 10 cltl1 cztl3 c3ti5 . . . ~ ~ - ~ t ~ ~ +

+

+

+

+

+

+ ~ t the subfamily

+

+ + + + t +~~ ~ e ++ ~ ~ +t ~~ e~ ++ ~~+ t~ ~ ~ ~~

f x ( t )= (t4 ,t 10 + ~

~+ ~ t

X = ( X i , A2, X3, X4, As) E C5,X 1 characteristic (4,10,2C+ 9 ) .

+

++ ~c,+4t2e+21), ~~ t ~ ~

+

~+ ~ ~ t ~ ~+~~+ t ~~+~~ +~ t ~t ~~ ~ + + ~

# 0, is the symplectic normal form for the

Example 2.3. (E12,m = 3, ,& = 7 ) . The quotient O l / f *O2 has the monomial basis t , t 2 ,t 4 ,t 5 ,t', t". The symplectic normal form is given by

f x ( t ) = ( t 3 t, 7

+ At",

(A E C).

Moreover f x is symplectomorphic to fx/ if and only if there exists C' E C satisfying p(t) E we see

(h)*rnp),

qJ=

(1"5

- - -3);

x lt15 +

(h)*rng).

($ $) -

Xzt16

+

+

;( $) -

X3t17

+. ..

7

E (K)*rn?) and therefore, by belongs t o In particular t15 t )see ~ t29+. . . E ( T ~ ) * m pThus ) . we see v = (0,t k ) multiplying it with ~ ( we is soluble up to higher order terms for k = 23. On the other hand, multiplying $(t) with z(t)= t 6 , we see

82

G. Zshikawa

belongs to

(y~)*mp). Multiplying $(t) with p ( t ) = t7 + Xlt9 + . . . , we see (A)*m), we see t21 + 3Xlt23 + . - . E (A)*m?).

). from ~ ( tE ) ~ belongs to ( T ~ ) * m rMoreover,

Thus, if

# 0, namely, if

(A)*m?).

then t21+. . . , t22+. . ,t23+. belong to Thus we see w = (0,t k ) is soluble up to higher order terms for k = 15,16,17. In this case, we have the normal form under contactomorphisms 3.

+ t 9 + XtlO + ptll). + g. The condition becomes p # -A2 If A1 # 0 but 17408X:: + 11475Xi - 10240A1x3 = 0, then just t21 + . ,tZ2+ . belong to (f;)*mf) but not for t23 + . . . . Then w = (0,t k ) is (t6,t7

soluble up to higher order terms for k = 15,16 but not for k = 17. Thus we have the normal form ( t 6 , t7

+ t9 + XtlO + (2048 2295x2 + 10

Other cases can be treated similarly. Finally we obtain Table 7. beginproof[Proof of Theorem 1.5:] Let 'p : C6 + C6 be the linear isomorphism define by

I

'03

1-

'03

Then 'p maps the decomposition Z)djff of C6 by diffeomorphism equivalence to the decomposition Dcontof C6 by contactomorphism equivalence. Thus a homeomorphism : M d i ~ ( 6 , 7+ ) Mcont(6, 7) is induced as required.


8. Open questions. Lastly we pose two natural open questions, based on the results in this paper; a speculative one and a fundamental one: Question 1: In general, are the differential moduli space h f d i f f ( f ) and the contact moduli space Mcont (f)homeomorphic? Question 2: In general, d o plane branches f and g have the same Puiseux characteristic if their Legendrisations ?and 5are contactomorphic?

Acknowledgements. T h e author would like t o express his gratitude t o the organisers of JARCS 2005 for giving him the chance t o give a talk in Sydney and t o write this extended survey based on that talk. T h e author was partially supported by Grants-in-Aid for Scientific Research, No. 14340020.

References

1. V.I. Arnold, First steps of local contact algebra, Canad. J. Math. 51-6 (1999), 1123-1134. 2. J.W. Bruce, T. Gaffney, Simple singularities of mappings C ,0 -+ C2,0, J. London Math. SOC.,26 (1982), 465-474. 3. D. Eisenbud, W. Newmann, Three-dimensional link theory and invariants of plane singularities, Ann. of Math. studies 110, Princeton Univ. Press, 1985. 4. G. Ishikawa, Classifying singular Legendre curves by contactomorphisms, Journal of Geometry and Physics, 52-2 (2004), 113-126. 5. G. Ishikawa, Singular Legendre Curves and Goursat Systems, Proceedings of Szdal symposium 2004. 6. G. Ishikawa, S. Janeczko, Symplectic bifurcations of plane curves and isotropic liftings, Quart. J. Math., 54 (2003), 1-30. 7. G. Ishikawa, S. Janeczko, The complex symplectic moduli spaces of uni-modal parametric plane curve singularities, preprint. 8. G. Ishikawa, P. Mormul, Classification of uni-modal Legendre curve singularities. in preparation. 9. D.T. L6, Sur les noeuds alge'briques, Compositio Math., 25 (1972), 281-321. 10. M. Lejeune-Jalabert, Sur l'e'quivalence des singularir6s des courbes alge'briques planes, Travaux en Cours 36,Hermann (1988), pp. 49-124. 11. J.N. Mather, Stability of Coomappings III: Finitely determined map-germs, Publ. Math. I.H.E.S., 35 (1968), 127-156. 12. R. Montgomery, M. Zhitomirskii, Points and curues in the monster tower,

preprint. 13. A.N. Varchenko, Local classification of volume forms in the presence of a hypersurface, Funct. Anal. Appl. 19-4 (1984), 269-276.

84

G. Ishikawa

14. C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. SOC.,13 (1981), 481-539. 15. C.T.C. Wall, Singular Points of Plane Curves, Cambridge Univ. Press (2004). 16. 0. Zariski, General theory of saturation and of saturated local rings 11 : saturated local rings of dimension 1, Amer. Jour. Math., 93 (1971), 872-964. 17. 0. Zariski, Le p r o b l h e des modules pour les branches planes, Cours donne au Centre de mathematiques de 1’Ecole Polytechnique, 1973, (ed. F. Kmety, M. Merle, with an appendix of B. Tessier), Hermann, Paris (1987). 18. M. Zhitomirskii, Germs of integral curves in contact 3-space, plane and space curves, Isaac Newton Inst. Preprint NI00043-SGT, December 2000.

85

Introduction to algebraic theory of multivariate interpolation Shuzo Izumi

Department of Mathematics Kanki University Kowakae Hagasha-Osaka 577-8502, Japan email: izumi9math. kindai.ac.jp

The theory of multivariate polynomial interpolation has made a big progress at the beginning of the nineties. The interpolation of Hermite type reduces t o the theories of holonomic systems of PDEs with constant coefficients and Grobner basis. There is another scheme of interpolation: the theory of the least interpolation spaces. This method allow us to treat more general linear functionals such as the integrations (by compactly supported Radon measures).

1. Introduction This is an algebraic introduction to the theory of multivariate interpolation. Usually “interpolation” means the task to seek a polynomial p ( z ) ( x := (21,. . . ,z,)) such that its value or, more generally, some linear combinations of its higher derivatives take prescribed values bi at prescribed points 8i E C”, both finite in number:

alaIp

xaia-(8i) = bi (i = 1,.. . ,t). axa We call the points 8i nodes and the linear span of the total symbol C of the differential operators at each 8i space of interpolation type. We introduce a sesqui-linear form on the product of the space of total symbols of differential operators and the function space F applying differential operators to the complex conjugates of functions and evaluating them at the origin. If the spaces of interpolation types are stable under partial derivations with respect to & (i = 1,.. .,n) we call them D-closed and say that the interpolation is of Hermite type (see ’$5).In this case, we can canonically associate each of the spaces of interpolation types a polynomial ideal whose radicals are just the ideal of nodes, using the sesqui-linear form.

86

S. Izumi

In general, it is productive to consider a polynomial ideal as a system of linear partial equations (PDEs) with constant coefficients (cf. [18]). Since nodes are finite in number the system is a holonomic system, a system with finite dimensional solution space. Section 4 is devoted to recall some known facts about this system. The readers can refer to the book of Strumfels [18] for wider perspective. Applying duality theory with respect to the sesqui-linear form (or the bilinear form in the real case), the theory of multivariate interpolation has made a big progress at the beginning of the nineties. We can distinguish two major tendencies and hence the readers can skip the corresponding sections (stated below). One group of the algebraic stream developed the theory of Grobner bases introduced by Buchberger [7], which treat generators of an ideal of a polynomial ring and representative system of its residue algebra. Interpolation problem of Hermite type is the case of ideals of 0-dimensional subvarieties (in the scheme theoretic sense) of @”. The interpolating functions are spanned by a certain set of monomials. A merit of this method is that there is a choice of the monomial sets, which is possible by the choice of monomial order of the multi-exponent. Hence this method is convenient for calculation. This theory, together with its algorithmic aspects, is found in the famous papers Marinari-Moller-Mora [12], [13]. We introduce this method in $3-56. Another group consists of proper researchers of interpolation theory. In particular, Dyn-Ron [9] started from approximation by exponential polynomials and got the idea of switch of the roles of differential operators and polynomial functions (see Boor-Ron [4] also). This is described in the elementary formula 6.2. This leads to interpolation with D-closed type i.e. Hermite type interpolation. Further, Boor-Ron [4], [5], [6] found the space of “least” interpolating functions. In this theory, we have no need to assume D-closedness and can treat a set of general linear functionals on polynomial space. This space has a number of natural properties. For example, the space is invariant under affine transformations. This theory is explained in $7 and $8. We introduce “filtered vector space” and “associated graded vector space” in $7. This resembles the blow-up in algebraic geometry and will simplify the description. We may replace CC by R in these sections. Only the positivity of the bilinear/sesqui-linear form is important. The examples 9.1,. . . ,9.4 in the final section have no important meaning but they will help understanding of the schemes of interpolation. In this note we do not treat the algorithmic part of multivariate interpo-

Introduction to algebraic theory of multivariate interpolation 87

lation. We tried not to use special terms of interpolation theory. Instead, we often use non-degenerateness of a sesqui-linear form or a bilinear form. We assume only standard knowledge of algebra, namely, the elementary part of sesqui-linear map found in the beginning part of Bourbaki's Algdbre Chaptre 9 [ 2 ] . To understand the interpolation theory of Hermite type in §3-§6, we need the Chinese remainder theorem, Hilbert's Nullstellensatz and the primary decompositions of ideals of a polynomial ring which are found in the book of Kunz [lo]or Matsumura [ll]. Interpolation theory has been important in the study of smooth functions. The author has studied it in order to attack the problems in such a fields. The references of multivariate interpolation are scattered in various fields of mathematics. In particular, the recent ones are written putting emphasis on algorithmic aspects. The author hopes that this note would save the time for access to the theoretical part of multivariate interpolation. This is a note of the lectures for graduate students in Kinki University (2004) and talks of workshops in Hakodate (2003), Bqdlewo (2004) and Wuhan (2006). 2. Sesqui-linear maps Let us recall the basic properties of sesqui-linear maps and forms. Let C be a complex number field and II, F and L be vector spaces over C. A map

(, ):IIxF-L is called sesqui-linear m a p if it satisfies the following ("P

+ bq, f) = a ( p ,f) + b(q, f),

( P ,a f

+ b9) = Z(P, f) + b(P, 9 )

for a , b E C, where - denotes the complex conjugate. If L = C, a sesquilinear map

(, ):IIxF-C

-

is called sesqui-linear form. We know the following (cf. [2], $1).

Lemma 2.1. Let ( , ) : II x F L be a sesqui-linear map. For a subset V c F , we define its orthogonal space by

V' := {f E II : ( p , f) = o f o r a n y p T h e n the space V'

is a vector subspace and V

E

c V",

v). V' = VLLL.

88 S. Izumi

We can also define the orthogonal space q1 of q

c II by

q1 := { p E II : (p,f) = o for any f E 4). We do not repeat the properties of q* similar to those of VL. We call a sesqui-linear form ( , ) : II x F -+ C non-degenerate if III = 0, F* = 0.

Lemma 2.2. Let ( , ) : 11 x F -+C be a sesqui-linear form. Suppose that this is non-degenerate. Then we have the following. (1) If V C F is a vector subspace and if one of dime V and dime II/V* is

finite, then the sesqui-linear form ( , ) induces a non-degenerate one o n II/V* x V and we have dime V = dime H/V' and V** = V . (2) If V, W c F are subspaces, we have (V W ) l = VL n W*. (3) If V, W c F are finite dimensional vector subspaces, (V n W)' = V' W*.

+

+

-

Corollary 2.3. Let ( , ) : II x F C be a sesqui-linear form and . . , p d and f 1 , . . . , f e be the bases of II and F respectively. Then the following conditions are equivalent. PI,.

, ) is non-degenerate. (2) The equality d = e holds and the matrix ( ( p i ,fj)) is invertible. (1) (

In this note, linear differential operators are always assumed to be those with complex constant coefficients. Hence they are commutative and we can identify a differential operator

with the polynomial p ( z ) (z:= ( 2 1 , . . . ,zn)), which is often called the total symbol of p ( D ) . (Here we adopt the reversed usage of the variable 6 and the corresponding differential symbol x because their roles will switch below.) The space of total symbols are denoted by II := C[z]. Solutions of a system of differential equations

are always sought in F := C [ [ [ ] ] the , C-algebra of formal power series. There is a natural sesqui-linear map

IIxFW

( P ( Z ) ,f 1

-

F W

(P,f ) := P ( D P

Introduction t o algebraic theory of multivariate interpolation 89

over @. Here, 7 of f = CaENo c a t a is defined by 7 := CaENo Eata. Let q c ll be a subset. Then the space of solutions of the system is expressed as the orthogonal space .ql Since the system above is equivalent to

P ( D ) f = 0 (P E

a),

we may always assume that q is an ideal. This defines an algebraic subvariety 2 ( q ) := {zE

cn: v p E q, p ( z ) = O}.

Let us put

a!:= ( a l ! .. .an!) (a:= (a1,.. . ,a,)

E

rq).

The following is easy to see. Proposition 2.4. (1)

(otherwise). (2) The sesqui-linear form ( , ) i s non-degenerate. (3) The partial di'erentiation by and the multiplication by the dual variable xi are adjoints: (xCip(z),f) = ( P ( Z ) ,

af ) ' K

f) = ( d z ) ,

MZ)P(Z),

If V c F is a vector subspace such that, for any f E V ,its partial derivatives belong to V, we say that V is diflerentially closed (abbreviated to D-closed). By 2.4, (3), we have the following. Corollary 2.5. (1) For a subset V c F , VL is a n ideal. (2) For a subset q c II,q1 is a D-closed vector subspace.

Define a sesqui-linear form

IIxFW

(P,f )

-

C UJ

(P,f ) o

90

Corollary 2.6. Ifp, q E I3 = C[z] and f E CK], we have

aP

(%'f)O

= (P,Jif)o,

M D ) P ( Z ) ,f ) o = ( P ( Z ) , q ( O f ( t ) ) o .

We can also define the orthogonal spaces qLO and V'o with respect to this evaluation sesqui-linear form for subsets q C ll and V c F respectively:

qLO := {f E F :

vLO:= { p E rI : Obviously, q' and V'

(p,f)O

= 0 ( p E q)},

( p ,f ) o = 0

(f E V ) } .

are vector subspaces of 4'0

and V'o respectively.

Proposition 2.7.

(1) If q C I3 is an ideal, then 4'0 = q L and these are D-closed. (2) If a subspace V C F is D-closed, then V ' O = V' and these are ideals. Proof.

(1) I f f E qLo and p E q,

for any q E It.This implies that all the higher order partial derivatives of p ( D ) f ( J ) vanish at 0 and hence p ( D ) T = 0 and (p,f) = 0. This proves f E q' and oql = 4'. (2) If p E V'o and f E V,

q ( D ) P ( D ) T ( o )= ( P ( Z ) ,3 D ) f ) o = 0 for any q E IT by 2.4. This implies that ( P ( Z ) ,f ( 0 )= P(D)T(S)= 0

just in the same way as (1).Hence V ' O

= V'.

0

Introduction. to algebraic theory of multivariate interpolation

91

3. Zero-dimensional subset of Cn

In this section we recall a standard fact, the correspondence of polynomial ideals and affine algebraic varieties, in 0-dimensional case. Let J(8) of 8 c C" denote the ideal of all the polynomials vanishing on 8. Conversely, if q c II is an ideal, the zero set of q means the intersection 2 ( q ) of zero loci of all the elements of f E q.

Lemma 3.1. Let

q

c II := C[z] (z:= ( X I , .. . ,xn), n 2 1)

be a n ideal. The following conditions are equivalent. (1) q is maximal (among the proper ideals). (2) q is a n ideal of a point 8 := (81,. . . E C" : q = 3(8). (3) There exists a point 8 := (81,. . . ,8") E Cn such that

,en) n

j=1

Proof. First assume that q is maximal. Suppose that X := 2 ( q ) is empty. Then 1 vanishes on X = 0 and hence 1 E q by Hilbert's Nullstellensatz. This contradicts to the fact that q is a proper ideal. Thus X is not empty. Take a point 8 E X. Obviously q c (z- 8 ) . By the maximality of q, it coincides with the ideal (z- 8 ) of 8. This proves (1) =+ (2). To see (2) rj(3), practice division by zj - 8, ( j = 1,.. . ,n). The condition (3) implies that lI/q is isomorphic to the field C , which implies (1). 0 In the below, we use the condition that the associated primes of q are maximal. Usually this condition is expressed as the residue class algebra I I / q is of Krull dimension 0 and denoted by dimlI/q = 0. On the other hand, dim@II/q expresses the vector space dimension of the residue class algebra II/q of II := C[z].

Lemma 3.2. Let q

c II := C[z] (z:= (XI,. . . ,zn),n 2 1)

be an ideal. Then following conditions are equivalent. (1) dimSl/q = 0. (2) There exists a finite subset 0 C C such that

=3(8).

92

S. Izumi

(3) There exist non-zero polynomials (pi 1,..., n). (4) dimcII/q < 00.

C[A] such that cpi(xi) E q (i =

Proof. (1)===+ (3): Let q = q1 f ~* .n q t be the shortest (reduced) primary decomposition, which implies that the associated primes ti := fi are distinct. Such a decomposition a.lways exists and is unique by the condition (1) (see [ll],[lo]). By the definition of Krull dimension, every tiis maximal. By Lemma 3.1 above, n

ti = (Z -

ei)=

-

j=1

eij)n (ei:= (Bill. . . ,e,,)).

Hence some power of (xi- O l i ) . (xi- B t i ) belongs to q. (1):The condition (2) implies that every prime component of (2) Jri are maximal and hence dimII/q = 0. (2) ==+ (3): If fi is an ideal of the set

pi := (eil,. . . , ein) : i = 1,.. . ,ti, a certain power of (xi- 81i). . . (xi- dti) belongs to q by Hilbert’s Nullstellensatz. These powers are cpi. (3) ==+ (2): The condition (3) implies that the i-th coordinates of points of 2(q) are contained in 2(cpi). Hence 0 := 2 ( q ) is a finite set. Then (2) follows from this by Hilbert’s Nullstellensatz. (3) ===+ (4): If there are polynomials cpi(xi) E q with degrees di respectively, the residue classes are represented by polynomials of degree less than di in xi ( i = 1 , . . . ,n). (4) ===+(3): Since the residue classes of 1, xi, x:, x:, . . . are not linearly independent in II/q, there is a linear dependence relation. This implies the existence of cpi. 0 4. Holonomic systems

Ehrenpreis and Palamodov has completed the theory of systems of linear differential equations with complex constant coefficients on @” (see Bjork [l]). We need only the very easy case: the system with a finite dimensional solution space. Such a system is called a holonomic system (HS). In the previous section, we have defined a sesqui-linear map defined on the product IIx F . (Different choices of the spaces of symbols and functions are possible cf. Mourrain [14]). We adopt simply duality between the space of total

Introduction to algebraic theory of multivariate interpolation

93

symbols and formal power series. Our treatment of HS in this section is much suggested by the last part of Sturmfels's stimulating book [18].

Theorem 4.1. Let V c F be a finite dimensional vector subspace. Then the following conditions are equivalent. (1) V is the solution space of an HS. (2) V is D-closed.

If these holds, V' c II is an ideal such that dimII/V' = 0 , dim@V = dim@IT/V' and V" = V. Hence V is the solution space of the H S P ( D ) f = 0 (P E v 9 . Remark 4.2. The theorem below, 4.3, (3), implies that a D-closed subspace consists of exponential polynomials. It is easy to see that, if V is a finite dimensional vector space consisting of holomorphic functions on C", it is D-closed if and only if it is translation invariant. For the case of infinite dimensional spaces, see Ben-Artzi-Ron [3], (1.3) and Schwartz [17] (real case). Proof. Since the operators are assumed to be with constant coefficients, ( 1 ) 3 ( 2 ) easily follows. Suppose that (2) holds. By 2.5, V' c ll is an ideal. Let f := t(fl,. . . ,fk) ( t : transposed) denote the bases of V as a vector space. Then there is a matrix Mi with complex elements such that Dif = Mif. If pi E @[A] denotes the characteristic polynomial of Mi we have cpi(Di)f = cpi(Mi)f = 0

(1 F i 5 n)

by the Cayley-Hamilton formula. In other words, the ideal V'- includes monic polynomials cp1(21),. . ., (~"(2").This proves that dimll/V' = 0 and dimcIT/V' < 00 by 3.2. By 2.7, we know that VlO = V' and VlO'O = V l l . Applying 2.2 to the non-degenerate sesqui-linear form ( , ) o : IT x F C and the finite dimensional subspace V c F , we see that VlO'o = V and dim@V = dimc II/V'o. Hence we have V" = V and dim@V = dimcIT/V'. Of course, V" = V implies (1).

-

In the below, take q c C[z] for the ideal generated by the total symbols of a system of linear partial differential equations with constant coefficients. Let us restrict ourselves to the holonomic case, the case when the equivalent conditions of 3.2 hold for q.

94

S. Izumi

Theorem 4.3. Let q

c Il := C [ x ]be

an ideal such that dimII/q

=0

and

q = q1 n ... n q t

its unique shortest primary decomposition. Then we have the following. (1) The radicals of the primary ideals qi are expressed as n

f i = (a:- e i )

=

CcXj- eij)n. j=1

( e i := (Oil,. . . , ein), Bi #

Bj

(i # j ) ) .

(2) There exist decompositions

q=q1...4t, II/q = n/q1 a3 * * * a3 n / q t of the ideal, the ring respectively and

q I = q 11@ ” ’ @ q t I of the D-closed subspace such that

dim@qf = dim@II/qi< 00,

(3) If we define the shift

ti

qf’

= q 2, .

q l l =q.

of qi by

ti := { P ( Z

+ 0,) :p E qi} c II,

then ti is a primary ideal whose radical is the ideal (x):= Cy=lxjII corresponding to the origin and t’ consists of polynomials. W e have also

-

I exp(Oi. E ) , qiI = ti

where

(82 v(cp(f)) and this power 2 is the least possible number ( [29]). 5. Inequality of the orders of products: (CI-1) The inequality (CI-2) is reduced to a simple inequality of the order ofproduct of elements of analytic integral domain ( [23]). Rees [47] generalises this to general local domains whose completions are integral domains as follows: Let A be a local integral domain whose completion is also a n integral domain and let m be its maximal ideal. Then we have the following la

> 1 3 b 2 OVf

E

AVg

E

A:

(40+ 4 7 ) 5 ) v(fL7)I 4.f)+ a . 4 7 ) + b.

(CI-1)

This enables us to generalize ((21-2) to homomorphisms of excellent local domains with characteristic 0 ( [28]).Excellence is needed to define r1 using Kahler differential. The proof of (CI-1) reduces to linear comparability of valuations defined by the irreducible components of the exceptional fibres of resolution of singularity. Hubl-Swanson [19] has given a valuation theoretic proof of such comparability. These inequalities are useful in algebraic singularity theory (commutative ring theory) until today ( [63], [20], [21], [lo], . . .), which is beyond the scope of this note. 6. Local boundedness of the constants in CIS

The subject of this survey has the root in the geometry and the analysis in real smooth category. One important step is Glaeser’s composite function theory ( [14]) which asserts the closedness of the ring of pullbacks of Whitney functions with respect to a certain good real analytic map. A prototype of ((21-2) appears in the proof of [58], IX, Lemme 1.3 which is used to prove Glaeser’s theorem. The composite function theory is related to local boundedness of constants in (CI-2) and they are also equivalent to many other important

116 5’. Izumi

properties of closed subanalytic set [5]. Wang [65] reduces uniformity of constants in (v) to those of the inequality of order of product and obtained a partial proof of them. Adamus-Bierstone-Milman [2] completes the proof of uniformity of constants for general analytic maps which satisfy the condition r1 = 7-3 everywhere. They also obtained the uniformity result on (CI-3) in Section 9 for ”Nash subanalytic subset” ( [2]).

7. Zero estimate The complementary inequality (CI-2) can be also applied to transcendental aspect of singularity. In the theory of transcendental numbers, we often come across the problem of zero estimate. The problem of zero estimate at a single point is the following. Given f l , . . . ,f, E C{z}, find the maximum O(k) := Ofl ,...,f , ( k ) of the order of a non-identically-vanishingpolynomial of degree k in them. This function O(k) measures the stiffness of f1,. . . , f,. The most stiff functions are polynomials and the stiffness is expressed by Bezout’s theorem. In general, we have the following [27]. Let K be a field of characteristic 0, A a local K-algebra with maximal ideal m and let { f l , . . . ,f,} c A contains a system of generators of an

m-primary ideal. Then the following conditions are equivalent. (1) trdegKK(f1,. . . , f,) = dimA, where K(f1,. . . ,f,) denotes the field extension of K . (2) O(k) 5 ak b for some a , b E R. (3) a = a ( f l ,. . . , f,) : = l i m ~ u p ~ - , ~ l o g ~ O = I( .k )

+

For example, let f E R{xl,.. .,xn}.Then a(x1,.. . ,xn,f) = 1 if and only if f is algebraic over the rational function field Q(z1, . . . ,xn). Hence Q is a measure of transcendence. The case of exponentials in linear functions is important in number theory and closely studied. For example, it is known that

+

a ( x , y, expx, expy, exp(d3x 4%)) =3 a(expx, expy, e x p ( h x h y ) ) = 1.5

+

in c{~,Y}, in C { x ,y } .

We can generalise the zero estimate for the case of exponentials in algebraic (Nash) functions with a certain additional condition using (CI-2) ( [29]). For example we have

a(%,expx2 ) = 2,

a(%,e x p m ) = 2 in ~ { x } , in C { x } . a(exp &Ti, exp 1/2.?-2)= 2

FIILndamental properties of germs of analytic mappings

117

Khovanskii and Tougeron [62] introduce a quite general category of functions called Noetherian functions. Gabrielov-Khovanskii [13] obtained a theorem on intersection multiplicity, an effective bound of the multiplicity of “Noetherian complete intersection”. This yields a certain effective zero estimate and a certain upper bound for the quantity Q in the Noetherian category ( [32]).

8. Analogy between Gabrielov’s theorem and (CI-2) Gabrielov’s theorem (0) (ii) and the complementary inequality (0) (v) are analogical in the sense that both imply that, if the condition r1 = 7-3 is satisfied, the original function germ has the similar property to its pullback modulo a kernel element. Furthermore, these two theorems have parallel variants. They can be interpreted in terms of analytic properties of holomorphic (or formal) functions on projective varieties (or Moishezon spaces). The assertion (0)

(ii) is equivalent to each of the following

(1) Let S c C” be a reduced and irreducible afine algebraic variety S C”. Then

PK(z):= sup{If(z)ll’d

: d E

c

N,deg(f) I d such that l l F l l ~= 1)

is locally bounded o n S , where 11 I ( K denotes the uniform norm: Degree regulates growth, Sadullaev [55]. (2) Let S be a thin connected Moishezon subspace of a complex space X such that X is reduced and irreducible along S and 02 the completion of the structure sheaf of X . If a global formal function f” E r(S,0 2 ) is convergent at a point E S , it is convergent o n i.e. J E r(s,0x1: Simultaneous convergence theorem, Izumi [26].

s

Original Sadullaev’s theorem (1) is a little more sharper and (2) can be generalised to rnorphisms ( [30]). The assertion (0)-

(v) is equivalent to each of the following

(1’) Let S c C” be a reduced and irreducible afine algebraic variety S C”. Then we have the following Vc E S 3a > 0 : v s , c ( f )5 adeg(f) or (f = 0 o n S ) : Degree regulates vanishing order, Izumi [26].

c

118

5’.Izumi

(2‘) Let S be a thin connected Moishezon subspace of a complex space X such that X is reduced and irreducible along S defined by a coherent ideal sheaf Z c Ox. Then we have the following

s 3a > o ~j E r(s,0x1 : (v;,,(f) I)vx,& i a . v;,,(f). (see 52 for the symbol v;,,): Simultaneous vanishing theorem, Izumi V< E

1261.

*

(ii) with the estimate Tougeron [61] proves Gabrielov’s theorem (0) of the convergence radii, from which he deduces (CI-2). Conversely Hubl’s algebraic proof [18] of the generalization of the Gabrielov’s theorem stated in $3 uses the generalised (CI-2) stated in 55. Thus we see that Gabrielov’s theorem (0) logical.

* (ii) and (CI-2) are ana-

9. Geometric flatness along subsets

The inequalities (CI-1) and (CI-2) are equivalent to another linear complementary inequality (CI-3) below. Let S c X c R” be a closed subset of an analytic subset X and p s , ~ ( f (f ) E O X , € denote ) the order of vanishing of 1 f 1 along S at E S: ps,,t(f) := inf{p : 3 c > 0 , 3neighbourhood

U of < s.t. If(z)lI cl 0 V ’ ~ > 0 3 f ~ ( ~ = ( z l , z 2 , ~~ ~3 }a : . v ( f ) ~ < p S , € ( f )

holds and we can not replace the power 2 by a smaller number ( [31]). This set S is constructed as an image of a semianalytic set by Osgood map and using the zero estimate introduced in 57. 10. Artin approximation theorem

The inequality (CI-1) for orders of product has close relations to approximation theorem for analytic equations. Artin [l]has proved the following.

Fundamental properties of germs of analytic mappings

119

Consider simultaneous functional equations fi(Z,Y) = 0

(i = 1,.. . ,P),

where fi E C { x , y}, x := ( X I , . . . ,x,), y := ( y l , . . . ,ym). W e consider y as unknown functions in x with y(0) = 0. Then, for any natural number k and for any 8 with fi(z,g(x))= 0 and g ( 0 ) = 0 (formal solutions), there exists a system of analytic solutions y(x) with y(0) = 0 , such that v ( y j ( x ) - y?i(x))2 k, where v is the order defined by the maximal ideal of

~“41. This is called Artin approximation theorem (AT). If we replace the equa) some tions fi(x,g(z))= 0 by the inequalities v(fi(x,g(z)))2 ~ ( k for function ~ ( k )we , have a strong approximation theorem (SAT). Artin has found that SAT is valid if the equation are algebraic i.e. fi are all polynomials. Wavrik has proven that it is valid still in the case f i E C{x}[y]. These are very general assertions and very useful. Lejuene-Jalabert [35] treats SAT for the system of the equation of constraining curves t o a hypersurface with isolated singularity. Hickel [16] deletes the condition of isolated singularities. These problem are related to Nash’s problem [42]. Spivakovsky [57] poses a general conjecture: a linear approximation theorem (LAT), namely, SAT with affine function ~ ( k=) ak b for algebraic equations. The simpelest nontivial example of a LAT is the following.

+

Let p C C { x } be a prime ideal and a1,. . . ,aq be its generators. Then it is easy to see that (CI-1) mentioned above is equivalent to the following.

( f l , f 2 , S l , * * . 797-E C{zc)>.

Recently Rond [50] finds an example for which LAT does not hold. He shows also some sufficient conditions under which LAT is affirmative and some applications [49], [51], [52], [53]. In the process, he has obtained an analogue (for formal functions) of Diophantine inequality in number theory.

120

S. Izumi

11. Arc space versions Recently, Hickel [17] has shown arc space versions of the following results on germs of general analytic mappings (see $1): analytic mappings:

(1) Tougeron’s theorem [59] on Lojasiewicz inequality for analytic mappings.

(2) (0) (ii) (The case when the analytic closure of t h e geometric image is smooth). (3) (0) (v) ((CI-2))(The case when the analytic closure of the geometric image is smooth). Further Hickel [17] introduces the notion of Nash sequences and proves its relation t o the motivic integration.

References 1. Artin, M., On the solutions of analytic equations. Invent. Math. 5 (1968), 277-291. 2. Adamus, J., Bierstone, E., Milman, P., Uniform linear bound in Chevalley’s lemma, preprint 2005. 3. Abhyankar, S. S., van der Put, M., Homomorphisms of analytic local rings, J. Reine Angew. Math. 242 (1970), 26-60. 4. Bierstone, E., Milman, P., Relations among analytic functions I, 11, Ann. Inst. Fourier Grenoble 37-1 (1987), 187-239, 37 2, 49-77. 5. Bierstone, E., Milman, P., Geometric and differential properties of subanalytic sets, Ann. Math. 147 (1998), 731-785. 6. Bierstone, E., Milman, P., Subanalytic geometry, in: Model theory, algebra, and geometry (ed. by Haskell, D. et al.), MSRI vo1.39, Cambridge U. P., 151172, 2000. 7. Bierstone, E., Milman, P., Pawhcki, W., Composite differentiable functions, Duke Math. J. 83 (1996), 607-620. 8. Becker, J., Zame, W. R., Applications of functional analysis to the solution of power series equations, Math. Ann. 243 (1979), 37-54. 9. Eakin, P., Harris, G., When @(f) convergent implies f is convergent, Math. Ann. 229 (1977), 201-210. 10. Ein, L., Lazarsfeld, R., Smith, K., Uniform approximation of Abhyankar valuation ideals in smooth function filds, Amer. J. Math. 125 (2003), 409-440. 11. Gabriblov, A. M., The formal relations between analytic functions. Funkcional. Anal. i Priloihen 5 (1971), 64-65 (Functional Anal. Appl. 5 (1971), 318-319). 12. Gabriblov, A. M., Formal relations between analytic functions. Izv. Akad. Nauk. SSSR 37 (1973), 1056-1088 (Math. USSR Izv. 7 (1973), 1056-1090). 13. Gabriblov, A. M., Khovanskii, A., Multiplicity of a Noetherian intersectioin, Amer. Math. SOC.Transl. 186 (1998) 119-130.


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14. Glaeser, G., Fonctions composees differentiables, Ann. Math. 77 (1963), 193209. 15. Grothendieck, A., Technique de Construction in Geometrie Analytique VI, Etudes local des Morphismes, Sem. H. Cartan, 1960/61, exp. 13. 16. Hickel, M., Fonction de Artin et germs de courbes tracees sur un germ d’espace, Amer. J. Math. 115 (1990), 1299-1334. 17. Hickel, M., Sur quelques aspects de la geometric de l’espace des arcs traces sur un espace analytique, Ann. Facult’e Toulouse XIV (2005), 1-50. 18. Hubl, R., Completions of local morphisms and valuatons, Math. Z. 236 (2001), 201-214. 19. Hubl, R., Swanson, I., Discrete valuations centered on local domains, J. Pure Appl. Algebra 161 (2001), 145-166. 20. Ishii, S., The asymptotic behavior of plurigenera for a normal isolated singularity, Math. Ann. 286 (1990), 803-812. 21. Ishii, S., Extremal functions and prime blow-ups, Com. Alg. 32 (2004), 819827. 22. Izumi, S., Linear complementary inequalities for orders of germs of analytic functions, Invent. Math. 65 (1982), 459-471. 23. Izumi, S., A measure of integrity for local analytic algebra, Publ. RIMS Kyoto Univ. 21 (1985), 719-735. 24. Izumi, S., Gabrielov’s rank condition is equivalent to an inequality of reduced orders, Math. Ann. 276 (1986), 81-89. 25. Izumi, S., The rank condition and convergence of formal functions. Duke Math. J. 59 (1989), 241-264. 26. Izumi, S., Increase, convergence and vanishing of functions along a Moishezon space. J. Math. Kyoto Univ. 32 (1992), 245-258. 27. Izumi, S., A criterion for algebraicity of analytic set germs. Proc. Japan Acad. 68 Ser.A (1992), 307-309. 28. Izumi, S., Note on linear Chevalley estimate for homomorphisms of local algebras, Communications in Algebra 24 (1998), 3885-3889. 29. Izumi, S., Transcendence measures for subsets of local algebras, in: Real analytic and algebraic singularities (ed. T.Fukuda et al.) Pitman Res. Notes Math. 381, Longman, Edinbcrgh Gate 1998, 189-206. 30. Izumi, S., Convergence of formal morphisms of completions of complex spaces, J. MathSoc. Japan 51-3 (1999), 732-755. 31. Izumi, S., Flatness of differentiable functions along a subset of a real analytic set, J. Analyse Math. 86 (2002), 235-246. 32. Izumi, S., Local zero estimate, in: Several topics in singularity theory, Suurikaiseki Koukyuuroku 1328 (2003), 159-164. 33. Izumi, S., Restrictions of smooth functions t o a closed subset, Annales de 1’Institute Fourier, Grenoble 54 (2005), 1811-1826. 34. Khovanskii, A. G., Fewnomials, Transl. Math. Monographs, 88, American Mathematical Society, Providence, 1991. 35. Lejune-Jalabert, M., Courbes t r a c k s sur un germes d’hypersurface, Amer. J. Math. 112 (1990), 525-568. 36. Lejeune-Jalabert, M., Teissier, T., Cloture integral des ideaux et

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Bquisingularite, Univ. Sc. et Medicale de Grenoble, 1974. 37. Malgrange, B., Frobenius avec singularit&, 2, Le cas ghnkral, Invent. Math. 39 (1977), 67-89. 38. Morimoto, T., Thhor&me d’existence de solutions analytiques pour des systmes d’hqations aux derivees partielles non-linbres avec singularits, C. R. Acad. Sci. Pairs 3 2 1 S rie I, (1995), 1491-1496. 39. Muhly, H. T., A note on a paper of P. Samuel, Annals of Math. 60 (1954), 576-577. 40. MOUSSU, R., Tougeron, J. C., Fonctions composBes analytiques et differentiables, C. R. Acad. Sci. Paris 282 21 (1976), 1237-1240. 41. Nagata, M., Note on a paper of Samuel concering asymptotic properties of ideals, Mem. Coll. Sci. Univ. Kyoto, Ser. A 30 (1957), 165-175. 42. Nash, J., Arc structures of singularities, Duke Math. J. 81 (1995), 31-38. 43. Osgood, W. F., Lehrbuch der Funktionentheorie, Erster Band, Chelsea publishing (1965). 44. Rees, D., Valuation associated with ideals (11), J. London Math. SOC. (3), 31, 221-228 (1956). 45. Rees, D., Valuation associated with a local ring (11), J. London Math. SOC. (3), 31, 228-235 (1956). 46. Rees, D., Lectures on the asymptotic theory of ideals. London Mathematical Society Lecture Note Series, 113, Cambridge University Press 1988. 47. Rees, D., Izumi’s theorem, in: Commutative algebra (ed: Hochster, M. et al.), MSRI vo1.15, Springer 1989. 48. Rider, J. J., Les exposants de Lojasiewicz dans cas analytique &el, Appendix of [36], 1974. 49. Rond, G., A propos de la fonction de Artin en dimension N 2 2, C. R. Math. Acad. Sci. Paris, 340, 8 (2005), 577-580. 50. Rond, G., Sur la linharith de la fonction de Artin, Ann. Sci. Ecole Norm. Sup. 38, 4 (2006), 979-988. 51. Rond, G., Lemme d’Artin-Rees, thhorerni: d’Izumi et Fonction de Artin, J. Algebra 299, 1 (2006), 245-275. 52. Rond, G., Exemples de fonctions de Artin de germes d’espaces analytiques, Proceedings of the Third Franco-Japanese Symposium on Singularities, Sapporo 2004, to appear. 53. Rond, G., Approximation diophantienne dans le corps des series en plusieurs variables, Ann. Institut Fourier 56 2 (2006), 299-308. 54. Samuel, P., Some asymptotic properties of powers of ideals, Annals of Math., 56 (1952), 11-21. 55. Sadullaev, A., An estimate for polynomials on analytic sets, Math. USSR Izv., 20 (1983), 493-502. 56. Shiota, M., Relation between equivalence relations of maps and functions, in: Real analytic and algebraic singularities (ed. T.Fukuda et al.) Pitman Res. Notes Math. 381, Longman, Edinburgh Gate 1998, 114-144. 57. Spivakovsky, M., Valuations, the linear approximation theorem and convergence of formal functions, Proceedings of the second SBWAG, Santiago de Compostela, Alxebra 54 (1990), 237-254.


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58. Tougeron, J . C., Idhaux de fonctions diffhrentiables, EM 71, Springer 1972. 59. Tougeron, J. C., An extension of Whitney’s spectral theorem, Pub. Math. IHES 40 (1972), 139-148. 60. Tougeron, J. C., Courbes analytiques sur un germ d’espace analytique et applications. Ann. Inst. Fourier 26 (1976), 117-131. 61. Tougeron, J. C., Sur les racines d’un polynome a coefficients series formelles, in Real analytic and algebraic geometry (LNM 1420). Berlin, Springer 1990, 325-363. 62. Tougeron, J. C., Alghbles analytiques topologicment Noetheriannes, Th orie de Khovanskii, Ann. Inst. Fourier 41 (1991), 823-840. 63. Tomari, M., Watanabe, K., On L2-plurigenera of not-log-canonical Gorenstein isolated singularities, Amer. Math. SOC.109 (1990), 931-935. 64. Yoshinaga, E., Fukui, T., Izumi, S., Analytic functions and singularities (in Japanese), Kyoritsu, Tokyo 2002. 65. Wang, T., Linear Chevalley estimates, Trans. Amer. Math. SOC.347 (12) (1995), 4877-4898.

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Singularity theory of smooth mappings and its applications: A survey for non-specialists Shyuichi Ieumiya Department of Mathematics, Hokkiado Uniwersity, Sapporo, 060-0810,Japan E-mai1:immiyaOmath. sci.hokudai. ac.jp This is an elementary survey on the results of the singularity theory of smooth mapping and its applications. Keywords: Singularity theory, Smooth mappings, Applications of singularity theory

1. Introduction: Elementary calculus

There are two roots of the theory of singularities of smooth mappings. One is the Morse theory the other is the theory of immersions and embeddings of manifolds. However, the both theory are originated by the theory of smooth functions of one-variable. Here we start to review some results on functions of one-variable from elementary calculus. In the course of elementary calculus, one of the most exciting results is that we can solve the extremal problem as an application of differential calculus. By using the method of differential calculus, we can recognize the shape of graphs of smooth functions of one-variable. How can we recognize the shape of the graph? The algorithm is as follows: 1) Calculate the first derivative f’(z) of f(z) and find a point z o such that f’(z0) = 0. 2) Calculate the second derivative f”(z),then the point

20

is

minimal if maximal if

f”(z0) f”(z0)

>0 0 non-extremal if det 'H(f)(zo,yo) < 0

'

If det 'H(f)(zo, yo) = 0, the situation depends on the individual cases. For the singular point (z0,yO) with det'H(f)(xo,yo) # 0, the local shape of the graph z = f (x, y) looks like z = f ( x 2 y2) or z = & ( x 2 - y2). (See Fig. 2.1 and Fig. 2.2). This is the reason why we can distinguish the point is extremal or not in the case det%(f)(xo,yo) # 0. The above arguments might be called the "local Morse theory" for functions with two variables.

+

Fig. 2.1.

Fig. 2.2.

We can generalize the above arguments to functions with n-variables as follows: We say that 20 E Rn is a singular point (critical point) of a smooth function f ( ~ 1 , .. . , z), if d f /ax:,(zo) = 0 for i = 1,.. . ,n. In this case the Hessian matrix is a n x n matrix defined by

20 of f(s1,. . . , z), is non-degenerate if det 'H(f)(zo)# 0. Then we have the following Morse lemma (cf., Milnor [55])

A singular point

Theorem 2.1. ( T h e Morse lemma) Let 0 E R" be a non-degenerate singular point of f(x1,. . .x,). T h e n there exists a diffeomorphism g e r m

128

-

S. Izumiya

cp : (R", 0)

(R",0) such that

f 0 cp(x1,. . . ,xn)= *xcq f . * . fx:

+ f(0).

By the Morse lemma, we can recognize the shape of the graph y = f(z1,. . . ,2), around a non-degenerate singular point very well. If we consider the global situation, we consider smooth functions on manifolds. Locally the shape of the graph (i.e., the manifold) is well understood by the above reason. Moreover we can obtain global information by Morse functions (i.e., smooth functions with only non-degenerate singular points and the critical values are different). This is known as the Morse theory on manifolds. We now arrange the terminology of the local theory of singularity for smooth functions. Let f,g : (Rn, 0) R be smooth function germs. We say that f and g are R+-equivalent if there exists a diffeomorphism germ cp : (R", 0) (Rn, 0) and a real number c such that focp(x1,. . . ,x,)+c = g(x1,.. . ,x,). Under this terminology, the Morse lemma asserts that the smooth function germ f at the non-degenerate singular point 0 is R+equivalent to a non-degenerate quadratic form. This means that the nondegenerate quadratic forms give the normal forms of functions with nondegenerate singular point. On the other hand, we say that 0 E R" is a degenerate singular point if af/&:i(O) = det ( a 2 f / ~ x & , ( 0 ) )= 0 i , j = 1,.. . ,n. The next question is as follows:

-

Problem:

-

How can we study degenerate singular points?

In order to consider this problem, we introduce the following notion : corank(f)(O) = n - rank'H(f)(O). We call it the corank off at 0. A singular point 0 of f is non-degenerate if and only if corank (f)(O) = 0. We can interpret that the Morse lemma is a classification theorem on smooth functions around corank zero singular point. The following theorem is a stepping stone to the next development of the singularity theory of functions with several variables. Theorem 2.2. (Thorn's splitting lemma) Let 0 E Rn be a singular point of f ( x 1 , .. . ,xn)with corank(f)(O) = r. Then there exist a diffeomorphism germ cp : (Rn,O) --t (R",O)and a smooth function germ g(x1,.. . ,z,) at 0 such that 2 f 0 cp(z1,. . - 2 , ) = g(x1,.. . ,2,)f x,+1 f . . . f x:

+ f(0)

Singularity theory of smooth mappings and its applications

129

and rank'Fl(g)(O) = 0. We call g a residual singularity of f. By the above theorem, it is enough to consider the residual singularity in order to classify the function germs by the %?,+-equivalence. If corank(f)(O) = 1 , the residual singularity g is a smooth function of one-variable, so that we can classify f by Theorem 1.1. For the case corank(f)(O) 2 2, we need the notion of unfoldings. Let f : (Rn,O) (R,o) be a smooth function germ. We say that a smooth function germ F : (R" x RT, 0) (R, 0) is an r-parameter unfolding of f if F(x,O) = f ( x ) . We denote Em the local ring of smooth function germs (Rm,O) R with the unique maximal ideal ?lJlm = { h E Emlh(0) = 0). Let F, G : (R" x RT,0) (R,0) be unfoldings. We say that F and G are P-R+-equivalent if there exists a diffeomorphism germ : (R" x RT,0) (R" x RT,O)of the form @ ( x , u )= (%(x,u),cp(u))and a function germ h : (RT, 0) R such that G(z, u)= F ( @ ( xu))+h(u). , For any F1 E ?lJln+r and F2 E ?lJln~+T, Fl, F2 are said to be stably P-R+ -equivalent if they become P-R+-equivalent after the addition to the arguments xi of new arguments yi and to the functions Fi of nondegenerate quadratic forms Qi in the new arguments (i.e., F1 Q1 and F2 Q2 are P-R+-equivalent). Let F : (R" x RT,0) (R, 0) be an r-parameter unfolding of f. We say that F is an R+-versal unfolding of f if

-

- -

where

Jf =

-

-

+

+

(g,...,

For classification for R+-versa1 unfolding, we need extra machinery such asl'finite determinacy" and "versality theorem" etc. The reader who is interested in the detail of the story can refer to the books ( [2,8,48]). We only refer the following classification theorem which is called the Thorn's seven elementary catastrophes. Theorem 2.3. Let f : (Rn,O) --t (R,O) be a smooth function germ with En

1I p(f) = dimw- < 5. Jf The ( p (f) - 1)-parameter R+-versal unfolding of f is stably P-R+equivalent to one of the following unfoldings: ( 1 ) x; u1z1,

+

130 S. Zzumiya

We will return to this subject in $4 from the different (more sophisticated) view point.

3. Singularities of smooth mappings

The inverse function theorem (Theorem 1.2) asserts that the inverse function around the singular point is also a single valued smooth function. We now consider the vector valued version of the inverse function theorem. It can be formulated as the implicit function theorem (cf., [15], Theorem 2.4). Let f : R" -+ RP be a smooth mapping. We consider the Jacobi matrix at x E R" is the p x n-matrix defined as follows:

where f(z)= ( f l ( x ) , .., . f p ( z ) and ) z = ( X I , ...,x,). As a corollary of the implicit function theorem, we have the following theorem:

-

Theorem 3.1. Let f : (Rn,O) (Rp,O) be a C"-map germ. If rankJf(0) = min (n,p), then there exist diffeomorphism germs cp : (Rn,O) (Rn,O) and $J : (RP,O) ( I w p , O ) such that the following conditions hold: (1) I f n 5 p then $J o f o cp-l(x1,. . . , x n ) = ( x i , .. . ,zn,O,.. . , O ) . (2) ~ f >np then .IC, o f 0 cp-'(xi,. . . , x n ) = ( 2 1 , . . . ,xp).

-

-

We say that f is an immersion at 0 if the condition (1) in the above theorem holds and a submersion if the condition (2) holds. We also say that 0 is a singular point o f f if rankJf(0) < min ( n , p ) . We have the following typical examples of singular points: ( 4 f : (R2,0) (b) f : (R2,0) (c) f : (R2,0)

--

(R3,0) ; f(.,v) = (x2,Y,2Y), (a2,0) ; f(x,Y) = (x,y2), (R2,0) ; f ( X , Y ) = ( S , Y 3 ZY>.

+

Singularity theory of smooth mappings and i t s applications

131

Example (a) The germ of example (a) is called the Whitney's umbrella (or, cross cap). The Jacobian matrix is given by

It follows that the singular point is the origin (0,O). The image of this map-germ is depicted in Fig. 3.1.

Whitney's umbrella Fig. 3.1.

This map germ was discovered by Whitney [78] in the process of the research of immersions and embeddings of manifolds. Actually, Whitney has shown the following theorem [78]. Theorem 3.2. (Whitney's immersion theorem) Let N be a n ndimensional manifold. The set of immersions

Imm (N,R2")= { i : N

-

W2"

I rankdi,

= n at any 5 E N }

is an open and dense subset of the space of all C"-mappings C"(N,RZn) equipped with the Whitney CO"-topology. The natural question after the above immersion theorem was proven has been given as follows: Question How about the set Imm ( N ,R2"-1)?

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S. Izumiya

The answer is that the set of immersion is not dense in C“3(N,R2”-11. Whitney [78] discovered that there are Cw- mappings which are written 2

f(z1,. . . ,%) = (21,22,. . . ,xn,z1x2,.* .212,)

under suitable local coordinates on N and R2”-’ around a singular point. The above singular point cannot be changed under any small perturbation of f. Such a singular point is called a stable singular point. The case n = 2 is the Whitney’s umbrella. This is the first stable singular point of Cm-mappings appearing in the history of Singularity theory of smooth mappings.

Example (b) The germ of example (b) is called the fold. The Jacobian matrix is given by

It follows that the set of the singular points is the x-axis (z, 0). The critical value set is also the X-axis (X,O). We describe how we can observe the singularities. Define a mapping

If we consider the canonical projection T ( X ,Y,2 ) = ( X ,Y ) ,then we have f ( z , y ) = T o Gf(z, y). The map germ Gf is an embedding, so that the image is a surface without singular points. This surface can be considered as the graph of f. We can draw the picture of the image of Gfin Fig.3.2.

fold Fig. 3.2.


133

Example ( c ) The germ of example (c) is called the cusp (or pleat). The Jacobian matrix is given by

It follows that the set of the singular points is the parabola (-3y2,y) (cf., Fig.3.3). The critical value set is the 3/2-cusp (-3y2, -2y3) (cf., Fig. 3.3).

parabola

3/2-cusp Fig. 3.3

We also define a mapping

Gf : (R2, 0)

+

(a3,0 ) ; Gf(Z,Y) = (z,y3 + XY,Y>,

then we have f (x,y) = 7~ o Gf(x, y), We can also draw the picture of the image of Gf in Fig.3.4 which looks like a pleat.

pleat Fig. 3.4.

134 S. Izumiya

In both cases, Gf can be considered as a unfolding of the folded singularities. This is the germ of the notion of unfoldings due to Thom. For folds and pleats, there is a theorem of Whitney.

Theorem 3.3. ( Whitney’s plane to plane mappings) Let N , P be two dimensional manifolds. 0 = { f E C m ( N ,P ) I f has only folds or cusps as singularities }

is a n open dense subset of C”(N, P ) . Here, C“(N, P ) is the space of all Cm-mappings from N to P , equipped with Whitney C” -topology. For the detailed description, we refer books [15,48]. Here, we only remark that the fold and the cusp are “stable” singularities. Therefore, we may say that Coo-mappings with only folds or cusps are “generic” in the space of Coo-mappings between two dimensional manifolds. For general dimensional manifolds, Thom proposed some problems (for example, see [74]). Around 1970, J. Mather solved the main part of Thorn’s problems (see the celebrated series of Mather’s papers [14,49-541). However, there are still many open problems in the topological theory on singularities of smooth mappings (cf., du Plessis-Wall [64]). On the other hand, if we consider a surface N c W3 and an orthogonal projection 7r : R3 P on to a plane P in R3,then the restriction f = TIN : N P can be considered as a “camera”. By Theorem 3.3, the contour of f ( N ) looks like “folds” or “cusps”. This fact leads the “Mathematical theory of solid shape” (see, Koenderink [44]). We will come back to this topic later.

- -

4. Lagrangian and Legendrian singularities

In this section we give a brief review on the theory of Lagrangian and Legendrian singularities due to [2,79] which are the geometric version of the theory of Thorn’s elementary catastrophes. First, we describe the theory of Lagrangian singularities. We consider the cotangent bundle 7r : T*R” + Rn over Rn. Let ( z , p ) = ( X I,..., z n , p l ,...,p,) be the canonical coordinate on T*R”. Then the canonical symplectic structure on T*R” is given by the canonical two form w= dpi A dui. Let i : L T*R” be an immersion. We say that i is a Lagrangian immersion if dim L = n and i*w = 0. In this case the critical value of 7r o i is called the caustic of i : L T*R” and it is denoted by CL. We call i ( L ) a Lagrangian submanifold. The notion of Lagrangian

cy=,

-

-


135

-

submanifolds is a natural generalization of the notion of C"-functions as follows: Let F : (Rkx R", (0,O)) (R, 0 ) be a function germ. We call

the catastrophe set of F and

BF = {x

E

(R",O)l 3 ( q , x ) E C ( F ) such that rank

-

(Exn, 0 ) be the canonical the bifurcation set of F . Let 7rn : (Rk x Rn, 0 ) projection, then we can easily show that the bifurcation set of F is the critical value set of .rr,lC(F). We call 7 r q ~ = ) 7r,lC(F) : ( C ( F ) , O ) R" a catastrophe map of F. We say that F is a Morse family of functions if the map germ AF=

(g

,...,

")

dqk

-

:(RkxR",O)-(Rk,O)

is non-singular, where ( q , x ) = (41,. . . ,qk,x1,. . .,X n ) E (Rk X R", 0 ) . In this case we have a smooth submanifold germ C ( F ) c (Rkx Rn,O) and a map germ L ( F ) : (C(F),O) T*R" defined by

-

We can show that L ( F ) is a Lagrangian immersion. Then we have the following fundamental theorem ( [2], page 300). Proposition 4.1. All Lagrangian submanifold germs in T*R" can be constructed b y the above method.

Under the above notation, we call F a generating family of L ( F ) . By definition, C L ( F= ) BF. We define an equivalence relation among Lagrangian immersion germs. Let i : ( L , x ) (T*Rn,p) and i' : (L',x') (T*Rn,p')be Lagrangian immersion germs. Then we say that i and i' are Lagrangian equivalent if there exist a diffeomorphism germ u : ( L , s ) (L',x') , a symplectic diffeomorphism germ T : (T*R",p) (T*Rn,p')and a diffeomorphism germ ';i : (Rn,7r(p)) (Rn,7r(p'))such that T o i = i' o u and 7r o T = 7 o 7r, where 7r : (T*Rn,p) (Rn, n ( p ) ) is the canonical projection and a symplectic diffeomorphism germ is a diffeomorphism germ which preserves symplectic structure on T*Rn.In this case the caustic CL is diffeomorphic to the caustic C L by ~ the diffeomorphism germ 7.

-

-

--

-

136

S. I m m i y a

A Lagrangian immersion germ into T*Rn at a point is said to be Lagrangian stable if for every map with the given germ there is a neighborhood in the space of Lagrangian immersions (in the Whitney C"-topology) and a neighborhood of the original point such that each Lagrangian immersion belonging to the first neighborhood has in the second neighborhood a point at which its germ is Lagrangian equivalent to the original germ. We can interpret the Lagrangian equivalence by using the notion of generating families.

Theorem 4.2. Let FI E Z!Jlk+" and F2 E Z!Jlk'+" be Morse families. Then we have the following: (1) L(F1) and L(F2) are Lagrangian equivalent if and only if F1, F2 are stably P-R+-equivalent. (2) L ( F ) is Lagrangian stable i f and only if F is an R+-versa1 deformation o f F ( R k x (0). For the proof of the above theorem, see ( [2], page 304 and 325). By Theorem 2.3, the generating family of a generic Lagrangian stable map germ is stably P-R+-equivalent to one of the germs in the list of Theorem 2.3 for n 5 4. Especially, we can draw the picture of caustics for n=3:

cuspidal edge

swallowtail Fig. 4.1

-

pyramid

purse

On the other hand, we now give a quick survey on the Legendrian singularity theory. Let 7r : PT*(Rn) Rn be the projective cotangent bundle over R". This fibration can be considered as a Legendrian fibration with the canonical contact structure K on PT*(R"). We now review geometric properties of this space. Consider the tangent bundle T : TPT*(Rn)-+ PT*(Rn) and the differential map d7r : TPT*(R") ---i TR" of 7 r . For any X E TPT*(IW"),there exists an element Q E T*(Rn)such that T ( X )= [a]. For an element V E T,(R"), the property a ( V )= 0 does not depend on the


137

choice of representative of the class [a]. Thus we can define the canonical contact structure on PT*(Rn)by

K

=

{ xE TPT*(Rn)1T(X)(d7r(X)) = 0).

For the canonical coordinates ( 2 1 , . . . ,z,) on R", we have a trivialization PT*(R") --" R" x P(Rn-')* and we call

. . zn),[s-type singular point at u, where v E EVM.For a function germ f : (Wn, ZO) W,f has Ak-type singular point at zo if f is R+-equivalent to the germ x:" f. . .&xi. We say that two function germs f i : (Rn,zi) W (i = 1,2) are R+-equivalent if there exists a diffeomorphism germ @ : (an, 21) (Rn,z 2 ) and a real number c such that f2 o @(z)= f z ( z ) c. The notion of ridge points was introduced by Porteous [65] as an application of the singularity theory of unfoldings to the evolute and the geometric meaning of ridge points is given as follows: Let F : R3 ----+ R be a function and X : U R3 a surface. We say that X and F-l(O) have a corank r contact at p = X ( u ) if the Hessian of the function g ( u ) = F o X ( u ) has corank r at u.We also say that X and F-'(O) have an Ak-type contact at p = X ( u ) if the function g ( u ) = F o X ( u ) has the Ak-type singularity at u.By definition, if X and F-l(O) have an Ak-type contact at p = X ( u ) ,then these have a corank 1 contact. For any r E IR and a0 E W3, we consider a function F : R3 W defined by F ( z ) = I(z- a ~ ( r(2 .~We define that

h i

-

-

S 2 ( a , r )= F - ~ ( o = ) {u E

111~: - all 2 = 2 ) .

It follows that S 2 ( a , r )is a sphere with the center a and the radius Irl. We put a = Ev,(u) and r = l / ~ ( u )where , we fix a principal curvature ~ ( u ) on U at u,then we have the following simple proposition: Proposition 5.6. Under the above notations, M = X ( U ) and S 2 ( a , r ) have either corank 1 or two contact at u.

In the above proposition, S 2 ( a ,r ) is called a focal sphere of M = X ( U ) . We also call a the focal center ~(u). By Proposition 5.6, M = X ( V ) and the


147

focal sphere has corank 2 contact at an urnbilic point. Therefore the ridge point is not an umbilic point. We also consider another geometric meaning of ridge points. A curve -y(t)= X ( u ( t ) , v ( t )on ) a surface M = X ( U ) is a line of curvature if the tangent vector q(t)is a principal direction for any t. We have the following proposition 165,661.

Proposition 5.7. Let -y(t) = X ( u ( t ) , w ( t ) be ) a line of curvature on a surface M = X ( U ) and K ( t ) the principal curvature with respect to the principal direction q(t). Then p = X(u(to),v(to))is a ridge point if and only i f i c ( t 0 ) = 0. Porteous discovered the notion of ridge points when he investigated the singular points of the evolute EVM[65]. By the general theory of unfoldings of function germs, the bifurcation set 1 3 is~ non-singular at the origin if and only if the function f = FIR" x (0) has the A2-type singularity (i.e., the fold type singularity). Therefore we have the following proposition:

Proposition 5.8. Under the same notations as in the previous proposition, the evolute EVM is non-singular at a = Evn(u) i f and only i f M = X ( U ) and S 2 ( a ,r ) have A2-type contact at u.

All results mentioned in the above paragraphs on the evolute were shown by Porteous and Montaldi [57,65]. We also define a family of functions fi : U x ( S 2 x R) R by

-

E(u,z1, r ) = ( ~ ( uv)) ,- r. We call it the extended height function of M = X ( U ) . By the previous calculations, we have

Vg

= {&CPeM(u)

IuEU }

and 130 = EVM.

) ) &n(u) = Moreover, the catastrophe map of H is n q ~ ) ( u , & n ( u= f G ( u ) . Therefore, we can identify the Gauss map of M = X ( U ) with the positive component of the catastrophe map T C ( H ) . For a surface X : U R3,we consider the distance squared function D and the height function H . We have the following propositions(cf., [39]):

-

-

-

Proposition 5.9. Both of the distance squared function D : U x R" R and the height function H : U x S2 R of M = X ( U ) are Morse families of functions. Moreover, the extended height function of M = X ( U ) is a Morse family of hypersurfaces.

148 S. Izumiya

By this proposition, the evolute is a caustics and the pedal is a wave front. Therefore, we can apply the theory of Lagrangian or Legendrian singularities to the study of the landmarks of surfaces [39]. On the other hand, we now consider the image of orthogonal projection of a surface in R3.For orthogonal projection in the unit direction k we can take an image plane through the origin 0, that is an image plane with equation x . k = 0. Then k is called the view direction and the line through p in this direction is called the visual ray. For p E M , the corresponding point q of the image plane satisfies p = q Xk, where X is the signed distance from the image plane to the point p . It follows that X = p . k, so that we have q = p - (pa k)k. The interesting point on M is the point p at where the visual ray is tangent to M . These are the points where, viewed in the direction k, the surface appears to be folded, or to have a boundary or occluding contour. The contour generator r on M is the set of points of M for which n k = 0, where n is the normal vector field of M . The corresponding apparent contour y is the set of points q of the image plane forming the projection of I? in the direction k to the image plane. In other word, the contour generator is the singular set of the orthogonal projection restricted on M and the apparent contour is the critical value set. As an application of the standard jet transversality theorem and the theorem of Whitney’s plane to plane mapping, we can show that the apparent contour has only 3/2-cusps as generic singularities. We say that the direction of a tangent vector v of M is asymptotic if it is contained in the kernel direction of the second fundamental form. Then we have the following proposition.

+

+

Proposition 5.10. The apparent contour y is smooth at q except when the view direction k is asymptotic at the corresponding point p of r. The apparent contour has a 312-cusp when the line through q in the view direction k is asymptotic and in fact has exactly 3-point contact with M at 9.

We can recognize the cusp point of the apparent contour on the picture of a mountain (cf., Fig. 5.5 : Mt. Moiwa in Sapporo). Around a smooth point of the apparent contour, we have the following formula for Gauss curvature of Koenderink [43].

Theorem 5.11. Assume that the apparent contour is smooth at q. Let I C ~ be the curvature of the apparent contour at q and I C ~is the normal curvature of M at p in the direction of the tangent vector k. Then the Gauss curvature K of M at p is given by K = K Q ~ .


149

This theorem says that if the surface looks convex (respectively, concave) around a point p , then the Gauss curvature K at p is positive (respectively, negative). We do not need to touch the surface to discern between positively curved parts and negatively curved parts (cf., Fig. 5.5, Fig. 5.6).

Fig. 5.5

Fig. 5.6

We also have several results on submanifolds of pseudo-spheres in Minkowski space as applications of the theory of singularities [29,31-38, 40,411.

6. Singularities of solutions for first order partial differential equations

In this section we consider the following two similar Cauchy problems :

where H , f, cp are C’”-functions. The equation (H) is called a HamiltonJacobi equation which plays an important role in geometric optics, calculus of variations, optimal control theory and classical mechanics. The equation (C) is called a single conservation law which plays also an important role in gas dynamics and oilreservoir problems. In this article we do not mention how these equations are used in each field. We now solve these Cauchy problems by using the classical method of characteristics. The integration of a first order partial differential equation reduces to the integration of a

150 S. Izumiya

system of ordinary differential equations, so-called the characteristic equations. We now have a system of ordinary differential equations :

t’(7)= 1, d ( 7 ) = H ’ ( p ( 7 ) ) ,

Y W

= -H(P(T))

p ’ ( 7 ) = 0,

S’(7)

+ P(7)H’(P(7)),

= 0.

This is called the characteristic equation for (H). We consider the initial condition t(0) = 0, x(0) = u, y(0) = cp(u),p(O) = ~ ’ ( us(0) ) , = -H(p(O)) corresponding to the initial curve y(u) = I’(0,u). In this case we have the exact solution

t(7,U ) = 7,

Z(7, U )

=U

+ TH’((P’(U)), +

d7 .

= 7{--H((P’(uZL))+(p’(u)H’((p’(u))} du),

p ( 7 , u ) = c p ’ ( 4 , s ( 7 , u ) = -H(cp’(u)).

We call the solution curve of the characteristic equation the characteristics. We now study the properties of the characteristics. We have an embedding L, : R2 R5 given by

-

L,(T, ). = (7,477u), Y(7, u ) ,--H(P(T, U ) ) , P ( T , u)). We also consider the canonical projection n z ( t , x , y , s , p ) = ( t , ~and ) the composition @ ( T ,u)= r z o L , ( ~ ,u ) = (7,x(7,u ) ) .Then the Jacobian matrix of @ is degenerate at the points ( t ,u ) where 1 ~H”((~‘(u))(p’’(u) = 0. Assume that 1 ~H”(cp’(u))cp’‘(u) # 0 at a point (70,uo).Then there exists an inverse mapping Q ( t ,x) of @ around the point (TO, uo ~oH’(cp’(u0)) by the inverse mapping theorem. In this case the inverse mapping has the form Q ( t , x ) = (t,$(t,z)). Moreover we define a function g by g ( t , x) = y ( t , $(t,x)). The partial derivatives of g ( t , x) can be calculated

+

+

+

ag as = cp’(u), where ( t , x ) = (7, u ~ H ’ ( p ’ ( u ) ) )It. is as = -H(cp’(u)), at ax clear that g is a solution of Hamilton-Jacobi equation (H). For sufficiently short time from the initial time, there always exists the inverse of @, so that the smooth (classical) solution of (H) exists. On the other hand, we also have the characteristic equation for (C). In this case the characteristic equation is the system of equations on 3dimensional space R3 with the coordinates (t,x,y) given as follows :

+

t’(7)= 1, d(7)= f’(y(T,Z(T)), y’(7) = 0. The corresponding initial condition is ~ ( 0 = ) u,y(0, x ( 0 ) )= cp(u).We also have the exact solution as follows :

t(7,u ) = 7,

x(7,u ) = 21 -k T.f‘((P(u)),Y(7,

= Y(0, x ( 0 ) )= P(u).


151

We can also consider an embedding Z : R2 -+ R3 given by Z(T,U ) = (7,‘ZL

+ Tf’(cp(Ul)), c p ( U ) ) .

+

Like as the case for (H), we consider the mapping @ ( T , u )= ( T , U T ~ ’ ( ( P ( u )The ) ) . Jacobian determinant is given by 1 ~f”(cp(u))cp’(u). As) a point (TO, U O ) . By the inverse mapsume that 1 T ~ ” ( ( P ( u ) ) ( P#’ (0uat ping theorem, there exits the inverse mapping 9 ( t ,$) of @ around the point (TO,uo ~of’(cp(u0)) with the form 9 ( t ,x) = (t,$(t,z)). We can also calculate the partial derivatives of $ and show that g ( t , z ) = cp o $ ( t , z ) is a solution of (C). We also have the smooth (classical) solution for sufficiently short time from the initial time. Both of these arguments for (H) and (C) are usually written in the first part of the text book on the theory of partial differential equations [3]. However, there exists a critical time at when the Jacobian determinant vanishes in general. After the critical time, the characteristics on the ( t , z )plane cross. This means that the solution y is multi-valued. The situation is depicted in Fig. 6.1 and Fig. 6.2. Fig. 6.1 is the picture of the image of ll o L, for (H) with the Hamiltonian function is H ( p ) = p 2 , where H ( t , z, y, s,p ) = (t,2,y). Fig. 6.2 is the picture of the image of Z for (C) with f ( y ) = y2. We adopt cp(u) = sinu as the initial condition in both cases. These are the pictures of graphs of multi-valued solutions solved by the characteristic method. We can observe that the multi-valued solutions appear after some critical times. Moreover, the graph of multi-valued solution for (C) is a smooth surface in IR3 and the graph of multi-valued solution for (H) has singularities. Therefore, we can understand the difference of these similar equations by observing the multi-valued solutions for both equations.

+

+

+

Fig. 6.1

Fig. 6.2

152 S. Izumiya

As we mentioned in the above sentences, we have a multi-valued solution if we solve the first order partial differential equation by the characteristic method. Why these solutions are different? Since the multi-valued solution for (C) is a smooth embedded surface in R3.The projection of such a surface onto the (z, t)-plane a(., u)is the Whitney’s plane to plane mapping for generic initial data cp(u). However, for the multi-valued solution for ( H ) , we have the canonical contact structure 6 = dy - sdt - pdx on R5. The surface L, is a smooth embedded surface in R5 with L$O = 0, so that L, is a Legendrian submanifold of (R5,S). In this case the graph of the multi-valued solution 7r(L,) is the wave front of L,. By Theorem 4.8, generic singularities of the wave fronts in R3 are the cuspidaledge or the swallowtail. We can observe the swallowtails in Fig. 6.1. Therefore the theory of Legendrian singularities is useful for the study of the singularities of multi-valued solutions for first order partial differential equations.

-1.5 -2

t

Fig. 6.3

-1

-2

Fig. 6.4

On the other hand, we need single-valued solutions for applications in some area. In this case, we cannot expect the differentiability of the solutions. We call such a solution a weak solution. We call the set of non-smooth points of a weak solution a shock wave of the solution. The notion of viscosity solutions [12] (respectively, entropy solutions [SO]) has provided the right weak setting for the study of (H) (respectively, (C)). Existence and uniqueness of the solution of both solutions hold. Although the existence of both solutions can be proved by the common method (i.e., so called the vanishing viscosity method), their features are quite different. It has been known that the viscosity solution for (H) is a piecewise smooth continuous function and the entropy solution for (C) is a piecewise smooth discontinuous function. We can observe their difference on the picture of graphs of multi-valued solutions. Fig. 6.3 and Fig. 6.4 are the pictures of the sections of multi-valued solutions in the plane (x,t ,y ) (t = constant) after the criti-


153

cal time. We can easily choose a continuous single valued function from Fig. 6.3 (i.e., then minimum branch of the graph of the multi-valued solution). We cannot, however, choose such a function from Fig 6.4. It is very interesting to study how shock waves appear and propagate. It is obvious that shock waves are deeply related to singularities of multivalued solutions solved by the characteristic method. The singularity the0y of smooth mappings provides the method for studying singularities of multivalued solutions. In fact, Guckenheimer [16] assumes that the first singular point of the multi-valued solutions for the single conservation law is the Whitney’s pleat ( the cusp ), then he describe how shock waves appear for entropy solutions. It is, however, a quite new result that the Whitney’s pleat is the first singular point of the multi-valued solution for generic initial data cp ( [20]). Classifications of generic singularities for multi-valued solutions for general dimensions have been given in the same paper [20]. The history of the study for entropy solutions is much longer than that of viscosity solutions. However, properties of viscosity solutions are well-understood more than those of entropy solutions. One of the reason is that we can easily recognize singularities of the graph of the multi-valued solution for (H). 7. Ruled surfaces

A surface in Euclidean space is called a ruled surface if it is given by a one-parameter family of lines. Ruled surfaces are classical subjects in differential geometry which have been studied since the 19th century. It has been considered that almost all interesting properties have been already known until the middle of the 20th century. It is, however, paid attention in several areas again [17,68,73].Moreover, the situation is quite different if we consider the case where ruled surfaces have singularities. Generally ruled surfaces have singularities. The first modern study of singularities of ruled surface is given in Cleave’s paper on a classification of singularities of developable surfaces of space curves [13] which has appeared incredibly new. After that, there appeared several articles concerning on singularities of developable surfaces in R3 (c.f., [18,19,22,23,56,71]).On the study of singularities of general ruled surfaces, the authors’ paper [24] might be the first result so far as we know. We now review some basic concepts on classical differential geometry of space curves and ruled surfaces in Euclidean space. For any two vectors z = ( 2 1 , 2 2 , 2 3 ) and y = ( y l , y 2 , y 3 ) , we denote z . y as the standard inner product. Let y : I R3 be a unit speed curve (i.e., IIy’(s)II = 1,

-

154 S. Izumiya

where ~ ’ ( s )= d y / d s ( s ) ) .Let us denote t ( s ) = y‘(s) and we call t ( s ) a unit tangent vector of y at s. We define the curvature of y by K ( S ) = If ~ ( s # ) 0 , then the unit principal normal vector n ( s )of the curve 7 at s is given by y”(s) = n(s)n(s). The unit vector b(s) = t ( s ) x n ( s ) is called the unit binormal vector of y at s. Then we have the Frenet-Serret formulae:

d m .

{

t ‘ ( s ) = fi(s)n(s> n’(S) = -K(S)t(S) T(S)b(S) b’(s) = -7(s)n(s),

+

-

where ~ ( s = ) det (y’(s), y”(s), y”’(s))/lly”(s)l12 is the torsion of the curve y at s. For any unit speed curve y : I R3,we call D ( s ) = - r ( s ) t ( s ) r;(s)b(s)the Darboux vector field of y (cf., [44], Section 5.2). We define a vector field E(s) = ( T / K ) ( s ) ~ ( s ) b(s) along y under the condition that K ( S ) # 0 and we call it the modified Darboux vector field of y. On the other hand, a ruled surface in IR3 is (locally) the map F(?,&): I x IR --+ R3 defined by F(y,6)(t,u) = 7 ( t ) uS(t),where y : I --+ R3, 6 : I --+ R3 \ ( 0 ) are smooth mappings. We call y the base curve and 6 the director curve. The straight lines u H y ( t ) u 6 ( t ) are called rulings. We say that a ruled surface F(T,z)is a developable surface if the Gaussian curvature K of the regular part of F(?,z) is identically zero. The following theorem is the classification theorem of non singular developable surfaces which has been classically known (cf., [77]):

+

+

+

+

Theorem 7.1. Let F(?,z) be a developable surface. Then we always have one of the following situations: (1) F(?,;i) is a part of a cylindrical surface (i.e., 6 ( t ) is constant). (2) F(?,z) is a part of a conical surface ( i e . , y ( t ) is constant).

-

-

(3) F(?,z) is a part of a tangent developable ( i.e., S ( t ) is parallel to the tangent line of y ( t ) ) . (4) The glue of the above three surfaces.

We have the following examples of developable surfaces:

Example 7.2. (Tangent developables of space curves) Let y : I + IR3 be a regular curve (i.e., y’(t) # 0 ) . If we choose 6 ( t ) = y’(t),then we call F(?,6) the tangent developable of y. It has been classically known that the tangent developable of a space curve has the cuspidal edge along the curve y ( t ) if the torsion T ( t ) # 0 (cf., Fig. 7.1). It is incredible that the generic classification of the singularities of tangent developables was shown quite recently. Cleave [13] showed that the tangent developable of a space curve

Singularity theory of smooth mappings and its applacataons 155

is locally diffeomorphic to CCR (cf., Fig. 7.1) at a point ?(to) if T ( t 0 ) = 0 and T ’ ( t 0 ) # 0. These conditions are generic for space curves. Here, CCR = { ( 5 1 , 2 2 , 5 3 ) I 2 1 = u3 , 5 2 = uv3,5 3 = v2 } is the cuspidal crosscap.

cuspidal crosscap Fig. 7.1

On the other hand, even if there exists a point to E I such that y‘(t0) = 0, we can smoothly extend the tangent vector field along the curve

under a certain condition (cf., [19]). Here, we only consider an example given by y ( t ) = (t2,t3,t 4 ) .In this case y’(t) # 0 except at the origin. The direction of y ’ ( t ) is equal to the direction of the vector 6 ( t ) = (2,3t,4t2) which is also smooth at the origin. So the ruled surface F(,,s) is called a tangent developable surface of the singular curve y ( t )= ( t 2 t, 3 , t 4 ) Arnol’d . [l] observed that this surface has a swallowtail at the origin. It is, however, known that the curve y ( t ) = ( t 2 , t 3 , t 4is) deformed into a regular curve under a sufficiently small perturbation. Therefore, the swallowtail is not a generic (stable) singularity of tangent developable surfaces of space curves. For the curve y ( t ) = ( t 2 ,t3,t 4 ) ,the tangent developable is given by F(,,6)(t,u) = (t2 2u,t3 3tu,t4 4tu). If we slightly perturb the curve into y,(t) = ( t 2 ,t3 - ~ t 4,) ,the corresponding tangent developable is F(,,,&)(t, u)= (t2 2tu, t3 - E t u(3t2 - E ) , t4 4t3u), which has a cuspidal crosscap. The situation is depicted in Fig. 7.2. The left picture is F(?,a)(t,u) and the right one is F(Y0,5,6)(t,u).

+

+

+ +

+

+

-

Example 7.3 (Rectifying developables of space curves). W e again R3 with non vanishing curconsider a unit speed regular curve y : I

156 S. Izumiya

Fig. 7.2.

vature ~ ( s ) .There is another important developable surface along y with respect to the Frenet frame. The envelope of the family of rectifying planes along y is called the rectifying developable of the curve y. Here, the rectifying plane at y(s) is defined to be the plane generated by the tangent vector y’(s) and the binormal vector b ( s ) . I n [22] we studied the singularities of rectifying developables of space curues and their geometric meaning. I n classical treatises of differential geometry, the Darboux vector of y is defined by D ( s ) = ~ ( s ) y ’ ( s ) ~ ( s ) b ( s )However, . we define a vector B(s)= (~/n)(s)y’(s) b ( s ) which is called a modified Darboux vector of y. W e can show that the rectifying developable of a unit speed space curve y is F(,,D)(S, u)= y(s) u f i ( s ) ,which has already been given in $2. In [22] we studied the singularities of the rectifying developable of a space curve y(s) with the condition that K ( S ) # 0 and ~ ( s # ) 0 for all s. I t was shown that the singularities of the rectifying developable of a generic curve with the condition that K ( S ) # 0 are the cuspidal edge or the swallowtail. The swallowtail point of the rectifying developable corresponds to a point y(s0) where the conditions

+

+

+

are satisfied.

-

Example 7.4. (Focal developables of space curves).Let y : I R3 be a regular curve such that the curvature and the torsion of the curve do not vanish at any point. The envelope of the family of normal planes along y is called the focal developable (or the polar developable) of the curve y. In order to represent the focal developable in our form, we now


157

consider the arclength parameter s, so that the tangent vector y’(s) is a unit vector. The principal normal of y is n ( s ) = ’”(’) ~

Ilr’’(s)II

and the binormal

is b ( s ) = y(s) A n ( s ) . We denote the torsion of y(s) by T ( s ) . We now define 1 new curves a ( s ) = y(s) -n(s) and 6(s) = - K ’ ( S ) b ( s ) .Then the

+

4s)

T ( S )K2 (s)

focal developable is the surface F ( n , ~(cf., ) [44]).The set of singularities is the locus of the centers of osculating spheres. We remember that the osculating sphere of a curve is the sphere which has at least four point contact with the curve. Porteous [65] showed that the singularities of the focal developable of the generic space curve are the cuspidal edge or the swallowtail. The swallowtail of the focal developable corresponds to the point y(s0) at which

Under the assumption that K ( S ) # 0 and T ( S ) # 0, the curve y is a spherical curve if and only if T ( s ) / K ( s ) = ( K ’ ( s ) / K ~ ( s ) T ( s ) ) ’ . Moreover the swallowtail point of the focal developable corresponds to the point on the curve y at where the curve has exactly five point contact with the osculating sphere.

Example 7.5. (Darboux developables of space curves).We consider the Darboux developable F Q , , ~ ) )u) ( S ,= b ( s ) uy’(s) of y with K ( S ) # 0. A point (SO,uo) is a singular point of the Darboux developable of y if and only if uo = ( T / I C ) ( S O ) .In [22], it was shown that the Darboux developable of a generic space curve is locally diffeomorphic to the cuspidal edge or the swallowtail around a singular point under the condition that T ( S ) # 0. The swallowtail point of the Darboux developabIe corresponds to a point ?(so) where the conditions

+

are satisfied. On the other hand, a curve y(s) with constant ( T / K ) ( s ) is a cylindrical helzx. So the singularities of Darboux developables describe how the shape of y(s) is similar to cylindrical helices. In fact, Fig. 7.3 is the picture of the Darboux developable of y ( t ) = ( t ,t 2 ,t 4 ) .We can observe the cuspidal crosscap. It is known that the Darboux developable is locally diffeomorphic

to CCR at the point s where

T(S)

= 0 for generic y.

158 5’. Izumiya

Fig. 7.3. We now give examples of ruled surfaces which are not developable surfaces. Let y(s) be a unit speed space curve with K ( S ) # 0.

Example 7.6. (The principal normal surface of a space curve) A ruled surface F(7,n)(s1.1

= Y(S) 3- 4 s )

is called the principal normal surface of y. By the Frenet-Serret formulae, we can show that y’(s) x n ( s )

Therefore

(SO,U O ) is

+ un’(s) x n ( s ) = (1 - U K ( S ) ) b ( S ) - .r(s)ut(s). a singular point of F(7,n) if and only if

T(SO)

=0

and

1

Consider the space curve defined by y ( t ) = ( t , t 2 , t 4 )In . this case, we can calculate that the principal normal direction is given by

5(t)= (-24t5

- 2 , l - 32t6, 6t2

+ 16t4).

Therefore, the principal normal surface is given by

F ( t ,u ) = (t - u(24t5 + 2), t2 + ~ ( -1 32t6), t4 + u(6t2 + 16t4)). We can draw the picture of the surface in Fig. 7.4. The singular point looks like a cross cap.


159

1 5

1 5

Fig. 7.4

For principal normal surfaces, we have the following classification theorem.

-

Theorem 7.7. For a unit speed curve y : I R3 with ~ ( s )# 0 , the principal normal surface F(7,n)(s,u) of y is a cross cap at ( s 0 , u o ) if and only i f 1

uo = 4so)

T(SO)

= 0 and

so) # 0.

We now consider singularities of developable surfaces. We have the following simple lemma.

Lemma 7.8. Let F(?,b) be a noncylindrical ruled surface (i.e., 6 ( t )x6’(t) # 0 ). Then F(?,6) is a developable surface if and on13 i f there exist smooth lR such that y’(t) = p ( t ) d ( t ) X(t)G’(t). functions p , X : I

-

+

By the lemma, we can control noncylindrical developable surfaces by using p ( t ) ,X(t) and 6 ( t ) . We define the space of noncylindrical developable surfaces as follows:

D e v ( I , IW~) = { ( p , A, 6)

I

(pl A, 6) : I

-

IW’ x ( I B ~\ ( 0 ) )

C”-map with 6 ( t ) A 6’(t) # 0 ) with the Whitney C” topology. As an application of the classification of singularities of the tangent developable of a (not necessary regular) space curve (cf., [13,18,19,56,71]),we can state a classification theorem of singularities of generic developable surfaces.

160 S. Izumiya

-

Theorem 7.9. Let F(y,6): I x J -i R3 be a noncylindrical developable surface. We f i x smooth functions p, X : I R with r’(t)= p ( t ) b ( t ) X(t)b’(t). Let ( t o , u o ) E I x J be a singular point of F(?,6) and set 20 = r(to) + uoWo) = F(-,,6)(tO,’LLO). (1) Suppose that det(b(t0) &’(to) b”(t0))# 0. Then ( a ) The g e m of F(-,,a)(Ix J ) at 20 is locally diffeomorphic to C x R if uo = W O ) and p(t0) # X ’ ( t 0 ) . (b) The germ of F(y,g)(I x J ) at 20 is locally diffeomorphic to SW if uo = X ( t o ) , p(to) = X‘(to) and p’(t0) # A“(to). ( 2 ) Suppose that det(b(t0) d ’ ( t 0 ) b‘’(t0))= 0. Then the germ of F(-,,q(I x J ) at 20 is locally difleomorphic to CCR ifuo = X ( t o ) , p(t0) f X’(to) and det(b(t0) # ( t o ) d 3 ) ( t o ) # ) 0.

+

On the other hand, we now review recent results on singularities of general ruled surfaces (cf., [24]). The Gaussian curvature of the regular part of a ruled surface is generally non-positive. So the developable surface is a member of the special class of ruled surfaces. Therefore we have the following natural question:

Question. How different are singularities of developable surfaces from those of “general” ruled surfaces?

-

We give a classification of singularities of general ruled surface. Let Cm(I,R3 x S 2 ) be the space of smooth mappings (y,6) : I R3 x S2 equipped with Whitney CW-topology,where I is an open interval. In [24] we showed the following theorem which gives a “generic” answer to the above question.

Theorem 7.10. There exists an open dense subset 0 C Cm(I,R3x S 2 ) such that the g e m of the ruled surface F(-,,6) at any point ( t o , u o ) is an immersion germ or A-equivalent to the cross cap for any (y,6 ) E 0. It is well known that any singular point for generic smooth mappings from a surface to R3 is the cross cap (cf.,[1,5,13,19]). The set of ruled surfaces is a very small subset in the space of all CW-mappings. The above theorem, however, asserts that the generic singularities of ruled surfaces are the same as those of generic CW-mappings. We can summarize the results of the above theorems as the following relations :

{ Singularities of generic developable surfaces} # { Singularities of generic ruled surfaces},


161

{ Singularities of generic ruled surfaces} = { Singularities of generic Coo-mappings}.

We remark that the cross cap is realized as a singularity of a ruled surface as 1 follows: Consider curves y ( t ) = ( t 2 ,0,O) and 6 ( t ) = then F(.r,6)(t, u)is the cross cap (cf., Fig. 4)which corresponds to the normal form.

0

Fig. 7.5

One of the examples of ruled surfaces with cross caps is the Plucker conoid which is given by r(8)= (0, 0 , 2 cos8 sin 8) and d(8) = (cos 8, sin 8 , O ) (0 5 8 5 27r) (cf., Fig. 7.6).

IV -2

-1

Fig. 7.6

0

1

2

162

S. Zzurniya

8. Gravitational lensings

Recently there appeared several articles considering gravitation lensing systems as applications of the theory of singularities for smooth mappings. The gravitational lensing is the deflection of light from a distant source (e.g., quasar) by an intervening matter distribution (e.g., a galaxy or a cluster of galaxies). The first gravitational lensed quasar was detected only in 1979. By now gravitational lensing is quite an active field in astrophysics [63,72]. On the other hand, singularity theory of Lagrangian varieties [2,79] is the best natural setting for discussing optical systems. In fact, A.O. Petters and his collaborators [61,63] pointed out that a single gravitation lensing can be described in the framework of symplectic geometry. Especially the caustics in a single gravitation lensing system coincide with caustics in the theory of Lagrangian singularities. Moreover, they also investigate multiplane gravitation lensing as an application of singularity theory [46,63]. In another paper [47] they pointed out that new, unexpected from the standard singularity theory point of view, caustics might appear for multiplane gravitational lensing. In [30] the symplectic framework for multiplane gravitational lensing based on the notion of symplectic relation was given. In the first place, we give a quick review of the basic concepts from the theory of gravitational lensing discussed already in [46,47,61+3,72]. (1) Single lensing (cf., [61,63,72]) Consider the typical single lens plane gravitational lensing as follows: source p l a n e

l e n s plane ( d e f l e c t o r )

'ver

QS

Fig. 8.1

We assume that the deflector is thin and apply the small angle approximation (cf., ISl]). The extra time with respect to the unperturbed ray gives


the time-delay map; Ts : R2 2 U

-

163

R defined by

Here, ZL is the red shift of the lens plane, dOL, d o s . d L s are angular diameter distances, r is the position on the lens plane where the ray hits, s is the position of the source, and @ ( r )is the two-dimensional potential of the deflector on the lens plane. The deflector potentials Q occurring in the time-delay map are given by

They are solutions of the two-dimensional Poisson equation A Q ( r ) = 87ra(r),where a ( r ) is the surface mass density. By suitable coordinate transformations, we can express the time-delay map in the convenient form:

Here y corresponds to the point on the source s -plane and x corresponds to the point on the lens plane r . Fermat’s principle yields the critical points of the time delay map Ty(x) with respect to variations in x determining those rays that are real light rays (cf., [Sl]). For this reason, a critical point of Ty(x) relative to x is called a n image of the point source at y. The magnification of an image x of a source at y is defined by

where T(x;y) = Ty(x) and Tzz(x;y) is the Hessian matrix with respect to x. A caustic point in gravitational lensing is a position y E R2 for which a source at y will have at least one image of infinite magnification. In other words, caustics are source positions y E R2 for which the time-delay map TY(x)has at least one degenerate critical point (i.e., detTxz(x;y) = 0). So, we may consider that the time-delay map is the generating family of a certain Lagrangian submanifold in T*R2 (cf., 53). (2) Multiplane gravitational lensing (cf., [47,62,63,72]) Although we can consider a general k-planes gravitational lensing, we now only consider the case when k = 2 (i.el a double plane gravitational lensing) for convenience. The typical double lens plane gravitational lensing situation is given as follows (Fig. 8.2):

164

S. Izumiya

source plane S

second lense plane f i r s t lense plane L2 L,

c

S

observer dl 2

dl s

0

/

dl

4

d2 dS

Fig. 8.2

There are two lens planes with "thin" deflectors in each plane. The deflectors are assumed to be independent, that is, the lens planes are sufficiently spaced so that they do not interact. Furthermore, the small angle approximation is assumed. We also parameterize all rays originating from the point source at s, deflected by two gravitational lens, using the 4-dimensional vectors ( T I , 7 - 2 ) . Relative to these approximations the extra time Ts to reach the indicated observer from s is given by the time-delay map. It is the function Ts : U1 x U2 c IR4 + IR with each domain Ui c IR2 being an open subset, defined by

Here, zi is the redshift of the ithe lens plane, di, is the angular diameter distance separating the ithe and jthe lens planes, di is the angular diameter distance from the observer to the ith lens plane with dk+l E ds the distance to the source plane, ci is the position on the ithe plane where the ray hits, T L + ~= s , and Q i ( r i )is the two-dimensional potential of the deflector on the ithe lens plane. By suitable coordinate transformations, the double plane time-delay map can be expressed conveniently as follows:


165

In [62], Fermat's principle is adapted exactly in the same way as it was used already for the single lens plane case, so that the image of a gravitational lensed point like light source at position y are identified to the critical points of TY e.g., the set of images is given as follows: ( ( 2 1 ,~ 2 ) l g r a d ~ ~ T y 3( 2z )l ,= 0 , = 1,2}. If we adapt this principle simply, we cannot distinguish the effect of individual lens planes. In fact, in [62] two lens planes are treated like a single lens altogether. It is, however, pointed out in [47] that double-folds or handkerchiefs might appear as the generic caustics for double plane lensings. These singularities do not appear as generic caustics under the above construction. In 1611 Petters pointed out that single gravitational lensing can be described in the framework of symplectic geometry (i.e, Lagrangian singularity theory). Now let us recall the time-delay map Ty(z). If we consider the family of functions F : R2 x R2 -+ R given by

we can easily verify that F is a Morse family, so CPF : C ( F ) + T*R2 is a Lagrangian immersion. Here, we have

W )= ((2,Y) I grad,T

=0

1,

so that the Lagrangian immersion is corresponding to those rays that are

actual light rays. The set of critical values of the Lagrangian map x o CPF is the caustic.

On the other hand, we present another symplectic framework for single gravitation lensing, which is essentially the same as the above framework. Our framework will be, however, much useful when we try to generalize this framework to the case of multiple planes gravitational lensing (cf., $4). We consider the product symplectic space

where WM, and WM, are the corresponding canonical symplectic forms, and Q(z,y) = WM, 8WM, = T ~ ~ W M-,T ~ ~ w Mwhere ,, T M ,, X M , are the canonical projections of the product T*Mx x T*My. The corresponding phase spaces (T*Mx,WM,) and (T*My,WM,) are called the observer space and the source space respectively. In our 2-dimensional case M, = R2 and My = R2.

166 S. Zzumiya

The concrete realized single lensing system is represented (following [Sl]) by the Lagrangian submanifold

This means that the generating function of L ~ isJ the time-delay map

By the previous arguments, light rays are given by {(z, y) I grad,T = 0 } and the set of point sources for light rays is the Lagrangian submanifold

LS = {(y,gradyT) E T*M,

I (z,y)E Mxx M,,

grad,T = 0 }

of T*M,. Then we have

n ((Mxx ( 0 ) ) x T*M,)))

ufv(L+b

= Ls.

Let us recall the basic notions of the theory of symplectic relations ( [42]). Let X1,Xz be smooth manifolds with the same dimension. We consider the product symplectic manifold

We define a symplectic relation from where wxze wxl= r;wx2- ~Twx,. T*X1to T*X2 as a Lagrangian submanifold R of (T*X1xT*X2,wxZewxl). If the restriction of the projection

xxl x rx2 : T*X1 x T*X2 --+Xi x Xz

to R is always non-singular, we call R the elementary symplectic relation. Let R be a symplectic relation in (T*X1x T*X2,wx2e wxl)and S be a subset of T*X1,then the symplectic image of S by R is defined as R ( S )= {p2 E T*X2 : 3pl E S such that ( p 1 , p ~E) R}. If S is Lagrangian submanifold in (T*Xl, wxl), then R ( S ) is a Lagrangian subset in (T*X2,wx2). Since both of S and R are Lagrangian submanifolds, we have their generating families at least locally. We only consider the local situation here, so that we assume that XI = X2 = R".Let FI : (Rkl x X1,O) R be a generating family of a Lagrangian submanifold germ S c T*X1 and FZ : (Rkz x (XI x X2),0) IR be a generating family of a symplectic relation R c T*X1 x T*X2. Then we have a function germ

-

-

F : ((Rkl x Xi x R k z )x

X2,O) +IR

Singulare'ty theory of smooth mappings and its applfcations

167

defined by

F ( ( t 1 ,Q 1 , e2>,Q 2 ) = Fl(e1,Ql)+ F2(l2,41,42). Hence we have the following lemma:

Lemma 8.1. If F is a Morse family, then F is a generating family of the Lagrangian submanifold R(S) C T*X2. In the case of single gravitational lensing, if SO denotes the observer Lagrangian submanifold of system of gravitational rays then the source Lagrangian submanifold of rays is an image

L$(So) c T * M y . In the standard setting (cf. 1611 and the previous arguments ) SOis the zero section of the cotangent bundle T*M,. Therefore we have

L$,(SO)= {(Y,gradyT) I grad,T

=0

1,

so that the generating family for L$(So) is given by

which is the same generating family as that of the source Lagrangian submanifold in the previous framework in [61]. We call the pair (S,R) a (single) lensing system if S is a Lagrangian submanifold of T * X 1 and R is a symplectic relation from T * X 1 to T*X2. If the projection ~ 1 : R1 ~ T * X 1 is non-singular, R is the graph of a symplectomorphism H : T * M , T*My. In this case we say that (S,R) is regular. Moreover, if S is the zero section of T * X 1 , we call ( S ,R) a special lensing system. Therefore, the single gravitational lensing is a regular special lensing system. Here we consider the problem how to construct a kind of the notion of generating families for double lensing systems. We already have a solution because a double plane gravitational lensing is described by the pair of time-delay maps. In which we only consider local properties, so that we assume that X1 = X2 = X3 = Rn. For any double lensing system germ (S,R1, Rz), we have generating families FO : (Rko x X I ,0 ) R of S, F1 : (ak1 x ( X I x X z ) ,0 ) R of R1 and F2 : (Wk2 x (X2 x X 3 ) , 0 ) R of R2. On the other hand, there always exists a symplectomorphism germ @ I : (T*X1,z1) (T*Xl,O)such that @ l ( S )is a zero section germ of T * X 1 , so that we might assume that S is a zero section germ of T * X 1 under the Lagrangian equivalence among double lensing system germs. In other words,

--

-

--

-

168 S. I m m i y a

it is enough to investigate special double lensing system germs. In this case FOcan be chosen as a constant function. We call (F1,F2) a generating pair of the special lensing system germ (S,R1, R2) if Fi is a generating family of R i (i = 1,2). Then F1 can be regarded as a generating family of R1(S) c T*X2 and F = F1+ F2 is a generating family of R2 o R1(S) c T * X 3 (cf., [30,42]). Since a double gravitation lensing is a regular double lensing system, we now pay attention to regular special double lensing systems here. In this case F1 : ( X I x X2,O) IR is a generating function of R1 and F2 : ( X 2 x X3,O) R is a generating function of R2. By the arguments in the previous paragraph, F1 is a generating family of the Lagrangian submanifold germ R l ( S ) c T*X2 and a map germ F : ( ( X I x X 2 ) x X3,O) IR defined by

-

-

F ( ( s l , 2 2 ) , Z 3 )= Fl(217Z2)

-

+

F2(22,23)

is a generating family of the Lagrangian submanifold germ R2 o R1(S) c T*X3. In other words, (F1,F2) is a generating pair of a regular special double lensing system germ (S,R1, R2) if (dF1( X I x X Z ) ,( z 1 , 2 2 ) )= (R1,( z 1 , z 2 ) ) and (dFa(X2 x X 3 ) , (22,~ 3 ) = ) (R2,( ~ 2 ~ ~ 3 ) ) . Since any elementary symplectic relation has a generating function at least locally, we have the following fundamental proposition:

Proposition 8.2. All regular special double lensing system germs are constmcted by the above method We can translate equivalence relations among double lensing systems into those of corresponding generating pairs.

9. Other topics We describe some topics related to the singularity theory other than the topics in the previous sections. 9.1. Ocean acoustics

Since electromagnetic waves in the water decay very quickly, radar systems are not useful in oceans. Therefore, sonars are used as usual. In the simplest mathematical model, the wave field is governed by the Helmholtz equation

where ~ ( zis) the refraction index. In general, the domain we consider has a boundary (the sea surface) and q ( z ) is a piecewise smooth function. In


169

this case, the trajectory of the sound is described by the projection of the characteristics flow of the Eikonal equation 1 H ( z , z , P , Q ) = s ( P 2 q2 - v2(z>>= 0

+

onto the (z, z)-plane. If ~ ( z is) a smooth function, we can apply our theory of graphlike unfoldings, so that the generic perestroikas of the fronts is the standard perestroikas of fronts. However, in general, ~ ( z is) a piecewise smooth function and we have the boundary (the sea surface) z = 0. At the boundary point, the sound reflected following by the Snell law. These cause that we have shadow regions and broken caustics these situation do not appear in our previous framework. There are mathematical theories for describing this rather general situation which are called the theory of the boundary caustics [69] and the theory of the reticular caustics [76]. However, these theories are not sufficient for describing our concrete case. For the previous results and the construction of the asymptotic solution for the Helmholtz equation, see [25]. 9.2. Lightlike hypersurfaces

in Minkowski space

We consider I%: = (R3, (,)) to be Minkowski 3-space with the pseudo inner product ( z , y ) = - z i y i 22y2 xgy3. The essential difference from the Euclidean case is the existence of the lightcone

+

+

In [23] we studied singularities of the envelope of the one-parameter families of Light cones. We call such a surface a lightlike surface or a lightlike developable. In this case, the family of the lightcones always parameterized along a spacelike curve y : I I%;. We may assume that the curve y(s) has a arc-length parameter (i.e., (y’(s),y’(s)) = l).Then we have the following F’renet-Serret type formula:

-

{

t’(s)= /G(s)n(s) n’(S)= -6(y( S ) ) K ( S ) t (S) b’(s)= 7 (s)n (s) ,

+

T (S)b(S)

where t ( s ) is the unit tangent vector, n ( s ) is the unit principal normal vector and b(s) is the unit binormal vector of y(s). Moreover, 6(y(s)) = ( b ( s ) ,b ( s ) ) ,~ ( s is ) the curvature and T ( S ) is the torsion of y(s). Under the above notation, we can show that n ( s ) f b ( s ) is lightlike and the lightlike surface is (at least locally) parameterized by L D , f ( s ,u ) = y(s) u ( n ( s ) f b ( s ) ) .In [18] we adopted the Lorentzian distance squared function on y

+

170 S. Izumiya

and showed that singularities of the lightlike surface L D ~ ( su) , correspond to points on y with the property ( ~ ' ( s )- T ( s ) K ( s ) ) = 0. We also gave a classification of singularities of the lightlike surface. Moreover, we showed that the Lorentzian invariant ( ~ ' ( s )- T ( s ) K ( s ) ) is constantly equal to zero if and only if the curve y is located on the lightcone. We can interpret that the one-dimensional analogue of the characterization for the conformal flat 3-dimensional manifold by Asperti-Dajczer [4]. We try to understand from the view point of the theory of partial differential equations. Consider the degenerate stationary Hamilton-Jacobi equation: - (aU)2 -

+ (au)2 + (8,)2 - =o. 8x2

8x3 The corresponding Hamiltonian function H : T*R; ax1

-& +

H(x~,x~,X~,P~, =P ~ , P ~p:)

-

R is given by + p i . The lightlike surface LD$(s,u)

--f

has a lift LD, ( s ,u)= ( L D , f ( su), , u ( n ( s )fb ( s ) ) )on the cotangent bundle --f

T*R;. We can easily show that the lift LD, ( s , u ) is an embedding and --f

satisfies the condition alLD, = 0. Since ( n ( s )fb(s),n ( s )fb(s)) = 0, the image is contained in H-l(O). It follows from these facts that the lightlike surface is a solution surface (the graph of the multi-valued solution) of the equation H = 0. Fig.9.1 is the picture of the lightlike surface along the ellipse y ( t ) = (0,2cost,sint) in R2.

Fig. 9.1.

We can easily recognize that there exist four swallowtails.


171

On the other hand, the lightlike surface is given as the tangent developable surface of a lightlike curve y, where y is lightlike if +(t)is always lightlike. In [ l l ]we , give the following classification of singularities of the lightlike tangent developable of a generic lightlike curve.

Theorem 9.1. For a “generic” lightlike curve y, the tangent developable of the curve y at a singular point is locally diffeomorphic to C x I%, SW or SB. Here, SB = {(q, z2, z3)(z1 = u,z2 = v 3 uu2,x3 = 12v5 10uv4} is the Scherbak surface.

+

+

Scherbak surface Fig.9.2. We remark that Scherbak [70] showed that S B is given as the irregular orbit of the finite reflection group H3 on C3.

Finally we may say:

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6. D. E. Blair, Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics 509 Springer (1976) 7. D. Bleeker and L. Wilson, Stability of Gauss maps. Illinois J. Math. 22, (1978), 279-289 8. Th. Brocker, Differentiable Germs and Catastrophes. London Mathematical Society, Lecture Note Series 17, Cambridge University Press, (1975) 9. J. W. Bruce, The dual of generic hypersurfaces. Math. Scand., 49 (1981), 36-60 10. J. W. Bruce and P. J. Giblin, Curves and singularities (second edition), Cambridge University press, (1992) 11. S. Chino and S. Izumiya, Lightlike developables in Minkowski 3-space, preprint 12. M.G.Crandal1 and P.-L.Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. AMS., 282 (1984), 487-502 13. J.P.Cleave, The form of the tangent developable at points of zero torsion on space curves, Math.Proc.Camb.Phi1. vol 88 (1980), 403-407 14. C. G. Gibson, K. Wirthmuller, A. A. du Plessis and E. J. Looijenga, Topological stability of smooth mappings. Lecture Notes in Math., 552 Springer-Verlag, (1976) 15. M. Golubitsky and V. Guillemin, Stable Mappings and their Singularities. Springer GTM. (1973) 16. J. Guckenheimer, Solving a single conservation law, Lecture notes in Mathematics 468 Springer Verlag, New York, (1975), 108-134 17. J. Hoschek and H. Pottman, Interpolation and approximation with developable B-spline surfaces , in Mathematical Methods for curves and surfaces, ed. by M. Drehlen, T . Lyche and L. L. Schumacker, Vanderbilt Univ. Press (1995), 255-264 18. G. Ishikawa, Determinacy of envelope of the osculating hyperplanes to a curve, Bull.London Math.Soc., vol 25 (1993), 787-798 19. G. Ishikawa, Developable of a curve and determinacy relative to osculation-type , Quart. J. Math. Oxford, vol46 (1995), 437-451 20. S.Izumiya and G.T.Kossioris, Geometric singularities for solutions of single conservation laws, Arch. Rat. Mech. Anal., 139 (1997), 255-290 21. S. Izumiya and T. Sano, Generic afine differential geometry of space curves, Proceedings of the Royal Society of Edinburgh , vol 128A (1998), 30 1-3 14 22. S. Izumiya, H. Katsumi and T. Yamasaki, The Rectifying developable and the Darboux indicatrix of a space curve, Geometry and Topology of Caustics-Caustics '98, Banach Center Publications, vol 50 (1999), 137149 23. S. Izumiya, D.-H. Pei and T. Sano, The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space, Glasgow Math. J., vol42 (2000), 75-89 24. S. Izumiya and N. Takeuchi. Singularities of ruled surfaces in It3. Math. Proceedings of Cambridge Philosophical Soc.,vol 130 (2001) ,1-11 25. S.Izumiya, G.Kossioris and G. Makrakis: Multi-valued solutions to the

Singularity theory of smooth mappings and its applications 173

26. 27. 28. 29. 30. 31.

32. 33. 34.

35.

36.

37.

38.

39.

40. 41. 42.

43.

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174 S. Izumiya

.

44. J. J. Koenderink, Solid shape, The MIT Press, Cambridge, Massachusetts, (1990) 45. E. E. Landis, Tangential singularities, Funct. Anal. Appli., 15 (1981), 103-114 46. H. I. Levine, A.O. Petters, J. Wambsganss, Applications of singularity theory to gravitational lensing I, Multiple lens planes, J. Math. Phys. vol. 34 (1993) ,4781-4808 47. H. I. Levine and A. 0. Petters, New caustics singularities in multiple lens plane grawitational lensing, Astron. Astrophys. vol. 272 (1993),L17-L19 48. J. Martinet, Singularities of Smooth Functions and Maps, London Math. SOC.Lecture Note Series, Cambridge Univ. Press,58 (1982) 49. J.N. Mather, Stability of Cm-mappings I:The division theorem, Ann. of Math., 87 (1968), 89-104, 50. J. Mather, II:Infinitesimally stability implies stability, Ann. of Math., 89 (1969), 254-291, 51. J. Mather, III:Finitely determined map-germs, Publi. Math. I.H.E.S., 35 (1968), 127-156. 52. J. Mather, IV:Classification of stable germs by R algebras, Pub.. Math. I.H.E.S., 37 (1970), 223-248. 53. J. Mather, V:Transversality, Adv. Math., 4 (1970), 301-336. 54. J. Mather, VI:The nice dimensions, Lecture Notes in Math., 192 (1972), 207-253 55. J. Milnor, Morse theory, Ann. Math. Stud. 51, Princeton University Press (1963) 56. D. Mond, O n the tangent developable of a space curve, Math. Proc. Cambridge Phil. SOC.91 (1982), 351-355. 57. J. A. Montaldi, Surfaces in $space and their contact with circles, J. Diff. Geom., 23 (1986),109-126 58. J. A. Montaldi, O n contact between submanzfolds, Michigan Math. J., 33 (1986), 81-85 59. J. A. Montaldi, O n generic composites of maps, Bull. London Math. SOC., 23 (1991), 81-85 60. 0. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. SOC.Transl. Ser. 2, 26 (1957) 95-172 61. A, 0. Petters, Arnol’d’s singularity theory and grawitational lensing, J. Math. Phys. vol. 34 (1993),3555-3581 62. A.O.Petters, Multaplane grawitational lensing I, Morse theory and image counting, J.Math. Phys. vol. 36 (1995), 4263-4275 63. A.O.Petters, H.Levine and J. Wambsganss, Singularity theory and Gravitational lensing, Birkhausr (2001) 64. A. A. du Plessis and C. T. C. Wall, The geometry of topological stability, Oxford University Press, (1995) 65. I. Porteous, The normal singularities of submanifold, J. Diff. Geom., vol 5, (1971), 543-564 66. I. Porteous, Geometric Differentiation second edition, Cambridge Univ. Press (2001)

Singularity theory of smooth mappings and its applications 175 67. M. C. Romero F'uster, Sphere stratifications and the Gauss map. Proceedings of the Royal SOC.Edinburgh, 95A (1983), 115-136 68. T. Sasaki, Projective Differential Geometry and Linear Homogeneous Differential Equations, Rokko Lectures in Mathematics, Kobe University, vol 5 (1999) 69. I. G. Scherbak, Boundary fronts and caustics and their metamorphosis, London Math. SOC.Lecture note series 201, Singularities, (1994), 363-373 70. O.P.Scherbak, Wavefront and reflection groups, Russian Math. Surveys 43-3 (1988), 149-194. 71. O.P.Scherbak, Projectively dual space curves and Legendre singularities, Trudy Tbiliss. Univ. 232-233 (1982), 280-336. 72. P.S.Schneider, J.Ehlers and E.E. Falco, Gravitational lenses, Springer (1992) 73. M. Schneider, Interpolation with Developable Strip-Surfaces Consisting of Cylinders and Cones, in Mathematical Methods f o r curves and surfaces 11, ed. by M. Dahlen, T. Lyche and L.L. Schumacker, Vanderbilt Univ. Press, (1998), 437-444 74. R. Thom and H. Levine, Singularities of differentiable mappings, Bonner Mathematische Schriften, vol 6, Bonn, (1959) 75. J. A. Thorpe, Elementary Topics in Differential Geometry, SpringerVerlag 76. T. Tsukada, Reticular Lagrangian singularities, Asian J. of Math., 1 (1997), 572-622 77. I. Vaisman. A first course in differential geometry. Pure and applied Mathematics,A series of Monograph and Textbooks, MARCEL DEKKER, (1984) 78. H. Whitney, The singularities of a smooth n-manifold in (2n - 1)-space, Annals of Math., vol45 (1944), 247-293 79. V. M. Zakalyukin, Lagrangian and Legendrian singularities, Funct. Anal. Appl., 10 (1976), 23-31 80. V. M. Zakalyukin, Reconstructions of fronts and caustics depending one parameter and versality of mappings. J. Sov. Math., 27 (1984), 2713-2735

176

Birational geometry and homological mirror symmetry Ludmil Katzarkov*

University of Miami E-mail: 1.kat.zarkovQmath.miami. edu

To Ross Street with admiration. In this paper we outline a program of connecting some old problems in algebraic and symplectic geometry with new methods coming from Homological Mirror Symmetry. We discuss several examples and applications.

1. Introduction Mirror symmetry was introduced as a special duality between two N = 2 super conformal field theories. Traditionally an N = 2 super conformal field theory (SCFT) is constructed as a quantization of a a-model with target a compact Calabi-Yau manifold equipped with a Ricci flat Kahler metric and a closed 2-form - the so called B-field. We say that two Calabi-Yau manifolds X and Y form a mirror pair X I Y if the associated N = 2 SCFTs are mirror dual to each other [9]. This paper contains an outline of a program for applications of ideas of Homological Mirror Symmetry (HMS) to algebraic geometry and algebraic topology we have announced in the Ross Street Conference in July 2005. Complete details will appear elsewhere. We emphasize the geometrization of categorical notions - a big part of Mathematics in which we all owe to Ross Street. We start with a quick introduction of Homological Mirror Symmetry. After that we introduce our construction of the Mirror Landau-Ginzburg Model which differs from constructions in [16]and is a natural continuation of our previous results [3] and of the works of Moishezon and Teicher. The construction underlines the connection of monodromy techniques with theory of categories. *Partially supported by NSF Grant DMS0600800 and by Clay Math Institute

Birational geometry and homological mirror symmetry 177

1. As a first application we outline an application to Birational Geometry. One of the main questions in classical Algebraic Geometry is if a smooth projective n dimensional variety X has a field of meromorphic functions CP1 Adding singular fibers Taking out singular fibers

2. We also outline some applications to algebraic topology. Using HMS we introduce symplectic invariants, which seem to be stronger than Gromov-Witten invariants. 2. Definitions We first recall a definition which belongs to Seidel [28]. Historically the idea was introduced first by M.Kontsevich and later by K.Hori. We begin by briefly reviewing Seidel's construction of a Fukaya-type ^loo-category associated to a symplectic Lefschetz pencil see [28], [29]. Let (y, w) be an open symplectic manifold of dimension diniu y = 2n. Let w : Y —> C be a symplectic Lefschetz fibration, i.e. w is a C°° complexvalued function with isolated non-degenerate critical points pi,...,pr near which w is given in adapted complex local coordinates by w ( z i , . . . , zn] =

178 L. Katzarkov

+ +. +

f ( p i ) 21" . . 22, and whose fibers are symplectic submanifolds of Y. Fix a regular value Xo of w , and consider arcs "yi c C joining XO to the critical value X i = f ( p i ) . Using the horizontal distribution which is symplectic orthogonal to the fibers of w , we can transport the vanishing cycle at pi along the arc -yi to obtain a Lagrangian disc Di c Y fibered above -yi, whose boundary is an embedded Lagrangian sphere Li in the fiber Exo = w-l(X0). The Lagrangian disc Di is called the Lefschetr thimble over -yi, and its boundary Li is the vanishing cycle associated to the critical point pi and to the arc -yi. After a small perturbation we may assume that the arcs 7 1 , . . . , -y, in C intersect each other only at XO, and are ordered in the clockwise direction around Xo. Similarly we may always assume that the Langrangian spheres Li c Exo intersect each other transversely inside CO. Definition 2.1 (Seidel). Given a coeficient ring R, the R-linear directed Fukaya category FS(Y, w ; {-yi}) is the following A,-category: the objects of FS(Y, W ; {ri})are the Lagrangian vanishing cycles L1,. . . ,L,; the morphisms between the objects are given by Hom(Li, Lj) =

CF*(Li, Lj; R) = RILlnLjl if i < j R . id ifi=j

{o

if i

>j ;

and the differential ml, composition m2 and higher order products defined in terms of Lagrangian Floer homology inside Exo. More precisely, m k : Hom(Li,,

mk

are

Li,) 8 ... @Hom(Lik-,,Lik) -+ Hom(Li,, Lik)[2- k ]

is trivial when the inequality io < il < ... < i k fails to hold (i.e. it is always zero in this case, except for m2 where composition with an identity morphism is given by the obvious formula). W h e n io < - . . < i k , m k is defined by fixing a generic w-compatible almost-complex structure o n Exo and counting pseudo-holomorphic maps from a disc with k + 1 cyclically ordered marked points o n its boundary to Cxo, mapping the marked points to the given intersection points between vanishing cycles, and the portions of boundary between them t o Li, , . . . ,Lik respectively (see [.9]).

It is shown in [29] that the directed Fukaya category FS(Y, W ; {Ti}) is independent of the choice of paths. We will denote this category by FS(Y, w ) and will refer to it as the Fukaya-Seidel category of w.

Bimtional geometry and homological mirror symmetry

179

Let Y be a complex algebraic variety (or a complex manifold) and let w : Y + C be a holomorphic function. Following [25] we define:

Definition 2.2 (Orlov). The derived category D b ( Y ,w) of a holomorphic potential w :Y C is defined as the disjoint union

o b ( y , w ) :=

JJ

@ing(yt)

t

of the derived categories of singularities D:ing(yt) of all fibers yt := w-’(t) of w.

The category D:ing(x) = Db(Y,)/Perf(yt) is defined as the quotient category of derived category of coherent sheaves on yt modulo the full triangulated subcategory of perfect complexes o n yt. Note that D h g ( X ) is non-trivial only for singular fibers Yt. In what follows we will use notions defined in [16] section 7.3. Let X be a manifold of general type, i.e. a sufficiently high power of the canonical linear system on X defines a birational map. We will use the above definitions to formulate and motivate an analogue of Kontsevich’s Homological Mirror Symmetry (HMS) conjecture for such manifolds as well as for Fano manifolds - manifolds whose anticanonical linear system defines a birational map. A quantum sigma-model with target X is free in the infrared limit, while in the ultraviolet limit it is strongly coupled. In order to make sense of this theory at arbitrarily high energy scales, one has to embed it into some asymptotically free N = 2 field theory, for example into a gauged linear sigma-model (GLSM). Here “embedding” means finding a GLSM such that the low-energy physics of one of its vacua is described by the sigma-model with target X . In mathematical terms, this means that X has to be realized as a complete intersection in a toric variety. The GLSM usually has additional vacua, whose physics is not related to X . Typically, these extra vacua have a mass gap. To learn about X by studying the GLSM, it is important to be able to recognize the extra vacua. Let 2 be a toric variety defined as a symplectic quotient of CN by a linear action of the gauge group G 21 U ( l ) k .The weights of this action will be denoted Qia, where i = 1,.. . , N and a = 1,...,k. Let X be a complete intersection in 2 given by homogeneous equations G,(X) = 0, a = 1,.. . ,m. The weights of G, under the G-action will be denoted daa. The GLSM corresponding to X involves chiral fields @iti = 1,.. . ,N and Q, a = 1,.. . ,m. Their charges under the gauge group G are given by

180 L. Katzarkov

matrices Qi, and d,,, respectively. The Lagrangian of the GLSM depends also on complex parameters t a , a = 1,.. . , k. On the classical level, the vector t, is the level of the symplectic quotient, and thus parameterizes the complexified Kahler form on 2. The Kahler form on X is the induced one. On the quantum level the parameters t , are renormalized and satisfy linear renormalization semigroup equations:

i

a

In the Calabi-Yau case all pa vanish, and the parameters t , are not renormalized. Here p denote the massive and massless vacua. According to Hori-Vafa [17] the Landau-Ginzburg model (Y, w) that is mirror to X will have (twisted) chiral fields A,, a = 1,. . . ,k,Yi,i = 1,. . . ,N and Ta,a = 1,. . .,m and a superpotential is given by

a

\ i

a

a

The vacua are in one-to-one correspondence with the critical points of w. By definition, massive vacua are those corresponding to non-degenerate critical points. An additional complication is that before computing the critical points one has to partially compactify the target space of the LG model. One can determine which vacua are “extra” (i.e. unrelated to X ) as follows. The infrared limit is the limit p + 0. Since t, depend on p , so do the critical points of w. A critical point is relevant for X (i.e. is not an extra vacuum) if and only if the critical values of e-% all go to zero as p goes to zero. In terms of the original variables @i, this means that vacuum expectation values of l@iI2 go to +m in the infrared limit. This is precisely the condition which justifies the classical treatment of vacua in the GLSM. Recall also that the classical space of vacua in the GLSM is precisely X . Now let us state the analogue of the HMS for complete intersections X of general type. We will write D b ( X )for the standard derived category of bounded complexes of coherent sheaves on X and by Fuk(X,w) the standard Fukaya category of a symplectic manifold X with a symplectic form w.

Conjecture 2.3. Let X be a variety of general type which is realized as a complete intersection in a toric variety, and let (Y,w) be the mirror LG model. T h e derived Fukaya c a t e g o y F’uk(X,w) of X embeds as a direct


181

summand into the category Db(Y,w) (the category of B-branes for the mirror LG model). If the extra vacua are all massive, the complement of the Fukaya category of X in D b ( Y ,w) is very simple: each extra vacuum contributes a direct summand which is equivalent to the derived category of graded modules over a Clifford algebra. There is also a mirror version of this conjecture: Conjecture 2.4. The derived category D b ( X ) of coherent sheaves o n X embeds as a full sub-category into the derived Fukaya-Seidel category F S ( Y ,w) of the potential w.

With an appropriate generalization of Fhkaya-Seidel category to the case of non-isolated singularities we arrive at Table 1 summarizing our previous considerations. The categories Dii,Ti(Y,w) and FS(A,,,.~)(Y, w, w ) appearing in the last row of Table 1 are modifications which contain information only about singular fibers with base points contained in a disc with a radius - a real number A, are introduced in order to deal with problems arising from massless vacua. In this formalism we may have to take a Karoubi closure on both sides - this will be discussed later. 3. The construction

The standard construction of Landau-Ginzburg models was done for smooth complete intersections in toric varieties in [16]. We describe a new procedure which is a natural continuation of our previous results [3] and of the works of Moishezon and Teicher. Our procedure underlines the geometric nature of HMS and several categorical correspondences come naturally. We will restrict ourselves to the case of three dimensional Fano manifolds X with the anticanonical linear system on it. For simplicity we will assume that I - Kxl contains a pencil. Step 1. Choose a Moishezon degeneration [3] X -+ A of X corresponding to a projective embedding of X given by a multiple of the ample line bundle - K x . This means that we choose a generic Noether normalization projection X 4 P3in the given projective embedding, and then degenerate the covering map t o a totally split cover. The central fiber XO of this degeneration is a configuration of spaces intersecting over rational surfaces and curves. Next we choose a pencil f : X --+ P1in the linear system I -Kx,al so that on each fiber X,,t E A , the base of the pencil ft : X t --+ P1intersects the discriminant locus Ax, in minimal number of points. As we will

L. Katzarkov

182

A side compact manifold, w - symplectic form on X . X

B side X

-

Fuk(X,w)

Li

=

Obj: ( L i , & ) Mor: H F ( L i ,L j )

- Lagrangian submanifold of

x, &

-

-

smooth projective variety

over C

D b ( X )=

Obj: C,? Mor: Ezt(Cj',Cj')

C,?- complex of coherent sheaves X

on

flat U(1)-bundle on Li

2 (Y,w)

-

open symplectic mani-

fold, w : Y -+ C - a proper C" with symplectic fibers.

F S ( Y ,w, w)=

map

Y - smooth quasi-projective variety over C , w : Y 4 C - proper algebraic map.

Obj: ( L i , E )

Mor: H F ( L i , L j )

Li - Lagrangian submanifold of yx, & - flat U(1)-bundle on Li. 7

F S x i , r i (Y W , w) -

the Fukaya-Seidel category of

(~t-Ail<Ti,W,4.

see below, such lines P1c I - Kx,I have very strong rigidity properties. Step 2. The degeneration X --+ A can be seen as a logarithmic structure (see [12]) Xi on Xo - roughly an union of toric varieties, intersecting over toric varieties plus monodromy data. The pencil f induces a pencil f t on the logarithmic space X i . Step 3. We apply a Legendre transform [12] to (XJ,ft) and obtain a new logarithmic structure ( Y i ,wt) with a pencil on it. Roughly we take the dual intersection complex and then for vertices we specify the normal fan structure and corresponding monodromy. Step 4. We use the logarithmic structure Y i to smooth y0 and get a mirror degenerating family Y --+ A and a mirror Landau-Ginzburg potential


w :y Y --+

183

P1.Basically, on the generic fiber Y of y -+ A, the potential w : P1is an anticanonical pencil of mirror manifolds to the anticanonical K 3 surfaces in X . As before we move the P1 c I - KyI corresponding to the pencil w : Y --+ P1 so that it intersects the discriminant locus Ay in a minimal number of points (0 and 00 among them - in Physics language 4

this is the point of maximal degeneration and the Gepner point). The construction above will allow us to see HMS in purely geometric terms. In many cases it could lead to a proof of HMS and in many cases it could lead to establishing an isomorphism between the K-theory of the categories involved. The construction suggests other categorical correspondences on a way. Similar procedure works in dimensions other than three and applies not only to Fano manifolds.

Remark 3.1. The construction above works not only on anticanonical pencils but for other pencils as well. We will demonstrate this on examples of three and four dimensional cubics. In all these examples the lines we chose to intersect Ay are rather rigidified - the number of intersection points is given by the eigenvalues of the operator of quantum multiplication by c 1 ( X ) on H ( X ) . The correspondences coming from the associated noncommutative Hodge structures put additional restrictions (see [21]). 3.1. Blow ups

Let X be a smooth projective variety and Z be a smooth subvariety. As a consequence of the weak factorization theorem [30]it is enough to study the Landau-Ginzburg mirrors of blow-ups and blow-downs with smooth centers. Recall the following result. Theorem 3.2. (Orlov 1.41) Let X be a smooth projective variety and X Z be a blow up of X in a smooth subvariety Z of codimension k. Then D b ( X z ) has a semiorthogonal decomposition ( D b ( X )D, b ( Z ) k - l , .. . ,Db(Z)0).Here D b ( Z ) i are corresponding twists by O ( i ) (see [ 4 ] ) .

This B-side statement has an A-model counterpart. In a joint work in progress with D.Auroux we consider the following: Conjecture 3.3. The Landau-Ginzburg mirror ( T , g ,W T ) of X z is the connected symplectic sum of the Landau-Ginzburg mirror (Y,w, w y ) of X and the Landau-Ginsburg mirror ( S ,f,w s ) of Z x (C*)k. On the level of categories this means that F S ( T ,g , W T ) has a semiorthogonal decomposition ( F S ( T , g , w T ) , F S ( S f, , W S ) k - l , .. ., F S ( S ,f r W S ) O ) *

184

L. Katzarkov

Here F S ( S ,f ,W S ) ~are categories of vanishing cycles located at k - 1 roots of unity around infinity. We discuss some simple examples in order to illustrate the above statement.

Example 3.4. We discuss the LG model mirror to CP3 blown up in a point. In this case k = 3. The LG mirror of the blown-up manifold is a family of K3 surfaces with 6 singular fibers. Four of these fibers correspond to the LG mirror of CP3 and they are situated near zero. The two remaining fibers are sitting over second roots of unity in the local chart around co see Figure 1.

Fig. 1.

LG model of CP3 blown-up at a point

Example 3.5. We discuss LG model mirror to CP3 blown up in a line. In this case k = 2. We get as LG model of the blown up manifold a family of K3 surfaces with 6 singular fibers. Four of these fibers correspond to the LG model of (CP3 and they are situated near zero. The two other fibers are very close to each other. We briefly describe the procedure in this case. We start with the LG model for (CP3 1

-.

w =z+y+z+

XYZ

We add the additional term pxy, with p corresponding to the volume of the blown up CP1. The critical points of 1

w

=x+y+z+

1

-+pxy XY

Birational geometry and homological miwor symmetry

Fig. 2.

185

LG model of UP3blown-up at a line

split in two groups described in Figure 2.

Example 3.6. We discuss the LG model mirror to CPn blown-up in CPk. As in the previous examples it can be seen as the gluing of several LandauGinzburg models mirroring to CPn and CPk x (C*)n-k.The result is a LG model with n 1 singular fibers located around zero (corresponding to CPn) glued to n - k - 1 other LG models of (CPkwith singular fibers located around k - 1 roots of unity far away from zero - see also [23].

+

Example 3.7. Let us look at the Landau-Ginzburg model mirror to the manifold X which is the blow-up of CP3 in a genus two curve embedded as a (2,3) curve on a quadric surface. It is well known that X is birational to the Fano threefold X4 - the intersection of two quadrics in CP5. The Landau-Ginzburg mirror of X4 can be computed as follows. Start with the variety

+ 22 = -1 23 + = -1 21

24

21""'

25

. 26

=A

with a superpotential w' = 2 5

+ 26.

The function w' defines a pencil on this variety. To find the LandauGinzburg mirror of X4 we need to compactify this pencil partially, resolve the singularities of the compactified total space and then extend w' to the resolution.

186 L. Katzarkov

Now we describe the mirror of X. It is the deformation of the image of the compactification D of the map 1; U I ; ~

2 213; ; ~ 1 . ~~ 22 ; . ~~ 3 1; . ~ 3

of C*3 in CP7. The function w gives a pencil of degree 8 on the compactified D. We can interpret this pencil as result of first taking the Landau-Ginzburg mirror of CP3 and then adding to it a new singular fiber consisting of three rational surfaces intersecting over degenerated genus two curve. This process is the A-model counterpart of the blowing up of the genus two curve in CP3 and according to Orlov’s theorem, stated above, can be used to define HMS for manifolds of general type. We can also easily chase the transition from D to Landau-Ginzburg mirror of X 4 which corresponds to sending singularities of two singular fibers of D to the singularities to the fiber at infinity in Landau-Ginzburg mirror of X4.

Remark 3.8. In the case when X is a manifold of general type one can actually apply a different procedure. We can realize X as a singular set of a Calabi Yau manifold, which on its own is a complete intersection in a toric variety. Then the Landau-Ginzburg mirror of X will be the singular set of the mirror Calabi Yau manifold. We give a brief example. Suppose X is a genus three curve realized as a plane quartic. We can view X as the singular locus of a Calabi-Yau threefold as follows: Consider a rank 3 quadric Q 2 in P5.The quadric Q 2 is singular along a plane P c Q 2 c P5. Choose a smooth quartic Q 4 in P5so that Q 4 n P = X. For a general such quartic is general the intersection V := Q 2 n Q 4 is a Calabi-Yau threefold which has a family of A1 singularities along X C V . Applying the procedure discussed above we get as a mirror LandauGinzburg model which is a family of K3 surfaces with a degenerated genus three curve in the singular fiber over zero. We notice here that above procedure allows us to look at the LandauGinzburg models in a different way. Namely we can see that a blow up of subvariety X in V is nothing else but moving the line in the dual space and creating a new pencil K y with one more singular fiber. As we have pointed out in remark 3.1 moving this line is rather obstructed. 3.2. HMS on the language of pencils and degenerations

In this section we will look at the procedure described above from the point of view of Lefschetz pencils. We will restrict ourselves to the case of


187

Fano manifold X. Consider a pencil on X associated with the anticanonical linear system. Consider a degeneration described in Step 1 of the procedure which is compatible with the pencil. We first consider a degeneration so that we have minimal number of singular fibers with two very singular ones (corresponding to the point of maximal degeneration at infinity and the cusp Gepner point at zero). In the language of pencils this means that we have considered a line in the dual space of anticanonical Calabi-Yau sections which passes through two very singular points of the discriminant loci AX • The Legendre transform brings us, after smoothing, to the pencils of mirror Calabi Yau manifolds. We will have again two very singular fibers at zero and infinity and may be others. The LG mirror model is a pencil, which in the dual space corresponds to a line intersecting the discriminant loci in some very singular points. A - side X line intersecting AX Singularities of AX

B - side w : Y -> CP1 line intersecting Ay Singularities of Ay

From now on we will call the smoothed fibrations | — KX \ and | — Ky \ pencils. Clearly the degenerations and the pencils are selected in a special way so that hypercohomology of the perverse sheaf of vanishing cycles (see section 3.4) of the | — /fx (-pencil is isomorphic as MHS to a rotation and a shift of the hypercohomology of the perverse sheaf of vanishing cycles of the | — /fy (-pencil. In this situation we can associate with each of the two pencils derived categories of singularities and then with each of these categories a noncommutative Hodge structure NCH(\ - Kx\) and NCH(\ - KY\) (for definitions and theory see [21].) We conjecture: Conjecture 3.9. The non-commutative Hodge structures NCH(\ — KX\) and NCH(\ — KY\) are isomorphic modulo shift and rotation. Conjecture 3.10. The Fukaya-Seidel category at the singular point at zero for the | — Ky\-pencil is equivalent to the derived category of singularities of the corresponding singular point on Ay. Conjecture 3.11. The Fukaya-Seidel category at the zero fiber for the pencil | — KX \ is equivalent to the derived category of singularities of the corresponding singular point on AX •

188 L. Katzarkov

With this said we have:

Here we discuss blow ups with smooth center. Clearly absorbing a singular fiber in the I - Ky I-pencil in the fiber at infinity, namely going to a deeper singularity in A y , mirrors to a blow down. Conjecture 3.12. The perverse sheaf of vanishing cycles associated with the I - Ky I-pencil is a n invariant of the birational geometry of X . Conjecture 3.13. The perverse sheaf of vanishing cycles associated with the I - Kx) is a n invariant of the symplectic structure of X . After the discussion above it becomes clear that we can formulate a different version of HMS on level of open pencils: On the side of the I - Ky I-pencil we have two singular fibers Y, and Yo - the section of deepest degeneration and the most singular one. We have also compactified Y partially and we have removed the section at infinity in order to get the pair ( Y , w ) . On the side of X we have gotten X , these divisors determine the potential w. In turn the potential : X --+ C is determined by Y,. We will need one more fiber on the X side. It comes naturally on the B side X induced by a natural transform from Serre functor S[-n] to identity functor idx. On the symplectic side Y this choice depends on the symplectic form. Conjecturally we should have a correspondence of the following form.

n HMS

I for pencils fl

R e m a r k 3.14. This correspondence should extend over fields of finite characteristics.

3.3. Degenerations As in the classical Moishezon-Teicher procedure the degeneration and the smoothing data are remembered by braid factorization and it is interesting


189

to study how braid factorization data recover HMS. We leave this as an open question and move to a rather basic example.

Example 3.15. We describe our procedure in the case of X = CP2 in the picture below.

Fig. 3. The mirror of @P2

Example 3.16. We also describe the above procedure in the case X = CP' x CP1 and L = (2,2). This is an instructive example for our purposes. Moishezon degeneration can be summarized by Figure 4.

Fig. 4. Moisheson's degeneration

190 L. Katzarkov

Now we apply Legendre transform to above degeneration replacing the two affine structures at the ends by C2, the middle six by CP1 x C1 and the central point by Del Pezzo surface of degree three in order to get the following degeneration of the Landau-Ginzburg model (cf. Figure 5).

Fig. 5. The intersection complex

Figure 6 illustrates the singular fiber over 0 in the Landau-Ginsburg model.

Fig. 6. The fiber over 0 of the LG potential

Birational geometry and homological mirror symmetry 191

The fiber over infinity (compare with [4])is a degenerate elliptic curve consisting of eight rational curves (cf. Figure 7).

Fig. 7.

The fiber over

cc)

Following Moishezon we regenerate and obtain well understood (compare with [4])Landau-Ginzburg mirror w : Y 4 C with 3 singular fibers in the finite part. Observe that under the mirror construction introduced above there exists a correspondence between lagrangian cycles on side A with the algebraic vanishing cycles on the B side. Therefore in the LandauGinzburg mirror w : Y + C (1,-1) lagrangian cycle in H2(CP1x CPl) survives as an element of H 2 of the central Del Pezzo surface. We outline the whole theory of the vanishing cycles in the next section but already now it is clear that we have the following correspondence between Hodge theory of X and singularities in the Landau-Ginzburg mirror fibration w : Y + C. Cycles X Transcendental Cycles Algebraic Cycles

I Homological Mirror Symmetry I w:Y+(cP1 Singularities in the finite part Singularities at co

-

3.4. Perverse sheaf of vanishing cycles and new technique f o r studying non-rationality We suggest a procedure to geometrize the use of category in the nonrationality questions - perverse sheaf of vanishing cycle. We start by re-

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L. Katzarkov

calling some results from [14]. Suppose M is a complex manifold equipped with a proper holomorphic map f : M -+ A onto the unit disc, which is submersive outside of 0 E A. For simplicity we will assume that f has connected fibers. In this situation there is a natural deformation retraction T : M + MO of M onto the singular fiber MO := f-'(O) of f . The restriction T t : Mt + MO of the retraction T to a smooth fiber Mt := f (t) is the "specialization to 0" map in topology. The complex of n e a r b y cocyc l e s associated with f : M + A is by definition the complex of sheaves Rrt,ZMt E D- ( M o ,Z). Since for the constant sheaf we have Z M ~= r f Z M o , we get (by adjunction) a natural map of complexes of sheaves

-'

The complex of v a n i s h i n g cocycles for f is by definition the complex cone(sp) and thus fits in an exact triangle

Z M% ~ Rrt*ZMt

-+

cone(sp) + Z M [l], ~

of complexes in in D-(Mo, Z)). Note that IHIZ(M0,R r t * Z M t ) FZ H i ( M t ,Z), and so if we pass to hypercohomology, the exact triangle above induces a long exact sequence

. . .H ~ ( M O , Z)5 H i ( M t ,Z)+ w ~ ( M O , cone(sp))

-+

~ i +( M l 0 , Z ) -+

.. .

(1) Since MO is a projective variety, the cohomology spaces Hi(M0, @) carry the canonical mixed Hodge structure defined by Deligne. Also, the cohomology spaces H i ( M t ,@) of the smooth fiber of f can be equipped with the Schmid-Steenbrink limiting mixed Hodge structure which captures essential geometric information about the degeneration Mt -Mo . With these choices of Hodge structures it is known from the works of Scherk and Steenbrink, that the map r ; in (1)is a morphism of mixed Hodge structures and that Wi(M0,cone(sp)) can be equipped with a mixed Hodge structure so that (1) is a long exact sequence of Hodge structures. Now given a proper holomorphic function w : Y + C we can perform above construction near each singular fiber of w in order to obtain a complex of vanishing cocycles supported on the union of singular fibers of w. We will write C c C for the discriminant of w, YE := Y X @ C for the union of all ' E D - ( Y z , Z ) for the complex of vanishing singular fibers of w, and 9 c C we can look at cocycles. Slightly more generally, for any subset the union YQ of singular fibers of w sitting over points of and at the


193

corresponding complex

9;= @9koE D - ( Y + , Z ) U€+

of cocycles vanishing at those fibers. In the following we take the hypercohomology W i ( Y z ,9')and Wi(Y+,9'; with ) their natural Scherk-Steenbrink mixed Hodge structure. For varieties with anti-ample canonical class we have:

Theorem 3.17. Let X be a d-dimensional Fano manifold realized as a complete intersection in some toric variety. Consider the mirror LandauGinzburg model w : Y -+ C. Suppose that Y is smooth and that all singular fibers of w are either normal crossing divisors or have isolated singularities. Then the Deligne i p t q numbers for the mixed Hodge structure on W'(Yz, 9') satisfy the identity iP,q((w'(yz,

9') = )hd-P,q-1 ( X ) .

Similarly we have:

Theorem 3.18. Suppose that X is a variety with an ample canonical class realized as a complete intersection in a toric variety. Let (Y,w) be the mirror Landau-Ginzburg model. Suppose that all singular fibers of w are either normal crossing divisors or have isolated singularities. Then there exists a Zariski open set U c C so that Deligne's i p ~ qnumbers of the mixed Hodge structure o n W'(YznU, 9;) satisfy the identity

ip"(Wi(Yznu, 9&,u)) = hd - p , q - i f l

(X).

We move to the main example of this section - the example of three dimensional cubic. First using the procedure discussed in previous sections we build Landau-Ginzburg model of three dimensional cubic. The theorem 3.2 allow us to recover the Intermediate Jacobian of the cubic. We will show that the perverse sheaf of vanishing cycles for the Landau-Ginzburg model for three dimensional cubic contains much more information. Applying the procedure described in the previous sections we get, after smoothing, the following Landau-Ginzburg Mirror:

x y u v w = (u

+

+ 21 + W ) 3 . t

with potential x y . Here u , v , w are in @P2and x , y are in C2. The singular set W of this Landau-Ginzburg model looks as in Figure 8.

194

L. Katzarkov

Fig. 8. The singular set for the LG of a 3d cubic

4 Blo wn-up ruled

surfaces

Fig. 9.

K3 and Arrows

These singularities are produced as intersection of six surfaces - see Fig. 9.

Topologically the sheaf of vanishing cycles is a fibration of tori over the singular set described above. We desingularize the singular set of 3? of rational curves with three points taken out. Then 9 restricted on such a rational curve produces an S1local system with nontrivial monodromy. Recall that [8]that three dimensional cubic is a conic bundle over CB2 with


195

a ramification curve a curve of genus five. The two sheeted covering corresponds to an odd theta characteristic and this is exactly the reason why 9 restricted on such rational curve produces a S1local system with nontrivial monodromy. We will briefly work out the case of a conic bundle over CP2 with a degeneration curve - a curve of degree five and two sheeted covering corresponding to an even theta characteristic. The first part of the Fig. 10 describes the structure of a conic bundle - two CP1 fibrations glued over the base. The second part of Fig. 10 describes Moishezon’s degeneration of the base.

Fig. 10. The mirror of the rational conic bundle

Following the procedure from the previous sections we get the Landau Ginzburg mirror. In this case 9 restricted on rational curves with 3 points taken out produces a S1 local system with trivial monodromy only - see Fig. 10. We arrive at:

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L. Katzarlcov

Conjecture 3.19. (Non-rationality criterion for three dimensional conic bundles) Let X be a three dimensional conic bundle and Y + C is its Landau-Ginzburg mirror. If there exists a rational curve in the singular set such that 9 restricted on it produces a local system with nontrivial monodromy, then X is non-rational. Remark 3.20. The above conjecture can be seen as an easy consequence of HMS and the construction in section 3. The proof amounts to chasing the images of some algebraic cycles under this construction. The idea behind this conjecture is based on studying all possible singular sets of 9 we can get by applying the blow up procedure. These curves correspond to singularities of 9 that come from infinity by moving the pencil in I - K y I away from the very singular point in A y . Such curves correspond to amoebas in CP3 and in dimensions three and four we have strong restrictions on possible singularities. We will briefly discuss one more example - a complete intersections of hypersurfaces of degree (3,O) and (2,2) in CP3 x CP2. This produces a three dimensional conic bundle X over a smooth cubic surface in CP3 with a curve of degeneration a curve of genus four. The intermediate Jacobian J ( X ) cannot be used to detect non-rationality in this case due to the fact that J ( X ) is isomorphic as a principally polarized abelian variety to a Jacobian of a curve (see [8] and section 3.5). From another point the above non-rationality criterion shows that X is not rational (for more details see [20].) We move to the case of four dimensional cubic. We start by degenerating the four dimensional cubic X to three CP4 and applying our standard procedure. After smoothing we get the following Landau-Ginzburg Mirror:

xyzuww = (u

+ w + w)3 . t

+

with a potential z + y z . Here u, w,w are in CP2 and 2,y, z are in C3. The singular set W of this Landau-Ginzburg model can be seen on Figure 11 it consists of twelve rational surfaces intersecting as shown.

Conjecture 3.21. The generic four dimensional cubic is not rational. The idea behind this statement is based on the following fact: There exists an open rational surface in the singular set such that 9 restricted on it produces a nontrivial local system with nontrivial monodromy. (For more details see [20].)

Birational geometry and homological mirror s y m m e t r y

197

Fig. 11. Singular set for the LG model of the 4d cubic

There is an interpretation of this in terms of the geometry of the pencil in 1 -K X I. The linear system of hyperplane sections on the four dimensional cubic produces a ten dimensional algebraically completely integrable system S with fibers Intermediate Jacobians of the hyperplane sections. The above conjecture has the following B - side interpretation:

Conjecture 3.22. If the algebraically completely integrable system (ACIS) S has a section - a globally defined normal function, then the f o u r dimensional cubic X , associated to ACIS S , is rational.

Remark 3.23. Using the construction described in section 3 the above conjecture can be restated in purely algebro - geometric terms. Take P to be a two dimensional cubic in X and Yt to be a family of three dimensional cubic passing through P. Consider all elliptic quintics in each of Yt. If they form a Del Pezzo surface of degree five in X then X is rational. This statement can be attacked without the use of HMS and it will be discussed in more details in [20].

198 L. Katzarkov

3.5. Connection with other methods for studying non-rationality and some ideas for proving HMS

In this section we outline a connection of well established techniques in studying rationality questions with the method introduced in the previous section. The notions and constructions discussed in this part are well known in Algebraic Geometry and Theory of Singularities - here we will only give references instead of complete definitions. 1) We start with technique of intermediate Jacobian introduced by Clemens and Griffiths [8]. As we saw above, the Landau-Ginzburg mirror of a three dimensional cubic is a family of K3 surfaces. To the partially desingularized singular set of w we associate a set of ten thimbles in the following way - they are S1x D2 defined by choosing a closed loop y entirely contained in the smooth part of the critical locus of w in the fiber above the origin and then the corresponding thimble is the set of all points for which w is real negative and the gradient flow of Re(w) converges to a point of y. They can be seen as images under Legendre transform of the Fano surface of lines on X. We chose a maximal isotropic subspace = H192(X).The intersection matrix of the above five in W1(Ycnv, thimbles in a maximal isotropic subspace of a chosen bases can be seen as follows:

The matrix above represents the well known unimodular system of vectors E5. A different way to describe this matrix is as a matrix of the map:

l-IZMt% Me, +

where are the moduli spaces of deformations of the thimbles described above and Me is the moduli space of deformations of the handlebody thimble associated with the partially desingularized singular set of w. The nonrationality criterion discussed above gives that the system of vectors represented in the matrix is not cographic. This is an A-side translation of the fact that the degenerations of Jacobians of curves produce only cographic matrices (see [2,13,26]) and the intermediate Jacobian at hand is not a Jacobian of a curve. 2) The technique of birational automorphisms of X introduced by Iskovskikh and Manin [15]. As a consequence of it, namely of the Noether


199

Fano inequality, Puchlikov and later Ein, Mustata, DeFernex have introduced the technique of log canonical thresholds [27], [ll],[lo]. To a pair K y and the fiber at zero YOwe can associate a log canonical threshold L C ( K y ,YO). We formulate the following: -

Conjecture 3.24. L C ( K y ,YO)= L C ( K x ,X O ) . Remark 3.25. The higher thresholds (other roots of the Bernstein-Sato polynomial) should coincide as well.

It is quite possible that theories of the arc spaces for ( K y ,YO)and for ( K x ,X O )are parallel and define a series of birational invariants of X. 3) The degeneration techniques introduced by Kollir [22]. KollAr deals with hypersurfaces of big degrees. It is easy to see that in this case in the Landau-Ginzburg mirror there always exists a rational curve in the desingularization of the singular set such that 9 restricted on it produces an S1 local system with nontrivial monodromy. The argument, we have briefly outlined, suggest that generic hypersurfaces of big enough degree is non-rational. Small degree hypersurfaces present a problem in dimension greater than four. 4) Artin's and Mumford's approach developed later by Saltman and Bogomolov. The examples of Saltman and Bogomolov are quotients of affine spaces by finite groups - so their mirrors are relatively easy to compute. We expect due to the orbifold nature of these examples one will be getting a quite serious singularities and Landau-Ginzburg mirror and there will always exist a rational curve in the desingularization of the singular set such that 9 restricted on it produces an S1 local system with nontrivial monodromy. All these techniques will be discussed in [20]. In the discussion above we have used HMS in full we will briefly discuss how one can try proving HMS for cubics - we will extend Seidel's ideas from the case of the quintic. We first have: Conjecture 3.26. HMS holds for the total space of Ocp4(3).

The next step is: Conjecture 3.27. The Hochschild cohomology HH'(Di(tot(Ocp4 (3))) are isomorphic to the space all homogeneous polynomials of degree three in five variables. Here (Dtstands f o r category with support at the zero section.

The Hochschild cohomology HH'(Dk(tot(Ocpr (3)))) produce all B side deformations of categories of the three dimensional cubic and we have similar

200 L. Katzarkov

phenomenon on the A side as well. For the opposite statement of HMS and more general case see [2O].

Remark 3.28. Clearly the above argument should work for any degree and for any dimension toric hypersurfaces. 4. Applications to symplectic topology

In this section X will be a two dimensional smooth projective surfaces (a four dimensional symplectic manifold) with fixed symplectic form w on it. The construction described in section 3 allows us to define new computable symplectic invariants for a four dimensional manifold underlying a projective algebraic surface. We first make the following observation. Let us denote by H2(X)"'g the lattice of algebraic vanishing cycles. It is clear from the construction that this lattice can be seen as generated by the integral points and on the edges of the degenerated polytope. After the Legendre transform the lattice H2(X)"'g corresponds to sheaves of the smoothing of w : Y --+ C. We define several invariants coming naturally from the construction.

Conjecture 4.1. The non-torsion discrete part of Ko(Db(Y,w)) maps onto H2(X)"l9. We briefly recall the constriction of the Landau-Ginzburg model for the two dimensional cubic. "Classically" it is obtained in the following way. We start with the mirror of P2 which can be viewed as (C*)2 as the surface {zyz = 1) c (C*)3, and wo = z y z . Compactifying (C*)3 to P3 leads one to consider the cubic surface {XYZ = T 3 } c P3, which has A2 singularities at the three points (1 : 0 : 0 : 0 ) , (0 : 1 : 0 : 0), and (0 : 0 : 1 : 0). After blowing up P3 at these three points, and taking the proper transform of { X Y Z = T 3 } c P3 we obtain a smooth cubic surface, in which the hyperplane sections d;lo = {X+Y+ Z = 0) and = {T = 0) define a pencil of elliptic curves with three base points. As before, GOis a smooth elliptic curve, and is a configuration of 9 rational curves (the proper transforms of the three coordinate lines where the singular cubic surface intersects the plane T = 0, and the six -2-curves arising from the resolution of the singularities). Blowing up the three points of 60n we again obtain a rational elliptic surface, and an elliptic fibration with twelve isolated critical points, nine of which lie in the fiber above infinity. We get the mirror of the cubic by pulling an configuration from infinity into the interior. We are left with a configuration of 3 lines at infinity.

+ +

em

em

em,

E6

Bzratzonal geometry and homologicnl mirror symmetry 201

Of course we can obtain this picture by directly applying the procedure described in the previous section of this paper. To carry out the computation we first degenerate the two dimensional cubic to three planes. Then we apply the Legendre transform and after that we do a smoothing. As a result we get xuuw = (u w ~ ) ~ Here t the . potential is x and u,u,w are coordinates in CIP", t is a constant determined by the anticanonical polarization. ~ 0. The E6 fiber occurs at x = 0 , where we get the fiber (u u w ) = The surface with an equation xuuw = (u u ~ ) ~has. singularities t when x = uuw = u u w = 0, i.e. three singular points, and these are A2 singularities. Resolving them produces a ,?& fiber. The procedure is recorded pictorially in Figure 12.

+ +

+ +

+ +

+ +

LT

t Fig. 12. Legendre Transform

The intervals connecting the dots in Figure 12 correspond to algebraic vanishing cycles. Using combinatorics of this situation we obtain that for two dimensional cubic algebraic vanishing cycles and -2 Lagrangian spheres are the same. Let us look at the E 6 fiber obtained above. We choose a basis of K o ( D b ( Y , w )as ) follows. Denote by O(Ei) the sheaves on the LandauGinzburg mirror of the two dimensional cubic obtained after the smoothing and considered as sheaves in Db(Y,w ) .We compute the lattice they generate in Ko(Fuk(X,f)). It is easy to see they generate the integral non-torsion discrete part of K 0 ( D b ( Y , w ) )To . compute the intersection form on nontorsion discrete part of Ko(Db(Y, w ) ) we proceed as follows. We find a free resolution of O(Ei)as follows:

OEi

--$

Oy(-Ei)

4

OY,

-+

OEi

Here the sequence is taken over the zero fiber YOof the Landau-Ginzburg mirror ( Y , w ) .We write such resolutions for all O(Ei) and compute their Ext's. As a result we obtain a rank six lattice isomorphic to H2(X)a'g.

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L. Katzarkov

Remark 4.2. The calculations above suggest a pencil to pencil HMS correspondence and will be discussed in detail in [5]. We move to the Horikawa surfaces X1 and X2. X1 = is a double cover p l of CP1 x CP1 branched along (6,12) divisor. X2 = is double cover p2 of - the Hirzebruch surface branched along the divisor 5Ao U Am in the standard notations. XI, X2 are minimal algebraic surfaces of general type, which are homeomorphic, not C deformation equivalent and have the same Gromov-Witten invariants. We have the following:

Lemma 4.3. ~ ~ ( ~ 1# )~ ~~ '( g~ 2 ) ~ ' g . Indeed H2(X1)"'g = pTH2(CP1x CP1) and H2(X2)a'g = pzH2(F6). It is clear that prH2(CP1x CP1) # &H2(F6). Note that p f H 2 ( F s )does contain an element with a self intersection -3 and p;H2(CP1x CP1) does not. The Horikawa surfaces are fibrations of genus two curves and their mirror Landau-Ginzburg models can be obtained in a similar way as explained in the previous sections. As in the case of the two dimensional cubic it is expected that non-torsion discrete part of K o ( D b ( xw)) , = H2(Xi)azg.As a consequence of conjecture 4.1 and HMS we get:

Conjecture 4.4. The four dimensional symplectic manifolds XI, w1 X2, w2 are not symplectomorphic. (In the statement above w1 and w2 are the symplectic forms corresponding to the canonical classes of X1 and XZ.)Indeed H2(Xl)a'g # H2(X2)azg and with the assumption of HMS we get that KO(hk(X1, WI)) # KO(Fuk(X2, ~ 2 ) ) .

Remark 4.5. In the calculation above we do not Karoubi complete K theories. Instead of looking at Horikawa surfaces, where the calculations are much more involved, we proceed by demonstrating calculation of Ko(Db(Y,w)) for the Landau-Ginzburg Mirror of a genus two curve. We demonstrate that taking Karoubi completion enlarges the categories substantially. In the following calculations we work with Karoubi completions. We realize the genus two curve as a hypersurface in a toric variety - the Hirzebruch surface PO= P ( 0 p (O)@Op).Then the mirror of C is given by the following torus action:

Bimtional geometry and homological m i w o r s y m m e t r y

203

We recall a different, LLclassicaYway of getting the Landau Ginzburg mirror following [17]. As a mirror of C we get a pencil given by the function W.XO = xi +.. . + x ~ + uover the threefold Wi in C*5 given by the equations: 51

53

. x2 = .u 3

. x4 . xo = .u 2 .

After resolving the singularities of W1 we get the following:

Lemma 4.6. (see [lS]) T h e set of critical points of w o n partially compactified W1 consists of one point t and three CB1’s - Q 1 , Q 2 , Q 3 passing through two points p and q. (See figure 13).

t 0

&I

Q~

Q3

Fig. 13. The critical points on the partially compactified W1.

We have :

Lemma 4.7. K 0 ( D b ( Y , w )=) Z4 x C*.

204 L. Katzarkov

In the above lemma we do not consider torsion - see also [l].The calculation is done using ideas of Weibel and geometry described above. The result is obtained by applying following standard exact sequence:

.. .

-+

Ko(Perf(Y0) -+ Ko(Db(Yo))-+ Ko(Db(Y,w))

-+

K-l(Perf(Yo)

-+

0

Here YO is the zero fiber of the Landau-Ginzburg mirror w : Y 4 @. The main effort goes t o dealing with the non-triviality of K-l(Perf(Yo). We also have the following Mayer Vietoris (homotopy) K-theory sequence for the central fiber YO= Y1 U Yz U Y3.

.. .

-+

Ko(Perf(Y1)) e K O ( P e r f ( Y 3 U Yz))

+

-+

K-I(Perf(Y0))

-+

Ko(Perf(Q1 U Q3))

0.

Here Y1 and Yz and Y3 are @IP2 blown up in three, three and fifteen points respectively. The last sequence allows us to compute K , (Perf(Y0)) and we get a proof of the lemma. The calculations for Horikawa surfaces follow a similar pattern but are much more involved. In conclusion we observe that previous considerations suggest stronger symplectic invariants. Indeed the whole theory of the arc runs for the pair K x , X o . According t o considerations in section 3.5 we should be getting a series of new symplectic invariants for X .

Conjecture 4.8. L C (Kx , Xo) as well as higher thresholds and invariants of arcs spaces are symplectic invariants. Acknowledgments We are grateful t o D.Auroux, M.Gross, T.Pantev, P.Seide1, D.Orlov, M.Kontsevich for many useful conversations. Many thanks go t o V.Boutchaktchiev without whom this paper would not have been written. Special thanks to the referee and to J. Starr for pointing to reference [13].

References 1. M. Abouzaid, On the Fukaya category of higher genus surafces preprint. 2. V. Alexeev, C. Birkenhake, K. Hulek Degenerations of Prym varieties. AG 0101241. 3. D. Auroux, L. Katzarkov, S. Donaldson, M. Yotov Fundamental groups of complements of plane curves and symplectic invariants. Topology 43 (2004), no. 6, 1285-1318.

Bimtional geometry and homological mirror symmetry 205 4. D. Auroux, L. Katzarkov, and D. Orlov. Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves, 2005, arXiv:math.AG/0506166, t o appear in Inventiones Mathematicae. 5. D. Auroux, L. Katzarkov, T . Pantev and D. Orlov. Mirror symmetry for Del Pezzo surfaces 11. in prep. 6. F. Bogomolov. The Brauer group of quotient spaces of linear representations., ) Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 485-516, 688. 7. H. Clemens. Cohomology and obstructions 11. AG 0206219. 8. H. Clemens, Ph. Griffiths The intermediate Jacobian of the cubic threefold. Ann. of Math. (2) 95 (1972), 281-356. 9. D. Cox and S. Katz. Mirror symmetry and algebraic geometry, volume 68 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999. 10. T. deFernex Adjunction beyond thresholds and birationally rigid hypersurfaces. AG/0604213. 11. L. Ein, M. Mustata Invariants of singularities of pairs AG/0604601. 12. M. Gross, B. Siebert. Mirror Symmetry via logarithmic degeneration data I. AG 030907. 13. T. Gwena. Degenerations of cubic threefolds and matroids. Proc. Amer. Math. SOC.133 (2005), no. 5, 1317-1323. 14. M. Gross and L. Katzarkov. Mirror Symmetry and vanishing cycles. in preparation. 15. V.A. Iskovskikh, Y.I. Manin Three-dimensional quartics and counterexamples to the Liiroth problem. , Mat. Sb. (N.S.) 86(128) (1971), 140-166. 16. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow. Mirror symmetry, volume 1 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa. 17. K. Hori and C. Vafa. Mirror symmetry, 2000, hep-th/0002222. 18. A. Kapustin, L. Katzarkov, D. Orlov, and M. Yotov. Homological mirror symmetry for manifolds of general type, 2004. preprint. 19. L. Katzarkov Homological mirror symmetry and Moishezon degenerations, preprint, 2006. 20. L. Katzarkov Mirror symmetry and nonrationality, in preparation. 21. L. Katzarkov, M. Kontsevich, T. Pantev From Fukaya categories t o Gromov Witten invariants, in preparation. 22. J. Kollar Singularities of pairs, Algebraic geometry-Santa Cruz 1995, 221287, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. SOC.,Providence, RI,1997. 23. G. Kerr Weighted Blow - ups and HMS for toric surfaces, preprint. 24. D. Orlov. Derived categories of coherent sheaves and equivalences between them. Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89-172; translation in Russian Math. Surveys 58 (2003), no. 3, 511-591. 25. D. Orlov. Triangulated categories of singularities and D-branes in LandauGinzburg models. Tk. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240-262, 2004.

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26. J.G. Oxley Matroid Theory. Oxford University Press, 1992. 27. A. Pukhlikov Birationally rigid Fano varieties. The Fano Conference, 659681, Univ. Torino, Turin, 2004. 28. P. Seidel Vanishing cycles and mutation., European Congress of Mathematics, Vol. I1 (Barcelona, 2000), 65-85, Progr. Math., 202, Birkhauser, Basel, 200 1. 29. P. Seidel Fukaya categories and deformations. , Proceedings of the International Congress of Mathematicians, Vol. I1 (Beijing, 2002), 351-360, Higher Ed. Press, Beijing, 2002. 30. J. Wlodarczyk Birational cobordisms and factorization of birational maps , J. Algebraic Geom. 9 (2000), no. 3, 425-449.

207

Desingularization and equisingularity at undergraduate level Tzee-Char Kuo and Laurentiu Paunescu

School of Mathematics, University of Sydney, Sydney, NS W, 2006, Australia, E-mail:tckOmaths.usyd.edu.au,1aurentOmaths.usyd. edu.au, www.maths.usyd. edu. au. Resolution of Singularities (Desingularization) and Equisingularities (Equivalence of Singularities) are neighboring research fronts in Algebraic/Analytic Geometry; they involve sophisticated mathematics and much jargons. However, a number of fundamental concepts can be fully explained to undergraduate students, using elementary examples in W2.This we try to achieve. Student readers are encouraged to do the exercises, most of which are easy.

Singularities never die, they just get blown away.

1. Introduction

This is the text of a mini-course for undergraduate students on Singularity Theory/Algebraic Geometry. The course is not a “standard” one, in the sense that the emphasis is not on stating theorems; instead, plenty of examples and exercises are presented in order to expose the essential meaning of Resolution of Singularities and the Equisingularity Problem, two closely related research fronts. The examples and exercises are all based on functions of two real variables. A student can see for him/herself how to formulate and prove the Resolution of Singularities in the case of two variables (real and complex). The general case, Hironaka’s Theorem, is (perhaps) too hard for a student. However, in the past two decades, several mathematicians have found simpler proofs. For example, Kolar’s “Seattle Lecture” (arXiv:math.AG/0508332, 17 Aug 2005) contains a proof for senior undergraduate students. He/she can also appreciate the Equisingularity Problem, for which a really good theorem is still to be found. The notion of blowing up is thoroughly illustrated by the exercises and examples. Blowing up is very similar to the polar coordinates. The former is a polynomial transformation (x = X Y , y = Y ) while the latter is tran-

208

T.C.Kuo and L. Paunescu

scendental (z = T cos 8, y = T sin 8). Both are of fundamental importance in mathematics at all levels. A collection of good examples is like a gold ore. One can see the sparkling gold (ideas). The extraction process (formalization of concept, theorem, etc.) is challenging and interesting, but can be very hard in some cases. (Some not so hard cases are explained in [22].) Attempts made in the process, successful or not, are always the most rewarding. For the Resolution of Singularities, it took several generations of mathematicians to finally arrive at Hironaka’s Theorem in 1964. (The case of non-zero characteristics is still open.) As for the Equisingularity Problem, in our opinion, the theory is still in its infancy. Our recent result, [25], gives strong indication that the notion of Morse stability will play a vital role in the theory. 2. Total and Strict Transforms.

This notion can be understood by high school students. Imagine we were in a high school mathematics class, our teacher had asked us to sketch the locus of the curve C defined by y2 = z2(z 1) in the plane R2. This is a typical example of what we do in Algebraic Geometry. We are interested in understanding the locus defined by a given set of polynomial equations in two, or more, variables. In Analytic Geometry, we study the locus defined by analytic functions (convergent power series). The locus is called a variety. Returning to the above curve C, some of us might pick up values for z at random, one at a time, and compute the corresponding values of y. Since we can live only a finite number of years, this approach would not go very far. A better idea, which is of fundamental importance in mathematics at all levels, is to make a substitution: y = tz, where t is a parameter. This yields t2x2 = z2(z l),and then a parametrization of C:

+

+

z = t 2 -1,

y=t

3

4,t E R .

(1)

It is then straightforward to sketch the locus, which is the Q -shaped curve in Fig.1. Intuitively, the straight line y = ta: is like a radar beam, emitted at time t to detect the unknown object C. It encounters C at the point (t2-1, t 3 - t ) , and the origin. In Algebraic Geometry, this parametrization is interpreted as follows.

Desingularization and equisingularity at undergraduate level

-

209

Consider the mapping (Fig.1) p1 :

R2

R2,

( X , Y ) H (x,y):= ( X , X Y ) ,

where the notation “:=” means “is, by definition, equal t o ” . Let us see what p1 does. Each horizontal line Y = c, c a constant, is mapped to the line y = cx. The entire Y-axis ( X = 0) collapses to the origin 0. Each other vertical line X = c‘, c‘ # 0, has image x = c‘. A rectangle with vertices (=tc’,fc) is mapped to a sector around the x-axis (Fig.1). Convention. In this paper, by a rectangle we shall always mean one centered at (0, 0), whose sides are parallel to the coordinate axes.

Y

Fig. 1. Total Transform of the a-Curve

+

The curve C is pulled back to X 2 Y 2 = X 2 ( X l),which consists of a parabola Y 2 = X 1, plus X 2 = 0 (the Y-axis, counted twice). Note that the parabola Y 2= X 1 admits an obvious parametrization Y = t , X = t2 - 1, which leads to the parametrization (1) The Y-axis is called the exceptional divisor, or exceptional locus, of PI; this is because p1 is a bijection except along the Y-axis. We call the pulling back by p1 a basic blowing-up, (which is similar to introducing the polar coordinate,) and call the pushing forward action a basic blowing-down. But, as we often abuse language, can mean either. Geometrically, the parabola

+

+

210


Y 2= X + 1 is called the strict transform of C, which, together with the Y axis ( X = O, counted twice), is called the total transform of C, as illustrated in Fig.1. Observe that ,& is useless for curves, such as x2 = y3, which are tangent to the y-axis. Therefore, we ought to complement ,& by the mapping p 2 : R2

-

R2,

-- -

(F,qH (z,y):=( X Y , Y ) ,

--

which magnifies a sector around the y-axis to a rectangle in the ( X , Y ) plane. This is shown in Fig.2. We also call ,& a basic blowing-up.

t

t

Fig. 2.

We shall see that, in $8, ,f31 and 8 2 can be put together as a single blowing-up. Let f (x,y) be a given analytic function defined near (O,O), f o ,f31 and f 0,62 are called the total transforms of f . (The strict transform of f will be defined in $5, Exercise 5.4.) We all appreciate the power of parametrization. In undergraduate mathematics, we have used the above parametrization to compute the area of the loop, the arc length, the curvature, etc., of C. Philosophically, the reason why a parametrization can be so powerful is that a given complicated object, like C, is decomposed into two simpler ones: the t-parameter space, which is a straight line in this case, and a simple mapping, like t H (t2 - 1,t3 - t). In many cases, this kind of decomposition gives us all we want. Resolution of singularities can be regarded as parametrizations of general varieties.

Deszngularization and equisingularity at undergraduate level 21 1

We now give more examples.

Exercise 2.1. Consider the cusp y2 = x3 in R2. Show that the strict transform under is again a parabola: Y2 = X , inducing the parametrization z = t 2 ,y = t3 (Fig.3).

Y

b

-

4

X

Fig. 3.

* X

Strict Tkansform of the Cusp

Example 2.2. Let us sketch the surface S defined by y2 = z2(x R3.

+ z 2 ) in

First, take the section by z = a, a # 0. This is an a-shaped curve crossing the z-axis at ( - a 2 , 0 ) and (0,O). If we take a = 0, the section is a cusp. The section by y = 0 is a parabola: z = - z 2 , plus the z-axis. Hence we can sketch S, as in Fig.4. Now, let us consider the following mapping:

,f31 x id : ( X ,Y,2 ) H (z,Y,z):= ( X , X Y , 2). The surface S is blown up to a smooth surface g:= { ( X ,Y, 2 ) I Y2 = X + Z 2 } , plus, of course, X 2 = 0, which is the (Y,Z)-coordinate plane (counted twice). Intuitively, we can treat R3 as a book, and blow up each page ( z = constant) by PI. In the new book, the strict transform, g, no longer has singularities.

212


Fig. 4. Surfaces

s and S

In the above examples, one blowing-up suffices to make the strict transform smooth. In general, a succession of them may be needed.

Example 2.3. (The Zig-Zag phenomenon.) Let us desingularize C : y17 = is a cusp tangent to the horizontal x-axis.

239,which

First, we can apply P1 o P 1

/31

twice:

: (X2,YZ)H (X1,K) H (ZlY>= (X2,X,2Y2).

The strict transform of C is C2 : Y217 = X:. Note that C2 is tangent to the vertical Y2-axis. A “Zig-Zag”has taken place; and PIis useless for C2. We then apply ,& three times:

P20P20P2:(X5,Y5)H(X4,P4)H(X3,Y~)H(X2,Y2)=(X5Y53,~5).

z5,

The strict transform is Cg : E2= tangent to the x5-axis (another Zig-Zag!). Finally, apply P1 twice, and the strict transform is smooth. Thus, a composition of seven basic blowing-ups, Pp o Pz o P f , is needed to make C smooth.

Exercise 2.4. Consider the general cusp ym = P ,m, n relatively prime. How to calculate the number of basic blowing-ups needed to make the cusp smooth? (By continued fraction.) 3. Normal Crossing.

This notion is based on that of changing variables, or the Inverse Function Theorem in Multivariate Calculus.

Desingularization and equisingularity at undergraduate level 213

In high school, we are familiar with linear change of variables, such as u=x+y v=x-y

with inverse (solving backward)

x = $(u+ v)

For example, consider the function fl(x, y) = x 2 - y2. With this substitution, fl(x,y) becomes u v u v g1(u, v):= f l ( 2 - - = uv, 2’2 2

+

-)

whose locus coincide with the coordinate axes. As another example, take f i ( z , y ) = (2x-y3)2(y+x-x100), u = 2x-y3, v = y x - x1O0.The Jacobian

+

Hence, by the Inverse Function Theorem, we can solve x, y in terms of u, v, in a suficiently small neighborhood of the origin (0,O). Thus, f z ( z , y ) becomes g2(u, w) = u2v, whose locus again coincides with the coordinate axes (in a neighborhood of (0,O)). Geometrically, the locus f2(x,y) = 0 consists of two smooth curves which cross each other at the origin, but are not tangent to each other. In the (u, v)-plane, they have been “straightened up’’ to the coordinate axes. This suggests the following definition of a normal crossing. Let u(x,y), v(x, y) be analytic functions defined in a neighborhood of (O,O), with u(0,O)= v(0,O) = 0, such that

a(u7v) # 0,

Y)

when x = y = 0.

Then, by the Implicit Function Theorem (the analytic version, [ S ] ) , the inverse

x

= x(u,v), y = y(u,v), with x(0,O) = y(0,O) = 0,

are defined and analytic in a neighborhood of (0,O). Thus, there exist neighborhoods N and U of (0,O) such that the mapping 7

:U

-

N,

(21,

v) H (x,Y) := (xc(u,v),Y(U, v)),

which sends (0,O) to (0, 0), is an analytic isomorphism. (Both are analytic.) We call T a local analytic coordinate transformation.

T

and

7-l

214

T.C. Kuo and L. Paunescu

Definition 3.1. Let f ( z , y ) be an analytic function defined near (O,O), f ( 0 , O ) = 0. We say f(z,y) is a normal crossing at (0,O) if there exists T such that f(z(u, v),~ ( u v)),= ~ ( u v)umwn, , m 2 0 , n 2 0, E(O,O)

# 0,

(2)

where E ( U ,w) is defined and analytic near (O,O), (called a unit). Take ( a ,b), f ( a ,b) = 0. We say f(z,y) is a normal crossing at ( a ,b) if the translated function f ( z + a , y + b) is a normal crossing at the new origin (z,B) = (0,O). We say f is a normal crossing if it is so at every (a,b), f ( a , b) = 0.

Examples and Remarks. First, as f ( 0 , O ) = 0, m, n cannot both be zero. In the definition, however, we do allow one of them to be zero. In this case, the (geometric) locus f ( z , y ) = 0 consists of only one smooth curve, which is straightened up to a coordinate axis (counted n times if, for example, m = 0). This is still called a normal crossing. Suppose m 2 1, e(0,O) > 0. Then E = u zi = v, is a coordinate transformation, for which f becomes P P . (In case E ( O , O ) < 0, and one of m, n is odd, f can still be transformed to this form; but if m, n are both even, then f can only be transformed to the form - - P P . ) Here is an easy exercise. We say ( a , b ) is a non-singular point if (fz(a,b),f,(a,b)) # (0,O). Use the Implicit Function Theorem to show that f(z,y) is a normal crossing at every non-singular point. For f(x,y) = z2, every (0, b) is a singular point, yet f(z,y) is a normal crossing. Take fl(z,y) = z2 - y2 as before. It has only one singular point (O,O), and is a normal crossing here (and elsewhere). Consider f3(z,y) = y(z2 - y2). The locus consists of three lines, y = 0, z = f y ; (0,O) is the only singularity. A coordinate transformation, being a homeomorphism, can never turn them into two coordinate lines. Hence f3 is not a normal crossing. The locus z2+y2 = 0 (in R2) consists of (0,O) alone. The function is not a normal crossing. But the complex function z 2 w2 = ( z i w ) ( z- iw)is a normal crossing (in C?). As another example, for -z2y2, no T can bring it to the form u2v2. But in the complex case, -z2w2 = ( i ~ ) ~ w(Indeed, '. for a normal crossing in the complex case, we can assume c = 1.) This shows that the complex case is more convenient than the real case. However, the real case is intuitively clearer for undergraduate students. For functions, normal crossings are the simplest possible type of singularities; they cannot be improved any more by blowing-ups. This is illustrated

vm,

+

+

Desingularization and equisingularity at undergraduate level

215

as follows. Consider f l ( z , y ) = x 2 - y2. The total transform ( f l o p l ) ( X , Y ) = X2(1- Y ' ) , apart from the exceptional divisor, has two singular points, (0, &l),which are normal crossings. No improvement has been achieved by .&, Any further blowing-ups would merely increase the number of normal crossing points. This phenomenon persists in general. On the other hand, for the above f 3 , which is not a normal crossing at (O,O), the total transform ( f 3 o,f?l)(X,Y)= X 3 Y ( l -Y2)has three normal crossings, at (0, 0), (0, &l).The singular point of f 3 has been improved, or, as we say, blown up, to three normal crossings.

Example 3.2. Take g(z, y) = y2 - x3. Let us study g o p1, 9 o 02 and goP1

oP2.

X2(Y2- X ) is not a normal crossing at

Note that (g 0 p l ) ( X , Y ) =

(070). As for g

,&, since g(Z,y)-=-0 has no locus in a (small) sector around the y-axis, the locus (g o /32)(X,Y )= 0 consists merely of the x-axis. This 0

is confirmed by the formula: (gop')(x,F)

=E(X,F)F',

--

-3-

€ ( X , Y ) : =l-x Y ,

--

where E(X,Y) is a unit at every point of the x-axis.Hence g normal crossing.

o

,& is a

Exercise 3.3. Study the composition g o p1 o p 2 in the same way. Is it a normal crossing? Are g o p1 o p1, g o p1 o pz o p1 and g o p1 o ,4o p 2 normal crossings? (Ans. No. Yes, Yes, Yes.) 4. The Analytic Arc Lifting Property. Take &

> 0, sufficiently small. Take two convergent power series 00

M

i= 1

We call the mapping

X : (-&,&)

i=l

-

R2, X ( t ) : = ( z ( t ) , y ( t ) ) ,

an analytic arc in R2 at (ao, bo), or with initial point (ao,bo).

Exercise 4.1. Let X be given. Suppose z-

= antn

+ ... ,

- bo = bmtm

+ ... ,

an

# 0 # bm.

216


What is the tangent line at (ao,bo)? (Ans. If n < m, tangent is y = bo. What if n = m?)

Exercise 4.2. (Important) Let X be a given analytic arc at ( O , O ) , not tangent to the y-axis. Show that there exists E’, 0 < E’ 5 E , and a unique analytic arc, p , in the ( X ,Y)-plane such that (Dl o p ) ( t ) = X ( t ) , I t E’. The initial point of p lies on the Y-axis, and is determined by the tangent line of A.

I
ll < 61. We can identify I (as we do below) with an open interval in R, ( l o , 1 as real numbers). The analytic structure on RP’ x R2is characterized as follows: A function f(1, x,y), defined on I x U , is analytic if, and only if, it is analytic in (Z,x,y ) E R3. The Mobius band sits naturally in RP1 x R2 as a 2-dimensional surface; namely,

I

M

= (1

I

:= { (1,x,y)

I (x,y ) E Z E RP’},

(1 as a line through (0,O)).

Exercise 8.1. See for yourself that M is, topologically, a Mobius band. Take ( l o , 0,O). Let us examine M near this point. Let ax - by = 0, (a,b) # (O,O), be the equation of lo. Without loss of generality, we can assume a = 1. Those lines 1 near 10 can be written as x - sy = 0, where s is a real variable, s near b. Hence, near ( l o , 0, 0), M is the graph of the function x = sy, s near b. It follows that M is a 2-dimensional analytic sub-manifold of RP1 x R2. The Mobius band is a fundamental object in mathematics. There are two important projections, 7r

:M

-

RP’,

(l,x,y) H 1 ;

p :M

-

R2,

(l,x,y) H ( 2 , ~ ) ;

the former is called the canonical line bundle over the real projective line, and the latter is called the blowing-up of R2 with center (0,O). (See Fig.6.)

Exercise 8.2. Find two coordinate charts on M , covering M , such that p reduces to /31 on one, and reduces to /32 on the other. Having combined /31 and /32 into a single mapping p, let us consider Exercise 5.3 again. There is no redundancy any more: p blows up (0,O) into m distinct points on M .

226


Fig. 6.

Canonical Line Bundle and Blowing-up

9. Global Desingularization

The passage from IR2 to M can be interpreted as replacing (0,O) by a copy of RPl, thus modifying R2 into a Mobius band. An important observation is that P is an analytic isomorphism in a sufficiently small neighborhood of every given point not of the form (1,0,0), that is, not on the center circle (the exceptional divisor). Let S be a given 2-dimensional analytic manifold. Take any P E 9. We can define P p : M + 9 in the same way as the passage from IR2 to M . The manifold M is constructed as follows. Take a coordinate neighborhood, U , of P , which is a copy of IR2, with origin at P. Remove U from 9, then glue back a copy of the Mobius band M . This amounts to replacing P in 9 by a copy of IRP'. The new manifold is M. The mapping PP is call the blowing-up of 9 at P , or with center P. Note that PP is an analytic isomorphism between M -RP' and 9- { P } . The way you did Exercise 5.8 actually proved the following theorem.

Theorem 9.1. (Global Desingularization) Let Mo be a given compact 2dimensional analytic manifold, and f : MO -+IR an analytic function. Then there exists a composition p P B:=P1 0 ... 0 P N : M N P N " ' 3 M 1 2 Mo,

-

where

Pi is the

blowing-up of

Mi-1

at some point, 1 5 i 5 N , such that

B o f is a normal crossing in MN. It follows that there also exists a (possibly different) composition of blowing-ups such that the strict transform of the locus f - l ( O ) is a smooth curve. This theorem, and its variations, were known toward the end of the 19th century. In 1935, R.J. Walker proved the Global Desingularization

Desingularization and equisinggzllarity at undergraduate level 227

for (2-dimensional) surfaces. In the 1940's, Zariski proved the Local Uniformization Theorem in all higher dimensional cases, and the Global Desinguarization Theorem for varieties of dimension up to 4. Then, in his celebrated work, Hironaka, [14], 1964, proved the General Global Desingularization Theorem for characteristic 0. Hironaka's proof is extremely hard. A more elementary proof, with canonical centers, was found by BierstoneMilman, [3]. Interested readers are referred to [13] and [31] for more information on this subject. 10. Blow-analytic Equisingularity

Definition 10.1. Let h : (R2,0 ) t (R2, 0) be a germ of homeomorphism. We say h is a blow-analytic homeomorphism if there exist two compositions of blowing-ups

~ ' = P ~ O * * * O P :M'Nt N )

-**.-M b: = u ' ,

where U , U' are sufficiently small neighborhoods of (0, 0), h ( U ) = U',and an analytic isomorphism : MN 2 ML,, such that B' o @ = h o B. (It then follows that N = N'.) That is to say, although h is merely a homeomorphism, it actually comes from an analytic isomorphism.

Exercise 10.2. (Important) Use the Analytic Arc Lifting Property to show that h is arc- analytic. That is, if X is an analytic arc in U , then h o X is also one in U'. Definition 10.3. Let f,g : (R2,0) -+ (R, 0) be analytic. We say f , g are blow-analytically equivalent if there exists a blow-analytic homeomorphism h such that f = g o h. An important fact is that the Whitney family (3) consists on only one blow-analytic equivalence class, in sharp contrast to its C1-classification. This was proved in [17],by integrating a naturally constructed vector field. We shall not give the details, which are simple, [21], but somewhat beyond the undergraduate level. For a more general result on non-degenerated weighted homogeneous forms, see [lo].

228


Suppose f, g are blow-analytically equivalent, do they have the same multiplicity? M. Suzuki was the first to give an affirmative answer to this question. His proof involves long computations and is not published. Fukui, in [7], gave a simple and elegant proof. Take f. Take any analytic arc A@), X(0) = 0. Let Ot(f(X(t)))denote the multiplicity of the power series f ( X ( t ) ) . The Fukui numerical set (see [7]) o f f is A ( f ) : ={Ot(f(X(t)))},for all A.

Exercise 10.4. Show that A ( f ) = A(g) if f and g are blow-analytically equivalent. The multiplicity o f f is the smallest number in the Fukui numerical set A ( f ) .Hence A(f) = A(g) implies m ( f )= m(g).More results on A ( f ) can be found in [12].

Exercise 10.5. (see [29]) Suppose f , g are blow-analytically equivalent under h. If ( a ,b) E U is a singular point of f (i.e., fac(a, b ) = f g ( a ,b) = 0), show that h((a,b ) ) is a singular point of g in U'. (Take any A, X(0) = (a, b). Compare the derivatives of f ( X ) and g(h o A).) Remark 10.6. It can be proved that a homeomorphism h is blow-analytic if the components of h and h-' are blow-analytic functions in the sense defined in Definition(6.1). The proof is not at the undergraduate level. Interested readers are referred to the survey article [8] and also [ll]for more information on the blow-analytic equisingularity theory. 11. Bi-lipschitz Homeomorphisms

Ever since the notion of blow-analyticity was defined, [18], experts had believed, and had tried to prove, that a blow-analytic homeomorphism was always Lipschitz. Then, in 1993, Koike, [16] made the astounding discovery that this was not true. Nowadays, there are many simple examples of this kind (see [9], [29], [301).

-

Exercise 11.1. Let q(z,y) =

h : (W2, 0)

$&$. (@, 01,

Find the inverse of the mapping (z, Y)

(w2, Yq).

Use Remark 10.6 to show that h is a blow-analytic homeomorphism. Compute the derivative & ( y q ) , to conclude that h is not Lipschitz.

Desingularizataon and equisingularitv at undergraduate leuel 229

The situation can be worse. The following example shows that a blowanalytic homeomorphism need not preserve the multiplicity of analytic arcs (hence is not Lipschitz). We move to R3.It is clearer to explain the situation in R3 than in R2. The definitions of blowing-up and blow-analyticity readily generalize to R3, with center of blowing-up either a point or a closed 1-dimensional analytic sub-manifold (a closed curve). The other notions also generalize.

Exercise 11.2. (see [30])Take p ( x , y) = mapping h : (R3,0)

-

$$ as in 56. Show that the

(E3,0), ( ~ , Y , z )

( Z , Y , Z - ~P(z,Y)),

is a blow-analytic homeomorphism. Take A ( t ) = (t3,t 2 ,t ) ,which is a smooth curve, having multiplicity 1. Show that the multiplicity of h o X is 2. (It follows that h-l is not Lipschitz.) This kind of examples also exists in R2,[15],but is beyond the undergraduate level. All these results, following Koike's discovery, point at the actual incompatibility of blow-analyticity with the Lipschitz condition. Therefore, some modification of the Lipschitz condition has to be found. At this stage, it is not clear which condition, or conditions, modified from the Lipschitz condition, ought to be chosen to supplement the blowanalyticity. Recent research suggests that it is more promising to consider blow-analytic homeomorphisms which preserve the contact order of analytic arcs. But it is still too early to report on the progress in this direction. In mathematics, one comes across definitions and theorems. For a student, theorems are much harder than definitions; for older people, however, definitions are much harder than theorems.

References 1. V.I. Arnold, Normal forms of functions in the neighborhood of degenerate critical points, Uspekhi Mat. Nauk 29 (1974), 11-49, English transl., Russian Math. Survay, 29 (1974), 19-48. 2. Abderrahmane J, Ould M., Newton polygon and trivialisation of families, J. Math. SOC. Japan, 54 (2002), 513-550. 3. E. Bierstone and P.D. Milman, Canonical desingularization in characteristic zero by blowing-up the m a x i m u m strata of a local invariant, InventionesMath. 128 (1997), 207-302.

230


4. E. Bierstone and P.D. Milman, Arc-analytic functions, Inventiones math., 101 (1990), 411-424. 5. E. Bierstone, P. D. Milman, and A. Parusiriski, A functin which is arc-analytic but not continuous, Proc. Amer. Math. SOC.,113 (1991), 419-423. 6. H. Cartan, Analytic theory of analytic functions of one or several variables, Dover, 1995. 7. T. F’ukui, Seeking invariants for blow-analytic equivalence, Compositio Math. 105 (1997), 95-107. 8. T. F’ukui, S. Koike, and T.-C.Kuo, Blow-analytic equisingularities, properties, problems and progress, in “Real analytic and algebraic singularities” , Pitman Research Notes in Mathematics Series, 381 (1997), Longman, 8-29. 9. T. Fukui, T.-C. Kuo, and L. Paunescu, Constructing blow-analytic homeomorphisms, Annales de 1’Institut Fourier, 51,No.4, (2001), pp 1071-1087. 10. T. F’ukui, and L. Paunescu, Modified analytic trivialization for weighted homogeneous function-germs, J. Math. SOC.Japan, 52, No. 2, (2000), 433-446. 11. T. F’ukui, and L. Paunescu, On blow-analytic equivalence, to appear. 12. S. Izumi, S. Koike, and T.-C. Kuo, Computations and stability of the Fukui invariant, Compositio Mathematica, 130 (2002), 49-73. 13. H. Hauser, J. Lipman, F. Ort and A. Quirbs, eds, Resolution of singularities. A research textbook in tribute to 0. Zariski, Progr. Math., 181,BirhauserVerlag, Basel, 2000. 14. H. Hironaka, Resolution of singularities of a n algebraic variety over a field of characteristic zero, I-I1 Ann. of Math., 79, No. 1, (1964), 109-326. 15. M. Kobayashi, and T.-C. Kuo, On Blow-analytic equivalence of embedded curve singularities, in “Real analytic and algebraic singularities” , Pitman Research Notes in Mathematics Series, 381 (1997), Longman, 30-37. 16. S. Koike, On strong Co- equivalence of real analytic functions, J. Math. SOC. Japan 45 (1993), 313-320. 17. T.-C. Kuo, The modified analytic trivialization of singularities, J. Math. SOC. Japan 32 (1980), 605-614. 18. T.-C. Kuo, On classification of real singularities, Inventiones Math., 82 (1985), 257-262. 19. T.-C. Kuo, Singularities of real analytic functions, in “Singularity Theory”, World Scientific, 1995, 276-283. 20. T.-C. Kuo, Stratification Theory, ibid, 284-290. 21. T.-C. Kuo, Truncation of Taylor series, ibid, 291-297. 22. T.-C. Kuo, Inspiring exercises for undergraduates, Int. J. Math. Sci. Technol., 30,No.6 (1999) 811-821. 23. T.-C. Kuo and A. Parusiriski, Newton polygon relative to an arc, in “Real and complex singularities” , Pitman Research Notes in Mathematics Series, 381 (1997), Longman, 76-93. 24. T.-C. Kuo and L. Paunescu, An elementary expose‘ on equisingularity, AMS/IP Studies in Advanced Mathematics, 20 (2001), 345-350. 25. T.-C. Kuo and L. Paunescu, Equisingularity in R2 as Morse stability in infinitesimal calcus, Proc. Japan Acad., 81,No. 6, Ser. A 20 (2005), 115-120. 26. K. Kurdyka, Ensembles semi-alge‘briques syme‘triques par arcs, Math. Ann.,

Desingularization and quisingularity at undergraduate level 231 282 (1988), 445-462. 27. J.N. Mather, Stability of C”-mappings 111, Finitely determined map-germs, Publications Mathematiques de 1’Institut des Hautes Etudes Scientifiques, 35 (1969), 127-156. 28. J. Milnor, Singular points of complex hypersurfaces, Ann. of Math Studies, 61,Princeton University Press, 1968. 29. L. Paunescu, Invariants associated with the blow-analytic homeomorphisms, Proc. Japan Acad., 78,Ser. A, (2002), 194-198. 30. L. Paunescu, An example of blow-analytic homeomorphism, in “Real analytic and algebraic singularities” , Pitman Research Notes in Mathematics Series, 381 (1997), Longman, 62-63. 31. 0. Villamayor, An Introduction To Constructive Desingularization, arXiv:math.AG/0507537, 2005. 32. R.J. Walker, Algebraic Curves, Princeton University Press, 1950 (Reprinted, Dover, 1962; Springer-Verlag, 1972). 33. H. Whitney, Local properties of analytic varieties, in “A Symposium in Honor of M. Morse”, (Ed. S.S.Cairns), Princeton Univ. Press, (1965), 205-244.

232

General self-similarity: An overview Tom Leinster Department of Mathematics, University of Glasgow, Glasgow G1.2 8QW, U.K. Email: [email protected] Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued t o several copies of another space Y , and Y looks like several copies of itself glued to several copies of X-or the same kind of thing with more than two spaces. Thus, we have a system of simultaneous equations in which ‘higher-dimensional formulas’ specify how to glue the spaces together. This idea is developed in detail elsewhere [1,2]. The present account explains the theory in outline. Keywords: self-similarity; higher-dimensional algebra; Julia set; F’reyd’s Theorem; fractal; coalgebra.

Introduction The word ‘self-similar’ has a t least two meanings. Local or ‘bottom-up’ selfsimilarity says something like ‘almost any small pattern observed in one part of the object concerned can be observed throughout the object’. (See for instance Milnor [3], where such statements are made about Julia sets of complex rational functions.) Global or ‘top-down’ self-similarity, which is what we will be concerned with, says something like ‘the whole object consists of several smaller copies of itself glued together’. For instance, the Cantor set is the disjoint union of two smaller copies of itself. More generally, we may have a whole family of objects, each of which can be expressed as the gluing-together of several other objects in the family. Self-similarity is most commonly associated with fractal spaces. However, the notion is not limited to geometry or topology, as the following examples demonstrate:

Integration The bounded linear operator

so : L1[0,1] 1

-

IR is uniquely

General

self-similarity: An overview

233

determined by the equation

so 1

and its value on constant functions. Thus, is described as a combination of two copies of itself. (This concerns the self-similarity of a morphism, rather than an object, of a category. However, the object L1[0,1] does have a closely related universal property in the category of Banach spaces [4].) Type theory Any recursive type definition can be regarded as an example of self-similarity. Consider, for instance, the type T of binary, rooted trees with leaves labelled by objects of type L , where L is the type of lists of such trees:

T=L+T~ L = 1 + T . L. In contexts such as this, 'recursive' may be a more natural term than 'self-similar'. Linear algebra A system of simultaneous equations such as 51

+ .' .+

=mll~l

x, = mn1x1

m1nXn

+ . . . + mnnz,

(mij scalars) is a typical example of self-similarity, although so familiar that it is not usually regarded as such; each member of the family ( X I , .. . , z n ) is expressed as a combination of zjs.

Even within the world of topology, self-similarity is not limited to fractals: many more mundane spaces, such as manifolds, also have self-similar structure, as we shall see. Here I concentrate entirely on set-theoretic and topological selfsimilarity, sketching a theory developed in detail elsewhere [1,2]. It will largely be explained by two examples: a Julia set ($1)and a real interval ($2). Statements of the main theorems can be found in $3. 1. First example: a Julia set

In this section we look at the self-similar structure of a particular fractal space and, led by this example, begin to set up the basic definitions of the general theory.

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T.Leinster

The space concerned is the Julia set of a certain rational function (Figure 1).Since the sole purpose of the example is to motivate the definitions,

2

2 3

1

(b)

(a) Fig. 1. (a) The Julia set 11 of z

H

(22/(1

-

4

3

4

1

(c)

+ z 2 ) ) ' ; (b), (c) subsets 1 2 , 13

we will not need to know what a Julia set is; nevertheless, here is the rough idea. Every holomorphic map f : S S on a Riemann surface S has a Julia set J ( f ) C S ; it is the part of S on which f behaves unstably under iteration. The best-explored case is where S is the Riemann sphere CU{oo} and f is a rational function with complex coefficients. In this case, J ( f ) is a closed subset of C U {oo}, and is almost always fractal in nature. (For the rational function f considered below, I do not know of a proof of the isomorphisms claimed. However, similar examples can be given in which f is a polynomial, and proofs of the analogous isomorphisms are then provided by the theory of laminations [5,6].) A warning: while Julia sets have conformal structure, we will only be interested in our Julia set as an abstract topological space. We will ignore the way in which it is embedded in the complex plane. This theory is about self-similarity as an intrinsic structure on an object: there is no reference to an ambient space, and in general no ambient space at all. This is like doing group theory rather than representation theory, or abstract manifolds rather than manifolds embedded in R". Write 11 for the Julia set in Figure l(a). It has reflectional symmetry in a horizontal axis, so by cutting at the four points shown we obtain a

General selJsimilarity: An overview 235

decomposition

where 1 2 is a certain space with four marked points (Figure l(b)). Now consider Iz. Cutting at the points shown gives a decomposition

where 13 is another space with four marked points (Figure l(c)). Next consider 13;it decomposes as

Since no new spaces are involved, it seems that we can stop there. However, at every step we have glued at a number of isolated points, so the one-point space plays a hidden role. It therefore makes sense to include the one-point space in our family ( I n ) of spaces; let us call it Io. Since it cannot be decomposed, we can do no better than the trivial equation

We now have a system of four simultaneous equations with the unusual feature that the right-hand sides are not ordinary algebraic formulas; instead, they are ‘2-dimensional formulas’ specifying how the spaces are to be glued together. (There is a conceptual link between this and the world of n-categories, where there are 2-dimensional and higher-dimensional morphisms which can be composed or ‘glued’ in various ways. Self-similarity and n-category theory are both a kind of higher-dimensional algebra. The cognoscenti will

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T.Leinster

see a technological link too: in both contexts the gluing can be described by pullback-preserving functors on categories of presheaves.) The simultaneous equations (1)-(4) can be expressed as follows. Firstly, the spaces I l , I2,Is together with their marked points form a functor from the four-object category

I

A=

3 to the category Set of sets (or a category of topological spaces, but let us be conservative for the moment). Secondly, the gluing formulas define a functor

G : [A, Set]

-

[A, Set]

where [A, Set] is the category of functors from A to Set: given X E [A, Set], Put

and so on. (The diagrams are drawn as if X O were a single point.) Then the simultaneous equations assert precisely that I GI: I is a fixed point of G. Although these simultaneous equations have many solutions (G has many fixed points), I appears to be the ‘universal’ solution, in a sense made precise below. This means that the simple diagrams (1)-(4) contain just as much information as the apparently very complex spaces in Figure 1: given the system of equations, we recover the spaces as their universal solution.

General self-similarity: An overview

237

Next we have to find a way of making precise the notion of ‘gluing formula’, expressed in pictures so far. We have a small category A whose objects index the spaces involved, and I claim that the gluing formulas are encapsulated by a functor M : AoP x A Set (a ‘2-sided A-module’). The idea is that for b, a E A,

-

M(b, a) = {copies of the bth space used in the gluing formula for the ath space}. For example, in the gluing formula for 12, the one-point space 10 appears 8 times, 11 does not appear at all, I2 appears twice, and 13 appears once, so

(M(O,2)(= 8,

IM(1,2)1 = 0,

IM(2,2)(= 2,

-

lM(3,2)1 = 1.

Note the distinction between arrows b a in A and elements of M ( b ,a). The arrows of A say nothing whatsoever about the gluing formulas, although they determine where gluing may take place. The elements of M embody the gluing formulas themselves. An alternative explanation is that each gluing formula is a formal colimit of objects of A; in other words, it is an object of [Aop,Set].So the whole system of equations amounts to a functor A [AOP, Set],or equivalently,

-

-

x A Set. A system of simultaneous equations of the type we have been discussing

AoP

is called a ‘self-similarity system’.

-

Definition 1.1. A self-similarity system is a small category A together with a functor M : AoP x A Set satisfying

-

(a) finiteness: for each a E A, the set & b E A A(c, b) x M(b, a) is finite (b) nondegeneracy: for each b E A, the functor M ( b , -) : A Set is nondegenerate (see 52). Part (a) says that in the system of simultaneous equations, each right-hand side is a gluing of a finite family of spaces. So we may have infinitely many spaces (A may have infinitely many objects), but each is described as a finite gluing. The condition is more gracefully expressed in categorical language: for each a, the category of elements of M ( - , a) is finite. Generalizing our example, for any self-similarity system (A, M ) there is an induced endofunctor G of [A, Set]. To see this, first note that if A is a not-necessarily-commutative ring, Y a right A-module, and X a left Amodule, there is a tensor product Y @ A X , a mere abelian group. Similarly, if A is a small category, Y : AoP Set a contravariant functor, and

-

238

X

T. Leinster

:A

-

Set a covariant functor, there is a tensor product

a mere set; see section IX.6 of Mac Lane [7] for details. So any self-similarity system (A, M ) gives rise to an endofunctor G = M @ - of [A, Set], defined by

)/

M ( b ,a ) x X b

(A4@ X ) ( a ) = M ( - , a ) @ X = (bEA

N

( X E [A, Set], a E A). If (A, M ) is regarded as a system of equations then the solutions of the system are the fixed points of G. We seek a solution that is universal; what exactly this should mean is suggested by the next example. 2. Second example: Freyd’s Theorem

The second example is a universal characterization of the real interval [0,1], discovered by Freyd [8] following work of Pavlovi6 and Pratt [9]. (See also Escard6 and Simpson [lo] for development in a different direction.) We will need some terminology. Given a category C and an endofunctor G of C, a G-coalgebra is an object X of C together with a map : X GX. For instance, if C is a category of modules and GX = X @ X then a G-coalgebra is a coalgebra (not necessarily coassociative) in the usual sense. A map ( X , c ) (X’,c’) of coalgebras is a map X X’ in C making the obvious square commute. Depending on what G is, the category of G-coalgebras may or may not have a terminal object, but if it does then it is a fixed point:

-

c

-

-

-

Lemma 2.1 (Lambek [ll]). Let C be a category and G a n endofunctor of C . If ( I ,L ) is terminal in the category of G-coalgebras then L : I GI is a n isomorphism. Here is Freyd’s result, modified slightly. Let C be the category whose 21

X I in which XO and X I are sets and u and

objects are diagrams X O 2)

v are injections with disjoint images; then an object of C can be drawn as


-

239

-

where the copies of XOon the left and the right are the images of u and Y respectively. A map X X’ in C consists of functions Xo Xh and X1 Xi making the evident two squares commute. Now, given X E C we can form a new object GX E C by gluing two copies of X end to end-

-

x 1

-or

formally, by taking a pushout: ((3x11

(GX)o =

xo

XO

XO .

This defines an endofunctor G of C. For example, the unit interval with its endpoints distinguished is an object

of C, and G I is naturally described as an interval of length 2 with its endpoints distinguished: 0

-

So there is a G-coalgebra structure L : I G I on I defined by multiplication by two. Freyd’s Theorem says that this is, in fact, the universal example of a G-coalgebra.

Theorem 2.2 (F’reyd [ S ] ) . (I,L) is terminal in the category of Gcoalgebras. In section 2.1 of Ref. [2], this is derived from a much more general result. A direct proof runs roughly as follows. Take a G-coalgebra (X,l) and an element 5 0 E XI. Then ( ( ~ 0 )E (GX)1 is in either the left-hand or the right-hand copy of XI, so gives rise to a binary digit ml E ( 0 , l ) and a new element z1 E XI. (If E ( z o ) is in the intersection of the two copies of XI, choose left or right arbitrarily.) Iterating, we obtain a binary

240

T.Leinster

-

representation O.mlm2... of an element of [0,1], and this is the image of the unique coalgebra map ( X ,E ) ( I ,6 ) . Note that although L is an isomorphism (as predicted by Lambek’s Lemma), this by no means determines (I,/,):consider, for instance, the unique coalgebra satisfying Xo = X I = 0, or the evident coalgebra in which X O = {*} and X I = [0,1]n {dyadic rationals}. The most striking aspect of Freyd’s Theorem is that it characterizes the real numbers using only some extremely primitive notions. We will generalize it dramatically, using the same primitive notions to characterize other non-trivial objects. Thus, self-similar objects will be realized as terminal coalgebras for different endofunctors. Let us put Freyd’s Theorem into the formalism of the previous section. Define a category A by

20 under

Then C is the full subcategory of [A, Set] consisting of the functors X such that X u and X r are injections with disjoint images. Define M : AoP x A Set as follows:

-

M(-,0) :

0

(Here M ( 0 , l ) is just a 3-element set and M ( 1 , l ) a 2-element set, but their elements have been named suggestively.) Then (A, M ) is a self-similarity system, and it can be checked that the induced endofunctor M @ - of [A, Set] restricts to an endofunctor of C, namely, the functor G above. Freyd’s Theorem will lead us to the appropriate definition of a ‘universal solution’ of an arbitrary self-similarity system. To make this definition we will need to generalize the condition ‘injective with disjoint images’ determining the subcategory C of [A,Set]. It is essential to restrict to C: if we use the whole category [A, Set] then the terminal ( M @ -)-coalgebra J is trivial, Jo = J1 = {*}. It turns out that what we need is a form of flatness. A module X over a ring is called flat if the functor - @ X preserves finite limits. There is an analogous definition when X is a Set-valued functor. Here we require preservation of only finite connected limits:

General self-similarity: An overview 241

Definition 2.3. Let A be a small category. A functor X : A nondegenerate if the functor - @ X : [AOP, Set]

-

-

Set is

Set

preserves finite connected limits. The full subcategory of [A, Set] consisting of the nondegenerate functors is Written [A, Setlnondegen. Equivalently, X is a sum of flat functors, or each connected-component of the category of elements of X is cofiltered. This last formulation is quite explicit: for instance, it can be used to show that if A is the category of Freyd’s Theorem then [A, Setlnondegen = C. In Freyd’s Theorem, the endofunctor M @I - of [&Set] restricts to an endofunctor of [A, SetInondegen. In general, this restriction is possible precisely when for each b E A, the functor M ( b , -) : h Set is nondegenerate; this is condition (b) in the definition of self-similarity system.

-

Definition 2.4. Let (A, M ) be a self-similarity system. A universal soIution of (A, M ) is a terminal coalgebra for the endofunctor A4 @ - of [A,Set]nondegen. Lambek’s Lemma implies that if ( I ,L ) is a universal solution then I M@I, as the word ‘solution’ suggests. Freyd’s Theorem describes the universal solution of a particular self-similarity system. Our self-similar ‘spaces’ have so far been mere sets, but everything we have done can be topologized. This amounts to not much more than replacing functors X : A Set by functors X : A Top, where Top is the category of topological spaces. M remains Set-valued. There is just one subtlety. In Freyd’s Theorem, if we make the change described then the terminal coalgebra ( I ,L ) has I = ({*} [0, l]),as expected, but with the indiscrete topology on [O,1]. To obtain the Euclidean topology, it turns out to be enough to add the condition that the maps XO X1 be closed. (A map u is closed if the direct image under u of any closed set is closed.) So let C’ be the category whose objects are

-

diagrams Xo

-

U

X1 in which X Oand X I are spaces and u and w are closed V

continuous injections with disjoint images. Mimicking the set-theoretic case, define an endofunctor G’ of C’ and a GI-coalgebra ( I ,L ) , with the Euclidean topology on I1 = [ O , l ] .

Theorem 2.5 (Topological F’reyd). ( I ,L ) is terminal in the category of GI-coalgebras.

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T.Leinster

-

For an indirect proof, see section 2.1 of Ref. 121; directly, show that the map X I described just after Theorem 2.2 is continuous with respect to the Euclidean topology on [O, 11. Now let us return to a general self-similarity system (A, M ) . Call a funcTop nondegenerate if its underlying Set-valued functor tor X : A is nondegenerate and X f is a closed map for every map f in A. Then M @defines an endofunctor on the category of nondegenerate functors A Top, and a terminal coalgebra for this endofunctor is called a universal solution of ( A , M ) in Top. So Theorem 2.5 describes the universal solution in Top of a particular self-similarity system. Having considered self-similarity in the categories of sets and topological spaces, we are naturally led to ask what the definitions should be in an arbitrary category. This is not clear, mainly because it is not clear how to generalize the nondegeneracy condition. In the topological case, the only justification for demanding that the maps X f be closed is pragmatic: without some such condition, we do not obtain the Euclidean topology on [0,1]; nor do we obtain the results laid out in the next section.

-

-

3. Results

Here I describe some of the main results of the theory, and some examples. Details can be found in Refs. [l]and [2]. Not every ordinary system of simultaneous equations has a solution; in the same way, not every self-similarity system has a universal solution. However, the existence of a universal solution is governed by a completely explicit condition, which I shall call S. It is defined in the Appendix. Theorem 3.1. Let (A, M ) be a self-similarity system. The following are equivalent:

(a) (A,M ) has a universal solution in Set (b) (A, M ) has a universal solution in Top (c) (A, M ) satisfies condition S .

If these hold, the universal solution in Set is obtained from the universal solution in Top by forgetting the topology. For the proof, see Ref. [l].In the case where M is the hom-functor of A (that is, M ( b , a ) = A ( b , a ) ) , S says that every connected-component of A is cofiltered; then the universal solution ( I ,L ) is trivial, la = {*} for every a E A.

General self-similarity: A n overview

243

There is a completely explicit construction of the universal solution ( I , L ) when , it does exist. It is similar in spirit to the construction of the m

a to mean that m E M ( b , a ) : real numbers as infinite decimals. Write b 4 then an element of I a is an equivalence class of diagrams m3

m2

ml

. .. 4 a2 4 a1 4 ao = a , two diagrams being equivalent if they belong to the same connectedcomponent of the category of such diagrams. This gives the underlying functor I : A Set of the universal solution in Set. Each set I a carries a canonical topology, giving the underlying functor I : A Top of the universal solution in Top. For details, see Ref. [l]. Suppose we have before us a family X of spaces that appear to be selfsimilar, and a self-similarity system (A, M ) for which we believe X to be the universal solution. To confirm our belief, we could in principle go through the construction of the universal solution I and check that I E X . However, a much more efficient method is to use a 'recognition theorem' [2], one of which is stated now. We will need some notation. By Lemma 2.1, if ( X , [ ) is the universal solution of ( A , M ) then E : X M @I X is an isomorphism. Given m E M ( b , a ) , there is a canonical map m 18- : X b ( M @IX ) a , and its Xu. composite with will be written (;l : X b

-

-

-

--

Theorem 3.2 ( C r u d e Recognition [2]). Let (A, M ) be a self-similarity system with A finite. Let ( X , [ ) be an M-coalgebra. Suppose that E is invertible, that each space X u is nonempty and compact, and that each X u can be metriaed in such a way that f o r all m E M ( b , a ) , the map E l 1 is a contraction. Then ( X ,[) is the universal solution of (A,M),

-

For example, Theorem 2.5 now reduces to the facts that [0,1]is compact metric and the halving map [0,2] [0,1] is a contraction. Call a space self-similar if it is homeomorphic to l a for some selfsimilarity system (A, M ) and some a E A, where ( I ,L ) is the universal solution of (A, M ) . Here are some examples of self-similar spaces, all explained in detail in Ref. [2]. (a) [0,1], by Freyd's Theorem. (b) S1, realized as [0,1] with its endpoints identified. (c) [0,1]" for any n E N,and the torus S1 x S1. More generally, the product of two self-similar spaces is self-similar, as is the coproduct. (d) The Cantor set, isomorphic to the disjoint union of two copies of itself.

244 T. Leznster

(e) Sierpiriski’sgasket, and many other spaces defined by iterated function systems. (f) The standard topological n-simplices An, by barycentric subdivision. More specifically, there is a self-similarity system embodying the combinatorial process of barycentric subdivision, and (An) is its universal solution. So the n-simplices can be defined as the universal family of spaces admitting barycentric subdivision. (g) The n-simplices An, by edgewise subdivision. (This is an alternative to barycentric subdivision, dividing An into 2n rather than (n+l)! parts.) The moral here is that a space may be self-similar in more than one way. We have seen that every self-similar space is compact and metrizable (or equivalently, compact Hausdorff with a countable basis of open sets). Perhaps surprisingly, the converse holds:

Theorem 3.3. For topological spaces, self-similar

compact metrizable.

This calls for some explanation. Firstly, the result is non-trivial: the classical result that every nonempty compact metrizable space is a continuous image of the Cantor set can be derived as a corollary. See Ref. [2] for this and for the proof of the theorem itself. Secondly, recall that in a self-similarity system there may be infinitely many equations (even though each individual equation involves only finitely many spaces). So there may be infinite regress: X I might be described as a copy of itself glued to a copy of X z , X z as a copy of itself glued to a copy of X 3 , and so on. This is essential to the proof of Theorem 3.3. Finally, this theorem does not exhaust the subject. It characterizes those spaces that are self-similar in at least one way, but as the example of An shows, the same space may carry multiple self-similar structures. Compare the result that every nonempty set admits at least one group structure, which does not exhaust group theory. There is a restricted version of Theorem 3.3. Call a space discretely self-similar if it is homeomorphic to one of the spaces l a arising from a self-similarity system (A, M ) in which the category A is discrete (has no arrows except for identities). The Cantor set is an example: take A to be the one-object discrete category and A4 to be the functor AoP x A Set whose value is the two-element set.

-


245

Theorem 3.4. For topological spaces, discretely self-similar

totally disconnected compact metrizable.

Totally disconnected compact Hausdorff spaces are the same as profinite spaces, and the metrizable ones are those that can be written as the limit of a countable system of finite discrete spaces. For instance, the underlying space of the absolute Galois group Gal(o/Q) is discretely self-similar. If you find the general notion of self-similarity too inclusive, you may prefer to restrict to finite systems of equations, that is, finite categories A. This gives the notion of finite self-similarity. While there are uncountably many homeomorphism classes of self-similar spaces, there are only countably many homeomorphism classes of finitely self-similar spaces.

Conjecture 3.5. The Julia set J ( f ) of any complex rational function f is finitely self-similar. Certainly J ( f) is compact and metrizable, that is, self-similar; the question is whether it is finitely so. In our first example, for instance, we used a finite category A with four objects. The conjecture says that we could have taken any rational function f and seen the same type of behaviour: after a finite number of decompositions, no more new spaces I , appear.

Acknowledgements

I thank those who have given me the opportunity to speak on this: Jifi AdAmek, Francis Borceux, Robin Chapman, Eugenia Cheng, Sjoerd Crans, Iain Gordon, John Greenlees, Jesper Grodal, Jiirgen Koslowski, Peter May, Carlos Simpson, Bertrand Toen, and Jon Woolf. I am very grateful to Jon Nimmo for creating Figure l(a), and to Mary Rees for useful discussions. This work was partially supported by a Nuffield Foundation Award NUF-NAL 04.

Appendix A. Criterion for existence of a universal solution Condition S on a self-similarity system (A, M ) , referred to in Theorem 3.1, is that:

246

T. Leinster

S1 given any commutative diagram

... 4 a; mi

4 a: m2

f;’ a;, ml

in (A, M), there exists a commutative square a0

in A, and S 2 given any serially commutative diagram

in (A, M ) , there exists a fork c

-T h

a0

fo

bo in A.

Commutative and serially commutative mean that

(M(1if n - l ) ) ( m n )= ( M ( f n ,1))(~n)i and similarly for f;, for all n > 0. Fork means that foh = f&. References 1. Tom Leinster, A general theory of self-similarity I, math.DS/0411344 at arxiv .org (2004). 2. Tom Leinster, A general theory of self-similarity 11: recognition, math. DS/ 0411345 at arxiv.org (2004). 3. John Milnor, Dynamics in One Complex Variable,Vieweg, 1999. 4. Tom Leinster, A universal property of the integrable functions, in preparation.

General self-simalapa'ty:An overuiew

247

5 . William Thurston, On the combinatorics of iterated rational maps, unpub-

lished (1985). 6. Jan Kiwi, Rational laminations of complex polynomials, in M. Lyubich et al. (eds.), Laminations and Foliations in Dynamics, Geometry and Topology (Proceedings of the Conference o n Laminations and Foliations i n Dynamics, Geometry and Topology, May 18-24, 1998, SUNY at Stony Brook), Contemporary Mathematics 269, American Mathematical Society, 2001. 7. Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, 1971. 8. Peter F'reyd, Real coalgebra, post on categories mailing list, 22 December 1999, available via www .mta. ca/-cat-dist 9. Dusko PavloviE, Vaughan Pratt, The continuum as a final coalgebra, Theoretical Computer Science 280 (2002), 105-122. 10. Martin Escard6, Alex Simpson, A universal characterization of the closed Euclidean interval (extended abstract), in Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, IEEE, 2001. 11. Joachim Lambek, A fixpoint theorem for complete categories, Mathematische Zeitschrift 103 (1968), 151-161.

248

Generalized Plucker-Teissier-Kleimanformulas for varieties with arbitrary dual defect Yutaka Matsui Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan, E-mail: you31 7arns.u-tokyo.ac.jp Kiyoshi Takeuchi Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki, 305-8571, Japan, E-mail: [email protected] By reformulating a result of Ernstrom [6] in terms of Chow groups, we obtain a class formula, i.e. a formula for the degrees of dual varieties, which is applicable also for projective varieties with positive dual defect. Various generalized Plucker-Teissier-Kleiman type formulas will be obtained. By our approach, topological invariants such as Milnor numbers along the singular locus of the original projective variety naturally appear in the class formulas. Keywords: Dual varieties; Euler obstructions; Milnor numbers; ChernSchwartz-MacPherson classes.

1. Introduction

Projective duality is a classical but active area in algebraic geometry. Various mysterious relations between projective varieties and their duals were studied by many famous mathematicians of each generation. In the 19th century, Plucker and Clebsch discovered the following remarkable result for plane curves over C.

Theorem 1.1 (Plucker-Clebsch). Let C be a plane curve of degree d with 6 ordinary double points and K. cusps. T h e n for the dual curve C* of C we have degC* = d(d - 1) - 26 - 3n.

(1)

Namely, if C is a smooth plane curve of degree d, we have deg C* = d ( d - 1). But if C has singular points, the above formula for degC* contains some

Generalized Plucker- ~eiSSieT-Xleima?lformulas f o r varieties 249

correction terms coming from the singularities of C. In the 20th century, the higher-dimensional analogue of this theorem was studied by Teissier [32], Kleiman [15,16], Piene [26,27] and Thorup [34] etc. However, t o the best of our knowledge, it seems that there is no satisfactory computable class formula for general singular projective varieties yet. The aim of this paper is to propose a new approach based on the recent development of the theory of Chern-Schwartz-MacPhersonclasses by Yokura [36], Parusinski-Pragacz [25], Schurmann [29] and Allufi [l]etc. In order to explain our motivation more precisely, let us now recall basic notions on dual varieties. Let PN be a projective space of dimension N over C. Denote by Pk the dual projective space. Then IPk is naturally identified with the set of hyperplanes H c PN in PN.For a projective variety X c PN, we define its dual X* c Pk by

X* = { H E P> 1 % E Xreg such that TxX c TxH}.

(2)

Since the degree of the dual variety X* is called the class of X, the formulas for the degrees of dual varieties obtained by the above mentioned authors are called class formulas (or Plucker formulas). Note also that the dual variety X* is a hypersurface in Pk unless X is a ruled variety. For this reason, we call the number 6*(X) := codimX* - 1

(3)

the dual defect of X. Namely in general the dual defect 6*(X)of a projective variety X is zero, but there still exist many projective varieties with positive dual defect which are duals of higher-codimensional subvarieties in P>. Since most of previous approaches to class formulas are based on the study of Segre classes of Jacobian ideals (as explained in Section 9.3 of [8]), known class formulas are not applicable to the case where 6*(X) > 0 nor to the case where the singularity of X is complicated or higher-dimensional (for example, see Example 4.2). For example, under the condition 6*(X) = 0, Teissier [31,32] and Kleiman [16] obtained a generalization of Theorem 1.1 to higher-dimensional projective varieties with only isolated singular points. Moreover Piene [26,27] studied the case of algebraic surfaces very precisely. Assuming also the condition 6 * ( X )= 0, Thorup [34] treats the most general case, but to apply his result to concrete cases we need to compute further the Buchsbaum-Rim multiplicities in his formula. Our starting point is the following purely topological class formula obtained by Ernstrom [6].

250

Y.Matsui and K. Xakeuchi

Theorem 1.2 (Ernstrom [ 6 ] ) . Let X be a projective variety of dimension n in PN and set r = codimX* = N - dimX*. Then we have

where Li is a generic linear subspace of codimension i in PN.

L(.)

(for the definition of the constructible function E u x and topological integrals

etc. see Section 2). In [22], we proposed a new proof of this

theorem which does not use Segre classes nor polar varieties. We used instead a result of [5], [21] and some elementary lemmas on constructible functions. In Section 3, we rewrite Theorem 1.2 in terms of Chern-Mather classes and obtain a universal class formula which naturally extends the DeligneKatz formula for smooth varieties X satisfying S*(X) = 0, see [4]. With this formula and the localization theorem (Theorem 2.10) proved by ParusinskiPragacz [25], Schiirmann [29] and Yokura [36] at hands, we can now explicitly write down the class degX* for various new cases. In fact, in Section 4 we give an algorithm which enables us to calculate degX* as long as all the subvarieties W c X appearing in each step of it are locally complete intersection and the Chern classes of the normal bundles T ~ P N are computable. In Section 5, we prove some explicit class formulas by applying this algorithm. By our method, topological invariants such as (slice) Milnor numbers along the singular locus of X (see Definition 2.5) naturally appear in the class formulas. Compare this with the fact that Kleiman [16] required some deep results by Gaffney [9] to rewrite his generalized Plucker-Teissier formula in terms of Milnor numbers. It would be an interesting problem to compare our topological results with the algebraic ones obtained by Kleiman [16] and Thorup [34] etc. Finally, let us mention that our approach is also useful to compute the degrees of the k-th associated varieties studied by Brylinski [3] and GelfandKapranov-Zelevinsky [lo] etc. For the detail, see Section 6.

2. Preliminary notions and results

In this section, we briefly recall some recent results in the theory of characteristic classes for singular varieties.

Generalized Plucker- Teissier-Kleiman formulas for varieties

251

2.1. Euler obstructions

-

Definition 2.1. Let X be an algebraic variety over C. We say that a Zvalued function cp: X Z on X is constructible if there exists a stratX , of X such that cp(x, is constant on any stratum X,. ification X =

u

,€A

-

We denote by C F ( X )the abelian group of constructible functions on X . If X is complete, then for any constructible function cp: X Z there exists a finite family of subvarieties Zi c X and ai E Z (i E I ) such that cp = ailz,. Here we denote by 12, the characteristic function of Zi. Let

c iEI

-

us recall basic operations on constructible functions.

Definition 2.2. Let f : X

Y be a morphism of algebraic varieties.

(i) Assume that X is complete. Then for a constructible function cp = ailzi E C F ( X ) on X we define the Euler (topological) integral of

c iEI

cp by

-

where x ( Z i ) is the topological Euler characteristic of Zi. (ii) Let cp E C F ( X ) and assume that f : X Y is proper. We define the direct image

J, cp

E C F ( Y )of cp by

(iii) Let $J E C F ( Y ) .We define the inverse image f*$ E C F ( X ) of

$J

by

We can easily check that the above definition (i) does not depend on the expression cp = ailz, of cp by using the fact that the groups of coniEI

structible functions C F ( X )and C F ( Y )are isomorphic to the Grothendieck groups of the derived categories of constructible sheaves on X and ,Yrespectively (see e.g. Theorem 9.7.1 of [13] etc.). Let X be an algebraic variety over C. Then the Euler obstruction Eux E C F ( X ) of X , introduced by Kashiwara [ll]and MacPherson [17] independently, is a constructible function on X satisfying the following properties.

252

Y.Matsui and K. Takeuchi

(i) On the smooth part Xregof X , the function E u x takes the constant value 1. X , be a Whitney stratification of X consisting of con(ii) Let X =

u

,€A

nected strata. Then E u x is constant on any stratum X,. There are some algorithms t o compute the Euler obstruction E u x E C F ( X ) . For the detail, see for example [2,11], and [12] etc. In particular, we have the following result.

Proposition 2.3 (Kashiwara [12]). Let X = {f = 0 } be a n algebraic hypersurface in CN with a n isolated singular point at 0 E X defined by a function f : CN -+ C . Then we have

Eux(0) = 1

+

(8)

where & denotes the Milnor number of a generic hyperplane section X nH at 0 E X n H by a hyperplane H c CN passing through the origin 0. In order to state a slight generalization of Proposition 2.3, we need some definitions. Let Y be a smooth algebraic variety over C and X c Y an algebraic hypersurface in Y . Assume that the singular part W = Xsing of X is connected and smooth and X = Xre,UW is a Whitney stratification of

x.

Definition 2.4. (i) Let p E W be a point. We say that a germ ( N ,p ) of a smooth subvariety N of Y at p is a normal slicing subvariety germ for X at p E W if dimW+dimN = dimY and N intersects with Xregand W transversally. (ii) For a normal slicing subvariety germ ( N ,p ) for X at p , the hypersurface X n N in N with an isolated singular point at p E W n N is called a normal slice of X at p .

Definition 2.5. In the situation as above, choose a normal slice X n N of X at a point p E W . (i) The Milnor number of the hypersurface X n N of N at p is called the Milnor number of X along W and denoted by p ( X , W ) . (ii) For a generic smooth hypersurface H of N passing through p , the Milnor number of the hypersurface X n H of H at p is called the slice Milnor number of X along W and denoted by p’(X, W ) .


253

If we denote by Fj,p the local Milnor fiber of the hypersurface X = {f = 0 ) of Y at p E W , then obviously we have

By Proposition 2.3 (and the remark just after Theorem 3.1 of [2]), we immediately obtain the following. Proposition 2.6 (Brasselet-Lb-Seade [2] and Kashiwara [12]). I n the situation as above, for any p E W we have Eux(p) = 1 + (-l)dimY-dimW P '( X ,W )

(10)

2.2. Chem-Schwartt-MacPherson classes Denote by V a r c (resp. d b ) the category of complete algebraic varieties over C (resp. abelian category of abelian groups). Then by Definition 2.2 (ii) we obtain a covariant functor

C F : Var,

-

db.

(11)

On the other hand, we have a natural covariant functor

which assigns to a complete variety X E Varc its (total) Chow group C H , ( X ) = @ C H i ( X ) E d b (for the detail, see [8] etc.). In the 1960's,

-

i€Z

Deligne and Grothendieck predicted the existence of a natural transformaC H , satisfying very natural properties. This conjecture was tion C F solved by MacPherson [17] as follows.

-

Theorem 2.7 (MacPherson [17]). There exists a natural transformation c* : C F C H , such that for any smooth complete variety X E V a r c we have c , ( l x ) = c * ( T X ) n [ X ]Here . c * ( T X )is the total Chern class ofthe tangent bundle T X of X (defined in the Chow cohomology class C H * ( X ) of X ) and [XI E CHdimX(X) is the fundamental class of X (see [a] etc. for the definition).

-

For the proof of this fundamental theorem, see [14,17], and [28] etc. In this paper, we call c* : C F C H , the MacPherson Chern class transformation.

254

Y. Matsua and K. Takeuchi

Definition 2.8. Let X be a complete variety over C. (i) We set

c f M ( X )= c,(Eux) E C H , ( X ) and call it the Chern-Mather class of X . (ii) We set

c * ( X )= c * ( l x ) E C H * ( X )

(14)

and call it the Chern-Schwartz-MacPherson class of X By a result of Sabbah [28], when X is a subvariety of a smooth algebraic variety Y ,the Chern-Mather class cF'(X) of X can be defined by using the closure TgregYof the conormal bundle TgregYc T*Y of Xregin Y (see also [14] and the proof of Lemma 3.2). In general, it is not so easy to calculate the Chern-Mather class c f M ( X ) .However thanks to the recent progress by Yokura [36], Parusinski-Pragacz [25], Schiirmann [29] and Allufi [l]etc., we have now a very powerful method (the localization theorem below) to compute the Chern-Schwartz-MacPherson class c * ( X ) = c , ( l x ) . In order to explain this result, we need the following definition. Let Y be an algebraic variety over C and X = {f = 0) a algebraic hypersurface in Y (f is the defining function of X ) .

-

Definition 2.9. We define a group homomorphism @f:

CF(Y)

CF(X)

(15)

by setting

Qj(V)(P>= Cai{X(FrlZ,,p)- I Z ~ ( P ) ) ,

(16)

iEI

a i l z , E CF(Y)( I is a finite set, ai E Z and Zi's are subvarieties

for 'p = iEI

of Y ) and p E X , where Fflzi,p is a local Milnor fiber of the function flz,: Zi -+ C at p E X = {f = 0 ) .

-

We can easily check that this definition does not depend on the expresa i l z i of 'p. Indeed, the group homomorphism Qpf : CF(Y) sion cp = iEI

C F ( X ) corresponds to Deligne's vanishing cycle functor for constructible sheaves. Note also that by the recent progress in the theory of Milnor fibers over singular varieties due to Massey [18,19]etc. (see [30] for a brief review on this subject), we have now an efficient method to compute the number

Generalized Plucker- Teissier-Kleiman formulas

for

varieties

255

-

~ ( F f l ~Now ~ , we ~ )are . ready to state the following localization theorem for Chern-Schwartz-MacPherson classes. Since i: X = { f = 0 ) Y is a regular embedding, the normal cone CxY is the line bundle TxY on X (see [8] for the detail). Theorem 2.10 (Yokura [36]-Parusinski-Pragacz [25]-Schiirmann [29]). In the situation as above, f o r any cp E C F ( Y ) we have C*

-

(i* 9) = C* (TXY)-' { i*c*(p) - C* (@f (cp)) } ,

(17)

where i*: C H , ( Y ) C H , - l ( X ) is the Gysin map associated with the regular embedding i : X L) Y (see Chapter 6 of 181). I n particular, if Y is smooth and cp = l y we have i*c*(cp)= i*{c*(TY) n [ Y ]= } c * ( X x Y T Y ) n [XI

(18)

and hence

c * ( X )= c , ( l x ) = c*(TXY)-'{c*(X x Y T Y )n [XI - ~ , ( @ ~ ( c p ) ) } .

(19)

3. Universal class formulas for dual varieties

Let X be a (possibly singular) projective variety of dimension n in Y = and X = X , a Whitney stratification of X . Then for a generic

BN

u

LYEA

hyperplane H in Y = we see that H intersects with X , transversally for any (Y E A. Let us fix such a hyperplane H . For a conic subvariety W of the cotangent bundle T*Y (resp. T * H ) ,we denote by B[W] its projectivization in the projectivized cotangent bundle P*Y (resp. P * H).

Lemma 3.1. There exists a natural isomorphism

where TiregY(resp. Ti;rnH, H ) is the conormal bundle of the regular part Xreg o f x in Y (resp. ( X nr3)rego f x n H in H ) .

-

Proof. Note that there exists a natural smooth morphism H x T*Y

T * H associated with the embedding H the commutative diagram

HxT*Y Y

Y

-

Y . Then the result follows from

T*'H .

0

256

Y. Matsui and K. Takeuchi

-

Let i : X n H ~f X be the embedding and i, : CH,(X n H ) C H , ( X ) the push-forward morphism of Chow groups associated with i (in the sequel we sometimes omit the symbol i, when i is a closed embedding).

Lemma 3.2. In C H , ( X ) , we have Z,C:~(X

h

n H ) = -C M ( X ) 1

+ hC*

where h E C H 1 ( X ) is the hyperplane class. Proof. By Lemma 3.1 we obtain a Cartesian square

0

X n H

X.

Considering H as an ambient space of X n H , we calculate the class i,cFM(X n H ) E C H , ( X ) as follows by Sabbah's construction of ChernMather classes (see [14]and [28] etc.).

Z,cfM(X n H ) dimH-dim(XnH)- 1

N--12-1-

= (-1)

1

1 +

hC*(TYlx)

c*(TH(xnH)

Generalized Plucker- Teissier- Kleirnan formulas for varieties 257

= (-1)

--

1

N-n-1

----*(TYIx) l+h

nc l ( O x ( l ) )

+ hC*C M ( X ) .

Here we used Lemma 3.1 in the forth equality.

0

Remark 3.3. As is stated in [23],Lemma 3.2 can be deduced also from Proposition 1.3 of [24] or Theorem 2.10 above (the most general version in [29]).Since Lemma 3.2 can be proved more directly and elementarily, we gave a short proof here for the reader's convenience. By Lemma 3.2, we can rewrite Ernstrom's topological class formula (Theorem 1.2) in terms of Chern-Mather classes as follows.

Theorem 3.4. Let X be a projective variety of dimension n in Y = P N . Set r = codimX* = N - dimX*. T h e n we have

where h is the hyperplane class. In particular, if X * i s a hypersurface in Pfv (i.e. r = l), we have

Proof. By the MacPherson Chern class transformation (Theorem 2.7) and Lemma 3.2, for a generic linear subspace Li c Y = P N of codimension i , we have the following relations.

258


Hence by Theorem 1.2 we obtain deg X * = (-l)n+r+l{

= (-l)n+r+l

S, S,{ r

Eux

r

-

-

(r+ 1)

ll + 1 Eux

Eux}

(33)

L + 1

(r

+ 1)-l + h + ( l y+ h+

l

}

C F M ( X ) (34)

Note that in the special case where X is smooth and X * is a hypersurface (i.e. r = l ) ,we hence reobtain the famous Deligne-Katz formula [4]:

S,&

degX* = ( - l ) n (see also Example 3.2.21 of [8]).

4. An algorithm for computing the degrees of dual varieties By the following algorithm, in many cases we can rewrite our class formula (29) in Theorem 3.4 more explicitly.

Algorithm 4.1. Step.1 By using closed subvarieties X j of X , we write the Euler obstruction of X as Eux = l x

+xajlxi

( a j E Z).

(37)

j€J

Step.2 Applying the MacPherson Chern class transformation c,(.) t o (37), we rewrite the Chern-Mather class c f M ( X ) b y Chern-SchwartzMacPherson classes as follows.

+

C y y X ) = c*(X) x a j .C*(Xj).

(38)

j€J

Step.3 By the localization theorem (Theorem 9-10), if X and X j 's are complete intersections in Y = PN,we can reduce the calculation of c * ( X ) (resp. c * ( X j ) ) to that of c * ( p ) for constructible functions p's on X (resp. o n X j ) such that suppp c Xsing (resp. suppp c (Xj)sing).For example, when X is a hypersurface in

-

Generalized Pllcker- Teissier-Kleiman formulas f o r varieties

259

Y = PN defined by f = 0 and i : X Y = PN is the regular embedding to the ambient space Y , we have c*(X) = c*(i*ly)

(39) = C * ( T ~ Y )n - ~{c*(xx Y TY) n [X]- C*(af(ly))}. (40)

Note that Supp(@f(Iy))c Xsing. By this algorithm, we can calculate the degrees of dual varieties for many singular varieties. Let us give an example.

Example 4.2. Consider a 3-dimensional hypersurface X in Y = P4 defined by

x =(2=

(20

: 21 : 2 2 : 2 3 : 2 4 ) E Ifp4

I f ( 2 ) = 2g21 + 2 0 2 2 x 4 + 232;

= 0).

(41) This is a ruled hypersurface and the codimension T of the dual variety X* in Ifp; is two (see p.78 of [7]). Namely the dual defect S*(X) = r - 1 of X is one in this case. It is easy to see that

Xsing= ( 2 = (ZO : 21 : 2 2 : 23 : 2 4 ) E B 4 If we define a conic C in the projective plane

c = (2' = (21 : 2 2 : 23) E Xsing

N

I 20 = 2 4 = 0) N P2. Xsing N

Pz I

2;

(42)

Pz by

- 42123 = o),

(43)

then the Euler obstruction E u x and the vanishing cycle @ f ( l y ) can be calculated as follows.

+ 2 . l x s i n g \ c+ 1 . IC= 1~ + 1xsing- Ic, lXsing\C+ 1 lc = -lxs,ng+ 2 1 c .

EUX= 1 * Ix @f(lY)=

-1.

*

(44)

(45)

Then we obtain CfM(X)

(46) (47)

= c*(Eux)

+

= ~ ( x )C*(Xsing) - G(C) = c*(TXY)-l n (c*(x x Y

T Y ) n [XI- c,(@f(iy)))

+c* (Pz) - c*( C ) ( l + h)3 --(' + h)5 n [XI (2 3 h ) 3h p,] - (3 3 h ) ( l + h)3 1 3h (1 3h)(l 2h) Therefore we have

+

+ +

+

+ +

+

(48)

" PI.

(49)

1

degX* = (-1)4

= 3.

(50)


260

Note that this result coincides with the one obtained from the explicit description of the dual X* given in [7].

5. Various class formulas In this section, by Theorem 3.4 and Algorithm 4.1 we shall derive various Plucker-Teissier-Kleiman type formulas. In what follows, we shall use a convention on binomial coefficients: = 0 unless a 2 b 2 0. Let X c PN be a hypersurface in Y = PN with degree d. We set r = codimX* = b * ( X ) 1.

(3

+

Definition 5.1. (i) For Z = ( e l , . . . , el) E ZL, - and k E Z, we set

Iml=k, mEZl>o -

j=1 1

(d - l)mO n(ej

7rk(Z) : = e l . . . e l . Iml=k, mEZ;+;

~

l)mj, (52)

j=1

where Z>O - denotes the set of non-negative integers. Note that by definition n k ( E J = nk(EJ= 0 for k < 0. (ii) For two integers 1 and k satisfying 1 5 1 5 N and k 5 N - 1, we set

where M(1) := N - T - 1

+ 1.

Note that a notation similar to nk(E') was introduced by Kleiman [16].

Theorem 5.2. In the situation as above, for the decomposition Xsing = X, of Xsinginto connected components, assume the following condi-

u

,€A

tions. (i) X , is a smooth complete intersection subvariety of codimension 1, in + PN with multi-degree d, = (d,,l,. . . ,d,,lo,),


(ii)

x = xreg u

u

261

X, is a mitney stratification of X.

,€A

Then we have degX* = d(d - l)N-Tdx where the correction term R, is defined by

for p, = p(X, X,), p& = p'(X, Xa) and M , = M(Z,) = N - T - 1,

+ 1.

Proof. First note that the assumption (ii) implies that the Euler obstruction Eux is constant along each stratum X,. Therefore by Proposition 2.6, we have

By (9) and (19), we have

Hence by Theorem 3.4 it is sufficient to calculate the following three terms.

262


,

r-1

X

(1

+

+

(1 h)N+l d l h) * * . (1 d l h ) n [Xal,

+

E (r +j l ) (r - j ) s, (1

r-1

3'

= (-l)N+r+z

(60)

hj h)r+1

+

j=O

+

(1 h)N+l (61) X ( l + d h ) ( l + d i h ) * * . ( l + d l h ) n [Xal, where h is the hyperplane class and we write di (resp. 1 ) instead of da,i (resp. l a ) . We can calculate the term S1 as follows. r-1 N - r j=O

k=O

00

r+l

N-r

i=O

r-1 N - r

C

r+l 2) ( j + 1 )

= ( - l ) N + r dj=O C k=O ( j

+

("I,') (-d)N-l--r+l+j-k

The term S2 is calculated as follows. Set M = N - r - 1 have

(64)

+ 1. Then we

Generalized Plucker- Teissaer-Kleiman formulas far varieties 263

= (-l)N+r+ldl..

.d1

T-1

j=max{O,-M}

where we used the following lemma (Lemma 5.3) in the fifth equality. We can calculate the term Ss in the same way.

0

Lemma 5.3. Let p , q be non-negative integers and s a positive integer satisfying the condition t :=p s - 1 - q 2 0. Then we have

+

This lemma can be easily proved by induction on t. Note that the correction term R, in Theorem 5.2 is expressed only by S * ( X ) = r - 1 , the (slice) Milnor numbers and the multi-degrees of the singular locus of X. By Theorem 5.2, we obtain Plucker-Teissier-Kleimantype formulas in various special cases as follows.

264


Corollary 5.4. (a) In the situation of Theorem 5.2, assume that X* is hypersurface. Then we have deg X* = d(d - l)N-l -

c{

luol(nL,

+ nX-1-1,

+

+

(doll) + P&XN--l,&)}

>

(73)

olEA

where we set pa = p(X, Xa) and p& = $(X, Xe), (ii) Assume that X is a hypersurface of degree d with only isolated singular points Xsing = { P I ,* . . ,pq}. Then we have degX* = d(d - l)N-Tdx q

+

(74)

+(-1ITr C(Pi Pi),

i= 1

where r = codimX* = 6*(X)

+ 1 and

we set p i = p(X, {pi}) and

PI = ~ ’ ( {xp i,} ) . (iii) (Teissier [32] (for the prooi see [15] and [16])) Let X be a hypersurface of degree d with only isolated singular points Xsing= { p l , . . . , p q } . Assume that X* is a hypersurface. Then we have 9

degX* = d(d - l)N-l - c ( p i i= 1

+ pi),

(75)

where we set p i = p(X, { p i } ) and p: = p’(X, { p i } ) . Remark 5.5. Even when X is a complete intersection subvariety of PN,we can obtain formulas for the degree of X* by using Algorithm 4.1 repeatedly. See the last part of Section 6 below. 6. Generalizations to k-dual varieties

We define the Grassmann manifold G N ,(1 ~ I k 5 N - 1) by

G N ,=~ {L’

C

CN+l I L’ is a (k

+ 1)-dimensional linear subspace in C N + l } .

(76) For a (k 1)-dimensional vector space L’ E G N , ~we, denote its projectivization by L. This L is a k-dimensional linear projective subspace of

+

Generalized Pliicker- Teissier-Klezman formulas for varieties

265

PN = G N , ~In. the sequel, we identify the set of k-dimensional linear subspaces of PN with ( 6 N . k . Definition 6.1. Let X be a subvariety of the N-dimensional projective space PN. We define the k-dual variety X ( k ) C G N , k of X to be the closure of the set of all k-dimensional linear subspaces L E G.;rN,ksuch that there exists a point x E Xreg n L at which L does not intersect with X t r ansversally. In the case where k = N - 1, the k-dual X(k) C G N , k pv P> is the classical dual variety of X . Gelfand-Kapranov-Zelevinsky [lo] call X ( k )the k-th associated variety of X. Let X c PN be a projective variety of dimension n. Assume that X ( k ) is a hypersurface in G N , k .

Definition 6.2 (Gelfand-Kapranov-Zelevinsky[lo]). Consider Plucker embedding:

W e call the degree of the defining polynomial of X(k)in P

N+I (k+l>-l

the

the degree

of X ( k ) and denote it by deg X ( k ) . We proved the following topological class formula for k-dual varieties PI.

Theorem 6.3 (Matsui-Takeuchi [22]). I n the situation as above, for generic linear subspaces L N - k + l ? Pk-1, L N - k P k , LN-k-1 21 Pk+l of PN we have deg X ( k ) - (-q(N-”+n+l

{I

Eux-21

LN-k+l

LN-k

Eux+/

Eux}.(78) LN-k-1

We can rewrite this formula in terms of Chern-Mather classes in the same way as Theorem 3.4.

Theorem 6.4. I n the situation as above, for a generic linear subspace L 21 Pk+l of PN we have

where h is the hyperplane class.

266

Y . Matsui and K . Takeucha

From Theorem 6.4, we can derive various more explicit class formulas by Algorithm 4.1. For example, we obtain the following theorem as in the same way as Theorem 5.2. Theorem 6.5. Let X is a hypersurface of degree d in PN such that X ( k )is a hypersurface in G;rN,k.For a generic linear subspace L I I & + I , we assume that the decomposition ( X n L)sing = W, of ( X n L)sing into connected

u

,€A

components satisfies the following conditions:

(i) W, is a smooth complete intersection subvariety of codimension 1, in -+ L 21 pk+l with multi-degree d, = (d,,l,. . . d,,ia), (ii) (X n L)= ( X n L)regu W, is a m i t n e y stratification of x n L.

u

,€A

Then we have degX(k) = d(d

- l)k

-

{p~a(rk+i-1, ( d a ) -k rk-ia

-

+

-k pu&rk+l-l, ( d o ) } 7

(80)

,€A

where we set pa = p ( X n L , W,) and p& = p’(X n L , W,). More generally, by Theorem 6.4and Algorithm 4.1, we obtain an explicit formula for deg X ( k )also in the case where X is a complete intersection in P N as fo1h’s. intersection subLet X := { f l = . . . = f i = 0 ) c PN be a complete -+ variety of codimension 1 in PN with multi-degree d = ( d l , . . . ,di). For simplicity, we assume that 2 := ( f 1 = . = f i - 1 = 0 ) is smooth. For a generic h e a r subspace L N pk+l of PN,we may assume that 2 nL is smooth and X nL is a hypersurface in 2 nL defined by fi I ( z n ~= ) 0. Moreover we assume that the decomposition ( X n L)sing= W, of

u

ffEA

( X n L)singinto connected components satisfies the following conditions: (i) W, is a smooth complete intersection subvariety of codimension 1, in -+ L with multi-degree d, = (d,,ll * ,d,,ia), (ii) ( X n L) = ( X n L)regU W, is a Whitney stratification of X n L.

u

,€A

Then regarding X n L as a hypersurface in the smooth projective variety 2 n L , we can define the Milnor numbers p, = p ( X n L, W,) and pu& =

Generalized Plucker- Teissier-Kleiman formulas for varieties 267

PYX n L , K).

Theorem 6.6. In the situation as above, if X(k)is a hypersurface in GN,k, then we have deg X (k) = rk+l-l(d

where we set r;(d:)

= &,I

C

.*.&,la

J m J = k ,mEZ'-+' 20

1,

(dl - l)mo n ( d a , j - 1)"j. j=1

(82)

Corollary 6.7. I n the situation as above, assume that X(k) is a hypersurface in GN,k and x n L has only isolated singular points (X n L)sing = { P I ,. . . , p 4 ) . Then we have Q

d % x ( k )= r k + l - l ( d - c ( p i i=l

+pi),

(83)

where we set pi = p ( X n L , { p i } ) and pi = p'(X n L , {pi)). In the special case k = N - 1, by our formula (83) we thus reobtain the Plucker-Teissier-Kleiman formula [16].

Remark 6.8. As is clear from Algorithm 4.1 and our arguments above, the assumption that 2 = { f1 = = fl-1 = 0) is smooth can be removed by using iterated vanishing cycles of constructible functions. Indeed, to treat the case where 2 is not smooth, we need Schurmann's generalization [29] of the localization theorem of Parusinski-Pragacz [25]. For example, let X = { f l = f 2 = 0 ) c PN be a complete intersection subvariety of codimension 2 in Y = PN with multi-degree (d1,dz). Set 2 := { f l = 0 ) and let i : X 2 and j : 2 ~f Y be the embeddings. We set f := f l and g := fzlz for the sake of simplicity. Then by using Schurmann's theorem (Theorem 2.10) repeatedly we obtain

-

c*(X) = c*(i*lz) = C * ( T ~ Z )n- {i*c,(iz) ~ - C*(Gg(iz))} = c * ( T ~ Y ) -n~{c*(x xy

-c*(Txz)-~

(84)

(85) TY) n [ X I- i*c*(af(iy)))

n c*((ag(iz))

(86)

268


where h is the hyperplane class. Note that by the recent results of Massey [18,19]etc. we can now express the iterated vanishing cycle (a,(@f(ly))E C F ( X ) in terms of intersection numbers of polar varieties, i.e. t h e generalized LB numbers.

References 1. P. Allufi, Computing characteristic classes of projective schemes, J. Symbolic Compt. 35,no.1, 3-19 (2003). 2. J.P. Brasselet, D.-T. L6 and J. Seade, Euler obstruction and indices of vector fields, Topology 39,no.6, 1193-1208 (2000). 3. J.L. Brylinski, Transformations canoniques, dualit6 projective, th6orie de Lefschetz, transformations de Fourier et sommes trigonom6triques. GBometrie et analyse microlocales, AstLrisque 140-141,3-134 (1986). 4. P. Deligne and N. Katz, Groupes de monodromie en g6om6trie alggbrique, SGA 7 11, 1967-1969, Lecture Notes in Math. 340, Springer-Verlag, Berlin (1973). 5. L. Ernstrom, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76,1-21 (1994). 6. L. Ernstrom, A Plucker formula for singular projective varieties, Communications in algebra 25,2897-2901 (1997). 7. G. Fischer and J. Piontkowski, Ruled varieties, Vieweg, (2001). 8. W. Fulton, Intersection theory, Springer, (1984). 9. T. Gaffney, Multiplicities and equisingularity of ICIS germs, Invent. Math. 123,no.2, 209-220 (1996). 10. I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhauser, (1994). 11. M. Kashiwara, Index theorem for maximally overdetermined systems of linear differential equations, Proc. Japan Acad. 49,803-804 (1973). 12. M. Kashiwara, Systems of microdifferential equations, Progress in Mathematics 34, Birkhauser, Boston, (1983). 13. M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss. 292, Springer-Verlag, Berlin-Heidelberg-New York, (1990).

Generalized Pkicker- Teissier-Kleiman formulas for varieties 269 14. G. Kennedy, MacPherson’s Chern classes of singular algebraic varieties, Comm. Algebra 18,no. 9, 2821-2839 (1990). 15. S. L. Kleiman, The enumerative theory of singularities, Real and complex singularities, Sijthoff and Nordhoff international Publishers, Alphen an den Rijn, 297-396 (1977). 16. S. L. Kleiman, A generalized Teissier-Plucker formula, Contemp. Math. 162, 249-260 (1994). 17. R. MacPherson, Chern classes for singular varieties, Ann. of Math. 100,423432 (1974). 18. D. Massey, Hypercohomology of Milnor fibres, Topology 35,no.4, 969-1003 (1996). 19. D. Massey, Numerical control over complex analytic singularities, Memoirs of the AMS 163, Nr. 778, (2003). 20. Y. Matsui, Radon transforms of constructible functions on Grassmann manifolds, Publ. Res. Inst. Math. Sci. 42, 551-580 (2006). 21. Y. Matsui and K. Takeuchi, Microlocal study of topological Radon transforms and real projective duality, Adv. in Math., t o appear. 22. Y. Matsui and K. Takeuchi, Topological Radon transforms and degree formulas for dual varieties, submitting. 23. T. Ohmoto, An elementary remark on the integral with respect t o Euler characteristics of projective hyperplane sections, Rep. Fac. Sci. Kagoshima Univ., 36,37-41 (2003). 24. A. Parusinski and P. Pragacz, Chern-Schwartz-MacPherson classes and the Euler characteristic of degeneracy loci and special divisors, J. Amer. Math. Soc., 8, no.4, 793-817 (1995). 25. A. Parusinski and P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Alg. Geom. 10, 67-79 (2001). 26. R. Piene, Polar classes of singular varieties, Ann. Sci. Ecole Norm. Sup. 11, no. 2, 247-276 (1978). 27. R. Piene, Some formulas for a surface in P3,Algebraic geometry (Proc. Sympos., Univ. lkomso, 1977), Lecture Notes in Math. 687, Springer-Verlag, Berlin, 196-235 (1978). 28. C. Sabbah, Quelques remarques sur la gBomBtrie des espaces conormaux, Astkrisque 130, 161-192 (1985). 29. J. Schurmann, A generalized Verdier-type Riemann-Roch theorem for ChernSchwartz-MacPherson classes, preprint available in arXiv:math AG/0202175. 30. K. Takeuchi, Perverse sheaves and Milnor fibers over singular varieties, Advanced Studies i n Pure Mathematics, to appear. 31. B. Teissier, Cycles Bvanescents, sections planes, et conditions de Whitney, Astkrisque 7 et 8,285-362 (1973). 32. B. Teissier, Sur diverse conditions numbriques d’Bquisingularit6 des familles de courbes (et un principe de specialisation de la dBpendance intbgrale), Centre de Math., Ecole Polytechnique (1975), preprint. 33. E. Tevelev, Projective duality and homogeneous spaces, Encyclopaedia of mathematical sciences 133, Springer, (2005). 34. A. Thorup, Generalized Plucker formulas, Recent progress in intersection

270

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and

K. Takeucha

theory, Trends Math., Birkhauser Boston, 299-327 (1997). C.T. C. Wall, Singular points of plane curves, London Math. SOC.Student Texts 63, Cambridge Univ. Press, (2004). 36. S. Yokura, On the characteristic classes of complete intersections, in Algebraic Geometry Hirzeburch 70”, Comtemp. Math. 241,349-369 (1999).

35.

271

Derived Picard groups and automorphism groups of derived categories Jun-ichi Miyachi Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184-8501, Japan E-mail: miyachiQu-gakugei. ac.jp We review the notion of derived Picard groups of algebras and the relations with automorphism groups of derived categories of coherent sheaves on some projective varieties and the non-commutative space which was introduced by Kontsevich-Rosenberg.

1. Derived Picard groups of derived categories The relation between tilting complexes and triangulated equivalences of derived categories was introduced by J. Rckard, and it is well known as Morita theory for derived categories [ [9]], [ [lo]]. In the case of algebras over a field k, a complex T' E Db(Mod B" @k A ) is called a two-sided tilting complex if it is a bounded complex of finitely generated projective modules

at both sides and if - 6gT* : Db(ModB) 3 Db(Mod A) is a triangulated equivalence (this is called a standard equivalence). Rckard asked if any k-linear triangulated equivalence is isomorphic to a standard equivalence. For this problem, it is sufficient to consider the relationship between the group of two-sided self-tilting complexes, that is the derived Picard group, and the automorphism group of triangulated self-equivalences. Let A be an algebra over a field k. We denote by Mod A (resp., mod A) the category of (resp., finitely presented) right A-modules, and denote by projA the full additive subcategory of Mod A consisting of finitely generated projective modules. For an abelian category A and its additive full subcategory B, we denote by D(d) (resp., D+(d), D-(A), Db(A)) the derived category of complexes (resp., bounded below complexes, bounded above complexes, bounded complexes) of A , denote by K(B) (resp., K-(B), Kb(B)) the homotopy category of complexes (resp., bounded above complexes bounded complexes) of B.

272

J. Mayachi

Definition 1.1. Let A be an algebra over a field k , A" the opposite algebra, and let Ae := A" @k A be the enveloping algebra. The Picard group Pick(A) of A (over k) is

Pick(A) :=

{invertible bimodules P E Mod Ae} isomorphism

1

with identity element A, product (Pi, P2) H Pi @ A P2 and inverse P Pv := HomA(P, A). The derived Picard group of A (over k) is DPick(A) :=

{tilting complexes T E Db(Mod Ae)} isomorphism

with identity element A , product ( T I T2) , w TI 6 tT2 and inverse T H TV := RHomi(T,A ) . For a k-linear additive category B,Outk(L3) is the group of isomorphism classes of k-linear self-equivalences of B. For a k-linear triangulated category D,O u t t ( D ) is the group of isomorphism classes of k-linear triangulated self-equivalences of V. Morita theory says Pick(A) 2 Outk(proj A) itary k-algebras, we had the following.

2

Outk(Mod A). For hered-

Theorem 1.2 ( [7]).Let A be a finite dimensional hereditary k-algebra. Then we have isomorphisms DPick(A)

% Outt(Db(Mod A)) S Outf(Db(mod

A)).

Moreover, we described the derived Picard group of them by using quivers, that is, oriented graphs.

Theorem 1.3 ( [7]).Let k be an algebraically closed field and A a finite dimensional hereditary basic k-algebra with quiver Then there is a n isomorphism of groups

A.

DPick(A) E

{

Outk(k(Z& Im))if A is Dynkin Outk(k(Z& Im))x Z otherwise.

Here k(Z& Im)is the mesh k-category of the quiver For any n L 2 let

fin be the following quiver:

A.

Derived Picard groups and automorphism groups of derived categories

273

Kontsevich-Rosenberg [4]introduced the noncommutative projective space N P Z , n 2 1, and showed that Db(cohNP;) M Db(mod kfin+l). By Beilinson’s results [l],there is an equivalence Db(coh Pk) M Db(mod kfia). Hence we got the next corollary.

Corollary 1.4 ( [7]). Let X be either NPZ (n 2 1) or Pr ( n = 1). Then Outf(Db(cohX))

S

Z x (Z K PGL,+l(k)).

2. Isotropy g r o u p s and derived P i c a r d groups Definition 2.1. Let D be a triangulated category and E a full subcategory which is closed under isomorphic objects. We define the isotropy group Outf(D)& = { [ F IE outf(D)I

FIE

is an self-equivalence of E )

where [ F ]means the isomorphism class of F . Then we have the canonical morphism 7r& : Out;(D)E

4

OUtk(E)

For a k-algebra A, let PA (resp., f P A ) be the full subcategory of Db(Mod A) (resp., Db(mod A)) consisting of X ’which is isomorphic to some finitely generated projective A-module in Db(Mod A) (resp., Db(mod A)). A ring A is called right coherent if the category mod A of finitely presented right A-modules is an abelian category. Proposition 2.2. For a k-algebra A, the following hold. ( I ) Outf(Db(ModA))pA = { [ F ]E Outf(Db(ModA))I F(A) E P A ) (2) If A is a right coherent k-algebra, then O u t f ( D b ( m o d A ) ) p A= { [ F ]E Outf(Db(modA))IF(A) E PA) Proof. Let F be a triangulated functor such that F(A) E PA (resp., PA). Since F(A) generates Kb(proj A), any finitely generated projective A-module P is isomorphic to a complex which is obtained by successive mapping cones of direct summands of finite direct sums of copies of F(A). Therefore P is a direct summand of finite direct sums of F(A). Hence the restriction Flp, (resp., Flfpa) is an equivalence. 0

Theorem 2.3. For the isotropy group, the following hold.

274

J. Mzyachz

( 1 ) The group morphism xp, is a split morphism, and Outt(Db(Mod A))?, is isomorphic to a semi-direct product of Pick(A) by Ker TP, :

Outt(Db(ModA))p, 2 Pick(A) ix Kerxp, (2) The canonical injection DPick(A) ~ - - Outf(Db(Mod t A)) is an isomorphism if and only if Kerxp, is trivial. (3) I f A is a right coherent k-algebra, then the group morphism xfp, is a split morphism, and hence Out~(Db(rnodA))fp, is isomorphic to a semi-direct product of Pick(A) by Ker xfp, :

0 u t t ( D b ( m o d A ) ) p A% Pick(A)

P(

Kerxfp,

(4) If A is a right coherent k-algebra, then the canonical injection DPick(A) cf Outf(Db(modA)) is an isomorphism i f and only i f K e r x p , is trivial.

-

Proof. (l),(3) Outk(projA) is equivalent to Pick(A), and then any element of Outk(projA) is obtained by tensoring of some invertible Abimodule. Then the inclusion L : Outk(projA) Outt(Db(ModA)) (resp., L : Outk(proj A) -t Outf(Db(mod A))) is obtained by tensoring of invertible A-bimodules. Since projA is equivalent to PA (resp., f p A ) , xp, 0 L (respa, x f P A o L ) is an isomorphism. (2), (4) For any k-linear equivalence F of Db(ModA) (resp., Db(mod A)), by results of Rickard [lo] there is a two-sided tilting complex T' such that A &I 2T' 2 F(A). Therefore RHomk(T',-) o F(A) S A, and R H o m i ( T ' , - ) o F is an element of Outf(Db(ModA))p, (resp., Outf(Db(rnod A))fPA) by Proposition 2.2. Therefore, Ker xp, (resp., Ker 7rfPa) is trivial if and only if there is an invertible A-bimodule U such

' U , and F S -6:Uh:T'. Since U h i T ' that RHom>(T', -) 0 F - 6A is a two-sided tilting complex for A, we complete the proof. 0 Corollary 2.4. Let d be an abelian category which has a triangulated equivalence F : Db(ModB) 5 Db(d) (resp., F : Db(rnodB) 3 Db(d)), where B is a k-algebra (resp., a right coherent k-algebra). If the canonical A morphism x&F(B): Outk (Db(d))&,,,, --t OUtk(&F(B))is an isomorphism, then DPick(B) = Outf(Db(ModB)) % Outt(Db(d))

(resp., DPick(B) = Outf(Db(rnod B ) )

Outf(Db(d)))

where &F(B) is the full subcategory of Db(d) consisting of objects which are direct summands of finite coproducts of copies of F ( B ) .

Derived Pacard groups and automorphism groups of derived categories 275

Proof. A triangulated equivalence F : Db(ModB) following commutative diagram: Outf(Db(Mod B ) ) p ,

oUtt(Db(d))EP(B)

"pB

5 Db(d) induces the

Outk (PB)

"EF(B)

Outk(€F(B))

where all vertical arrows are isomorphisms. Then, by Theorem 2.3, (l), is an isomorphism if and only if KerrEF(,) = Kerwp, = 1. By Theorem 2.3, (2), we have the statement. Similarly, we have the statement 0 for the case of Out?(Db(mod B)). T&~(,)

It is known that many derived equivalences between the category of coherent sheaves over projective varieties and module categories over algebras are obtained by tilting objects, that is, tilting sheaves. Definition 2.5. Let d be an abelian category. An object M E d is called a tilting object if (a) E x t i ( M , M ) = 0 for all i > 0. (b) M generates Db(d), that is, Db(d) is the smallest full triangulated subcategory of Db(d) which is closed under isomorphisms and contains M. (c) EndA(M) is a right coherent k-algebra of which the right global dimension is finite.

Corollary 2.6. Let A be an abelian category, M a tilting object. If the canonical morphism 7rcM : Outt(Db(d))EM4 OUtk(€M)

is an isomorphism, then

DPick(B) = Outf(Db(modB))

Outt(Db(d)),

where B = EndA( M) .

Proof. According to [ [3]] or [ [9]], we have a triangulated equivalence F : Db(modB ) 5 Db(d). By Corollary 2.4, we are done. 0 Example 2.7. Let P = Pt be the n-dimensional projective space over a field k. By results of Beilinson [l],?; = @y=o O ( i ) , 12 = @y=o Q(-Z)

276

J . Mzyachi

are tilting objects of cohP, and B = E n d p ( Z ) , C = E n d p ( 3 ) are finite dimensional k-algebra of finite global dimension. Indeed, let be the following quiver:

0

and let pi, p2 be the following relations: pi : a;+la'. - a'.+la'. = o for 3 3 % p2

: a:+'at =

+

o 5 i < j 5 n, o 5 1 < n - 1

o for o 5 I < n - 1 o o 5 i <j

af+la'. aZ+la$ = for 3 3

5 n , 5~1 < n - 1

Then B and C are isomorphic to k(0, p1) and k ( 0 , p 2 ) as k-algebras, respectively. According to a result of Bondal-Orlov [2], Outt(Db(coh P))is generated by translations, twists and Autk(P), is isomorphic to Z2 x PGL,+l(k). Then we have Oute(Db(cohP))ETl = Outk(&z).Hence DPick(B) = Outt(Db(mod B)) (resp., DPick(C) = Outt(Db(mod C)) ) Ou tt(Db (co h P ))

Z2 x PGL,+l(k). Moreover, for any k-algebra A which is derived equivalent to B , we have DPick(A) = Outt(Db(modA)) 2 Z2 x PGL,+l(k). Example 2.8. For p = (po,pl,... , p n ) E Nn+', let P(p) = P;E.(p) be the weighted projective space over an algebraically closed field k. For a sequence A = (XO, XI, X2,. . . ,A), of pairwise distinct elements of Pk, we take XO = 00, A1 = 0, A2 = 1, and let X = X(p , A) be the weighted projective line defined by the equations in the weighted graded algebra ~ [ X O. ,. .,X,]:

By results of Geigle-Lenzing [ 5 ] , 7 = @o k,(t) + H(f,(U = C J t L

(1) (2)

(3)

with denoting 1 (~(,-~)(t)], 2 b,(t) = the p-th homogeneous term of cp(CL-l)(t)f(p-l)(t)l

a,(t) = the p-th homogeneous term of

c,(t) = the p-th homogeneous term of - ( h

and

cp(p-l)(t)

=

C"-' v=o

pv(t),

f(P-"(t)

-[cp(P-')(t),

+ Ic(,-')(t))o (fo + f ( p - l ) ( t ) ) ,

=

C;:;

fv(t), k("-l)(t) =

Mt). Under the condition H2(K;*) = 0, the equation in A:'(Nx/c.r)[t]

= R(b,(t))

(4)

is solvable for up@)E A Y ( N x l c ~ ) [ t ] ,where R is the quotient bundle homomorphism R : T1>OCfVx + N x / ~ NThen, . cpp(t),fp(t), k p ( t ) are de-

284

K. Magrajama

termined by solving

hold for some positive constants A, B independent of p , the convergence of IIfpIIp(t)follows by a standard argument using IIp,llrc(t) and the estimate by the dominant power series A(t) = $(tl+-. .+td), (cf. [lo]). We recall the sharp estimate of the standard solution u,(t) = -* d NR(b,(t)) of (4) (cf. [S]):

&

IIYB*Nullk I CllUllk

(8) holds if Y E I'(n,T1ioX)is an admissible vector field in the sense that it lies in " T ~ d R @ " T ~for d Rall x E d o where we denote "T'dR = T1loXlann CTdR and "TI' its complex conjugate. Therefore, if we have a solution of (5)-(7) satisfying

p,(t)pn

E r(dR, "T$'R)[t]

for all p 2 1,

(9)

then it may hold that

I Ib,l Idt> I

I Im

where we employed a norm 11 intermediate between 11 Ilk and 11 Ilk+l as follows; Let e l , . . . ,en be a local frame of T1ioX such that el,. . . ,en-l span the sub-bundle "?X := {v E T1yoXI v(CLl IziI2) = 0). If we take local (1,O)-type admissible vector fields ex(X E A) such that el, . . . ,en-l and ex(X E A) span T$OX for all x E R and generate all coefficients of p,(t) ( p 2 l ) , we define an intermediate norm

Since this definition of the new norm does not make sense unless #A is finite, we have to show it is possible to construct p,(t), f,(t), Ic,(t) ( p 2 1) so that p,(t) has this property with #A < +cm. By a careful inductive argument, we have a solution p,(t), f,(t), k,(t) ( p 2 1)having the following

Analytic approach to deformation of normal isolated singularities

285

description where 11 is a (1,O)-vector field on X which is complementary to " T ' X , 0 is an admissible vector field, 01 E A$'(T'l0X) such that O1 =0 and y is a non-vanishing differentiable function (cf. [15]).

where cpop(t) E A5'("FU)[t], fop E A$("FCN,,)[t], and locally on a fixed 8 neighbourhood U on which "T'Uis trivialized,

Where we denote cpg-"(t) = CEI:cpvo(t)and use the same notations for xp-"(t), &-jl)(t), xgf"(t), X$-')(t) and + ( p - ' ) ( t ) , and { c p ~ - l ) x ( t ) } l ~ ~ <are n - lgiven by cpfl)(t) = X = l ' p O(p-l)x (+A. Based on this description, we introduce the following intermediate norm

zn-l

286

K. Mayajima

with denoting en = 77: n

1;.

:=

n- 1

C IIGUII~ + C IIei(u)IIk + I I ~ ( ~ +) I I ~ i=

IIel(U)IIk

1

i=l

c

n- 1

+ ll(F(%-)))77('1L)Ilk +

i=l

II(.i(F(e(r))))77(U)Ilk

+ Il(e(wr)))77(U)I Ik + I I(~l(FW))77(41 Ik + I14Ik. Finally we have the following estimate which implies the convergence of

4 4 and f (t):

w

I If P I I 2 and of positive genus. The decomposition coincides

310

P. Norbury

with the JSJ decomposition so in particular it is canonical. The fibration p : Y - L --+ S1 equips Y - L with an element of H 1 ( Y - L, Z). An element of H 1 ( Y - L,Z) is sometimes known as a multilink structure on L since it assigns a multiplicity to each component by evaluating along a meridian circle. Any fibre F of the map Y - L + S1 has boundary the multilink L. Thus when the multiplicities of all components of L are 1, F is a Seifert surface for L, and if a multiplicity of a component Li is di > 1 then the boundary of F near Li gives di copies of Li. If di = 0 then one can attach a disk to the ith boundary component of F which meets Li transversally. The fibre F decomposes as in (1) and each F(') is the fibre of a fibration of the corresponding Seifert-fibred piece over the circle. The monodromy decomposes and since the monodromy of any Seifert-fibred piece is finite this gives (ii). Using the definition of a stable curve in terms of its homological monodromy being unipotent, (iii) follows from (i) and (ii), since on a cover where the lift of h is trivial on each F ( i ) , only twisting along separating circles remains. In the cover, the twisting is integral, not merely fractional. A precise description of how to calculate the fractional twisting of the monodromy along circles is given in [l].We will show this in calculations below and thus verify (iv). 0 The homology class of a polynomial map, i.e., the image of [P'] E

H2(P',Q) into H 2 ( P 1 , M g , n ) ,is trivial if and only if all generic fibres are isomorphic. Since the stable reduction of such a family is necessarily a trivial fibre bundle, a necessary condition for the generic fibres to be isomorphic is that the stable reduction of any exceptional fibre is smooth. Theorem 2.1 shows that this occurs precisely when the dual graph of the exceptional fibre has only one node of valency > 2 since this is equivalent to the monodromy having finite order. Applying this idea to the fibre over infinity of a polynomial p : C 2 4 C gives a severe restriction on polynomials with isomorphic general fibres. Kaliman classified all polynomials p : C2 + C with isomorphic generic fibres [5].It seems likely that the classification can be reproven beginning from the restriction just described. 2.3. Eficient dual graphs

Many calculations involving the dual graph require only some of the nodes. Examples are: (i) the Euler characteristic of the generic fibre; (ii) a description of branched covers of the dual graph;

Stable reduction and topological invariants of complex polynomials

31 1

(iii) determinants of intersection forms of subgraphs; (iv) the canonical class supported on the dual graph. An efficient dual graph hides valency 2 rational components, retaining components of valency # 2 or genus > 0. An efficient dual graph for the fibre over infinity of p = (z2 - y3)2 zy is given in Figure 5.

+

Fig. 5. The efficient dual graph hides valency 2 genus 0 nodes.

We have labeled the multiplicity of the polynomial p on each node. The calculation of the Euler characteristic of the generic fibre using the fibre at infinity gets a contribution of zero from the rational valency 2 nodes and hence one needs only the efficient dual graph. In Figure 5 a node of multiplicity m and valency v is replaced by m copies of a v-punctured 2-sphere and the Euler characteristic is 6 4 5 - 12 - 10 = -7 so the generic fibre is a genus 4 curve minus a point at infinity.

+ +

A cyclic branched cover along a singular fibre behaves quite generally as in the example in Section 2.1. A cyclic branched cover along a linear chain of genus zero components gives another linear chain of genus zero components. The example in Section 2.1 shows this explicitly. The most interesting behaviour occurs on components intersecting at least two other components, corresponding to nodes of valency > 2 in the dual graph, and on positive genus components. By simply understanding the behaviour of a cyclic cover on the nodes of valency > 2 and of positive genus, we can much more efficiently take cyclic covers of splice diagrams, hiding all chains arising from HirzebruchJung singularities. The long calculation shown in Section 2.1 is substantially shortened. An efficient dual graph that hides valency 2 rational nodes, and equipped with enough information to reproduce the full dual graph, is known as a splice diagram. Integer weights assigned to the graph supply the information required t o reproduce the full dual graph and in particular the multiplici-

312

P. Norbury

ties of each node. Determinants of intersection matrices of subgraphs have the property that they are unchanged by blowing up the subgraph. In particular, to each edge of the efficient dual graph there is a well-defined edge determinant given by the determinant of the intersection matrix of nodes in the full dual graph along that edge, excluding the endpoints. The efficient dual graph is called a splice diagram when we use a set of weights on half edges. The weight on a half edge is the determinant of the intersection matrix of the branch of the dual graph disconnected from the half edge if we cut along that edge. Figure 6 shows Figure 5 with weights on half-edges replacing multiplicities on nodes. The weights give back the whole dual graph. In particular, to get the multiplicity of a node take a path from the node to an arrow and multiply all weights adjacent to, but not on, the path. The black node has weights 3 and 2 adjacent to the path to the arrow so its multiplicity is 3 x 2 = 6. The closest node to the arrow has weights 5 and 2 adjacent to its path so its multiplicity is 5 x 2 = 10. When there are several arrows do this for each path joining a node to an arrow and add.

1

2 n 1 3

A

Fig. 6.

5 n1

b

2

A

Splice diagram

We will not go further into the calculation of the weights, nor will we emphasise the important fact that the splice diagram encodes a 3-manifold containing a link spliced together from Seifert-fibred pieces. See [l]for full details. In this particular case the multiplicities in Figure 5 are determined by, and determine, the weights in Figure 6. Given a polynomial map p : X -+ B, we may assume the canonical class K x is supported on fibres of p and sections. This is because any irreducible component of a divisor C c X is either contained in a fibre of p or it maps onto B and in the latter case we can pass to a cover B' -+ B so that the pull-back of C is a collection of sections, introducing new support only in fibres. Let UiDi denote the support of K and put K = kiDi. Define D = diDi where di = -ki - 1. Although D resembles a divisor it is not really well-defined since we can add zero times any curve to K and this will change

xi

xi


313

D . Nevertheless, D is a useful tool to study the support of K. The important property of

D is that for each Di

D . Di = x ( D i ) - valency Di where the valency of Di is its valency in the dual graph of UiDi. This expression vanishes on rational curves of valency 2, precisely those curves hidden in the efficient dual graph. In particular K . D uses only the multiplicities on the nodes of the efficient dual graph. For a polynomial p : (C2+ (C the multiplicities of D are easily calculated from the splice diagram. The node corresponding to the line at infinity in P2, indicated by the black node in the diagrams, has multiplicity 2 since K has multiplicity -3 there. From any other node v take a path toward the black node to get to the previous node v' of valency > 2. Then the multiplicity of D is given recursively by m(v)= w . m(v') edge determinant of the edge E joining v and v', and w is the product of the edge weights adjacent to v and not on E. See [9] for a thorough description of the calculation of D from weights in the splice diagram. In Figure 6 the multiplicities of D on the two valency 3 nodes are 5 and 3 respectively. Another feature of D is that it behaves well under branched covers. Locally along a curve of ramification x = 0 the canonical divisor is given by xmdx A dy, and hence a d-fold cover zd = x transforms zmd+d-'dz A dy, and along unramified curves the multiplicity remains the same. Thus D is given locally by x-m-ldx A dy transforming to zd(-m-l)dz A dy so the multiplicity simply multiplies by d.

+

2.4. Decomposing dual graphs.

Figure 7 shows how to decompose Figure 5 into pieces and how the Euler characteristic decomposes. Each node of valency > 2 corresponds to a Seifert-fibred piece of the JSJ decomposition of Y - L. The (2) on the arrow indicates that a fibre of the left piece consists of two identical disjoint components. The greatest common divisor of the (0) on an arrow and the default value of (1) on the right arrow is 1 which is the number of components of the fibre of the right piece. The numbers (2) and (0) are calculated using the weights of the splice diagram in the same way as multiplicities are calculated, by taking a path from the edge to an arrow and multiply all weights adjacent to, but not on, the path. Thus the left piece has fibre two genus 1 surfaces each with one boundary component and the right piece has fibre a genus 2 surface with three boundary components. The genus 4 generic fibre with one point removed decomposes along two circles as shown

314 P. Norbury

in Figure 8. By pinching along these circles, or vanishing cycles, one gets the stable curve in Figure 4.

b4

65

x=-2

x=-5 Fig. 7. JSJ decomposition

order 10 monodromy

order 12 monodromy

Fig. 8. The monodromy acts with finite order on each piece and fractionally twists along the decomposing annuli.

This demonstrates (i) of Theorem 2.1. The decomposition in Figure 7 gives the JSJ decomposiiton of Y - L into pieces Y ( i )each fibring over the circle and containing a multilink L(i).The numbers (2) and ( 0 ) in Figure 7 are the multiplicities of the multilink components. The monodromy on the fibre of Y - L decomposes to give the monodromy of the fibre of Y(2)- L(i). The latter has finite order given by the multiplicity of the valency > 2 component. This uses the fact that the monodromy on Seifert-fibred spaces is completely understood. Thus the multiplicity is 12 on the left component this consists of an order 6 map on the once-punctured torus and a swapping of the two components - and the multiplicity is 10 on the right component. A 60:l cover will pull back the monodromy to be trivial on each component and thus this will give the stable reduction of the fibre over infinity as seen in the calculation in Section 2.1. Thus (iii) and part of (ii) Theorem 2.1 are


315

seen. It remains to understand the twisting of the monodromy along the annuli in Figure 8. Upstairs, in the 60:1 cover we expect some whole number of Dehn twists along the annuli, so downstairs we expect fractional twists. This calculation is subtler and it is completely solved in terms of the JSJ decomposition as Theorem 13.1 in [1]. It is subtle only because we are choosing to emphasise the more intuitive multipicities of the polynomial p on each component of the efficient dual graph, rather than the weights of the splice diagram. The fractional twisting requires the weights of the splice diagram. It is given along an annulus corresponding to the edge E by

where ds is the number of annuli corresponding to E, I and I1 are the multiplicities of the neighbouring nodes, and AE is the determinant of the intersection matrix of the nodes from the full dual graph lying on the edge. In the example above, ds = 2, coresponding to the 2 annuli, / • I' = -2 1 0 12 • 10 = 120 and As = 1 - 2 1 = -7 = 5 • 1 - 2 • 3 • 2 • 1. (We have 0 1 -2 given the calculation of the edge determinant twice, the second using the splice diagram weights in Figure 6, not explained here further.) Thus the fractional twisting is 7/60 and upstairs in the 60:1 cover it is a Dehn twist of order 7. This tells us that locally the node of the stable curve is given by xy = t7 and there is a Zy quotient singularity in the ambient surface. 3. Cohomology classes In this section we evaluate the homology class associated to p : X —> B on rational cohomology classes

Since we consider 1-dimensional bases B, we will abuse notation and treat the cohomology classes as numbers 5,Ki,\i,ipi & Q by pulling back to H2(B,Q). _ A non-trivial homology class in H2(M9tn) is detected by the nonvanishing of at least one of the cohomology classes above. This follows from work of Wolpert [13] where he shows that KI + 5Zi*=i ^ ^s a positive multiple of the Kahler class w. The integral of w over any non-trivial homology class is positive and hence at least one of KI or if>i is non-zero on the homology class.

316

P. Norbury

There are obstructions to a dual graph appearing as the fibre at infinity of a polynomial p : C2 -+ C. The obstructions are not complete so there are dual graphs where it is not known if they appear as the fibre at infinity of a polynomial p : C2 -+ C. We do not know if the positivity condition ~1 Cy=ly!~i > 0 gives a new obstruction on dual graphs.

+

3.1. Definitions

ag,n

The class 6 E H 2 ( m g , n )is represented by the boundary divisor Mg,n.Thus it counts the number of singularities of a polynomial p : X -+ B. The contribution from a stable fibre is its number of nodes, where a node locally given by xy = tk contributes k to the count. Equivalently, if we insist that the total space X be non-singular and thus allow semi-stable fibres then the contribution from a semi-stable fibre is simply its number of nodes. The contribution from an unstable fibre is calculated by first passing to the stable reduction. In particular, the contribution is not necessarily an integer.

Definition 3.1. Let p‘ : X‘ + B’be the semi-stable reduction of p : X 4 B equipped with n sections, for X’ a smooth surface that d-fold covers X. Define # singularities p’ 6= d 3a (XI) K,1 = 26 d 6 a(X’) XI=-+4 4d si (B’) * ~i (B’) $.z -- , i = 1,...,72 d

+

~

where o ( X ‘ ) is the signature of X‘ and si(B’).si(B’) is the self-intersection of the ith section of X’4 B’.

It is not hard to see that if we take a further cover the number of singularities multiplies by the degree of the cover and hence 6 is well-defined. It is subtler that the signature behaves the same way. The signature of a branched cover is in general not easily calculated. The definition above is given for simplicity whereas for the purpose of calculations we give a better definition below in which the classes I C ~and A1 in families IC, and ,A for m E Zf. The relation between the two definitions of A1 is due to Smith [ll]and the others easily follow.

Stable reduction and topological invariants of complex polynomials 317 -

The forgetful map T : Mg,n+l -+ Mg,nis defined by forgetting the ( n 1)st point and possibly blowing down rational components of some equipped with n sections fibres. It defines a universal bundle over M,,, si : M g , n + M g , n + l . Over Mg,n+l define the vertical canonical bundle to be the complex line bundle

+

-

8T * K A

y = KMg,n+1

4

Mg,n+l.

Mg.n

The bundle y is used to define the classes ~ Define the Hodge classes

Am = c ~ ( T * ( Y )E) H

1 A1, , $Q

2m

as follows.

(Mg,n).

The push-forward sheaf n,(y) is a rank g vector bundle over M,,,best understood in terms of the fibres which are the g-dimensional vector spaces of holomorphic 1-forms on the curve associated to a point of M,,,. Over a stable curve, one uses 1-forms holomorphic outside the singular set that have at worst simple poles at the singular points with residues summing to zero. Define the Mumford-Morita-Miller classes Ic,

= T ! [ C l ( y ) m + lE ]

H2"(Mg,n).

where T ! : H k ( M g , n + l ) + Hk-2(Mg,n) is the umkehr map, or Gysin homomorphism, obtained by integrating along the fibres. For each i = 1,..., n pull back the line bundle y to s f y = yi + Mg+ and define 2 -

7cli = ~ ( y iE)H (Mg,n).

The classes are related [2] by K1+6

A1 = 12 .

3.2. Calculations The main purpose of this paper is to enable calculations of the homology class of a polynomial map p : X + B with 1-dimensional base B and in particular, calculations of the rational numbers &,&I, X I , @i. We will calculate these rational numbers for infinite families of polynomial maps p : C2 -+ C. We first do the calculations for the specific example p = (z2- y3)2 zy. The contribution from the fibre over infinity to b comes from the two nodes of the stable fibre over infinity, each counted with multiplicity 7. The stable reduction is obtained from a 60:l cover hence the contribution t o b

+

318

P. Norbury

&.

is This is the fractional twisting of the geometric monodromy and we have given it a cohomological interpretation. There are 8 finite singularities so 7 30

6=8--. To calculate 6 1 we need to understand the canonical class on the surface X’ which gives the stable family X’ -+ B‘. The stable fibre over infinity is given in Figure 9 where the numbers in brackets are self-intersection

...........

g=1

Fig. 9.

Canonical divisor

numbers as usual and the multiplicities are now the multiplicities of the canonical class. We deduce the canonical class by the fact that it can be supported on the infinite fibre plus 59 copies of the fibre of p. This is because the canonical class can be supported on the divisor at infinity X - C2 since it can be supported on P2 - C2 and blowing up introduces support only on the exceptional divisiors. A branched cover introduces canonical class along the ramification set so the 60:l cover means 59 copies of the fibre of p contributes to the canonical class and the rest remains supported on the fibre over infinity. Apply the adjunction formula K . C = -x(C)- C . C to a fibre to see K F = -6 so the section is given multiplicity 6 in the canonical class. To calculate the other multiplicities in Figure 9, similarly apply the adjunction formula to irreducible components of the fibre over infinity. Thus

and


319

Note that A1 times the degree of the cover is an integer, so the relation A1 = (61 4 / 1 2 puts a mod 12 condition on nl 6. Finally $1 is given by K X ~ I Bs1 , [B']= -B' B' by the adjunction formula. Thus

+

+

3

6 60

$1 = - =

1 lo.

The self-intersection (-6) used to calculate $1 is encoded in the fractional twisting of the geometric monodromy. It gives the fractional twisting of the generic fibre on the annulus around the marked point. In terms of the efficient dual graph, or splice diagram, the annulus A corresponds to the edge E joining the fibre over infinity to the section. Thus $1 is given by $1 =

dE 1 -twist(hlA) = - = 1 10

where d E = 1 is the number of annuli corresponding to E , and 1 is the multiplicities of the node of E. The formula for the twist is Theorem 13.5 in [I]. 3.2.1. Calculations in

mg,l, +

The example p = (z2- y3)' zy is a particular case of the general class of polynomials with one point at infinity. The splice diagram for the general polynomial with one point at infinity is given in Figure 10.

pz

....

Fig. 10. Family in

._._

ng,l.

The stable reduction of the fibre over infinity is easily calculated from Theorem 1.1. We refer to the node with weights p i , qi as the ith node and the edge between the ith and (i 1)th nodes as the ith edge. The ith node corresponds to qj components of Euler characteristic pi+qi-piqi, each meeting qi of the qj components corresponding to the (i-1)th node. The monodromy permutes the qj components and acts internally with order pig,, so its total order on the components corresponding to the ith node is piqi q j which is the multiplicity of the polynomial on that node.

n,,, n,,,-,

n,,,

+

n,,,

320 P. Norbury

Thus the stable reduction uses a cover of order n p i q i (or more efficiently gcd(p1, ...,Pn) nqi.) From the fractional twisting along annuli we see that the quotient singularity at each annulus above the ith edge has order piqiqi+l - pi+l, and the self-intersection of the section in the stable reduction is - n,,,piqi. We calculate inequalities for the rational numbers 6,1c1,A1 since their maximum value is taken when the fibre of p over infinity is the only unstable curve, and the rational numbers 6, 1c1, XI are upper-semicontinuous. We have no good interpretation of the messy expressions for 6 and ~1 so we show them only for the cases m = 1 and m = 2. For m = 1 the Euler characteristic of the generic fibre is

x = Pl + 41 - Pl!ll,

( m = 1).

The stable reduction of the fibre over infinity uses a degree plql cover and results in a smooth fibre over infinity. To calculate 6 we use a result of Suzuki [12]

where xc = x(p-l(c)) which expresses the Euler characteristic of the generic fibre as a sum of the number of vanishing cycles near exceptional fibres. If all fibres over finite values c E C are stable, which is the generic case that gives an upper bound on 6, then xc - x counts the number of nodes in the fibre p-l(c). Thus

6 I 1 - x,

( m = 1).

The fractional twisting along the boundary annulus is given by -l/pmqm so 1 + 1 = -. Pmqm If all finite fibres are stable, the canonical class upstairs in the plql-fold cover is K = (-2 - x)F (-1 - x)C where F is the generic fibre and C is the section. The fractional twisting -l/plql shows that C . C = -1 in the cover. Thus K . K = - ( 1 + x ) ~2 ( 1 + x ) ( 2 x) = (1 x)(3 x) and

+

+

61

I

+

+

+

K . K - ~ ( x + ~-)~ ' - 1 -

PlQl

P141

, (m = 1).

When m = 2, the stable reduction of the fibre over infinity uses a cover of order plqlpzqz and consists of an irreducible component with Euler characteristic 1 +pz 42 -pzqz plus qz irreducible components with Euler characteristic 1 + P I + 41 -plql, joined along 42 nodes with quotient singularities

+


321

of order plqlq2 - p2. Now

x = QlQ2+ PlQ2+P2

- plqlqz - p2q2,

( m= 2 )

and

6 < l - x - L + % , PlQl

(m=2)

Pz

uses ( 2 ) to get an upper bound for nodes away from infinity, plus l / ( p l q l p 2 ~ 2 )times the q2 nodes over infinity, each counted with multiplicity Plqlq2 - P2.

To calculate n1 we need a more efficient method than has been used so far. We use the divisor D introduced in Section 2.3. The expression

K . K = -K

*

D - C ( x ( C i )+ C i . Ci)

(3)

i

consists of K . D which is independent of rational valency 2 nodes in the dual graph, and C,(x(Ci) Ci Ci) which does depend on rational valency 2 nodes in the dual graph, however it behaves well on the semi-stable reduction of a family. A semi-stable curve contains strings of ( - 2 ) rational curves in place of the quotient singularities at nodes of stable curves. The expression Ci(x(Ci) Ci . Ci) vanishes on ( - 2 ) rational curves and hence (3) uses only the multiplicities of K on the irreducible components of the stable fibres, and on sections. From this we calculate

+

+

n1

Ql+ P2 + -, ql Pl Q2

1 - p l q l - p 2 q 2 + Pl 5 2 -2x+ -

P2Q2

( m= 2 ) .

3.2.2. Calculations in M o , ~ .

+

When p : C2 + C has rational fibres X = 0 = n 6 and in particular we get equalities instead of inequalities for n and 6, i.e. the generic behaviour always occurs, since if the sum of two upper-semicontinuous functions is continuous then the two functions are both continuous. When the n points at infinity are sections, no branched covering is required for stable reduction. We use the classification in [lo] to see that either the homology class is trivial or n1 = 1 - n, 6 = n - 1, $i = 1, i = 1,..., n - 1 and Gn = n - 3. The homology class depends only on n. There are many inequivalent polynomials for each n showing that the homology class is a rather course invariant. The homological monodromy is trivial in both cases when n1 = 0 or n1 = 1 - n so the homology class is useful to distinguish these two cases. Kaliman [6] classified all polynomials with rational fibres and one fibre isomorphic to C*. We find that n1 = -1 and 6 = 1 on the whole family.

322

P. Norbury

Acknowledgments

I would like to t h a n k Brandeis University for its hospitality while this paper was written. References 1. Eisenbud, David and Neumann, Walter Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematics Studies 110. Princeton University Press, Princeton, NJ, 1985. 2. Harris, Joe and Morrison, Ian Moduli of curves. Graduate Texts in Mathematics 187.Springer-Verlag, New York, 1998. 3. Jaco, William H . and Shalen, Peter B. Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. SOC.21 (1979), no. 220, viii+192 pp. 4. Johannson, Klaus Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics 761. 5. Kaliman, Shulim Polynomials o n C2 with isomorphic general fibres Dokl. Akad. Nauk SSSR 288 (1986), 39-42. 6. Kaliman, Shulim Rational polynomials with a C*-fiber. Pacific J. Math. 174 (1996), 141-194. 7. Morita, Shigeyuki Characteristic classes of surface bundles. Invent. Math. 90 (1987), 551-577. 8. Neumann, Walter D. Complex algebraic plane curves via their links at infinity. Invent. Math. 98 (1989), 445-489. 9. Neumann, Walter D. and Norbury, Paul T h e Orevkov invariant of a n a f i n e plane curve. Trans. Amer. Math. SOC.355 (2003), 519-538. 10. Neumann, Walter D. and Norbury, Paul Rational polynomials of simple type. Pacific J. Math. 204 (2002), 177-207. 11. Smith, Ivan Lefschetzfibrations and the Hodge bundle. Geom. Topol. 3 (1999), 211-233. 12. Suzuki, Masakazu Propriitis topologiques des polyn6mes de deux variables complexes, et automorphismes alge'briques de l'espace C2. J. Math. SOC.Japan 26 (1974), 241-257. 13. Wolpert, Scott O n the homology of the moduli space of stable curves. Ann. of Math. (2) 118 (1983), 491-523.

323

Singularities appearing in a stable perturbation of a map-germ Mariko Ohsumi

Department of Mathematics, College of Humanities and Sciences Nihon University, Tokyo, 156-8550, Japan E-mai1:[email protected]. ac.jp Let f : (Rn,O) + (RP,O) be a Coomapgerm. We are interested in whether the number modulo 2 of stable singular points of codimension n that appear near the origin in a generic perturbation of f is a topological invariant. In this paper we concentrate on investigating the problem when p is 2n - 1, where stable singular points of codimension n are only Whitney's umbrellas.

Keywords: Singularity of smooth maps; topological invariants; Whitney's umbrella; stable perturbations.

1. Introduction

In [25] the author showed that the number modulo 2 of Whitney's umbrellas that appear in stable perturbations of a generic Coo map-germ f : (R",0) 4 (R2"-l,0) is a topological invariant. In this paper we will survey problems surrounding the result; we give a history of the problem, give examples to explain how Whitney's umbrellas appear in perturbations and give further problems. A C" map-germ f : (Rn,p ) -+ (R2"-', q) is called Whitney's umbrella if it is d-equivalent to the map-germ from (R", 0) to (R2"-l, 0) defined by

(a,.. . ' ~ " - - 2l , ~ " ' ... ~ l ,Z,-lZn). ~"' Here two C" map-germs f : (M1,pl) (N1,ql) and g : (M2,pz) (z1,

.. . '2,)

-+

4

-+

(N2,42) are said to be A-equivalent if there exist C" diffeomorphism-germs h:(Ml,pl)--t(Mz,p2)andk:(Nl,ql)-,(N2,~2)suchthat k o f = g o h . - Let f : (R",O) -+ (R2"-l,0) be a generic C" map-germ and let f : U -+ R2*-l be a C" representative of f, U being a small open neighborhood of the origin 0 in R". By Whitney's theorem( [31], [33]), 7 can be approximated by a stable mapping f : U 4 R2"-' whose singularities are only Whitney's umbrellas. We call such f : U 4 R2"-l a stable

324 M. Ohsumi

perturbation of f : (R",O) (R2"-l,0). We are interested in the number of Whitney's umbrellas of f. Let En be the ring of C" function-germs of (R",O) to R . Let f : (R",O) -+ (R2"-l,0) be a C" map-germ. Let Z(C1(f)) be the ideal in &, generated by n x n minor determinants of the jacobian matrix of f . --f

Main Theorem 1 (M.Ohsumi, [ 2 5 ] ) . Let f : (R",O) -+ (R2"-l,0) be a generic C" map-germ such that dimR&,/Z(Cl(f)) < +oo. The number of Whitney's umbrellas that appear in a stable perturbation off is equal to dimR&/Z(C'(f)) (modulo 2) and it is a topological invariant o f f . Here we call a map-germ generic map-germ in a strong sense. See Definition 4.4 in 54 for the precise definition.

Remark 1.1. The statement that the number of Whitney's umbrellas that appear in a stable perturbation is equal to dimR E,/Z(C1 (f))(mod 2) is a consequence of [4], [5], [20], [23], [24]. Our assertion in the above theorem is that it is a topological invariant o f f . 2. History of the problem

The problem of counting isolated singular points in stable perturbations of a degenerated map-germ is old and new. The case of complex holomorphic functions is rather classical. Let f : (C", 0 ) 4 ( C ,0 ) be a holomorphic function-germ which defines an isolated singularity at 0. It is well known that the Milnor number p ( f ) o f f is the number of critical points of a Morse function near f and it is a topological invariant o f f (J.W.Milnor [HI). In the real case also, it is known that for a C" map-germ f : (R",0) --f (R,O) with p(f) < +co, p ( f ) modulo 2 is a topological invariant of f (C .T.C .Wall [29]). The problem in the case of map-germs was investigated first by T.F'ukuda and G.Ishikawa [3]. Let f : (R2,0)--f (R2,0)be a generic C" map-germ, let U be a sufficiently small neighborhood of the origin and let f : U ( C R2) R2 be a representative mapping o f f . Then we may suppose that has no degenerate singular points except for the origin. By Whitney's theorem [34], 7 can be approximated by a C" stable mapping f : U 4 R2. The degenerate singularity of 7 at the origin of R2 bifurcates into stable singular points o f f . Again by Whitney's theorem [34], the singular points of f are A-equivalent to one of the following two map-germs from (R2,0 ) to (R2,0): --f

7

Szngulallities appearing in a stable perturbation of a m a p - g e m

(1) ( G Y ) (2) ( % , Y )

(Z,Y2),

325

fold

(w3 +XY),

CUSP.

Suppose that f : (R2,0) -+ (R ,0) is generic and U is a sufficiently small neighborhood of the origin so that 7 : U -+ R2 has only fold singular points off the origin. The cusp singular points of f are isolated. Let f1 and f 2 denote the component function germs of f:

f

= ( f l , f 2 ) : (R2,0)

+

(R2,0).

Let J f = J ( f 1 ,f 2 ) denote the Jacobian determinant o f f :

Set

Throughout this paper we use the following notation.

( a ,b, . . - )

denotes the ideal generated by a, b,

. ...

Theorem 2.1 (T.Fukuda and G.Ishikawa, [3]). Let f : (R2,0) -+ (R2,0 ) be a C" m a p - g e m such that dimR&/(Jf, J1 f , J2 f ) < +oo. Then the following holds for any stable perturbation f : U ( C R2) -, R2 o f f . (1) The number of cusps off that appear near the origin is less than or equal to dimR&2/(Jf,Jlf,Jzf). (2) The number of cusps o f f that appear near the origin is equal to d i m ~ & / ( J f J, l f , J 2 f ) modulo 2. (3) The number modulo 2 of cusps of is a topological invariant of

f. In the complex case, T.Gaf€ney and D.Mond [8] showed that Theorem 2.1 holds more precisely. Let f : (C2,0) -+ (C2,0) be a holomorphic mapgerm, let U be a sufficiently small neighborhood of the origin in C2 and let : U(c C2) + C2 be a representative mapping of f. Then Whitney's theorem [34] also holds in the complex case, and can be approximated by a stable holomorphic mapping f : U C2 that has only fold and cusp type singular points.

7

7

-+

326

M. Ohsumi

Theorem 2.2 (T.Gaffney and D.Mond, [ 8 ] ) . Let f : ( C 2 , 0 ) 4 ( C 2 0, ) be an analytic map-germ such that dimc O 2 / ( J f ,J l f , J z f ) < +a. Then the following holds for any stable perturbation .f : U(C C 2 )+ C 2 of

f. The number of cusps off that appear near the origin is equal to dime O z / ( J f ,J l f , J 2 f ) . (2) The number of cusps off that appear near the origin is a topological invariant of f . (1)

Remark 2.3. The conditions that d i m ~ & z / ( J fJ1 , f,J z f ) < +co and that dimc O z / ( J f , J l f , J 2 f ) < +co are generic conditions in a strong sense. That is, the set of map-germs which do not fulfill this condition is of 03codimension in the set of all map germs. Apart from the problem of topological invariance, the study on the number of 0-dimensional singular points in generic perturbations of a degenerate map-germ is recently widely developed. For a k-tuple of integers I = ( i l ,i z , . . ,i k ) with il L i 2 L . . . L i k L 0, there is a submanifold C' of J1(Cn,CP)(Z 2 k) called Thorn-Boadman singularity set with symbol I . We will not give the definition of C', see [l] and [21] for the definition. If codim C' in J'(C", CP) = n,then for a generic mapping f : C" + CP, singular points o f f with type C' appear isolatedly. D.Mond [20] investigated the number of C1 type singular points, that is, Whitney's umbrellas, for a holomorphic map-germ ( C 2 ,0) 4 ( C 3 0). , A generalization of Theorems 2.1, 2.2 and 1201 on the number of codim n Thorn-Boardman singular points was first done by J .Nuiio Ballesteros and M.Saia [23], then was followed by T.fikui, J.Nufio Ballesteros and M.Saia [4],J.Nuiio Ballesteros and M.Saia [24],T.Fukui, J.Nuiio Ballesteros and M.Saia [5], T.Fukui and J.Weyman [6]. For a holomorphic map-germ f : (Cn,O)4 (CP,O),we suppose that j ' f ( O ) ~ $ in J'(C", CP). Let Z(C') denote the defining ideal of the setgerm C' in (J'(C",CP),j'f(O)):

Z(C') = {(. E OJ'(Cn,CP),jlf(O)I(.Is= 0)

c 0J'(Cn,CP),j'f(O)

and we define an ideal Z(C' (f )) in On by

Z(C'(f))= (j'.f)*(Z(c')). For example, the Thom-Boardman singularity of cusp singularity ( C 2 0, ) + ( C 2 0, ) is C1ilio and we have

C1AO = C l J .

Singularities appearing in a stable perturbation of a m a p - g e m 327

And for a holomorphic map-germ f : ( C 2 0, ) ( C 2 0, ) ,the ideal Z(C1il(f)) is the ideal (Jf,J1f , J2 f) appeared in Theorems 2.1 and 2.2. The Thom-Boardman singularity of Whitney's umbrella (C", 0 ) -+ (C2"-l,0) is and we have -+

And for a holomorphic map-germ f : (C",0 ) + ( C P0, ) with n I P and for CZ' = { j ' g ( q ) E J'(C", CP)IcorankJg(q) = il},

+

Z(Cil(f)) is the ideal generated by ( n - il 1) x ( n - i l + 1) minor determinants of the jacobian matrix of f and for a map-germ f : (Cn,O)+ (C2"-', 0 ) ,Z(C1( f)) is the ideal generated by n x n minor determinants of the jacobian matrix of f , which appeared in our main theorem. Theorem 2.4 (T.Fukui, J.Nuf~oBallesteros and M.Saia, [ 5 ] , [24]). Let f : (C",O) (CP,O) be a holomorphic m a p - g e m such that dimcO,/Z(C'(f)) < +m. Then the following properties hold for any generic perturbation f : U ( C C") + CP off. (1) The number of singular points of type C' off is equal to or less than dimc O,/Z(C'(f)). (2) The number of singular points of type C' of f is equal to d i m c o n / Z(C'(f)) if and only i f the Zarislci closure of C' is CohenMacauZay at a point j k f ( 0 )E c'. (3) When the length is equal to 1, the Zarislci closure of C' is always Cohen-Macaulay at j1f (0). ---f

Remark 2.5. T.Fukui and J.Weyman [6,7] investigate when the Zariski closure of C' is Cohen-Macaulay and proved that the defining ideals of the Zariski closure of some C i ~ jfor , example C 2 i 1 ( ( n , p )= (3,2)), C 3 > 1 ( ( n , p= ) (4,2)), are Cohen-Macaulay. In the real case, for a C" map-germ f : (R",0) -+ (RP,0), the defining ideal Z(C'( f)) can be defined in the same way as in the complex case. From Theorem 2.4, we have

Theorem 2.6. Let C' have codimension n. Let f : (Rn,O)-+ (RP,O) be a C" m a p - g e m such that dimR&,/Z(C'(f)) < fm. Let U be an open neighborhood of the origin 0 in R" and let f : U(cR") + R P be a generic perturbation off. Then the number of singular points of type C ' that appear in f is equal to dimR€$(C'(f)) modulo 2.

328 M. Ohsumi

As seen in the above, the numbers of singular points that appear in generic perturbations of map-germs are well investigated. However, strangely enough, the topological invariance of these numbers is not considered after [3], [8]. Thus, the following natural problems arise.

3. Problems

(1) Let C' be a Thom-Boardman singularity with codimension n. (i) Is the number of singular points of type C' that appear in a generic perturbation f : U ( C C") -+ CP of a holomorphic map-germ f : (C",0) t (CP,0) a topological invariant of f ? (ii) Is the number (modulo 2) of singular points of type C' that appear in a generic perturbation f : U ( c R") -+ RP of a Coomap-germ f : (Rn,O)-+ (Rp,O) a topological invariant of f ? As special cases of these problems, we have the following questions. (2) Do the assertion of our main theorem in [25] also hold in the complex case ? That is, is the number of Whitney's umbrellas that appear in a stable perturbation of f : U(CC") + C2"-l of a holomorphic map-germ f : (C", 0) -+ (CZn-l,O)a topological invariant of f ? (3) Let f : (R4,0) (R4,0) be a degenerate C" map-germ. Then there are two kind of Thom-Boardman singularities C2 and C1>l9l>lwhich may appear in a stable perturbation f of f . Are the numbers (modulo 2) of C2 singular points and C1ilil>l topological invariant of f ? (4) Let f : (R",0) --+ (RP,0) be a A-finite Coomap-germ. For any integer k with k 2 0 such that 5 = dirnREn/Z(C'(f)) (mod 2), does there exist a stable perturbation f of f such that the number (mod 2) of C' singularities that appear in f is k ? We can generalize Problem (1) as follows: (5) Let CI be a Thom-Boardman stratum with codimension less than or equal to n and f : (Kn,O) -+ (Kp,O) be a degenerate map-germ. For a stable perturbation f of f , what can we say about the topology of C'(f) in terms of invariants o f f ? --f

4. A generic property of map-germs We recall T.F'ukuda's theorem [2] on generic properties of C" map-germs. Let C"(Rn, Rp;0,O) denote the set of all C" map-germ from (R",0) to (RP,0). Let 7rT : C"(Rn, RP;0,O) t J'(R", RP) be the canonical projection defined by 7r,.(f) = j'f(0). A subset C of Cm(Rn,RP; 0,O) is said to be m-codimensional in C"(Rn, RP;0, 0), if for any positive integer k, there

Singularities appearing in a stable perturbation of a map-germ 329

exist a positive integer r and a semi-algebraic subset Ck in J'(Rn, RP) with codimension 2 Ic such that C c 7r;l(Ck). SincedimRE,/Z(Cl(f)) < +m ifandonlyifZ(C'(f)) 3 ( x i , . . . , Z m ) k for some k, we have Lemma 4.1. The set

C* = {f E C"(Rn,Rp;O,O)IdimR€,/Z(C'(f))

= +OO}

is an m-codimensional subset of Cm(Rn,Rp; 0,O). Theorem 4.2 (T.F'ukuda [2], Theorem 1). Let X be a semi-algebraic submanifold of the multi-jet space ,Jk(Rn,RP). Then there exists an m-codimensional subset C , of Cm(Rn,RP;O,O)such that any f E C"(Rn, Rp; 0,O) - C, has a C" representative 7 : U -+ RP that satisfies the following two properties; (I) for any m-tuple S = {XI,.. . ,x m } of distinct points of U - { 0 } , the multi jet extention ,jk7 : U(") -+ , Jk(Rn,RP) is transversal to X at

.. .

(21, lxrn),

(2)

if codimX 2 mn, then , j k f ( ( u- { o ) ) ( ~ )n) x = 0.

As an easy Corollary of Theorem 4.2, we have Corollary 4.3. There exists an m-codimensional subset C , of Cm(Rn, R2n-1;0,O) such that any f E Cm(Rn,R2n-1;0,O) - C, has a C" representative 7 : U -+ R2n-1 that satisfies the following properties. (1) 7 has no singular points except for the origin, . . ,x, are distinct points in U - (0) such that T(x1) = (2) if x1,x2,. f ( x 2 ) = . . . = f (xm),then the images of the germs of 7 at x l , x 2 , . . . ,x, meet transversally at y = f(x1) = 7 ( x 2 ) =_:.. = f(x,), (3) as a consequence of (1) and (2), f : U - ( 0 ) --t R2n-1as A-stable. '

Definition 4.4. A map-germ f : (Rn,0) + (R2,-', 0) is said to be generic if dirnR€,/Z(C1( f)) < +m and f has a representative 7 : U -+ RZn-' that satisfies conditions (l),(2) and (3) in Corollary 4.3. Such a representative f : U + R2n-1is called a proper representative of f . Lemma 4.5. Let f : (R", 0) -+ (R2n-1,0) be a generic C" map-germ and let 7 : U -+ R2n-1 be a proper representative of f , U being a suficiently small neighborhood of the origin 0 E Rn. Let U' be an open neighborhood of 0 such that 0E

u' c T7' c u.

330

M. Ohsumi

Let f : U + R2n-1 be a stable perturbation of 7 suficiently close to T h e n the restricted mappings 71u-vt and f"lv-otare A-equivalent.

7.

Proof. By Corollary 4.3 (3), we have that 71~-{0} : U - (0) 4 R2n-1is A-stable. Thus the restricted mapping 71v-~lis d-stable. Since f : U -+ R2n-1approximates 7 : U 4 R2n-1sufficiently closely with respect to the Whitney topology of C"(U, R2n-1;0, 0), f[v-ut is also sufficiently close to flu-ot with respect to the Whitney topology of C"(U - R2n-1;0,O). Since 71u-o,is A-stable, f"lv-a and 7Iv-a are d-equivalent from the definition of stability. 0

v',

Remark 4.6. Even when f : U -+ R2n-1 approximates 7 : U 4 R2n-1 sufficiently closely with respect to the Whitney topology of C"(U, R2n-1;0, 0), it is not necessarily that f(v-{o} : U - (0) -+ R2n-1 approximates flu-(o) : U - (0) + R2n-1sufficiently closely with respect to the Whitney topology of C"(U - {0), R2n-1;0,O). Therefore even if T~,TJ-{~} is d-stable, we can not claim that Tlv-co} and flv-{o} are d-equivalent. 5. Double points of a mapping The key of the proof of our main theorem is an observation of double points of a mapping.

Definition 5.1. A double point of a mapping f : X + Y is a point x for which there exists a different point y from x such that f(3)= f ( y ) . We denote by D(f) the set of double points o f f . Example 5.2. The double point set of Whitney's umbrella f : Rn R2n-1 7 f(x17

.. .

4

2

7

Zn) = (XI?... 7xn-1,xn7x1x~,* . * 7xn-1xn)7

is given by

D(f) = { ( 0 7 * . . 7o,%)Ixn #o). The singular . . . ,0)) of Whitney's umbrella f is coincident - point set { (0, with D ( f ) - D ( f ) , where D ( f ) is the topological closure of D ( f ) .See Figure 1. From Corollary 4.3, we have

Singularities appearing in a stable perturbataon of a map-germ

331

Singular point

f

Fig. 1. Whitney's umbrella.

7

Lemma 5.3. For a proper representative : U ( c R") R2"-l of a generic m a p - g e m f : (R",O) -+ (R2"-',0), D ( f ) is a smooth curve and consists of a finite number of connected components. -+

Definition 5.4. Let f : (R",O)+ (R2"-',0) be a generic C" map-germ and let 7 : U ---f R2"-l be a proper representative o f f . Then, D ( 7 ) consists of E v e n number of connected smooth curves which we call half branches of D ( f ) .For every half branch y of D ( f ) ,there exists a distinct half branch y* of D ( 7 ) such that for every point z of y,there exists a point y of y* with f(z)= f ( y ) , which we call the partner branch of y. We call the union of a half branch, its partner branch and the origin, y U y*U (01, a branch of D ( 7 ) . Lemma 5.5. Let f, g : (R",0 ) + (R2n-1,0) be generic C" map-germs and let 7 : U -+ R2"-l and ij : V -+ R2"-l be their proper representative mappings respectively. If and g are topological equivalent, that is, if there exist homeomorphism&: UV and h2 : R2"-l + R2"-l such that h2 o = g o h l , then D ( f ) and D ( g ) are homeomorphic and the number of branches of D ( 7 ) and the number of branches of coincide.

7

-+

7

From Lemma 5.5, to prove the main theorem, it suffices to prove Theorem 5.6. Let f : (Rn,O)+ (R2"-',0) be a generic C" map-germ and let 7 : U -+ R2"-l be a proper representative mapping off, U being a suficiently small neighborhood of the origin 0 E R". Then the number of Whitney's umbrellas of a s t a b a r t u r b a t i o n f : U --+ R2"-l off is equal to the number of branches of D ( 7 ) modulo 2. Theorem 5.6 was proved in [25].

332

M. Ohsumi

6. Some examples

In this section, we observe d-simple map-germs (R2,0) --t (R3,0) classified by D.Mond [19], and see that Theorem 5.6 holds for them. Theorem 6.1 (D.Mond, [19]). Each of the germs in the following list is d-simple, and every d-simple germ of a map from a 2-manifold to a 3-manifold is equivalent to one of the germs o n the list.

d-codimension 0

Name Immersion Cross-cap (So)

2

s:

k+2 k+2 k+2

Blcf

c#t

6

F4

k+2

Hk

Example 6.2. We consider the normal form S,f :

frcf

: (R2,0)--+ (R3,0)

given by f$(ZIY) = ( ~ , Y ~ , Y ~ * ~ ~ + ' Y k) ,2 1.

Since Z(C'(f,f)) is the ideal generated by 2 x 2 minors of the jacobian matrix 0

( +: f(k

1)zky 3y2 f 2Yzk+'

)

of f , we have

Z(C'(fkf)) = ( Y J k + l ) and we have dimR €2/Z(C1(f,')) = k

+ 1.

On the other hand,

O ( f t ) = {(z, Y)I

Xk+l

4-Y2 = 0, Y # 01,

Thus we see that -

the number of branches of O(fk+)=

{::

ifk+l-1 ifk+l=O

mod2 mod2

Singularities appearing in a stable perturbation of a map-gem 333

the number of branches of D(&) = Hence we have

ifk+l=l ifk+l=O

1, 2.

-

the number of branches of D(f,f) = dimR&/Z(C'(fk*)) as Theorem 5.6 asserts. For any integer 1 with 0 >

are stable perturbations of f . f& has no Whitney's umbrellas and Whitney's umbrellas of are the points (el,O), ( E Z , ~ ) .Thus the number of Whitney's umbrellas of and f,& is exactly 0 and 2 respectively. See Figure 8, 9, 10.

ft2

!to

Fig. 8.

B2

: fT(z,y)

= (z,y2,z2y -y5), d i m ~ & / Z ( x ( f T ) )= 2.

Singulnrities appearing in a stable perturbation of a map-germ 337

Example 6.4. We consider the normal form C,f : f : : (R2,0) + (R3,0) given by fkf(5,y) = ( X , Y ~ , Z Yf ~z 'Y),

k 2 3.

Since Z(C1(ft)) = (y, zk),we have dimR &z/Z(Cl(f:))= k. On the other hand,

Wk+)= {(Z,Y)I

ZY2

+ xk = 0, Y # 01,

-

the number of branches of D(fkf) =

the number of branches of D ( f L ) =

(3: {i:

ifk=1 ifkEQ

mod2 mod2

ifkE1 ifkEO

mod2 mod2

Hence we have

-

the number of branches of D ( f t ) E dirn~&2/Z(Cl(fk*)) (mod 2) as Theorem 5.6 asserts. For any integer 1 with 0 5 1 5 k and with 1 E k (mod 2), we have a stable perturbation of f

f,$ : U ( C R2) -+

R3

338

M. Ohsumi

such that the number of Whitney's umbrellas of f& is exactly 1, constructed as follows. Let €1,. . . ,€1 be sufficiently small distinct real numbers. Set k-1 m=and let 61, . . . , dm be small positive numbers. Then, 2 $;l(~,

Y) = (2,y2, zy3

*

+ 61)

. . . (x -

~ (-2~ 1 )

* * *

(x2

+

Jm))

is a stable perturbation of f. Whitney's umbrellas of f;l are the points is exactly ( ~ 1 , 0 ) ,. . , ( Q , 0). Thus the number of Whitney's umbrellas of 1.

ftl

Example 6.5. Now we consider the normal form F4 : f : (R2,0) 4 (R3,0) given by

f(x,Y) = (x,Y2, Z3Y + Y5). Since Z(C'(f)) = (y,x3),we have dimR&/z(Cl(f)) = 3.

On the other hand,

q f ) = ((2,Y)I

x3

+ Y4 = 0, Y # 0).

Thus we see that

the number of branches of D ( 7 ) = 1.

Hence we have

-

the number of branches of D ( 7 ) E d i m ~ & / Z ( C l ( f ) ) (mod 2) as Theorem 5.6 asserts. For any integer 1 with 0 5 1 5 3 and with 1 = 3 (mod 2), (that is, 1 is 1 or 3), we have a stable perturbation of f

fi : U ( c R2)

-+

R3

'6

such that the number of Whitney's umbrellas of is exactly 1, constructed as follows. Let ~ 1 €2, , € 3 be sufficiently small distinct real numbers. Let 61 be a small positive number. Then, $l(GY) = ( 2 , Y 2 , Y 5 +Y@

-

+Sd),

$3(5,Y) = ( 2 , Y 2 , Y 5 + Y ( 5 - E 1 ) ( 2 - - 2 ) ( 5 - - 3 ) )

are stable perturbations of f. Whitney's umbrella of $1 is the point ( E ~ , O ) and Whitney's umbrellas of f3 are the points ( ~ 1 , 0 )(, ~ 20), , ( ~ 3 ~ 0Thus ) . the number of Whitney's umbrellas of $1 and $3 is exactly 1 and 3 respectively.

Singularities appearing in a stable perturbation of a map-gem

Example 6.6. Now we consider the normal form (R3,0) given by 3

H k : fk :

339

(R2,0) -+

k 2 2.

fk(z,Y) = (zc,Y ,zY+Y3"'), Since z(c'(fk)) = (s,y2), we have dim~&2/z(C'(fk))= 2. On the other hand,

Thus we see that the number of branches of

D(5)= 0

Hence we have the number of branches of D ( 5 ) = dirnR&z/I(C'(fk))

(mod 2)

as Theorem 5.6 asserts. For any integer 1 with 0 5 1 5 2 and with 1 = 2 (mod 2), (that is, 1 is 0 or 2), we have a stable perturbation of f

such that the number of Whitney's umbrellas of fk,l is exactly 1, constructed as follows. Let E be a sufficiently small positive real number and the ci are real numbers with c1 = -2~&, c2 = 2e&, lcil < 2 f i ~ f i for i 2 3, and ci # cj for i # j. Let b be a small positive number. Then,

ZY -k !/3k-1), fk,2(z,Y) = (2,Y3 - 3 EY,zY -tY2 fk,O(z,

Y)

= (2, !I3 -k

k-1

3-

(Y

3 EY - Ci)))

are stable perturbations of f . Where fk,2 is cosidered in [9]. fk,O has no Whitney's umbrellas and Whitney's umbrellas of fk,2 are the points (0, &), (0, -&). Thus the number of Whitney's umbrellas of fk,O and fk,2 is exactly 0 and 2 respectively. Thus, for the map-germs f in the list of Mond's normal form, we have the following table and we see that dim~&2/2(C'(f)) E the number of branches of D ( T ) (mod 2) as Theorem 5.6 asserts.

340

M. Ohsumi

-

k+l 2 2

k k 3 2

number of branches ifk+l-l ifk+l=O ifk+l-l ifk+l=0

{i: {i:

of D ( f ) mod2 mod2 mod2 mod2

0,

2,

{i:

ifkEl if k - 0 ifk-1 ifkEO

mod2 mod2 mod2 mod2

1, 0.

Acknowledgments The author would like to thank Professors T.F'ukuda and T.Fukui for their helpful advices.

References 1. J.M.Boardman, Singularities of differentiable maps, Inst. Hautes Etudes Sci. Publ. Math. 33 (1967) 21-57. 2. T.Fukuda, Local Topological properties of differentiable mappings. I, Invent. Math. 65 (1981) 227-250. 3. T.Fukuda and G.Ishikawa, On the number of cusps of stable perturbations of a plane-to-plane singularity, Tokyo J.Math. 10 (1987) 375-384. 4. T.Fukui, J.Nuiio Ballesteros and M.Saia, Counting singularities in stable perturbations of map-germs, Siirikaisekikenkydsho k6kyiiroku 926 (1995) 1-20. 5. T . h k u i , J.Nuiio Ballesteros and MSaia, On the number of singularities in generic deformations of map germs, J.London Math. SOC.(2) 58 (1998) 141152. 6. T.Fukui and J.Weyman, Cohen-Macaulay properties of Thom-Boardman strata I, Morin's ideal, Proc. London Math. SOC.(3) 80 (2000) 257-303. 7. T.Fukui and J.Weyman, Cohen-Macaulay properties of Thom-Boardman strata 11, The defining ideals of Ci9j,Proc. London Math. SOC.(3) 87 (2003) 137-163. 8. T.Gaffney and D.Mond, Cusps and double folds of germs of analytic maps C2 + C2, J. London Math. SOC.(2) 43 (1991) 185-192. 9. W.L.Marar and D.M.Q.Mond, Real map-germs with good perturbations, T O ~ O 35 ~ O(1996) ~ Y 157-165. 10. J.Mather, Stability of Cm-mappings I. The division theorem, Ann. of Math. 87 (1968) 89-104. 11. J.Mather, Stability of Cm-mappings 11. Infinitesimal stability implies stability, A n n . of Math. 89 (1969) 254-291.

Sangularities appearing an a stable perturbation of a map-genn 341 12. J.Mather, Stability of Coo-mappings 111. Finitely determined map-germs, Publ. Math. Inst. Hautes Etudes Sci. 35 (1968) 127-156. 13. J.Mather, Stability of Coo-mappings IV. Classification of stable germs by R-algebras, Publ. Math. Inst. Hautes Etudes Sci. 37 (1969) 223-248. 14. J.Mather, Stability of Coo-mappings V. Transversality, Advances in Math. 4 (1970) 301-336. 15. J.Mather, Stability of Coo-mappings VI. The nice dimensions, Springer Lecture Notes in Math. 192 (1971) 207-253. 16. J.Mather, Stratifications and mappings, Proceedings of the Dynamical Sytems Conference, Salvador, Academic Press (1971). 17. J.Mather, How to stratify mappings and jet spaces, Springer Lecture Notes in Math. 535 (1976) 128-176. 18. J.W.Milnor, Singular Points of Complex Hypersurfaces, Priceton University Press (1968). 19. D.M.Q.Mond, On the classification of germs of maps from R2 to R3, Proc. London Math. SOC.(3) 50 (1985) 333-369. 20. D.M.Q.Mond, Vanishing cycles for analytic maps, Singularity theory and its applications, Lecture Notes in Math. 1462 (Springer, 1991) 221-234. 21. B.Morin, Calcul jacobien, Ann. Sci. &ole Norm. Sup. 8 (1975) 1-98. 22. T.Nishimura, Singular points and Mather’s theory, Mathematical of singular points vol. 2, Singularities and bifurcationpart 1, Kysritu Publisher (in Japanese) (2002). 23. J.Nufio Ballesteros and M.Saia, An invariant for map germs, (preprint, 1995). 24. J.Nuiio Ballesteros and M.Saia, ‘Multiplicity of Boardman strata and deformations of map germs’, Glasgow Math. J. 40 (1998) 21-32. 25. M.Ohsumi, Whitney’s umbrellas in stable perturbations of a map germ to appear in Tokyo J.Math.. (Rn,O) + (RZn-’,0), 26. R.Thom, Les singularit& des applications diffkr6ntables, Ann. Inst. Fourier 6 (1955) 43-87. 27. R.Thom, Un lemme sur les applications diffbrkntiables, Bol. SOC. Math. M e d c . 2nd series 1 (1956) 59-71. 28. C.T.C.Wal1, Finite determinacy of smooth map-germs, Bull. London Math. SOC.13 (1981) 481-539. 29. C.T.C.Wal1, Topological invariance of the Milnor number mod 2, Topology 22 (1983) 345-350. 30. H.Whitney, Differentiable manifolds, Ann. of Math. 37 (1936) 645-680. 31. H.Whitney, The general type of singularity of a set of 2n-1 smooth functions of n variables, Duke Math. J. 10 (1943) 161-172. 32. H.Whitney, The self-intersections of a smooth n-manifolds in 2n-space, Ann. of Math. 45 (1944) 220-246. 33. H.Whitney, The singularities of a smooth n-manifold in (2n- 1)-space, Ann. of Math. 45 (1944) 247-293. 34. H.Whitney, On singularities of mappings of Euclidean spaces I. Mappings of the plane into the plane, Ann. of Math. 62 (1955) 374-410.

342

Existence problem for fold maps Kazuhiro Sakuma

Department of Mathematics, Brigham Young University, Provo, Utah 84602, U.S.A. 8 Department of Mathematics, Kinki University, Osaka 577-8502, Japan E-mail: sakuma0math. byu.edu, sakumaamath. kindai.ac.jp This is a survey article on the existence problem of fold maps. Let M" be a closed n-manifold and f : M" + WP a generic smooth map into the p dimensional Euclidean space with n 2 p . If f has only fold singularities as its singularities, then f is called a fold map. When p = 1, a fold map is nothing but a Morse function. Then we will consider a problem belonging t o global singularity theory: Find the necessary and/or s u f i c i e n t condition(s) for the existence of fold maps. In the problem for the cases where p = 3 and p = 7, the existence of fold maps has a special position if we assume that the Euler characterisitc of the source manifold should be odd. The main purpose of the paper is to discuss the problem in the case that p = 3 , 7 respectively by summarizing recent results in relation with the Thom polynomials, the Eliashberg-Ando h-principle theorem, etc. We will also give a new result on the necessary and sufficient condition for the existence problem of fold maps of odd dimensional manifolds into W3.

Keywords: Fold map; Thom polynomial; Stiefel-Whitney class; Pontrjagin class; stable span; Postnikov invariant

Contents 1. Introduction 2. Thom polynomials 3. Eliashberg's h-principle for fold maps 4. Non-existence of fold maps and the Hopf invariant one problem 5. Fold maps for p = 3 6. Fold maps for p = 7 7. Remarks and Problems Appendix A Appendix B References

343 344 348 352 358 366 369 372 380 383

Exastence problem for fold m a p s

343

1. Introduction As is well known, every smooth n-manifold M admits a Morse function, f : M 4 R.A Morse function is a smooth function with only non-degenerate critical points. According to the Morse Lemma (see e.g. [52]), f has the following normal form around a critical point q:

y o f =*z+.*z:, where (XI,. . . ,s,) and y are local coordinates centered at q E M and f (q) E R,respectively. If we try to generalize the notion of non-degenerate critical points of a smooth function, we have only to consider a smooth map between manifolds and its singular points with a specific condition. For a smooth map f : M" --+ Np between manifolds with n 2 p , we define the set of singular points of f as

S(f)= ( 4 E M ; rankJ.f(q) < PI, where J t ( q ) is the Jacobian matrix of f at q E M". For q E S(f ), if we can choose local coordinates ( 5 1 , . . . , 2,) centered at q and ( y l , . . . ,y p ) centered at f ( q ) respectively such that f has the following normal form: Y i O f

=xi ( i '( f ) is the fold locus which is a 1-dimensional submanifold of M" and En-',' the cusp locus consisting of finite number of points (see Appendix A for the precise notations). In fact, Thom proved that for a stable map f : M n -, R2 we have

where [C"-'>'(f)]; stands for the Poincark dual of the mod 2 homology class [Cn-lt1(f)]2represented by the cusp locus C n - ' J ( f ) . From (2.1) we can immediately deduce that the number of cusp points is congruent modulo 2 to the Euler characteristic of M n , which is a direct generalization of Whitney's assertion mentioned above. The formula (2.1) is a typical example of a Thom polynomial which tells us the homological arrangement of the singular set for a given smooth map. Roughly speaking, the Thom polynomial is the Poincare dual of the homology class represented by the closure of the set C ( f ) of the singular set of f with type C. In fact, Thom gave the first result on the Thom polynomial, for instance, as follows [78]:

Theorem 2.1 (Thom). Let f : M" -+ RP be a smooth map of a closed n-manifold into the p-dimensional euclidean space (n 2 p ) such that j1f : M n -+ J ~ ( M "RP) , is transversal to C i ( M n , R P )= { j l g ( x ) E J 1 ( M n , R P ) ;rank dg, = i }

for 0 5 i 5 p . Then i f we regard the singular set S(f ) o f f as the mod 2 cycle in M", the Poincare' dual cohomology class [ S ( f ) ] ;E H"-P+l(M";Zz) coincides with the ( n - p + 1)-st Stiefel-Whitney class W , - ~ + ~ ( M "of) M " , i.e. [ S ( f ) ] a= W , - ~ + ~ ( M "E) Hn-''l(Mn;Z2).

(2.2)

For instance, due to (2.2), for a stable map f : M 4 --+ R3 of a closed orientable 4-manifold M 4 we see that the singular set of f represents an obstruction t o a spin structure of M 4 . Now we set

W f )= ( f f ) - W

Existence problem for fold maps 347

for a generic smooth map f : M -+ N such that j1f is transversal to the Thom-Boardman singularities C' c J'(M, N) (see Appendix A). Then it follows that the closure V(f)carries a Z2-homology class of M .

Theorem 2.2 (Thorn [78]).If M is a closed manifold, the Poincare' dual of the Z2- homology class of M represented by C r ( f ) can be expressed b y a polynomial of wi(M) and f*w.j(N), where i = l , 2 , ...,dimM or j = 1 , 2,...,dimN. Moreover, the polynomial expression does not depend o n the choices of M , N and f. This polynomial is called the Thorn polynomial of the singular set C ' (see [30] more precisely), which is determined only by a given singularity type and tells us the information on the homological location in the source manifold. To determine the Thom polynomial expression for a given singularity type is one of the most important but difficult problems by itself in global singularity theory. Recently, a breakthrough has been established by Richard Ftimaliyi ( [59]) using the generalized Pontrjagin-Thom construction (see [58]) skillfully. See also recent progress in [25], [26] and [35]. Here let us explain a Morin singularity for the later convenience. Let f : M n + NP be a smooth map with n 2 p and S ( f ) the singular set of f. Then a point x E S ( f ) is called a Morin singularity of type A k of f (1 5 k 5 p ) if there exist local coordinates (21,. . . ,2,) centered at z and ( y l , . . . ,y p ) centered at f (2)E Np such that f has the following form

( [541, [281): yi 0 f = xi Yp 0 f

= X kP+ l

(i = 1 , . . . , p - 1)

+

c

k-1

Xixi-i

f

f . . . f2,2.

a=1

In particular, 2 E S ( f ) is called a fold singularity of f if k = 1, a cusp singularity if k = 2 and a swallowtail singularity if k = 3. If a smooth map f : M" 4 NP has only Morin singularities as its singularities, then we call f a Morin map. Note that a Morin map satisfies the jet transversality condition in Theorem 2.1. In particular, a fold map is a special case of a Morin map. We may say that a Morin map is a smooth map with only the simplest family of singularities in some sense. Also, we can consider the analogous problem for Morin maps (see our problem posed in the Introduction) but in this paper we concentrate mainly on fold maps as possible as we can for avoiding divergence of our discussion.

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K. Sakuma

3. Eliashberg's h-principle for fold maps

The following problem was posed by J. Mather around 1970, which combines singularity theory with homotopoy theory: Does an arbitrary element of the homotopy group 7rn(SP)contains o fold map of S" into SP provided that n 2 p ? In 1201 Y. Eliashberg completely solved the problem by giving the affirmative answer with his h-principle theorem for fold maps. Here we shall explain the h-principle. Let J k ( M ,N ) --f M be a k-jet bundle over M and r ( M ) the space consisting of all continuous sections of the bundle. Then we obtain a continuous map

jk : C - ( M , N ) -+ r ( M ) defined by jk(f(x))= j k f ( x ) for any 2 E M , where j k j is the k-jet extension of f (see Appendix A). Let R be any open subset of J k ( M ,N ) and then we say that R is a differential condition of order k. Further, we set

C,-(M, N ) = {f E c - ( M , N);j'f (MI

c GI,

If the induced embedding

is a weak homotopy equivalence, then we say that R satisfies the w.h.e.principle; if jk induces a surjection

(P)* : .o(Cg(M,

N ) ) + no(m(M)),

then we say that the condition R satisfies the h-principle. If the h-principle holds, one can discuss the problem in homotopy theory. For example, let R be the differential condition of order one consisting of all the injections in Hom ( T M ,T N ) = J 1 ( M ,N ) . Then the condition R satisfies the w.h.e.-principle if dim M < dim N , which is nothing but the Smale-Hirsch immersion theorem ( [32]). There are many other results in the case that dim M < dim N or M is an open manifold, however, only a few results are known if dim M 2 dim N for a closed manifold M . Now, let us explain Eliashberg's h-principle for fold maps. Let M and N be smooth manifolds of dimensions n and p respectively with n 2 p . Let F ( M ,N ) be the space of all fold maps of M into N endowed with the Cm-topology. Of cource, F ( M ,N ) may be an empty set depending on the choices of M and N .

Existence problem f o r fold maps

349

For Q, E Hom(TM,TN), x E M is called a singular point of Q, if for the continuous map cp induced by Q, we have rank (a, : TM, -+ TN,(,)) < p. We denote by M(M, N ) the space of all Q, E Hom(TM, T N ) endowed with the compact open topology such that for each singular point x E M of a, there are a neighborhood U c M around x and a fold map fu : U -+ N satisfying dfu = Q,.(TU.For a singular point of Q, E M ( M ,N ) , an index X E A = {0,1,. . . , [(n- p 1)/2]} is defined and we denote the set of all singular points of Q, by S(Q,)and the set of all singular points of Q, with index X by S A(a). Let V,, i E A, be non-vacuous mutually disjoint ( p - 1)-dimensional closed submanifolds of M. We denote by F ( M , N ; V , , i E A) (resp. M(M,N ;K, i E A)) the subspace of F ( M , N) (resp. M ( M , N ) ) consisting of the smooth maps f : M -+ N with S x ( f ) = VA (resp. the homomorphisms Q, E Hom(TM,TN) with SA(Q,)= VA)for all X E A. Then we have the map

+

d : F ( M , N ; &, i E A)

+ M ( M ,N ; V,, i

E A)

defined by d(f) = d f , the differential of f. Then Eliashberg has proved the following: Theorem 3.1 (Eliashberg [20]). If p >_ 2, then the induced map

d, : ro(F(M, N ; V,,i E A))

--+

ro(M(M, N ; V,,i E A))

is surjective. I n other words, the h-principle for fold maps holds. An element of M ( M , N ; V,, i E A) can be regarded as a formal solution to the problem; Given a collection of disjoint nonempty ( p - 1)-dimensional closed submanifolds FA, X E A, of M , is there a fold map f into N such that SA(f) = FA for all X E A ? In that sense, Eliashberg's theorem asserts that we can deduce the existence of a real solution up to homotopy with all the pre-assigned indices from any formal solution. See also EliashbergMishachev [21]- [24]. Based on his own h-principle theorem, Eliashberg also proved Theorem 3.2 (Eliashberg, [19,20]). I f M" is stably parallelizable, then there always exists a fold map f : M" --+ IWP for all p 5 n. Moreover, let Mn be a closed orientable n-manifold. Then, M" admits a fold map into R" if and only if M" is stably parallelizable. This is the first clear answer in a general dimension pair to our problem as one of the sufficient conditions for the existence of fold maps. Only in

350 K. Sakuma

the equidimensional case, this gives a complete answer when the source manifold is orientable. In the dimensions greater than or equal to four, stably parallelizable manifolds form a very small class in all manifolds. Moreover, the converse implication in Theorem 3.2 is not valid for the case where n > p. For example, one can easily construct a fold map f : (CP2$@p2+ RP for p = 2,3. Note that (CP2#(cp2is not spin and hence it is not stably parallelizable. Let f : M" 3 N 2 be a stable map of a closed n-manifold M n with n 2 2 into an orientable 2-manifold N2.Then by Thorn's result (see [78]) we have

Hence every stable map f : M" + N 2has nonremovable cusp singularities if x ( M n ) is odd. So in contrast, it naturally arises a question:

"Does there exist a smooth map homotopic to f which has only fold singularities i f x ( M n ) is even ? I n other words, can we always eliminate cusp singularities o f f by smooth homotopy if x ( M n ) is even 1 " The first mathematician who treated this problem explicitly was Harold Levine ( [40]). He solved the problem just in the way as it was expected:

Theorem 3.3 (Levine [40]). Let M" be a closed n-manifold, N 2a n orientable 2-manifold and g : M" + N 2 a generic smooth map. I f n 2 3 and x ( M n ) is even, then cusp singularities of g can always be eliminated by smooth homotopy, that is, there always exists a fold map f : M" ---t N 2 homotopic to g . In his original proof, the case for n = 2 was excluded simply for technical reasons. But Eliashberg ( [19]) covered the case in the framework of his theory on the homotopy principle for fold maps. We also have elementary and intuitive proofs which work only for n = 2; e.g. see Millett [51]. Furthermore, Levine's cusp elimination theorem for maps into the plane has been extensively generalized to Morin maps for almost all dimension pairs ( n , p ) by Ando [4] in the following.

Theorem 3.4 (Ando [4]). Let M" and NP be orientable manifolds and f : M" + NP a Morin map with p >_ 2. Suppose that codim XIr = n f o r r 2 2.


(i) A Morin map f is homotopic to a smooth map g such that CJr(9)= 8 if and only if the Thom polynomial of % vanishes. (ii) The Thom polynomial of % vanishes in the following cases: 0

0 0

n > p and r 3 1 (mod 4), n > p , r - 2 , 3 o r 0 ( m o d 4 ) a n d n -p o d d , n = p and r = r(r 1)/2 (mod 2).

+

His proof is based on his own h-principle theorem for Morin maps. Let Rk C J ' ( M , N ) be the subbundle with the fiber consisting only of the regular point germs and the Morin singular point germs with the ThomBoardman symbol Ij for j 5 k. Then, Ando proved the following Theorem 3.5 (Ando [4]). Let M n and Np be smooth orientable manifolds and f : M" + NP a continuous map with n 2 p 2 2. Then the homotopy class of f contains a smooth map with only Morin singularities of type A j , j 5 k, if and only if there exists a section of the bundle flk.

Further, Ando has recently obtained a stronger result for fold maps by using the h-principle in the 2-jet space, which plays an important role. Theorem 3.6 (Ando [6,8]). Let M" be a closed n-manifold and NP a parallelizable p-manifold. If there exists a fiberwise epimorphism TM" @ E~ 4 E P with n 2 p 2 2, then there is a fold map f : M" -+ N p , where E~ denotes the trivial k-plane bundle over M". If n - p is even, then the converse also holds.

It is easy t o see that Theorem 3.2 immediately follows from Theorem 3.6. In [8], Ando has actually proved that there exists a fiberwise epimorphism TM" @ + E P if and only if there is a fold map f : M" -+ RP such that S ( f ) is orientable. Note that if n - p 1 is odd, then S ( f ) is always orientable for a fold map f : M n + R P (see 164, Lemma 2.21) but S(f)is not necessarily orientable if n - p 1 is even (see [75]). Therefore, it seems that the aspect for the case n - p being even is considerably different from that for the odd case in our problem. For example, Saeki had already indicated the following fact before Ando's h-principle theorem appeared:

+

+

Proposition 3.7 (Saeki [64]). Let f : M" + NP be a fold map. If the normal bundle of the immersion fls(f)is trivial, then there is a bundle isomorphism

TM"

@ E~ E f * T N p@ r]

352

K. Sakuma

f o r some ( n - p

+ 1)-plane bundle q over M"

We can immediately deduce several observations from this. For simplicity, we shall consider the case where NP = RP. Moreover, suppose that M" is a closed orientable n-manifold. If M" admits a fold map into R", then it follows from Proposition 3.7 that M" is stably parallelizable since q should be trivial, which is the only if part in Theorem 3.2 for n = p . Furthermore, if M" admits a fols map into RP with n - p being even, then it follows that w i ( M n ) = 0 for i L n-p+2 (see [64,Theorem 21). Hence we can deduce, for instance, that RP2" does not admit a fold map into for any m 2 k because it is knwon that W ~ , ( R P ~#~0) (see e.g. [53, § 4 ] ) , where note that RP2" is nonorientable. In this case the Thom polynomial of cusp singularities would be in Hn-p+2(Mn;Z2) (not in H n ( M n ;Z2) if p 2 3 ) and this suggests that the top Stiefel-Whitney class must be another obstruction to eliminating singularities other than the Thom polynomials. 4. Non-existence of fold maps and the Hopf invariant one

problem

Let f : M" -+ RP be a fold map. For x E S ( f ) we can choose local coordinates ( X I , . . . , X n ) centered at x and (yl, . . . ,y p ) centered at f ( x ) respectively so that f has the following normal form: yi = xi

(i 5 p

2

- l), y p = x p

+ . .. +

-

- * *

+

-

x; ,

where the integer X is called the reduced index if n - p 1 is even, denoted by i ( x ) = A. Note that n - p 1 - X = X (mod 2) if n - p 1 is even, and hence the parity of the reduced index is independent of choices of local coordinates. Therefore if n - p 1 is even, then we have the well-defined subsets of S ( f ) as follows:

+ +

+

S + ( f )= { x E S ( f ) ;i ( x ) 3 0 (mod 2 ) } ,

S - ( f ) = { x E S ( f ) ;i ( x ) = 1 (mod 2 ) ) . From the above normal form we see that the Jacobian matrix of f at x E S ( f )is 1 Jf ( x ) =

1 0 ... 0 2

~

.p -22, * *

Existence problem for fold maps

353

where all the vacant entries are 0. Thus we see that S ( f ) is defined by the equation z p = ... = z, = 0 locally and hence S ( f ) is a ( p - 1)dimensional submanifold of M " , and further it follows that the restricted map f Is(f): S(f) -+ RP is a codimension one immersion. This fact gives topological restrictions to S(f). For example, RP4 cannot be realized as one of the components of S(f) when p = 5 because RP4 cannot be immersed in R5. In [27] Fukuda gave fascinating results, which were the starting point of our global study treated in this paper. He observed in the following way: Let f : M -+ N be a generic smooth map and C'( f ) the singular set correspong to the Thorn-Boardman symbol I . Then the Thom polynomial [C1(f)]2E H * ( M ; Z 2 ) tells us the homological location of C 1 ( f ) in M . However, if H i ( M ;Z2) = 0 for i = dim E'( f ) , then it gives no information on C'( f ) though C'( f ) may be nontrivially embedded in M like knots or links in general. In that reason we need to take care of choosing a generic map which we try to study because under the most general situation it is usually difficult to determine the Thom polynomial for a given singularity type and read off the geometrical meanings from only the expression itself. From such a viewpoint Fukuda began to study the topology of Morin maps and proved the following formula involoving the Euler characteristics: Theorem 4.1 (Fukuda [27]). Let M be a closed n-manifold and f : M --+ RP be a Morin map. Then we have P

X(M)

X ( A k ( f 1)

k=l Moreover, iff is a fold map and n - p

(mod 2).

(4.3)

+ 1 is even, then we have

We give an outline of the proof. It is essentially treated inside the framework of Morse theory. First take a generic projection 7r : RP -+ R so that the composite map T O f might be a Morse function. The existence of a required projection is guaranteed by the result of Mather: Lemma 4.2 (Mather [49]). Let S be a submanifold of RP and 7 r :~ RP -+ L an orthogonal projection to a linear subspace L of RP. Let I be a ThomBoardman symbol. Then for almost every k-dimensional linear subspace L of RP, the jet extensionj"(nL1s) : S -+ J'(S, L ) is transversal to the ThomBoardman singularity E'(S, L ) in J"(S,L ) , where a property is said to hold

354

K. Sakuma

f o r almost every L C RP if the set of L c Rp so that the property does not hold for L has Lebesgue measure zero in RP.

lm

Proof. We may also assume that 7r o f are Morse functions for all Ic. We denote by # C ( g ) the number of elements in the critical point set C ( g ) of a Morse function g. By the Morse inequality (see e.g. [52]) we have

X ( M ) = #C(T0 f ) (mod 21, X W f )) = #C(TO f lm) (mod 2). Since it is easy to see that C(7r o

#C(X

0

f> = nC(n 0 f l A l ( f ) ) ,

flm)= HC(.

#c(7r

flm)n A k + z ( f ) ) = 0, we have flAk(f))

+ #c(7r f

IAk+i(f)).

Combining these facts, we have the required result.

0

Since S(f)= A l ( f ) ,we have an immediate corollary of the formula (4.3). Corollary 4.3. Let M" be a closed n-manifold and f : M" map. Then we have

X(M") f X ( S ( f ) ) (mod 2).

-+

BP a fold (4.4)

See also [65]. Motivated by the F'ukuda congruence (4.4), we have the following

Theorem 4.4 (Kikuchi-Saeki [36],Saeki-Sakuma [SS]). Let M be a closed n-manifold with odd Euler characteristic, N a n almost paralleleizable p-manifold ( n 2 p ) and f : M -+ N a generic smooth map.

(i) Iff has only fold singularities, then p = 1 , 3 or 7. (ii) Iff has only fold and cusp singularities, then p = 1 , 2 , 3 , 4 , 7 or 8. (iii) Iff has only fold, cusp and swallowtail singularities and p is even, then p = 2 , 4 or 8. It arises a question what geometric structure causes such a dimensional restriction in the target space. Note that in the theorem n must be even since x ( M ) is odd. In the context of the proof, we have shown that these phenomena are caused by the (non-)existence of the Hopf invariant one element as is explained below, where note that such existence problem has been completely solved by J. F. Adams [l].In other words, it claims that the nontriviality of the top Stiefel-Whitney class w ', E H " ( M ; Z 2 ) 2i Zz


relating to the Hopf invariant one element gives dimensional restrictions for the existence of certain Morin maps. Let M" be a closed n-manifold and f : M" + Rn+l an immersion with normal crossings. Then we set

for k = 2,3,. . . ,n follows:

+ 1 and then the self-intersection set is stratified as

I(f) = I 2 ( f ) U I 3 ( f ) U . " U l n + l ( f ) .

+

In particular, I n + l ( f ) , the set of (n 1)-tuple points of f , consists of a finite number of points since M" is compact. Then Eccles proved ( [18])that if #In+l(f) is odd, then n must be 0, 2 or 6. He also clarified that this restriction would come from the existence of the Hapf invariant one element. More precisely, he has shown that for a given p , a codimension one immersion of a closed manifold into RP with an odd number of ptuple points exists if and only if there exists an element of Hopf invariant one in the p-stem T:. Let us give an outline of the proof of Theorem 4.4.

Proof. Suppose that there is a fold map f : M" -+ RP such that x ( M n )is odd. Then S(f)is a ( p - 1)-dimensional submanifold of M n , the restricted map fls(f): S(f)-+ E%P is an immersion, and by Corollary 4.3 S(f)has odd Euler characteristic3).Modifying f slightly if necessary, we may assume that the restricted map is an immersion with normal crossings. Then a result of Herbert [31] implies that the number of ptuple points of the immersion with normal crossings is odd. Thus Eccles's theorem quoted above implies Theorem 4.4 (i). The proof of (ii) and (iii) proceeds in a similar spirit and we omit it here (see [SS]). We have another proof: The normal bundle of the restricted immersion fJs(f) is a line bundle, denoted by vf, and then we can set w ( v f ) = 1 Q for some Q E H1(S(f);Z2). It follows from the bundle isomorphism T ( S ( f )@ ) vf E (fls(f))*TRP that w(T(S(f)) = 1 a -t . . r 9 - l since

+

+

3)Therefore,p must be odd.

+

356

K. Sakuma

the bundle (f Is(f))*TRPis trivial. According to the assumption and the F'ukuda congruence, we see that

and hence it follows that w l ( S ( f ) ) = a is a nontrivial element.

0

Here we should refer to the Brown-Liulevicius theorem for cobordism class of codimension one immersion.

Theorem 4.5 (Brown [15]). Suppose that a closed n-manifold M" is i m mersed in T h e n M" is zero cobordant o r unoriented cobordant t o RPO, I W Po~r RP'. Applying this theorem to our situation, S (f ) should be cobordant to RPO, RP2 or RP', which completes the proof. We have proved our theorem but at this stage we still have the possibility we may have studied an 'empty set' under the assumption that the Euler characteristic of the source manifold should be odd. However, in [64] Saeki has shown that there is a fold map of a closed 4-manifold4) with odd Euler characterisitc into R3.Imitating his construction, we can actually construct an explicit example of a fold map of a closed even dimensional manifold with odd Euler characteristic into R3. It is evident that for p = 1 there always exists a fold map without assumption of the Euler characteristic of the source manifold. By a similar construction, it is not so difficult to construct a fold map of a certain closed 2n-dimensional manifold with odd Euler characteristic into R7 for n 2 4. For p = 2 we have already seen that every closed manifold M" with n 2 2 admits a smooth map with at most fold and cusp singularities. For p = 4 we can construct a smooth map f : M4 --+ R4 with only fold and cusp singularities for a closed nonorientable 4-manifold M4 with odd Euler characteristic (see [70]). For p = 8 by using an immersion of RP' into Rs we can imitate this construction in order to obtain a smooth map f : M s -+ R8 with only fold and cusp singularities for a closed 8-manifold M 8 with odd Euler characteristic. Summarizing, all the possible values in Theorem 4.4 are in fact realized as the target dimensions of the required Morin maps. In particular, if n = p and x ( M ) is odd, then n = p must be even. Thus we have an immediate corollary: 4)He actually gave such an explicit example for a closed non-orientable 4-manifold which is the total space of an WP2-bundle over P P 2 .


Corollary 4.6. Let MP be a closed p-dimensional manifold with odd Euler characteristic. If there exists a smooth map f : MP -+ R P with at most fold, cusp and swallowtail singularities, then p = 2 , 4 or 8.

Here note that p # 2 , 4 if MP is orientable. When p = 2, we need no explanation. When p = 4 and M4 is orientable with odd Euler characteristic, one cannot eliminate the umbilics because the first Pontrjagin class of M 4 does not vanish (see [70]).We do not know whether we can deduce that p # 8 or not under the assumption that M S is orientable. When the target dimension p is odd, we can prove the following result: Theorem 4.7 (Saeki-Sakuma [68]). Let M n be a closed n-dimensional manifold with odd Euler characteristic. Suppose that n 2 p and p is odd. If there exists a smooth map f : M" -+ R P with at most fold, cusp and swallowtail singularities such that [ A s ( f ) ] z= 0 in H p - 3 ( A 2 ( f ) ;a,), then p = 1 , 3 or".

Here it is appropriate that we state the Chess conjecture and its affirmative solution by Sadykov. Let us first refer to the result for the Thom polynomials of Morin singularities proved by Chess: Theorem 4.8 (Chess [17]). Let M", NP be orientable manifolds and f : M" t NP a Morin map with n - p odd.

(i) For k 2 4, the inclusion i : A k ( f ) A k - z ( f ) is cobordant to zero, i.e., there exist a n ( n+ 1)-manifold W with d W = M" and a smooth map F : W 4 NP with F ~ M = . f. (ii) i*[Ak(f ) ] z = 0 in H , ( M n ; Z,)for k 2 4. I n particular, the T h o m polynomial of A k (f) vanishes for k 2 4. -+

His proof was based on extensive use of the intrinsic derivative according to Boardman's argument ( [13]).Moreover, based on this result, Chess conjectured in [17]that every Morin map f : M" -+ NP with n - p odd would be homotopic to a Morin map with at most fold, cusp and swallowtail singularities, which is called the Chess conjecture (see also [ l l ,Chapter 4, $1.51). Recently, the Chess conjecture has been affirmatively solved by R. Sadykov ( [60]).He actually proved a stronger result as follows: Theorem 4.9 (Sadykov [60]). Let M", NP be orientable manifolds and f : M" -+ NP a Morin map with n - p odd. Then the T h o m polynomial of swallowtail singularities vanishes and the map f is homotopic to a smooth map with only fold and cusp singularities.

358

K. Sakuma

5. Fold maps for p = 3

Studying our problem, we infer from Theorem 4.4 (i) that there may be exceptional phenomenon when p = 3 or 7. The purpose of this section is to discuss the problem for p = 3. For a stable map f : M n R3,all possible singularities are folds, cusps and swallowtails, which means that any stable map into R3 is a Morin map. On the other hand, according to Theorem 3.2, there always exists a fold map of a closed orientable 3-manifold because all such 3-manifolds are (stably) parallelizable. Let g : BP2 + R2 be a stable map such that g has three cusps. Then we define a smooth map f as the composite --f

RP2 x

gxid

s1-It2

x

ernb s1-R3,

where note that f has three cusp circles in BP2 x S1which represents a nontrivial homology class in H1(BP2 x S1;Z2) E Z2. This implies that [ A z ( f ) ]= ; w 2 ( R P 2 x 5’’) # 0. Thus, studying our problem for the case ( n , p ) = (3,3), it is necessary to observe the Thom polynomial of cusp singularities as a primary obstruction when M is a closed nonorientable 3-manifold. We can actually obtain the following: Theorem 5.1. Let M be a closed nonorientable 3-manifold. If the 2-nd Stiefel- Whitney class of M vanishes, then there is a fold map f : M + B3. Hence there is a fold map f : M -+ R3 if and only if w 2 ( M ) = 0.

The proof will be given after we introduce the notion of the span of a manifold as a special case of a more general result (see Theorem 5.8). What about the case for n = 4 ? In that case, the Thom polynomials of both cusps and swallowtails vanish (see [5], [75]).Nevertheless, Saeki proved: Theorem 5.2 (Saeki [64,65]). Let M 4 be a closed 4-manifold provided with H i ( M 4 ;Z) N= Hi(@P2;Z). Then every stable map f : M 4 --t R3 should have nonremovable cusp singularities although the T h o m polynomial vanishes. Hence there exists n o fold map f : M 4 + B3.

To the author’s knowledge, this is the first discovered example in which the elimination is impossible although the Thom polynomial vanishes and in addition it is the smallest dimension pair in lexicographic order for which such a phenomenon occurs. We sketch here the original proof due to Saeki.


359

Proof. Suppose that there is a fold map f : M4 -+ R3. Fix an orientation of M4 so that the signature of M4 is equal to +1, i.e. a(M4) = +l. The singular set S(f) is an embedded surface and each component S c S(f ) consists of indefinite folds or definite folds. If S is an indefinite fold component, then the structure group of the normal bundle vs of the embedding S L) M4 is reduced to the dihedral group of order 8 and it follows that the self-intersection number of S in M4 vanishes, i.e. e(vs) = 0 (see [64]).If S is a definite fold component, then it is orientable and hence carries an integral homology class of M 4 , i.e. [S]E H 2 ( M 4 ; Z ) Z. On one hand, the author has shown in [74] that the self-intersection number of S(f) is congruent modulo 4 to the Pontrjagin number p l [ M 4 ]of M4 under the condition that H 1 ( M 4 ; Z )= 0. In this case, p l [ M 4 ]= 3 a ( M 4 ) = 3. Thus it follows that there must be an element E H 2 ( M 4 ;Z) whose self-intersection number is congruent to 3 modulo 4. This implies that k 2 = 3 (mod 4) for some k E Z, which is a contradiction. 0 This result suggests at least that the case where the source manifold is four dimensional is specific in studying our problem for p = 3 (see Akhmetiev-Sadykov 131 also). Recently, Saeki has obtained the following remarkable result which gives necessary and sufficient conditions for the existence of fold maps of closed oriented 4-manifolds into R3.

Theorem 5.3 (Saeki [SS]). Let M be a closed oriented 4-manifold. Then the following three are equivalent.

(i) There exists a fold map of A4 into R3. (ii) The intersection form of M is not isomorphic to

-

u= (iii) There exists a cohomology class v E H 2 ( M ; Z ) such that u p l ( M ) E H 4 ( M ;Z), where p l denotes the first P o n t j a g i n class of M . R. Sadykov has also obtained the same result by a different approach (see [61]).To the author’s surprise, we see from the conditions (ii) and (iii) that almost every closed orientable 4-manifold admits a fold map into R3 and such existence is determined only by the cohomological information. For instance, every closed spin 4-manifold admits such a smooth map. As far

360 K. Sakuma

as the author knows, we have no such characterization5) on the existence of a fold map of a closed nonorientable 4-manifold into R3. In order to prove Theorem 5.3 we need the following key result, which is an essential part of the theorem. Theorem 5.4 (Saeki [SS]). Let M be a closed oriented 4-manifold and F a closed surface embedded in M. Suppose that F is decomposed into a disjoint union of non-empty connected components; F = F+ u F-. Then there exists a fold map f : M -+ R3 such that S+( f) = F+ and S-( f) = Fif and only i f the following jive conditions hold. X(M) = X P + ) - X(F-1. F is a characteristic surface of M . F+ is orientable. The self-intersection number of each connected component of F- in M vanishes. (v) The self-intersection number of F in M coincides with the first Pontrjagin number of M .

(i) (ii) (iii) (iv)

Thus we have only to verify that the existence of an embedded surface F = F+ U F- satisfying the conditions (i)-(v) in the above theorem is equivalent to the conditions (ii) and (iii) in Theorem 5.3, which is not a difficult task (see the proof of Theorem 4.1 in [66, pp. 638-6411). We have already discussed the necessary conditions (i)-(iv) for the existence of a fold map. We will verify the condition (v) in the next section (not only for a fold map but also for a stable map). Conversely, suppose that the required embedded surface F = F+ U F- is given. Let N ( F ) be a closed tubular neighborhood of F in M . Then it is not so difficult to construct a fold map f : N ( F ) --f R3.Hence we also have a homomorphism T N ( F ) 4 TR3. To apply the h-principle here, we have only to extend this homomorphism to TM so that it is non-singular- on the complement of N(F). In other words, since the restricted map f l N ( F ) - F is a submersion and hence we have the appropriate framing, i.e., the nowhere vanishing four vector fields ti (i = 1,2,3,4) satisfying our convenient properties on N ( F ) - F , we have only to extend the framing 'p to linearly independent vector fields on M -F. He actually has found that the obstruction is p i ( M - I n t N ( F ) , V ) = P I [ M ]- F . C 5)For example, the author does not know whether or not there exists a fold map of R P 4 into R ~ .


361

where p l ( M - Int N ( F ) ,cp) E H4(M- Int N ( F ) ,d ( M - Int N ( F ) ) ;Z) E Z is the relative Pontrjagin number compatible with the framing cp. We need a delicate argument for the proof of this part and then we omit it here (see the original paper [66, pp. 633-6371 for more details). In addition, Sadykov has also treated the existence problem for fold maps of higher dimensional manifolds. In fact, he has obtained a significant result in the following:

Theorem 5.5 (Sadykov [ 6 2 ] ) . Let M" be a closed orientable n-manifold and NP an orientable p-manifold. If n - p 1 is even and n > 3p - 3, then any Morin map f : M" --f NP is homotopic to a fold map. I n particular, i f n = 2m > 6 , then there always exists a fold map f : M" 4 R3.

+

His proof of Theorem 5.5 is similar to that of Theorem 4.9 in strategy. In [60],he had already proved that any Morin map f should be homotopic to a smooth map g at most with only fold and cusp singularities. So, we have only to eliminate cusp singularities. By virtue of Theorem 3.6 it suffices to show that there exists a non-degenerate symmetric bilinear form 'p : Ker (df Is(g))8 Ker (df Is(g)) --$ Coker (df Is(g)) over the singular set S(g). We will see later (Theorem 5.7) that the condition that n - p 1 is even is essential in Theorem 5.5. In Theorem 5.5 only the case ( n , p ) = ( 6 , 3 ) is excluded. By applying the result on the span of a manifold discussed later on, we can at least prove the following

+

Theorem 5.6. Let M 6 be a closed 6-manifold with Wf, = 0. If M6 is orientable, then there always exists a fold map f : M 6 -+ R3.If M6 is nonorientable and w4(M6) = 0 , then there always exists a fold map f : M6 .+ R3. In conclusion, as far as we study the existence of fold maps of a closed even dimensional orientable manifold into R3, the case that n = 4 is exceptional in the sense that there is another obstruction other than the Thom polynomial, which means that the dimension pair ( 4 , 3 ) is specific and rather fascinating. Now, let us consider the case where n - p 1 is odd which is not covered by Theorem 5.5 for p = 3. First note that there exists a fiberwise epimorphism TM" @ E~ -+ E P if and only if spanO(Mn) 2 p - 1, where spanO(M)is the stable span of M . The span of a vector bundle E over M is the maximal number of everywhere linearly independent cross sections on 6, denoted by span(J). In particular if E is the tangent bundle of M , we call it the span of the manifold M, denoted by span(M) := span(TM).

+

362 K. Sakuma

The number span(TM @ E ~ -) r , which does not depend on the number r for r 2 1, is called the stable span of M , and we denote it by spanO(M) hereafter6). Thus, by Theorem 3.6 there is a fold map f : M n + RP if and only if spanO(Mn)L p - 1, when n - p 1 is odd (see [66, $51 also), and particularly, there is a fold map f : M" + R3 if and only if spanO(Mn)2 2. As examples with the lowest dimension, let us consider the case where n = 5. For instance, we can immediately see that there exists a fold map of a closed spin 5-manifold M into R3 if x(2)(M)= 0, where x ( ~ ) ( ME)2 2 denotes the Kemraire semi-characteristic of M defined by

+

This follows from the fact that span(M) L 2

-

X(2)(M)= 0

for a closed spin 5-manifold M due to Thomas [83, Theorem 31, where note that we always have spanO(M) span(M), and the Thom polynomial of cusp singularities defined in the 4-th cohomology group vanishes (see also Remark 5.9 below) because it holds that w4 = w2 w2 = 0 for a spin 5manifold. On one hand, there is no fold map of RP5 into R3 since it holds that spano(RP5) = 1 as we shall verify later. Note that R P 5 is orientable but not spin, and for a closed orientable 5-manifold M it always holds that w4 = w2 w2 by the Wu formula (see e.g. [53]). Furthermore, we can drop off the condition on the semi-characteristic for the existence of fold maps, that is, we can deduce that span0(M5) 2 for any closed spin 5-manifold M 5 as follows. First, taking the product with a circle as in [38, Example 20.181 we have

-

-

+

span(S1 x M ~=)spano(S1 x M ~ =)span0(m5) 1. On the other hand, according to Thomas [82], it follows that span(M6) 2 3 if and only if w4 = 0 for a closed orientable 6-manifold M 6 with x(M6) = 0. In our case, x(S1 x M 5 ) = 0 and w4 = w i = 0 since M 5 is spin. Thus we have span(S1 x M 5 ) 2 3, which means that spano(M5) 2 2 for a closed spin 5-manifold by the above equality. This argument can be generalized t o a closed orientable (4m 1)-manifold X with w ~ ~ ( = X 0, ) and we see that spanO(X) 2 2 for such a manifold. Here we should only note that, for a closed orientable (4m 2)-manifold X with x(X) = 0,

+

+

6)We refer the reader to [37] for more details on the span and stable span.


363

span(X) 2 3 if and only if wqm(X) = 0 by Thomas [82] again. Conversely, suppose that there exists a fold map f : X -+ R3.Then we see that wq,(X) = 0, which is deduced from the result of Thom ( [78]): for a smooth map f : M" --f RP as in Theorem 2.2, the Thom polynomial of cusp singularities coincides with the ( n- p 2)-nd Stiefel-Whitney class of M", W " - ~ + ~ ( M "E) Hn-p+2(Mn;&), under the condition that n - p 1 is odd. Moreover, a closed orientable n-manifold M" with n = 4m 3 always admits a fold map into R3 since it holds that span(Mn) 2 2 for such a manifold according to the result of Thomas [80, Theorem 1.11. In contrast we can prove that the (4m 1)-dimensional real projective space RPqm+' does not admit a fold map into R3 as follows. First note that spano(RPn) = span(RPn) for odd n (see [34]). According to Atiyah's theorem [12, Theorem 1.21, if span(M") 2 2 for a closed orientable nmanifold M", then xw(Mn) = 0. Here xw(Mn) E Z2 denotes the real Kervaire semi-characteristic of M" defined by

+

+

+

+

+

It is easy to see that xw(RPn) = 1 if n = 4m 1, and hence we have the required result span0(RPqm+') = span(RPqm+l)= 1. Summarizing these arguments, we have proved the following:

Theorem 5.7. Let M" be a closed orientable odd dimensional manifold.

+ +

(i) When n = 4m 3, there always exists a fold map of M n into R3. (ii) W h e n n = 4m 1, there exists a fold map f : M" + R3 i f and only if wn-l(Mn) = 0. I n particular, there always exists a fold map of a closed spin 5-manifold M 5 into R3. We can also obtain the following result for non-orientable manifolds.

Theorem 5.8. Let M n be a closed non-orientable odd dimensional manifold. Then, there exists a fold map f : M" -+ R3 i f and only if w,-1 ( M " ) = 0. Proof. For the proof of the 'if' part, we have only to repeat a similar argument as in the proof of Theorem 5.7 (ii) if n = 4m 1 and to apply the results in [57] if n = 4m 3, which is also proves Theorem 5.1 as a special case. As for the 'only if' part, we have only to recall Thom's theorem in [78], which asserts that for a smooth map f : M" + RP as in Theorem 2.1, the

+

+

364 K. Sakuma

+

Thom polynomial of cusp singularities coincides with the ( n - p 2)-nd Stiefel-Whitney class of M n if n - p 1 is odd. We are in a position to prove Theorem 5.6 with similar techniques already mentioned above. We have only to apply the tables for the existence of 3-fields in [57, pp. 2322331. The rest of detailed discussion will be left for the reader. We have another proof of Theorem 5.7 and Theorem 5.8 by the Postnikov tower argument a s follows. By Ando's h-principle theorem mentioned above, M" admits a fold map into R3 if and only if the bundle T M " @ c1 admits a 3-frame. So, we let cp : M" + BO(n+ 1)be a continuous map classifying the (n+l)-plane bundle TM" @ c1 and consider a natural fibration

+

(see 1841)

V3(Wf1)

-

BO(n - 2) -% BO(n

+ l),

(5.5)

where V3(JWnS1) is the Stiefel manifold consisting of all orthonormal 3frames in RWnfl. Then, T M " @ c1 admits a 3-frame if and only if there is a lift + : M" + BO(n 2) such that cp = P o + .

-

BO(n - 2)

Since 7ri(V3(Rn+l)) = 0 for i 5 n - 3, the first Postgikov invariant (primary obstruction) of the fibration (5.5) is the ( n - 1)-st Stiefel-Whitney class, w,-1

E H"-l(BO(n

+ 1);7rn-z(V3(Rn+1))) = H"-l(BO(n + 1); Zz) = [BO(n+ l),K(Z2,n - l)].

Let K(Z2,n-2) A E L B O ( n + l ) be the principal fibration induced by ~ " - 1 , the ( n - 1)-st Stiefel-Whitney class of the universal bundle y over BO(n 1). Suppose that w,-1 = 0, and hence wn-l(Mn) = cp*w,-1 = 0. In the second stage of Moore-Postnikov towers of the fibration (5.5), we can choose a map q : BO(n - 2) + E such that its restriction qlV3(Wn+i): V3(Rn+') 4 K(Z2, n - 2) is given by the characteristic classes of &(IW"+l).

+


Since it hold that ~ ~ - 1 ( h ( R " + ~E)Z, ) we see that the second Postnikov invariant kz belongs to Hn(E;Z) = [ E ,K ( Z ,n)].

BO(n - 2)

-L

M"

Wn-1

' PB O ( ~

K ( z ~n, - 1)

On the other hand, it is well-known (see e.g. [53, Problem 15.c] or [14]) that

where p i E H4i(BO(2m);Z) is the i-th Pontrjagin class and p denotes the Bockstein operator. Noting that n is odd, it follows that k2 = 0 since wn = 0 and there is no nontrivial cohomology class a E H"(BO(n+l); Z) such that ~ *=ak2 E H"(E; Z), from our dimensional reason that 2m = n 1. This means that there is no other obstruction for the existence of the lift (p under 0 the assumption that wn-l = 0. This completes the proof.

+

Remark 5.9. In Theorem 5.7 (ii), we have proved that there is no closed spin 5-manifold M5 such that span0(M5) = span(M5) = 1 (which implies x ( ~ ) ( M=~1). ) In other words, we do not have a 5-dimensional analogy of Theorem 5.2 which describes the nonremovability of cusp singularities in spite of the vanishing of its Thom polynomial.

+

Remark 5.10. For a closed orientable (4m 1)-manifold M , the Thom polynomial of cusp singularities for a stable map f : M -+ R3 coincides with w ~ ~ ( MSadykov ). suggested a relevant observation to the author in the following ( [62]): Let g : CP2 4 R2 be a stable map with exactly one cusp. Then we define a smooth map f as the composite

@ P 2x

gxid s1+R2

x

emb s1+R3

Then f has a cusp curve in @P2x S1which represents a non-trivial homology class in H1((CP2x S1;Z2) 2 Z2, which means that [A~(f)]z= w4 # 0. We can generalize this example to a smooth map f : @P2" x S1 -+ R3 with nonremovable cusp locus if we replace @P2by @P2" for m 2 2. This is because there exists a stable map g : R2 with exactly one cusp --f

366

K. Sakuma

since the Euler characteristic of @P2mis odd (see Levine [40]). Note again that [A2(f)] = w4m # 0.

a

Remark 5.11. The connection between the real and the usual Kervaire semi-characteristics is explained by a formula (see [42]) which, for a closed orientable (4m 1)-dimensional manifold M , asserts that

+

X R ( M ) - X ( 2 ) ( M )=w2

-

W4m-l[M].

In particular, these two semi-characteristics coincide for a spin manifold, where note that for example RP4m+1 is not spin (w2 # 0) but x ~ ( l W P ~ ~=+Xl )( ~ ) ( I W P ~=~1. + The ~ ) number w2 W ~ ~ - I [EM2 ]2 is often called the de R h a m invariant of a closed orientable (4m 1)dimensional manifold and it coincides with V ~ ~ S Q[MI ~ which V ~ , plays an important role in surgery theory, where v2, denotes the 2m-th Wu class. In [16],for a fold map into the plane, Chess defined a mod 2 invariant by using the information on the triviality of the normal bundle of fold curves and the number of double points of the immersion of fold curves which is independent of the choice of a fold map into the plane. Then he showed that it coincides with the de Rham invariant of the source orientable manifold.

-

+

A fold map f : M 4 + R3 is said to be special generic if S-(f) = 8. Then we can similarly consider the existence problem for special generic maps. It has been done by giving the complete list of such closed 4-manifolds up to diffeomorphism only when the fundamental groups of 4-manifolds are trivial, free and surface groups, respectively (see [69], [71], [72]). These results suggest that the existence problem of special generic maps would be closely related to the smooth structure of a given closed 4-manifold. This must be of independent interest in the four dimensional differential topology: however, we do not touch the problem any more in this paper. 6. Fold maps for p = 7

Let M be a closed manifold and let us consider the diagonal set, A = { ( p , q ) E M x M ; p = q } . Then it is easy to prove (see e.g. [53, Lemma 11.51) that the normal bundle of the embedding of A in M x M is canonically isomorphic to the tangent bundle of M , which implies

x(M)=A - A , where A . A denotes the self-intersection number of A in M x M . This formula implies that there exist nonremovable singularities of a vector field over M if x ( M ) # 0. Namely, the formula gives an obstruction to the


367

existence of a non-zero vector field related to the Euler characteristic. Is there such a formalism in the Pontrjagin numbers ? We may say that such a question is supposed to be natural because it is known that p k ( < ) = W2k( dim M" = n. Then any smooth map f E C w ( M n ,NP) can be approximated by g E Cw(M, N ) such that j k g ( M " )n S = 8. We shall give an easy application of Corollary A.6. First note that we can identify J 1 ( n , p )with the set of p x n matrices, denoted by M(n,p), by associating to j' f (0) the Jacobian matrix (afi/azj(O)) and we have J~(R",WP)

= R" x RP x M ( n , p ) .

Moreover, we set

Ci= {A E M(n,p);rank A = i}, C'(n,p) = R" x Rp x Cic J1(R",RP). Then we can easily verify by linear algebra that Ci(n,p)is a submanifold of J1(R", R P ) with codimension ( n - i ) ( p - i). So we set

C i ( M " , N P )= { j ' f ( p ) E J 1 ( M " , Np);rank dfp = i}. Then we can also verify that C i ( M n ,NP) is a submanifold of J1(Mn, NP) with codimension (n-i)(p-i). If we apply Theorem A.5 to this submanifold C i ( M n ,N p ) , then we immediately have Corollary A.7. Any smooth map f : M" --t Np can be approximated by g : M n 4 Np such that j l g is transversal to Ci(Mn, Np).

376

K. Sakuma

Now for a smooth map f : M n

+ NP

we set

Si(f ) = { p E M"; rank df,

= i}.

If f is a smooth map such that j1f is transversal to Ci(M", Np), then we have

S i ( f ) = (j' f ) - ' ( E i ( M n , Np)) and Si( f ) is a submanifold of M" with codimension ( n - i ) ( p - i ) . Therefore:

Proposition A.8. A n y smooth map f : Mn + Np can be approximated by a smooth map g : M" + NP such that Si(g)is a submanifold of M" with codimension ( n - i) ( p - i) . As is stated at the beginning of this Appendix, we can prove the Whitney immersion theorem by jet transversality theorem as follows. Let f E Cm(Mn,N2"). Then we know that f is an immersion if and only if Si(f)= 8 for all i 5 n - 1. By Proposition A.8 the codimension of S n - k ( f )in M" is codim S"-'(f) = { n - ( n - k ) } { 2 n- ( n - k)} = k(n

+ k) 2 n + 1

for k 2 1 as is required. To prove the Whitney embedding theorem we need the more elaborate version of the jet tranversality theorem, namely, the multi-jet transversality theorem due to Mather (see [28, Chapter 11, 541).

Appendix A.2. Thom-Boardman singularities Now we will introduce another notation mainly on the singular set of a smooth map for later convenience. The reader should be careful with a slightly different use of notations. For a smooth map f : M" -+ Np, first we set

ci(f)= { p E M n ;dimKer (df,)

= i},

which is the singular set of f unless i = 0. For almost every map f E C M ( M , N )we may assume that Ci(f) is a submanifold of M by Proposition A.8. Note that codim Xi(f ) = i(p - n i) in M". Then Thom proposed the following program ( [78]). First consider the subset

+

@(f)

= { p E Ci(f);dimKerd(fIci(f,)P = j } .


377

If it chanced that E"j(f) was also a submanifold of C i ( f ) ,we could define Ciyj>'(f) in an analogous way. Thus if we could succeed in this procedure, we would obtain the sequence of submanifolds

M

3

Ci(f)3 ciyj(f)2 E i " , k ( ( f ) 2 . . . 2 Ei>j>k,... (f1.

We experientially know that we can read off the behavior of singularities of f efficiently if such a sequence exists. However, the problems were that we could not always ensure that Ci>j?'9...(f)should be a submanifold and that it was not clear that t o what extent such a smooth map should exist in C m ( M ,N ) . Towards this problem Boardman gave a systematic answer. Let us summarize his results here. For the purpose, we have t o explain the ThomBoardman singular set defined in the jet space J ' ( M , N ) . There are two methods for the explanation; one by the Jacobian extension of the ideal of the formal power series ring and another by the intrinsic derivatives. Here we will employ the former and the latter will be discussed later related to Morin singularities. Let (21,. . . ,z), and (PI,.. . ,yp) be the coordinates of R" and RP respectively. Let R [ [ z l , .. . ,z,]] be the ring of formal power series defined over R and M its maximal ideal. Set A,(r) = R [ [ z l , .. . ,z:n]]/MT+l. For a smooth map germ f : (R,, 0) -+ (RP, 0), set z = j'f(0) and let Z(z) be the ideal of An(r) generated by f1, . . . , fp, where fi = yi o f. Clearly Z(z) is independent of the choice of the representative f of a jet z . Let f ( z ) be the ideal of A,(r) generated by the elements of A,(r) determined by all s x s minors of (afi/azj).Then we define the ideal A,Z(z) := Z(z)

+Z(Z),

which is called a Jacobian extension of Z(z). Note that this is independent of the choices of the coordinates of R" and RP. For example, if we set Zi = Z(zi) for zi E J ' ( n , p ) , we can immediately obtain the following: (1) Z(z) = A,Z(z) if s > n, (2) Z(Z) c AJ(2) c . * . C A ~ Z ( Z ) , (3) A1 (21 2 2 ) = A121 A122 and (4) Ai(Tiz2) = z i ( A i z z ) + z 2 ( A i z i ) . Next we define the rank of an ideal Z of A,(r) by

+

+

rank 2 := dim(M2

+Z/M2).

If rank Z(z) = s, then we set bZ(z) = A,+,I(z) simply and define Sk+lZ(z)= S ( S k 2 ( z ) )inductively.

378

K . Sakuma

Definition A.9. Let z E J T ( n , p ) . For Z(z) we define the sequence of nonnegative integers I = (il, . . . ,ir) by il = n - rank

Z(z), iz = n - rank SZ(z),

. . . ,i,

= n - rank SrZ(z).

We call this sequence the Boardman symbol of z. Let C'(n,p) be the subset of J T ( n , p ) consisting of those r-jets with Boardman symbol I = (il, iz, . . . ,2,) and we call it the local ThomBoardman singular set with symbol I (see [50]).

For example, let z = j r f ( 0 ) E J r ( n , p ) be the r-jet such that

yiof=xi Yn-il+lo

(1 < i < n - i 1 ) ,

2

f = xn-il+l

yi o f

=0

+ . . + x:-z2 + *

+ . . . + %-is 3

23,-i,+1

+... + x;-ir4+1 + . . . + X L i T ( n - il + 1 < i 5 p ) .

Then the Boardman symbol of z is I = ( i l ,. . .,i,).

Theorem A.10 (Boardman [13]). (i) C'(n,p) is not wacuous only i f the following three conditions are satisfied:

(a) n 2 il 2 i 2 2 2 i, 2 0, (b) ii 2 n - p , (c) ij il = n - p , then il = i z = . . . = i, = n - p . (ii) C'(n, p ) is a n analytic submanifold of J r ( n ,p ) and its codimension is given by VI =

(p-n

+ i l ) p ( i l , .. . , i,.)

- (il - iZ)p(iz,.

. . ,i,)

VI

- ...

-(i,-l - i,)p(i,), where p(ik, . . . ,i,) = dimw(A,(r - k

+ l)/Zk)

- 1.

Consider the jet bundle 7r : J'(M", NP) + M" x Np. We can define the local Thorn-Boardman singular set C'(n,p)(z,yl with symbol I in the fibre J r ( n , p ) over ( 2 ,y ) E M n x NP and set

C' =

u

Z'(n,P)(z,y)

(Z,!/)EMXN

in J T ( M n ,N p ) , which is called the Thom-Boardman singular set with symbol I. For a smooth map f : M --+ N , C'(f) = ( j " f ) - ' ( C ' ) is called the Thorn-Boardman singular set of f with symbol I. Now we are ready to state the main theorem of Boardman:


Theorem A . l l (Boardman [13]). (i) C' is a smooth submanifold of J T ( M n ,NP) and its codimension is equal t o VI. (ii) The subset C'(M, N ) = { f E C"(M, N ) ; j r f is transversal to C'} is dense in C" ( M ,N ). (iii) For f E C'(M, N ) we have

c('qf)= Cj(flCI(f)). His proof is based on the notion of intrinsic derivatives. Next we shall give another explanation of Morin maps in terms of the intrinsic derivatives due to Porteous ( 1561). Let E and F be smooth vector bundles over a smooth manifold M . Then we can define the bundle

C"(E,F ) = {a E Hom(E, F);rank Ker(a)

=T},

and we have two vector bundles K = Ker(a),C = Coker(a) over C'(E, F ) . Let N ( E ,F ) be the normal bundle of the embedding C T ( EF, ) L-) Hom(E, F ) and then there is a canonical bundle isomorphism

N ( E ,F ) N Hom(K, C). Take a section s : M --+ Hom(E,F), i.e., a bundle homomorphism L E Hom(E, F ) and set C'(L) = s-l(C'(E, F ) ) , which are the singular sets of L for r > 0. Set C ( L ) = U,.,l- C T ( L ) .For any 2 E C T ( L )we have the map

dl,, : T M ,

-+Hom(Ker(L,),

Coker(L,))

as the composition

TM,

% THom(E,F ) L ,

-

N

N ( E ,F ) L ,

N ( E ,F)L,,

where the second map is the natural projection THom(E,F ) I Z . ( ~ ,N ~ )TC'(E, F ) @I N ( E ,F ) + N ( E ,F ) . This is the intrinsic derivative. Now let us apply this to Morin maps. Suppose that we are given a Morin map f : M" -+NP. Then we have the bundle homomorphism df : T M " --+ f*TNP and the singular set C r ( f ) = CT(df).C " - p + l ( f ) is a closed ( p - 1)dimensional submanifold of M" and C " - p + k ( f ) = 0 for k 2 2. Moreover, we have the intrinsic derivative of f as defined above,

dl(df):TM"

+ Hom(K,C),

and we set dz(f)=dl(df)J~:K--+Hom(X,C),

380 K. Sakuma

+

where note that K = Ker(df) is an ( n - p 1)-plane bundle and C = Coker(df) is a line bundle both over C"-p+l(f). Further, we set Ker(dz(f)) = K nTC"-p+l(f). Note that Cn-p+'?j ( f ) = Cj(dz(f)) and C"-p+'~j(f) = 0 for j 2 2 since f is a Morin map. Similarly, we have the bundles K1 = K n TC"-p+l(f) = Ker(dz(f)), C1 = Coker(dz(f)) over Cn-p+lyl(f), and we can define d3(f) = dl(d2(f))lK1 : K1

+

HOrn(K1,Cl).

The definition of C'r(f) for the sequence I , = (n - p + 1,1,.. . , l ) ,2~3, proceeds by induction on r. That is, we have bundles K1,Cl over EIr(f) and a map d,+l(f) = dl(d,(f)) : K1

+

HOm(@.T-lKl,Cl),

and C1r((f)= E1(d,(f)) for T 2 3. As an easy application, we see that every stable map f : M" Morin map. By Proposition A.8 we have codim C"-'(f)

= 2n - 2

-+

R3 is a

> n 2 3,

so we may assume that C"-l(f) = 0. Further we have codim C"-2>o(f)= n - 2, codim C"-"'(f)

= n - 1,

codim Cn-2,1,1(f) = n, and codim Cn-2t1,1,1( f ) = n

+ 1>

72.

Thus we may assume E"-2i1i1)1( f ) = 0 and hence f has only fold (En-2>0 (f)),cusp (Cn-2,1j0(f)) and swallowtail (C n - z ~ l ~ (f)) l ~ O singularities. Then S(f) is a disjoint union of closed surfaces, Az(f) consists of a finite number of circle components and A3(f) a finite number of points.

Appendix B. Secondary Thom polynomial Here we restate the problem mentioned in 5 7. PROBLEM 5. Find other cases relating to the existence of the secondary obstruction other than the Thom polynomials, especially when the source manifold is not 4-dimensional. We add some comments to the PROBLEM 5 on the reason why we refer t o the 4-dimension. Let M 4 be a closed, oriented 4-manifold and f : M4 + R6 a stable map. Note that the set of stable maps is open dense in C"(M4, R6)


381

according to the structural stability theorem by Mather ( [48], [28]). Let C i ( f ) = {x E M4; dimKer dfz = i } . Then we have codim C1(f)

= 3,

codim C 2 ( f )= 8.

Thus we may assume that C z ( f ) = 8 by the jet transversality theorem ( [28]). The singular set S ( f ) = C1(f) is a 1-dimensional submanifold of M4 and consists only of Whitney umbrellas. We can easily see that the must vanish as follows: First note Thom polynomial [S(f)];E H3(M4;ZZ) that the Thom polynomial should have the form

[ S ( f ) ]=f xlwlwz

+ xzw; + x3w3,

We have assumed that M4 is orientable, i.e. where each coefficient X i E ZZ. w1 = 0. Thus, in order to see that the Thom polynomial actually vanishes, we have only to show that w3 = 0. By using the Wu formula (see [53, S ll]),we have w3 = sq 1 212

= w1vz = 0

as is required. Thus we have seen that the primary obstruction to the existence of a jet extension j r f : M 4 + Imm(M4,R6)(c J"(M4,R6))vanishes, where Imm(M4,R6) is the subbundle consisting only of regular jets. Here note that Imm(M4,R6) is homotopy equivalent to %(R6). Therefore, for any stable map of a closed oriented 4-manifold M4 into R6, we cannot quickly decide whether the singularities will be eliminated or not. However, fortunately by applying Smale-Hirsch theory ( [32]),we can obtain the characterization definitely on the existence of an immersion f : M4 + R6 for a closed oriented 4-manifold M 4 as follows (see [32]): A closed oriented 4-manifold M4 can be immersed in R6 if and only if there exists a n integral cohomology class e E H2(M4;Z) such that (i) e (mod 2) = w2(M4) E H2(M4;Z2),the 2-nd Stiefel-Whitney class of M4, and (ii) -e v e = p l ( M 4 ) E H4(M4;Z), the first Pontrjagin class of M4. From these conditions we can immediately deduce the removability of Whitney umbrellas of a stable map of a closed oriented 4-manifold into R6. Before discussing this, let us consider several cases for concretely specified 4-manifolds. Example B . l . Let M4 = CP2 or (cp2. For any stable map f : M4 -+R6 we can never eliminate Whitney umbrellas of f:

382

K. Sakuma

Proof. Suppose that there is an immersion f : @P2 4 R6.Then there should be an integral class e E H 2 ( @ P 2 ; Z S ) Z satisfying conditions (i) and (ii). By condition (i) we can set e = (2m 1)a for some m E Z since w2(@P2)= a (mod 2 ) , where a is a generator such that a a = 1. Then by condition (ii) we have

+

-

+ 1)2 = - p l ( @ p 2 ) = -3c(@p2)= -3,

e v e = (2m

which has no solution m E Z.Note that the similar argument works also for P. 0

Example B.2. Let M4 = CP2#@P2. For any stable map f : M4 -+ R6 we can never eliminate Whitney umbrellas of f : Proof. Suppose that there is an immersion f : CP2#@P2-+ R6.Then there should be an integral class e E H2((CP2#@P2; Z)S Z69 Z satisfying conditions (i) and (ii). By condition (i) we can set e = (2m+l)a+(2n+l)p for m, n E Z since w2(CP2#(CP2) = a p (mod 2), where we choose two generators a and p such that a v a = p v ,6 = 1,a v p = 0. Then by condition (ii) we have

+

e v e = (2m

+ 1 ) 2+ (2n + 1)2 = -3a(@P2#@P2)= -6,

which has no solutions m, n E

Z.

0

Example B.3. Let M4 = CP2#@p2.There exists an immersion f : M 4 -+ R6:

-+

z z

-

Proof. If we set e = a ,B E H ~ ( (Z c ) PCB~for# two P generators ; a and p such that a a = 1,p ,B = -1 and a v p = 0, then we can 0 easily verify that e satisfies conditions (i), (ii). Comparing these two cases, we see that the orientation of the source 4manifold is essential for the existence of an immersion or the elimination of Whitney umbrellas. Moreover, there is no immersion of CP2 into R6 but there should be an immersion of U P 2 # @ into R6, in other words, Whitney umbrellas can be eliminated by blowing-up on the source 4-manifold. It is very interesting to clarify the elimination method geometrically. Suppose that there is an immersion f of a closed oriented 4-manifold M4 into R6. Then by the condition (ii) we have

3

4

~= p~ l ) ( ~ =~-e)

v

e

(B.1)

Exastence problem for fold maps 383

for some integral cohomology class e E H 2 ( M 4 Z). ; Further, the condition (i) shows that e is an characteristic element. Then by the well known result on the theory of unimodular symmetric bilinear forms

4 M 4 )= e

-

e

(mod 8).

(B.2)

Combining (B.l) and (B.2), we have

a ( M 4 )= 0 (mod 2). Since it holds that x ( M 4 )3 4 M 4 ) (mod 2), we have shown that the top Stiefel-Whitney class w4 E H 4 ( M 4 ;Z2) must vanish if there is an immersion f : M 4 -+R6 for a closed oriented 4-manifold M 4 . Equivalently, every generic map of a closed oriented 4-manifold M 4 into R6 has nonremovable Whitney umbrellas if the top Stiefel-Whitney class of M 4 is nontrivial. Hence summarizing these arguments, the Thom polynomial of Whitney umbrellas is originally defined in the 3-rd cohomology group and identically vanishes, and the only nontrivial element w4 E H 4 ( M 4 ;Z2) Z2 is surely a secondary obstruction to the existence of the jet section. Thus we may say that the top Stiefel-Whitney class w4 is the secondary Thom polynomial of Whitney umbrellas for a generic map of a closed orientable 4-manifold M 4 into R6. Here note that w4 is not an obstruction any more if we suppose that M 4 should be nonorientable. For example, consider any immersion cp : R P 2 -+ R3 (e.g. one may consider Boy's surface.). Then the smooth map f = cp x cp : RP2 x RP2 + R6 is also an immersion and it holds, however, that x ( R P 2 x RP2) = 1 and hence w4 # 0. In the paper [77], Szucs has discussed the same subject, a stable map of a closed oriented 4-manifold into R6, and elimination of singularities by cobordism principle. All these mentioned phenomena on the secondary Thom polynomial are only involving the 4-dimensional manifold theory. Thus it is natural to ask whether we have another situations not involving the 4-dimension or not. The potential candidate has been discussed in [55]. References 1. J. F. Adam, O n the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960), 20-104. 2. J. F. Adam, Vector fields on spheres, Ann. of Math. 75 (1962), 603-632. 3. P. M. Akhmetiev and R. R. Sadykov, A remark on elimination of singularities f o r mappings of 4-manifolds into 3-manifolds, Topology Appl. 131 (2003), 51-55. 4. Y.Ando, O n the elimination of Morin singularities, J. Math. SOC. Japan 37 (1985), 471-487; Erratum, 39 (1987), 537.

384 K. Sakuma

5. Y. Ando, O n the higher T h o m polynomials of Morin singularities, Publ. RIMS 23 (1987), 195-207. 6. Y. Ando, Folding maps and the surgery theory o n manifolds, J. Math. SOC. Japan 53 (2001), 357-382. 7. Y. Ando, Fold-maps and the space of base point preserving maps of spheres, J. Math. Kyoto Univ. 4 1 (2001), 693-737. 8. Y. Ando, Existence theorems of fold-maps, Japan. J. Math. (N.S.) 30 (2004), 29-73. 9. Y . Ando, Smooth maps having only singularities with Boardman symbol (1, 0 ) , Topology Appl. 142 (2004), 205-226. 10. V. I. Arnol’d, Singularities of smooth mappings, Russian Math. Surveys 23 (1968), 1-43. 11. V. I. Arnol’d, V. A. Vasil’ev, V. V. Goryunov and 0.V. Lyashko, Dynamical systems VI: Singularities: local and global theory, Encyclopaedia Math. Sci., vol. 6, Springer-Verlag, 1993. 12. M. F. Atiyah, Vector fields o n manafolds, Arbeitsgemeinschaft fiir Forschung des Landes Nordrhein-Westfalen, Diisseldorf 1969, 200 (1970), 7-24. 13. J. M. Boardman, Singularities of differentiable maps, Publ. Math. I. H. E. S 33 (1967), 21-57. 14. E. H. Brown, T h e cohomology of BSOn and BOn with integer coeficients, Proc. Amer. Math. SOC.85 (1982), 283-288. 15. R. L. W. Brown, A note o n immersions u p to cobordism, Illinois J. Math. 21 (1977), 240-241. 16. D. S. Chess, A PoincarGHopf type theorem for the de R h a m invariant, Bull. Amer. Math. SOC.3 (1980), 1031-1035. 17. D. S. Chess, A note o n the classes [Sf(f)],Proc. Sympo. Pure Math. 40, Part I (Amer. Math. SOC.,Providence, RI, 1983), 221-224. 18. P. J. Eccles, Codimension one immersions and the Kervaire invariant one problem, Math. Proc. Camb. Phil. SOC.90 (1981), 483-493. 19. J. M. Elisberg, O n singularities of folding type, Math. USSR Izv. 4 (1970), 1119-1134. 20. J. M. EliaSberg, Surgery of singularities of smooth mappings, Math. USSR IZV.6 (1972), 1302-1326. 21. Y. Eliashberg’) and N. Mishachev, Wrinkling of smooth mappings and its applications, I, Invent. math. 130 (1997), 345-369. 22. Y . Eliashberg and N. Mishachev, Wrinkling of smooth mappings and its applications, II. Wrinkling of embeddings and K. Igusa’s theorem, Topology 39 (2000), 711-732. 23. Y. Eliashberg and N. Mishachev, Wrinkling of smooth mappings and its applications, III. Foliations of codimension greater than one, Topol. Methods Nonlinear Analy. 11 (1998), 321-350. 24. Y. Eliashberg and N. Mishachev, Introduction to the h-principle, Grad. Study in Math. vol. 48, Amer. Math. SOC.,Providence, RI, 2002.

l) J .

M. Elisberg = Y. Eliashberg

Existence problem for fold maps 385 25. L. Feh6r and R. R i m h y i , Classes of degeneracy loci for quivers: the T h o m polynomial point of view, Duke Math. J. 114 (2002), 193-213. 26. L. Feh6r and R. Rimhyi, T h o m polynomials with integer coeficients, Illinois J. Math. 46 (2002), 1145-1158. 27. T. Fukuda, Topology of folds, cusps and Morin singular ities, ‘A Fete of Topology’ ed. by Y. Matsumoto, T. Mizutani and S. Morita, (Academic Press, New York, 1987), pp. 331-353. 28. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math., vol. 14, Springer-Verlag, New York-Heidelberg-Berlin, 1973. 29. M. Gromov, Partail Differential Relations, Springer-Verlag, 1986. 30. A. Haefliger et A. Kosinski, U n the‘ordme de T h o m sur les singular it& des applications diffhentiables, SBminaire H. Cartan, E. N. S., 1956/57, Expos6 no. 8. 31. R. J. Herbert, Multiple points of immersed manifolds, Mem. Amer. Math. SOC.34,vol. 250, 1981. 32. M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. SOC.93 (1959), 242-276. 33. S.Izumiya, T. Sano, 0. Saeki and K. Sakuma, Geometry and Singularity (in Japanese), Kyoritu-shuppan, 2001. 34. I. James and E. Thomas, An approach to the enumeration problem for nonstable vector bundles, J. Math. Mech. 14 (1965), 485-506. 35. M. E. Kazaryan, Hidden singularities and Vassiliev’s homology complex of singularity classes (in Russian), Mat. Sb. 186 (1995), 119-128; English translation in Sb. Math. 186 (1995), 1811-1820. 36. S. Kikuchi and 0. Saeki, Remarks o n the topology of folds, Proc. Amer. Math. SOC.123 (1995), 905-908. 37. J. KorbaS and P. Zvengrowski, T h e vector field problem: a survey with emphasis o n specific manifolds, Expos. Math. 12 (1994), 3-30. 38. U. Koschorke, Vector fields and other vector bundle isomorphisms - A singularity approach, Springer-Verlag, Lecture Notes in Math., vol. 847, 1981. 39. H. I. Levine, Singularities of differentiable mappings, Springer-Verlag, Lecture Notes in Math., vol. 192, Proceedings of Liverpool Singularities, 1970, pp. 1-89. 40. H. I. Levine, Elimination of cusps, Topology 3 (suppl. 2) (1965), 263-296. 41. H. I. Levine, Stable maps: An introduction with low dimensional examples, Bol. SOC.Brasil Mat. 7 (1976), 145-184. 42. G. Lusztig, J. Milnor and F. Peterson, Semi-characteristics and cobordism, Topology 8 (1969), 357-359. 43. J. Mather, Stability of CM mappings I, the division theorem, Ann. of Math. 87 (1968), 89-104. 44. J. Mather, Stability of Coomappings 11, Infinitesimal stability implies stability, Ann. of Math. 89 (1969), 254-291. 45. J. Mather, Stability of Cm mappings 111, Finitely determined map germs, Publ. Math. I. H. E. S., 35 (1968), 127-156. 46. J. Mather, Stability of CM mappings I V , Classification of stable m a p germs

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by R-algebras, Publ. Math. I. H.E. S.,37 (1969), 223-248. 47. J. Mather, Stability of Coo mappings V, Transversality, Adv. in Math., 4 (1970), 301-336. 48. J. Mather, Stability of Coomappings VI, T h e nice dimensions, Springer Lecture Notes in Math., vol. 192 (1971), 207-253. 49. J. Mather, Generic projections, Ann. of Math., 98 (1973), 226-245. 50. J. Mather, O n Thom-Boardman singularities, Dynamical Systems, Academic Press, New York, 1973, pp. 233-248. 51. K. C. Millett, Generic smooth maps of surfaces, Topology Appl. 18 (1984), 197-215. 52. J. Milnor, Morse theory, Ann. of Math. Studies vol. 51, Princeton Univ. Press, 1963. 53. J. Milnor and J. Stasheff, Characteristic classes, Princeton Univ. Press, Ann. of Math Studies vol. 76, 1974. 54. B. Morin, Formes canonique des singularit6s d’une application diffhentiable, C. R. Acad. Sci. Paris 260 (1965), 5662-5665, 6503-6506. 55. T. Ohomoto, 0. Saeki and K. Sakuma, Self-intersection classes f o r singularities and its applications to fold maps, Trans. Amer. Math. SOC.355 (2003), 3825-3838. 56. I. R. Porteous, Simple singularities of maps, Springer-Verlag, Lecture Notes in Math. vol. 192, Proceedings of Liverpool Singularities, 1970, pp. 286-307. 57. D. Randall, O n indices of tangent fields with finite singularities, SpringerVerlag, Lecture Notes in Math. vol. 1350, Proceedings of the second Topology Symposium in Siegen, 1988, pp. 213-240. 58. R. R i m h y i and A. Sziics, Pontrjagin-Thom type construction f o r maps with singularities, Topology 37 (1998), 1177-1191. 59. R. R i m h y i , T h o m polynomials, symmetries and incidences of singularities, Invent math. 143 (2001), 499-521. 60. R. Sadykov, T h e Chess conjecture, Algebraic & Geometric Topology 3 (2003), 777-789. 61. R. Sadykov, Elimination of singularities of smooth mappings of 4-manifolds into 3-manifolds, Topology Appl. 144 (2004), 173-199. 62. R. Sadykov, A note on Morin mappings, preliminary note, 2004. 63. R. Sadykov, 0. Saeki and K. Sakuma, Existence problem of fold maps into I W ~ in , preparation. 64. 0. Saeki, Notes o n the topology of folds, J. Math. SOC. Japan 44 (1992), 551-566. 65. 0.Saeki, Studying the topology of Morin singularities f r o m a global viewpoint, Math. Proc. Camb. Phil. SOC.117 (1995), 223-235. 66. 0. Saeki, Fold maps o n 4-manifolds, Comment. Math. Helv. 78 (2003), 627647. 67. 0.Saeki, Topology of Singular Fibers of Differentiable Maps, Springer-Verlag, Lecture Notes in Math., vol. 1854, 2004. 68. 0. Saeki and K. Sakuma, Maps with only Morin singularities and the Hopf invariant one problem, Math. Proc. Camb. Phil. SOC.124 (1998), 501-511. 69. 0. Saeki and K. Sakuma, O n special generic maps into R3,Pacific J. Math.


184 (1998), 175-193. 70. 0. Saeki and K. Sakuma, Stable maps between 4-manifolds and elimination of their singularities, J. London Math. SOC.59 (1999), 1117-1133. 71. 0. Saeki and K. Sakuma, Elimination of singularities: Thom polynomials and beyond, London Math. Lecture Notes in Math. vol. 263, Proceedings in Honor of C. T. C Wall’s 60-th Birthday, Cambridge Univ. Press, 1999, pp. 291-304. 72. 0. Saeki and K. Sakuma, Special generic maps of 4-manifolds and compact complex analytic surfaces, Math. Ann. 313 (1999), 617-633. 73. 0. Saeki and T. Yamamoto, Singular fibers of stable maps and signatures of 4-manifolds, preprint. 74. K. Sakuma, O n special generic maps of simply connected 2n-manifolds into W3,Topology Appl. 50 (1993), 249-261. 75. K . Sakuma, O n the topology of simple fold maps, Tokyo J. Math. 17 (1994), 21-31. 76. S. Smale, Generalized Poincari’s conjecture in dimension greater than four, Ann. of Math. 74 (1961), 391-406. 77. A. Sziics, Elimination of singularities by cobordism, Contemporary Math. 354 (2004) 301-324. 78. R. Thom, Les singularitks des applications diffkrentiables, Ann. Inst. Fourier, Grenoble 6 (1955-56), 43-87. 79. R. Thom, Un lemme sur les applications diffirentiables, Bol. SOC. Mat. Mexicana 2-nd series, 1 (1956), 59-71. 80. E. Thomas, Postnikov invariants and higer order cohomology operations, Ann. of Math. 85 (1967), 184-217. 81. E. Thomas, The index of a tangent 2-field, Comment. Math. Helv. 42 (1967), 86-110. 82. E. Thomas, Fields of tangent k-planes on manifolds, Invent. math. 3 (1967), 334-347. 83. E. Thomas, Vectorfields on low dimensional manifolds, Math. Z. 103 (1968), 85-93. 84. G. W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math., vol. 61, Springer-Verlag, New York-Heidelberg-Berlin, 1978. 85. H. Whitney, Differentiable manifolds, Ann. of Math. 37 (1936), 645-680. 86. H. Whitney, The self-intersections of a smooth n-manifold in 2n-space, Ann. of Math. 45 (1944), 220-246. 87. H. Whitney, The singularities of a smooth n-manifold in ( 2 n - 1)-space, Ann. of Math. 45 (1944), 247-293. 88. H. Whitney, O n singularities of mappings of euclidean spaces I, Mappings of the plane into the plane, Ann. of Math. 62 (1955), 374-410.

388

On completely integrable first order ordinary differential equations Masatomo Takahashi

Department of Mathematics, Hokkaido University, Sapporo 060-0810, JAPAN e-mail: takahashiamath. x i . hokudai. ac.jp A complete solution of a first order ordinary differential equation is a certain one-parameter family of geometric solutions on the equation surfxe. In this survey, we shall discuss the existence conditions, properties, duality, a generic classification of completely integrable (regular) first order ordinary differential equations and of bifurcations of them. Moreover, we consider a general Clairaut type equations which is a generalized notion of the classical first order Clairaut equations, but the equations are not necessarily regular first order ordinary differential equations. We also give a generic classification and bifurcations of general Clairaut type equations as an application of singularity theory.

Keywords: Implicit first order ordinary differential equations; Completely integrable; (General) Clairaut type equations; Legendrian transformations; Bifurcations; Legendrian singularity theory.

1. Introduction

We consider an implicit first order ordinary differential equation given by

F(z,y,y’)

= 0,

where F is a smooth function of the independent variable 2 , the function y and its first derivative y’ = dy/d-z. The following are examples of first order ordinary differential equations (for short, first order O.D.E.s, or simply, equations):

( a ) y’ = 2 2 , ( b ) y

=

( y y , (c)

2 = (y’)2.

It is easy to solve these equations, giving 1 3 ( a ) y = --z + c, ( b ) y 3

=

G

--2

I- c

)2

,

y = 0, (c)

2

2 3

= t 2 , y = -t3

+ c,

O n completely integrable first order ordinary differential equations 389

. .%

-1 - 0 . 7 5 - 0 . 5 -0.25 -0.5

-0.75

-1

Fig. 1. Phase portraits of ( a ) ,( b ) and (c).

where c is an arbitrary constant and t is a parameter. Then we can draw the pictures of the solutions (phase portraits), see Fig. 1. We now consider the difference between these equations (cf. Remark 2.7 below). For the case (a), it is an explicit equation. If an equation F ( z ,y, y’) = 0 has the explicit form y’ = f(x,y) for some smooth function f,then at least locally there exists a classical solution around a point (ZO, yo). For the case ( b ) , it is an implicit equation and a Clairaut type equation (for the definition see Sec. 2). This equation has classical solutions (y = ( ( 1 / 2 ) ~ + c ) ~and ) a singular solution (y = 0) which is given by the envelope of the family of the classical solutions. For the case (c), it is not a Clairaut type equation but has geometric solutions (multi-valued solutions, y = f(2/3)zi c). For ( b ) and (c), it has a one-parameter family of classical solutions or geometric solutions like the explicit equations. Hence we have a natural question. When does there exist a one-parameter family of classical solutions or geometric solutions for an implicit equation? Here all manifolds and mappings considered are differential of class C”. In order to answer the question, we need to recall some notions and definitions. Let F(z,y,y’) = 0 be an equation. Replacing y’ by p , it is natural to consider the equation as being defined on a subset of the space of 1-jets of functions of one variable. Let (x,y, p ) be a local coordinate of J 1(R, R) and 7r : J1(R,R) R2 be the canonical projection 7r(z,y,p) = (z,y). We call F-l(O) c J1(R,R) the equation surface. The equation surfaces might have singularities. However, first we assume a regular condition. For an equation F ( z , y, p ) = 0, we shall assume that 0 is a regular value of F , unless otherwise stated. It follows that the equation surface is a surface in J1(R, R).

+

-

390 M. Takahashi

# 0. On the other hand, consider the corresponding definition of parametrized version for geometric solutions. Let r : (R x R, ( % , t o ) ) (F-'(O), ZO) be a one-parameter family of geometric solutions of F = 0. We call r a complete solution at zo if

-

O n completely integrable first order ordinary dzfferential equations

391

where r ( c , t ) = ( x ( c , t ) , y ( c , t ) , p ( c , t ) ) .This condition means that r is a diffeomorphism germ, that is, the equation surface is foliated by a oneparameter family of geometric solutions. We say that an equation F = 0 is smooth completely integrable (respectively, completely integrable) at zo if there exists a smooth complete solution (respectively, a complete solution) of F = 0 at ZO. Also we say that a geometric solution y : (R,O) (F-l(O),zo) is a singular solution of F = 0 at zo if for any representative ;i;: U F-'(O) of y and any open subinterval (c, d) c U at 0, Yl(,,d) is never contained in a leaf of a complete solution (cf. [3,21,25]). Around z E F-l(O) such that the contact plane EZ intersects T,F-l(O) transversally, it is easy to see that a complete solution exists by integrating the line field E n TF-'(O). We call points where transversality fails contact singular points and denote by C,(F) the set of contact singular points. Furthermore, we call a singular point of n : J1(R, R) R2 a n-singular point and denote it by C,(F) the set of n-singular points. Explicitly, contact singular set and n-singular set are given by

- -

C,(F) = {Z E J1(R, R) I F ( z ) = 0 , F,(z)+pF,(z) = 0, FP(z)= 0}, and

C,(F) = { Z E J 1 ( R , R ) I F ( z ) = 0 , F p ( z )= 0). These sets are useful to characterize the existence condition of complete solution and smooth complete solution. In section 2, we answer the question. Indeed, we give existence conditions of the complete solution (Proposition 2.1 and Theorem 2.2) and the smooth complete solution (Theorem 2.6). In section 3, we discuss the duality of completely integrable first order ordinary differential equations by using the Legendrian transformations. First order differential equations have been studied for a long time (cf. [1,5-8,14,17,28]). In [15,19,35],generic first order differential equations are studied as applications of singularity theory. In this paper, we consider generic completely integrable first order ordinary differential equations. We give not only a generic classification of completely integrable equations in Sec. 4, but also of bifurcations of them in Sec. 5. In sections 6 and 7, we consider the equation surfaces which might have singularities, namely, 0 is not a regular value of F . We say that an equation is a general Clairaut type equation if the equation surface is represented by the image of a map germ and foliated by a one-parameter family of classical solutions, that is, the equation has a smooth complete solution. In section 6, we give a generic

392

M. Takahashi

classification of general Clairaut type equations and an outline of the proof. In the proof, we use notions and results in singularity theory (especially, unfolding theory, versality, transversal theorem and Legendrian singularity theory), which can be found in [2,9,16,29]. In section 7, we give a generic classification of bifurcations of general Clairaut type equations. In this survey, we treat only implicit first order ordinary differential equations. However, almost every notion and theorem can be extended to first order partial differential equations or holonomic systems. Interested readers are referred to the original papers. Also higher order cases are studied as applications of geometric approach and singularity theory, see for example, [1,3,11,12,15,26,27,34]. 2. Existence conditions of complete solution and smooth

complete solution In this section we give existence conditions of the complete solution and the smooth complete solution of F = 0. To characterize the smooth complete solution, we introduce the Clairaut type equations. Moreover, we give a kind of the uniquness of complete solutions. First we give an existence condition of complete solution.

-

Proposition 2.1. For an equation F : (J1(R,R),z0) (IR,O), F = 0 is completely integrable at zo if and only i f there exist function germs a,P : (F-'(O), 20) R, which do not vanish simultaneously, such that

-

Q

. (F, + PFy)IF-'(O) + P - F,IF-l(O)

-

= 0.

Proof. Suppose that F = 0 is completely integrable at zo and let : (R x

R, ( % , t o ) )

(F-l(O),zo)

-

be a complete solution of the equation around zo. Then differentiating I' with respect to t yields a vector field Z : (F-'(O),zo) TF-l(O) given by Z ( r ( c , t ) ) = I't(c,t). Since Z ( z ) lies in the contact plane EZ for each z E (F-l(O),zo), it has the form 2 = ( a , p a , P ) for some functions a,/?: (F-'(O), ZO) IR which do not vanish simultaneously. Besides Z ( z ) also lies in T Z F 1 ( O )It. follows that the identity

-

Q *

(Fz+PFy)IF-'(O) + P . FpIF-l(O) = 0

holds. Reversing the argument yields the converse.

0

On the other hand, we can recognize by the contact singular set of an equation whether or not the equation is completely integrable.

-


Theorem 2.2 ( [21,25]). For an equation F : (J1(R,R),zo)

the following are equivalent. ( 1 ) F = 0 is completely integrable at ZO. ( 2 ) C , ( F ) = 8 or C,(F) is a 1-dimensional manifold around

-

Theorem 2.3 ( [21,25]). For an equation F :

(R,O),

ZO.

(J1(R,R),zo) (R,O) and a geometric solution y : (R,to) (J1(R,R),zo) of F = 0, the following are equivalent. ( 1 ) y is a singular solution of F = 0. ( 2 ) There exists a complete solution of F = 0 such that each leaf is transverse to y. (3) Image y C C,(F).

-

By Theorems 2.2 and 2.3, we have the following corollary:

(R,O)has a singular solution if and only if C,(F) is a 1-dimensional manifold around ZO. Moreover, C,(F) is a singular solution of F = 0.

Corollary 2.4 ( [21,25]). A n equation F : (J1(R,rW),zo)

-

The uniquness of the complete solution is dealt with in the following:

-

Proposition 2.5 ( [23]). Let rl : (R x R,(c1, t l ) ) (F-'(O), ZO) and I'2 : (R x R , ( c z , t z ) ) (F-l(O),zo) be complete solutions of F = 0 at ZO. Then there exists a difleomorphism germ @ : (R x R, (c1,t l ) ) (R x R,(c2,t2)) of the form @ ( c , t )= (cpl(c),cp2(c,t)) such that r2o @ = rl.

-

We now introduce the notion of Clairaut type equations as a generalized notion of the classical Clairaut equations (see, Example 2.8). The following definition is due to Dara [ l o ] .We say that an equation F = 0 is a Clairaut type equation at zo if there exist smooth function germs A , B : (J1(R, R), Z O ) R such that

-

F,

+p

*

F, = A . F

+B

*

Fp.

If F = 0 is a Clairaut type equation at ZO, then the contact singular set C,(F) coinsides with the .rr-singular set C,(F) and F = 0 is completely integrable at zo by Proposition 2.1. More precisely, we have the following theorem: Theorem 2.6 ( [23]). For an equation F

= 0 , the following are equivalent. ( 1 ) F = 0 is of Clairaut type at ZO. ( 2 ) F = 0 has a smooth complete solution at ZO.

394 M. Takahashi

Moreover, in this case, if C,(F) # 8 and hence C , ( F ) is 1-dimensional manifold around ZO, then .rr(C,(F)) is the envelope of the family of smooth solutions.

Remark 2.7. If (Y # 0 at zo in Proposition 2.1, then F = 0 is a Clairaut type equation by the regular condition of F = 0. Again, let us consider the first examples

( a ) F ( x ,Y ,P) = P - x2 = 0, 2

( b ) G ( X , Y , P ) = Y - P = 0, ( c ) H ( x , y , p ) = x - p2 = 0. In these cases, we have

+

) 0, & ( F ) = C,(F) = 0, ( a ) (Fx + ~ F , ) l ~ - l ( o ) 2 ~F p. l ~ - 1 ( 0E ( b ) (Gz +pG,)IG-'(O)

+ ( l / 2 ) *GpIG-l(O)

0,

C c ( G ) = C,(G) = {(X,Y,P) I Y = P = 01,

(4

+

2P. ( H x P%)IH-'(O) + HpIH-l(O) 3 0, C c ( H ) = 0, C,(H) = { ( x ,Y,P) I = p = 0).

Therefore, these equations are completely integrable by Proposition 2.1 or Theorem 2.2. Also G = 0 has a singular solution which is given by the envelope of the family of smooth solutions by Theroems 2.3 and 2.6. We present another example illustrating the notion of (smooth) complete solution. This example was due to A.C. Clairaut [6].

Example 2.8 (The first order classical Clairaut equations). Consider the first order ordinary Clairaut equation Y

dY

= x-d x

+f

(2),

where f ( p ) is a smooth function and p = d y / d x . In this case, Fx +pF, = 0 and Fp = -x - f ' ( p ) . I t follows that F = 0 is a Clairaut type equation. Hence the contact singular set C, coincides with the .rr-singular set C, and is given by C c ( F ) = { ( G Y , P ) I x = - f ' ( P ) , Y = -Pf'(P)

+ f (PI).

Notice that C,(F) is 1-dimensional manifold, there exist a complete solution and a singular solution by Theorems 2.2 and 2.3. I n fact, the (smooth) complete solution r : Iw x R +F-'(O) is given by r ( c , t ) = (t,ct f ( c ) ,c ) and the singular solution is given by y ( t ) = ( - f ' ( t ) , -tf'(t)+ f ( t ) , t ) . Then

+

On completely integrable first order ordinary differential equations 395

observe that the singular solution is given by the envelope of the family of smooth solutions by Theorem 2.6. Now we consider the classical Clairaut equations rather than the Clairaut type equations. We have the following theorem. Theorem 2.9 ( [23]). For an equation F = 0 at zo = (zo,go,pg),the following are equivalent. (1) There exists a function germ A : ( J 1 ( R R), , ZO) R such that

-

F, + p . Fy = A F, and Fu # 0 at ZO. (2) There exists a function germ f : ( R , p o )

-

R such that

F-l(o) = {@,Y,P) I Y = X P + f(P)l. 3. Duality among completely integrable first order 0.D.E.s

A notion of Legendrian transformation can be used to set up a dual relationship between equations. We adopt another coordinate system ( X ,Y,P ) of J1(R,R) by X = p , Y = z p - y , P = x . We refer to a smooth mapJ1(R,R) given by L ( z , y , p ) = ( p , z . p - y,z) ping L : J1(R,R) as a Legendre transformation (cf. [4,17,23]). By the definition, we have L - l ( X , Y , P ) = (P,X P - Y , X ) . If we apply the Legendre transformation to our equation F = 0, we obtain a new equation

-

-

F* ( X ,Y,P ) = F o L - ' ( X , Y,P ) = F (P ,X . P - Y,X ) = 0 in the coordinate system ( X ,Y,P ) . If we calculate the partial derivatives at the point (Xo,Yo,PO)corresponding to ( 2 0 ,yo,po), we can show the following:

F;(xo,yo,Po) = (Fp+z.Fy)(zo,Yo,Po), FXXO, yo, Po) = - w o , Yo, Po),

F;(xo,yo,Po) = (F, +P*Fy)(zo,Yo,Po). Since Legendre transformation is a contact diffeomorphism, we have the following lemma:

--

Lemma 3.1. (1) Let y : ( E l t o ) F-l(O) be a geometric solution of F = 0. Then L o y : (R, t o ) F*-l(O) is also a geometric solution of F* = O . (2) If t o is a geometric singular point of y,then t o is a geometric nonsingular point of L o y.

396

M. Takahashi

The following argument may be considered as the principle of duality among completely integrable equations. Let C~(zO,yo,po) be the set of germs corresponding to completely integrable equations at (50,yo,po). For each F E CI(Z0, yo,po), we have the unique complete solution r F : (R x R, 0 ) (F-'(O), ( 2 0 ,y0,po)). We denote it by r F ( C , t ) = ( Z F ( C , ~ ) , Y F ( C , ~ ) , ~ F ( CThen ,~)). we define three subsets of Cr(z0,y0,po) as follows:

-

C 1 1 ( ~ 0 , Y O , P O )=

C I Z ( ~ O , Y O , P O )=

{ {F

F

I I

dpF

cI(zo,Yo,Po)-&To,Llo,Po) dXF

E ~ I ( ~ O , Y O , P O )$ZO,YO,PO)

#O

1

,

# o}.

By the uniquness of the complete solution of F = 0 (cf. Proposition 2.5), these subsets are well-defined. We also denote by CJ*(Xo,Yo,Po)the set of completely integrable equations at ( X O YO, , PO)in the coordinate system ( X ,Y,P ) . Furthermore, we define subsets C;o(Xo,Yo,Po), C;, (Xo,Yo,Po) and C;z( X O YO, , PO)by exactly the same definitions as above. Then we have the following duality theorem.

-

Theorem 3.2 ( [23]). W e have a n one-to-one correspondence 23: C I ( z o , Y o , P o )

~;(xo,yo,~o)

defined by D ( F ) = F*. Besides, we have relations: ~(~,,(~O,YO,PO= ) ) ~;o(xo,yo,~o),

D ( G l (20,

Y0,PO)) =

GZ(XO,yo,Po),

~ ( ~ I , ( ~ O , Y O , P O= ) )~ ~ l ( x o , y o , ~ o ) .

The following is a corollary of Theorem 3.2. However, we also give a simple proof.

Corollary 3.3. Let F = 0 be a n equation at (zO,yo,po). T h e n F = 0 is completely integrable at (20,y0,po) if and only if F = 0 is of Clairaut type at ( z o , y o , p o ) or F* = 0 is of Clairaut type at (Xo,Yo,Po). Proof. Since F = 0 is completely integrable at ( z o , y o , p o ) , there exist function germs a,P : (F-l(0),(zo,yo,po)) + R, which do not vanish simultaneously, such that

+

a . (FZ +PF,)IF-~(O) P . F ~ [ F - ~ ( o0.)


By the Legendre transformation, we have

where a* = a o L-' and p* = ,B o L-l. Since either a(xo,yo,po># 0 or p * ( X o ,YO, PO)# 0, the consequence follows from Remark 2.7.

4. Completely inetegarble first order 0.D.E.s

In this section we give a generic classification of equations with complete integral. Denote by €(zl,,.,z,) the ring of all function germs on Rn at 0 with coordinates (XI,. . . ,xn) and by 9X(zl,...,z), the unique maximal ideal of qzl,...,z,)Since our concern is the local classification of differential equations, we can adopt the following formulation. Let F ( x ,ylp)= 0 be an equation and F-'(0) be a 2-dimension manifold around a point. Locally an equation surface is represented by the image of an immersion germ f : (R2,0) J1(R,R). We also say that f is an equation and completely integrable if there exists a submersion germ p : (R2,0 ) (R, 0) such that dp A f*O = 0. We call p a complete integral of f and the pair ( p , f ) : (R2,0) R x J'(R, R) is called an equation with complete integral. We observe that flp-l(c) is a geometric solution of the equation for each c E (R,O). Hence the equation surface is foliated by a one-parameter family of geometric solutions. Moreover, if ~ o f I p - ~ ( cis) a non-singular map for each c E (R, 0), then { f l p - ' ( ~ ) }is~the ~~ family of smooth solutions of the equation by the previous argument. We called such an equation the Clairaut type equation in Sec. 2. We introduce an equivalence relation among equations under the group of point transformations of R2. A point transformation cp on R2 is, by definition, a diffeomorphism of R2 onto itself. To define a lift of cp, let a contact manifold be given, which is a fiberwise compactification of J1(R,R). Let ?? : PT*R2 R2 be a projective cotangent bundle over R2 which contains T : J1(R,R) R2 as an affine part. Then we have the canonical contact lift @ : PT*R2 PT*R2 of cp. Let f and f ' : (R2,0) J1(R,R) be equation germs. Following S. Lie, we say that f and f' are equivalent as equations if there exist a diffeomorphism germ I) : (R2,0 ) (R2, 0) and a point transformation cp : (R2, ~ ( z ) ) (a2, ~ ( z ' ) such ) that the lift @ of cp satisfies that @(z) = z' and @ o f = f ' o I), where z = f(0) and z' = f'(O), ,

-

-

--

-

-

-

398

M. Takahashi

namely, the following diagram commutes:

f’

IF

-

The above consideration leads us to the following definition. Let ( p , g ) be a pair of a map germ g : (R2,0) (R2,0) and a submersion germ p : (R2, 0) (R, 0). Then the diagram

-

(R,O) 2-(R”0) -% (R2,O)

or, briefly, ( p , g ) is called an integral diagram if there exists an equation f : (R2,0) J1(R,R) such that ( p , f) is an equation germ with complete integral and r o f = g. Furthermore we introduce an equivalence relations among integral diagrams. Let ( p ,g ) and (p’,9‘) be integral diagrams. Then ( p ,g ) and (p’,9’) are equivalent as integral diagrams (respectively, strictly equivalent) if the diagram P

&I

*I P

.1 9

commutes for some diffeomorphism germs IE (respectively, IE = idw), $ and cp. The following theorem reduces the equivalence problem for completely integrable equations to that for the corresponding induced integral diagrams.

-

-

Theorem 4.1 ( [22]). Let ( p ,f ) : (R2,0) (R x Jl(R,R),(O,z))and (p’, f’) : (R2,0) (R x J1(R,R), ( 0 , ~ ’ ) )be equations with complete integral such that the set of singular points of r o f and r o f’ are closed sets without interior points. Then the following are equivalent: (1) f and f ‘ are equivalent as equations. ( 2 ) ( p , r o f ) and ( p ’ , r o f‘) are equivalent as integral diagrams. We now define the notion of genericity among equations with complete integral. Let U c R2 be an open set. We denote by Int(U, R x J1(R,R)) the set of equations with complete integral ( p , f) : U R x J1(R,R). The set Int(U, R x J1(R, R)) is a topological space equipped with the Whitney

-

O n completely integrable first order ordinary differential equations

399

C"-topology. A subset of the space is said to be generic if it is an open dense subset in the space. The genericity of a property of germs is defined as follows: Let P be a property of equations with complete integral ( p , f ) : U + IR x J1(IR,R). For an open set U c R2, we define P ( U ) to be the set of ( p , f) E Int(U, IR x J'(IR,IR))such that the germ at (u,w) whose representative is given by ( p ,f ) has property P for any (u,w) E U . The property P is said to be generic if for some neighbourhood U of 0 in R2, the set P ( U ) is a generic subset in Int(U,IR x J'(IR,IR)). We remark that in the assumption of theorem 4.1, the condition that the set of singular points of T o f is a closed set without interior points is satisfied for generic equations with complete integral. It follows that if we classify integral diagrams which are induced by generic equations with complete integral by the above equivalence, we have a generic classification of completely integrable equations.

-

Theorem 4.2 ( [18,22]).For a generic equation with complete integral

( p , f ) : (R2,0)

IR x J 1 ( R , q ,

the integral diagram ( p , g = T o f ) is strictly equivalent to one of the germs in the following list: (1) p = 'u, 9 = (u,v). (2) p = w (2/3)u3, g = (u2,w). (3) p = 21 - (1/2)u, g = (u, w". (4) p = (3/4)u4+(1/2)u2w+w+aog, g = (u3+uw, w), where a E 9X(z,y). (5) p = 'u a o g, g = (u, w3 uv), where a E m(z,y). (6) p = -(3/4)w2 - u a o g , g = (u, w3 uw2), where a E 9X(z,v).

+

+

+

+

+

We call a a functional moduli and the normal forms from (1) to (6) where a = 0 non-singular, regular fold, Clairaut fold, regular cusp, Clairaut cusp, mixed fold respectively. We draw the pictures of phase portraits { T o f I p - ~ ( c ) } c ~ ~ respectively, see Fig. 2. 5. Bifurcations of completely integrable first order 0.D.E.s The next problem is to classify one-parameter families of equations with complete integral. We want to know how to bifurcate the phase portraits {T o f I p - ~ ( c ) } c E ~ (geometric solutions and singular solutions). A one-parameter family of first order ordinary differential equation germs (or, briefly, a one-parameter family of equations) is represented

400 M. Takahashi

-0.25

Fig. 2. Phase portraits of non-singular, regular fold, Clairaut fold, regular cusp, Clairaut cusp, mixed fold

-

by a map germ f : (R2x R,O) J1(R,R) such that f t is an immersion germ for each t E (R,O), where f t ( u , v ) = f ( u , v , t ) . We also say that f is a one-parameter completely integrable if there exists a map germ p : (R2 x R,O) (R,O) such that pt is a submersion germ and dpt A fTe = 0 for each t E (R,O), where pt(u,w) = p ( u , w , t ) . The pair ( p , f) : (R2 x R,O) R x J1(R,R) is called a one-parameter family of equations with complete integral. The above situation leads us to the following divergence diagrams. Let ( p , g ) be a pair of germs of maps g : (R2 x R,O) (R2,0) and p : (R2 x R, 0) (R, 0) such that pt is a submersion germ for each t E (R, 0). Then the diagram

-

-

(R,O) 2-(R2 x R,O)

-

5(R2,0),

-

or, briefly, ( p , g ) is called a one-parameter family of integral diagrams if there exists a one-parameter family of equations f : (R2x R, 0) J 1(R, R) such that ( p , f) is a one-parameter family of equations with complete integral and T o f = g. In order to describe equivalence relations among one-parameter families of equations with complete integral and one-parameter families of integral diagrams, we consider the unfolding germs of them. Define F :


401

-

(R2xR,0) -+J 1 ( R , R ) x R b y F ( u , v , t )= (f(u,v,t),t)foraone-parameter family of equations f which is called a one-parameter unfolding of equations associated to f . We also define : (R2 x R,O) (R x R,O) by $(u,v,t ) = ( p ( u ,v,t ) ,t ) .The pair (p,F ) or ( p , F ) is called a one-parameter unfolding of equations with complete integral. Let F and F' be one-parameter unfoldings of equations associated to f and f' respectively. Then F and F' are equivalent as one-parameter unfoldings of equations if the diagram

commutes for some germs of diffeomorphism $,a and 'p of the form @ ( u , v , t )= ( $ l ( U , v , t ) , ( P ( t ) ) , @(GY,P,t)= (z(x7Y,P),'p(t)), 'p(X,Y,t)= ('pl(x,y,t),'p(t)), whereEistheuniquecontactliftof'p, = : (R2x {t),T(.) x { t ) > (R2 x {'p(t)),.rr(z')x {cp(t)))and 'p : @70) (RO) is a diffeomorphism germ. Let (p,G) be a pair of germs of maps G : (R2 x R, 0) -+ (R2 x R, 0) and p : (R2 x R,O) (R x R,O) such that pt is a submersion for each t E (R,O). Then the diagram

-

-

-

(R x R,O)

L (R2 x R,O) 5(R2 x R,O)

or, briefly, (j2,G) is called a one-parameter unfolding of integral diagrams if there exists a one-parameter family of equations f such that (j2, F ) is a one-parameter unfolding of equations with complete integral and (T x id) o F = G where F is one-parameter unfolding of equations associated to f . We remark that if a pair (p,G) is a one-parameter unfolding of integral diagrams, then G has the form G(u,v,t ) = (g(u,v,t ) ,t ) for some map germ g : (a2 x R,O) (R2,O). Now we introduce equivalence relations among one-parameter unfoldings of integral diagrams. Let (p,G) and (@',G') be one-parameter unfoldings of integral diagrams. Then (i;,G) and (@',G') are equivalent as oneparameter unfoldings of integral diagrams (respectively, strictly equivalent)

-

402

M. Takahashi

if the diagram

( R x R , O ) A (R2XR,O)

(RXR,O)

ii'

(R2XR,O)

G

(R2xR,O)

G' (R2XR,O)

commutes for some germs of diffeomorphism K , $ and cp of the form K ( s , t ) = ( K l ( s , t ) , c p ( t ) ) (respectively,tq(s,t) = s ) , $(u,w,t) = ($i(u, v, t ) ,cp(t>>, cp(z, Y , t ) = (cpi(x,Y , t ) ,cp(t)).If (P, G) and ( F ,G') are strictly equivalent, then we also say that ( p , G) and (p', G) are strictly equivalent. The following theorem reduces the equivalence problem for oneparameter unfoldings of equations with complete integral to that for the corresponding induced one-parameter unfoldings of integral diagrams.

- -

Theorem 5.1 ( [30]). Let ( p , F ) : (R2 x R,O) (R x J'(R,R) x R, (O,z,O)) and (p',F') : (R2 x R,O) (R x J1(R,R) x R, (O,z',O)) be one-parameter unfoldings of equations with complete integral such that the sets of singular points of T o ft and T o f i are closed sets without interior points except f o r isolated t. Then the following are equivalent: (1) F and F' are equivalent as one-parameter unfoldings of equations. (2) (ji,(Tx i d ) o F ) and (j.7,(T x i d ) OF')are equivalent as one-parameter unfoldings of integral diagrams. We also define the notion of genericity among one-parameter unfoldings of equations with complete integral. Let U x V c R2 x R be an open set. We denote by Int(U x V,R x J1(R, R) x R) the set of one-parameter unfoldings of equations with complete integral ( p , F ) : U x V R x J1(R, R) x R. This set is a topological space equipped with the Whitney CM-topology. A subset of the space is said to be generic if it is an open and dense subset in the space. The genericity of a property of germs is defined as follows: Let P be a property of one-parameter unfoldings of equations with complete integral ( p , F ) : U x V + R x J1(R,R) x R. For an open set U x V C R2 x R, we define P(U x V ) to be the set of ( p , F ) E Int(U x V,R x J'(R,R) x R) such that the germ at (u, w,t ) whose representative is given by ( p , F ) has property P for any (u, w,t ) E U x V . The property P is said to be generic if for some neighbourhood U x V of the origin in R2 x R, the set P(U x V ) is a generic subset in Int(U x v,R x J'(R, R) x R).

-

On completely integrable first order onlinary differential equations 403

We remark that in the assumption of theorem 5.1, the condition that the singular points of T o ft is a closed set without interior points except for isolated t is satisfied for generic one-parameter unfoldings of equations with complete integral. We give a generic classification of one-parameter unfoldings of equations with complete integral: Theorem 5.2 ( [33]). For a generic one-parameter unfolding of equations with complete integral

-

( p , F ) : (R2 x R,O)

R x J1(R,R) x R,

the one-parameter unfolding of integral diagrams (p,G = (T x i d ) o F ) is strictly equivalent to one of the germs in the following list: ( 1 ) p = ~ ,G = ( u , v , t ) . (2) p =v

+ (2/3)u3, G = (u’, V , t ) .

( 3 ) p = w - ( 1 / 2 ) ~G, = ( u , u 2 , t ) . ( 4 ) p = (3/4)u4 (1/2)u2v w a o G,

+ +

+

G = (u3+ uw,w,t ) , where a

E ?JJl(z,y,t).

+

( 5 ) p = w + a o G, G = (u, w3 uw,t ) , where a E 9X(z,y,t). 2 ( 6 ) p = -(3/4)v2 - u + a o G, G = (u,w3+ uw , t ) , where a E 9Jl(z,y,t). (7) p = -(3/4)v2 - + 0 G, G = (u,u3 + P(u, t ) v 2 ,t ) , where E r n ( z , y , t ) and P E r n ( u , t ) , ( W / W ( O ) # 0. (8) p =v

+a

0

G, G = ( u ,w4

+ uw+ P(u, t ) v 2 ,t ) ,

where a E r n ( z , y , t ) and P E r n ( u , t ) , ( W / W ( O ) # 0. (9) p = - ( l / 2 ) w 3 - (3/4)uw2- P(u,t)w- u a o G, G = (u,(3/4)w4 uw3 P(u, t ) w 2 , t ) ,

+

+

+

where a E r n ( z , y , t ) and P E r n ( u , t ) , (W/W(O) # 0. (10) p =,- ( l / 2 ) w 3 - (3/4)P(u,t ) w 2 - uv - u a o G,

+

G = (u,(3/4)w4+ P(u,t )w 3 where cx E qz,,,,,and

+ uw2,t ) ,

P E r n ( u , t ) ,( W / W ( O ) # 0.

+ + 21212 + - ( 2 / 3 ) p ( x u, , t)u + a + + 2uv - P ( X ,u,t ) ,w,t ) ,

(11) p = 4215 2tu3 G = (5u4 3tu2

21

where a E 2X(z,g,t), X = -5u4 - 3tu2 - 2uw and satisfying P ( X ,u,t ) = - 3 y ( x

G,

P E !IR(X,~,~)

+ P ( X ,u,t ) ,t ) u 2 , for y E t ~ ( , , ~ ) .

404

M. Takahashi

We call the function germ a a one-parameter functional moduli and p a second functional moduli. The normal forms from (1) to (6) are equations with complete integral which has already been classified in Theorem 4.2 (cf. [18,22]). We also draw examples of generic bifurcations of phase portraits {T o f t I p t - ~ ( c ) } c E ~ (t < 0,t = 0,t > 0 ) of the normal forms of the case (7),(8), (9), (lo), where a = 0 , p = t and (ll),where a = 0,p = 0 (y = 0). See from Fig. 3 to Fig. 7.

= 0 and

p = t,

Bifurcation of phase portraits of (8) where a = 0 and

p = t.

Fig. 3. Bifurcation of phase portraits of (7) where

Fig. 4.

01


-0.75

-1

Fig. 5 . Bifurcation of phase portraits of (9) where

a!

= 0 and

p =t.

Fig. 6. Bifurcation of phase portraits of (10) where a = 0 and p = t.

Fig. 7. Bifurcation of phase portraits of (11) where

a!

= 0 and

p = 0.

6. Generic classification of general Clairaut type equations

In this and the next section, we shall consider an equation surface F-l(O) which is not necessarily a surface in J1(R, R), that is, it might have singularities. Here we assume that the equation surface is the image of a map germ f : (R2,0) J1(R,R).

-

406

M. Takahashi

-

We say that f is a completely integrable if there exists a submersion germ (R, 0) such that d p A f*O = 0. We call p a complete integral of f and the pair ( p , f ) : (R2, 0) R x J1(R, R) is called an equation with ( ~ I map for each c E (R, 0), f is complete integral. If I T ~ ~ ~ ~ is- aI non-singular called a general Clairaut type equation. Moreover, iff is an immersion germ, then f is called a Clairaut type equation (cf. Sec. 2). The term “general” means that f is not necessarily an immersion germ. In this section, we give a generic classification of general Clairaut type equations and the strategy for a proof. The main idea of the proof of Theorems 4.2, 5.2 and 7.1 almost follows the same steps (cf. [22,32,33]). Let ( p , g ) be a pair of a map germ g : (R2, 0) (R2,0) and a submersion germ p : (R2, 0) (R, 0). Then the diagram p : (R2, 0)

-

-

-

(R,O) A (R2,O) -% (R2,O)

-

or, briefly, ( p , g ) is called an integral diagram if there exists an equation f : (R2,0) J’(R, R)such that ( p , f ) is an equation germ with complete integral and IT o f = 9. Also we introduce the equivalence relations among equations and integral diagrams by the same definitions as those in Sec. 4. Then we can show that two equations f and f‘ are equivalent if and only if the induced integral diagrams ( p ,7r o f ) and ( p ’ , IT o f’) are equivalent for generic ( p , f ) and (p’, f’) (cf. Theorem 4.1). Now we define the notion of genericity among general Clairaut type equations. Let U c R2 be an open set. We denote by Clair(U, R x J1(R, R)) the set of general Clairaut type equations (p,f): U R x J1(R,R). This set is a topological space equipped with the Whitney Cm-topology. A subset of the space is said to be generic if it is an open dense subset. The genericity of a property of germs is defined as follows: Let P be a R x J’(R, R). For property of general Clairaut type equations ( p ,f ) : U an open set U c R2, we define P ( U ) to be the set of (p, f) E Clair(U, R x J1(R,R)) such that the germ at (u,w) whose representative is given by ( p , f ) has property P for any (u,w ) E U . The property P is said to be generic if for some neighbourhood U of 0 in R2, the set P ( U ) is a generic subset in Clair(U,R x J1(R,R)). The main result in this section is the following theorem which gives a generic classification of general Clairaut type equations:

-

Theorem 6.1 ( [13,31]).For a generic general Clairaut type equation

( p , f ): (R2,0) --+R x J1(R,R),


407

the integral diagram ( p ,g = T o f ) is strictly equivalent to one of the germs in the following list: (1) p = v, 9 = (u,v). (2) p = 2, - (1/2)u, g = (u,v2). (3)* p = 2, f (1/2)u2, g = (u,v2). a o g, g = (u,v3 IN), where a E 9X(z,u). (4) p =

+

+

The normal forms (l), (2) and (3) are Clairaut type equations which have already appeared in Theorem 4.2 (cf. [24]). The normal form (3)* is a new one. If we take a diffeomorphism 6 = -idR, $(u, v) = (u, -v) and cp = idwz, then (3)+ and (3)- are equivalent. This type is a general Clairaut type but not Clairaut type equation. The normal form (3)- is called Clairaut cross-cap. We draw the pictures of equation surface (Image f ) and phase R )Clairaut cross-cap in Figure 8. portrait ( { T o f ~ , - I ( ~ ) } ~ ~of

Fig. 8.

The equation surface and phase portrait of Clairaut cross-cap.

We now give an outline of the proof of Theorem 6.1. Our strategy for classification is summarised from step 1 to step 5:

-

Proof of Theorem 6.1. Step 1. Construct the Legendrian immersion germs depending on general Clairaut type equations. R x R2 be a 1-jet bundle and (s,x , y , q , p ) Let II : J1(R x R, R) be the canonical coordinates on J1(Rx R, R). Then the canonical contact 1-form on J1(W x R, W) is given by 0 = d y - p d x - qds = 6 - qds. Suppose that ( p , f ) : (W2, 0) R x J1(R, W) is an equation with complete integral, then there exists the unique element h E &(u,v) such that f*6 = h . dp. Define a map germ

a,,,

: (R2,0)

-

J'(R x R,R)

408

M . Takahashi

by

l(,,f)(W) = M%'U),Z

O

f(u,v),y O f ( U , v ) , h ( % V ) , P o

f(%'U)).

If ( p , f ) is a general Clairaut type equation, then we can easily show that l,,,,) is a Legendrian immersion germ, that is, C(,,f) is an immersion germ with e(,,f)*Q = 0. We call C(,,f) a complete Legendrian unfolding associated to general Clairaut type equation (p,f).The general Clairaut type equations are characterized as follows:

-

Proposition 6.2 ( [22]). Let ( p , f ) : (R2,0) R x J1(R x R,R) be a n equation with complete integral. Then ( p , f ) is a general Clairaut type equation if and only if e(,,f) is a Legendrian non-singular, that is, IIol,,,,) is non-singular.

-

Let U be an open subset of R2. We define LR(U,J1(R x R, R)) to be the set of complete Legendrian unfoldings l,,,,) : U J1(R x R, R) such that II o is non-singular. This set is also a topological space equipped with the Whitney Cm-topology. By the construction, we have a well-defined continuous mapping

e(,,f) (HI)*

:LR(u,J~(R x

R,R))

-

-

Clair(U,R x J'(R,R))

defined by ( I I l ) * ( l ( , , j ) ) = TI1 ol,,,,, = ( p , f),where I t 1 : J1(Rx R,R) R x J 1 ( R , R ) is the canonical projection I l 1 ( s , z , y , q 7 p )= (s,x,y,p). Then we have the following theorem.

Theorem 6.3 ( [20]). The continuous map

(II~), : LR(u, J ~ ( xR R, R)) -+ Clair(U, R x J'(R,R)) is a homeomorphism. This theorem asserts that the genericity of a property of general Clairaut type equations can be interpreted by the genericity of the corresponding property of complete Legendrian unfoldings. Step 2. Consider the generating families of complete Legendrian unfoldings. Let ( p , f ) be a general Clairaut type equation. Since e,,,,) is a Legendrian immersion germ, there exists a generating family of l(,,,) by the theory of Legendrian singularities ( [2,36]).Let F : ((R x R) x Rk,0) (R, 0) be a germ of function such that d2F10xRxRk is non-singular, where

-

O n completely integrable first o d e r ordinary differential equations

Then C ( F )= dZF-'(O) is a 2-dimensional manifold and T F : ( C ( F ) 0) , W is a submersion germ, where T F ( S , z, q ) = s. Define germs of maps

409

-

-

C F : (C(F),O)+ J1(R,W) by

C F ( S , 2, q ) =

a

(z, F ( s ,z, q ) , a F -X( s ,

-

and

C F : (C(F),O)

2, q ) )

7

J1(W x W,R)

by CF(S,z,q)=

a

a

(s,z,F(s,z,q),aF-(s,z,q),aFZ(s,z,q))). S

Since dF/dqi = 0 (i = 1,.. . , k) on C ( F ) ,we can easily show that

-

(,cFi?FF1(S))*e = 0. By definition, C F is a complete Legendrian unfolding associated to (TF,&). By the same method of the theory of [2,36], we can also show the following proposition. Proposition 6.4. All complete Legendrian unfolding germs are constructed by the above method.

We say that F is a generalized phase family of the complete Legendrian unfolding C F . Furthermore, by Proposition 6.2, l(,,,, is a Legendrian nonsingular. Then we can choose a family of germs of functions

F : (R x R,O)

-

(R,O)

-

such that Image jlF, = f ( p - l ( s ) ) for any s E (R, 0) where F,(z)= F ( s ,z). We remark that the map germ j ! F : (R x R,O) J1(R,R) is not necessarily an immersion germ, where j ! F ( s , z) = jlFs(z). In this case, we have (C(F),O)= (W x W,O)and C F = j l F : (R

x R, 0 )

-

J1(W x W,W),

so that it is a complete Legendrian unfolding of general Clairaut type associated to ( r ~ , j : F ) Thus . the generalized phase family of a complete Legendrian unfolding of general Clairaut type C F is given by the above germ. We define

F : (Wx R x R,O)

-

(W,O)

410 M. Takahashi

-

by F ( s ,z, y ) = F ( s ,z) - y and call it a generating family of a complete Legengrian unfolding of general Clairaut type. Step 3. Corresponding equivalence relations among complete Legendrian unfoldings and generating families. The idea is to define an equivalence relation which can ignore functional modulus and to do everything in terms of generating families for complete Legendrian unfoldings of general Clairaut type. We consider an equivalence relation among integral diagrams which ignore functional moduli. Let ( p , g ) and (p’,g’) be integral diagrams. Then ( p , g ) and (p’,g’) are R+-equivalent if there exist a germ of diffeomorphism Q : (R x R2, 0) (R x R2, 0) of the form Q(s, z, y ) = ( s a(z,y ) , +(z, y ) ) and a germ of diffeomorphism cp : (R2,0) (R2,0) such that Q o ( p , g ) = (p’,g’) o cp. We remark that if ( p , g ) and (p’,g’) are R+-equivalent by the above diffeomorphisms, then we have p(u,v) a o g ( u , v) = p’ o cp(u,v) and o g(u,v) = 9’ o cp(u,v) for any (u,v) E (R2,0). Thus the integral diagram ( p a o 9, 9 ) is strictly equivalent to (p’,9’). We now define the corresponding equivalence relations among com(J1(Rx R,R),z) and plete Legendrian unfoldings. Let e ( , , ~ :) (R2,0) l ( , , , ~:, (R2, ) 0) (J1(Rx R, R), 2‘) be complete Legendrian unfoldings. We say that e(,,,) and &,,,I) are SP+-Legendrian equivalent (respectively, SP-Legendrian equivalent) if there exist a germ of contact diffeomorphism K : (J1(Rx R,R),z) (J1(Rx R,R),z‘), a germ of diffeomorphism Q, : (R2, 0) --t (R2,0) and a germ of diffeomorphism Q : (R x R2, II(z)) (R x R2, II(z’)) of the form Q(s, z, y) = (s a(z,y ) , +(z, y ) ) (respectively, Q(s, z, y ) = (s,+(z, y ) ) ) such that IIoK = Q o l 2 and K o l ( , , ~ = ) l(,t,~,~o@. where II : (J1(Rx R,R),z) (R x R2,11(z)) is the canonical projection. It is clear that if l ( , , ~and ) C ( , , , J ~ ) are SP+-Legendrian equivalent (respectively, SP-Legendrian equivalent), then ( p , o~f ) and ( p ’ , o~ f’) are R+-equivalent (respectively, strictly equivalent). By [36, Theorem 1.11, the converse is also true for generic ( p , f ) and (p’, f ’ ) . A complete Legendrian unfolding is said to be SP+-Legendre stable (respectively, SP-Legendre stable) if for every map with the given germ there is a neighborhood of the original point such that each complete Legendrian unfolding belonging to the first neighborhood has in the second neighborhood a point at which its germ is SP+-Legendrian (respectively, SP-Legendrian) equivalent to the original germ (cf. [2, Part 1111). On the other hand, we can interpret the above equivalence relations in terms of generating families. We now give a quick review of the theory of unfoldings of function germs (cf. [9,16,24,31]).

-

-

+

+

+

+

-

-

-

-

+

-


411

Let F and 3 : (R x R x R,O) --t (R,O) be generating families of complete Legendrian unfoldings of general Clairaut type, where F ( s ,z, y ) = F ( s , z )- y and p ( s , z ,y ) = F’(s,z) - y respectively. We say that F and F’ are PC+-equivalent (respectively, PC-equivalent) if there exists a diffeomorphism germ @ : (R x R x R,O) (Rx R x R,O) of the form

-

-

@(.,GY> = (s +(y(z,Y),cpl(z,Y),cp2(z,Y))

We say that p ( s ,2,y ) is C+ (respectively, C)-versa1 deformation of f =

+

(respectively, E, = (f)&, ((dF/dz)lwxo, By similar arguments like those of [2, Theorems 20.8 and 21.41, we can show the following:

-

Theorem 6.5. Let F and 3 : (R x R x R,O) (R,O) be generating families of complete Legendrian unfoldings of general Clairaut type CF and C p respectively. Then (1) CF and C p are SP+ (respectively, SP)-Legendrian equivalent if and only i f F and 9 are PC+ (respectively, PC)-equivalent. (2) CF is SP+ (respectively, SP)-Legendre stable i f and only i f is C+ (respectively, C)-versa1 deformation off = Flwxo. Step 4. Generic classification of generating families of complete Legendrian unfoldings. The set of SP+-Legendre stable complete Legendrian unfoldings is an open and dense subset in LR(U,J1(Rx R,R)). Therefore by Theorem 6.3, it gives a classification of SP+-Legendre stable complete Legendrian unfoldings of general Clairaut type under the SP+-Legendrian equivalence (or SP-Legendrian equiavelence). Let (p,f) be a general Clairaut type such that the corresponding Legendrian unfolding e,,,,) is SP+-Legendre stable. By the assumption and Theorem 6.5, the generating family F of l(,,f) is C+-versa1 deformation of f = F I w x 0 . By using Thorn’s transversality theorem, we can show the following theorem:

Theorem 6.6 ( [13,31]).For a generic general Clairaut type equation (p,f ) , the generating family F of Legendrian unfolding L(,,f) is P C -

412 M. Takahashi

equivalent to one of the germs in the following (1) s

+x -y,

(2) s2

or PC+-equivalent to (4) s3

+ xs - y ,

(3)f s2 f x 2 s - y ,

+ xs - y .

Step 5 . Detect the corresponding normal forms of integral diagrams. Finally, we detect the corresponding normal forms of integral diagrams in Theorem 6.6. However, the process is rather tedious. Therefore we only work on the normal form (3)*. In this case, the generalized phase family is given by F ( s ,x) = s2 f x2s. Then the complete Legendrian unfolding is given by &(s,

x) = (s, x,s2 fx%, 2s f 2 2 , f 2 x s )

and the corresponding integral diagram is strictly equivalent to p = s, g = (2,s2 f 2s).

We consider a local coordinate transformation by u = x, v = s f (1/2)x2, then ( p , g ) is strictly equivalent to p =w

(1/2)u2, g = ( u , v 2- (1/4)u4).

Again apply a local coordinate transformation defined by X = x, Y = y + (1/4)x4, we have the normal form (3)* in Theorem 6.1. 0 7. Bifurcations of general Clairaut type equations

The next problem is to consider the bifurcations of general Clairaut type equations. A one-parameter family of first order differential equation germs (or, a one-parameter family of equations) is defined to be a smooth map germ f : (R2 x R,O) J1(R,R).Denote a smooth map germ f t : (R2,0) J1(R,W) by ft(u,v) = f ( u , v , t )for each t E (R,O). We do not assume that ft is an immersion germ (cf. Sec. 5). We also say that f is a one-parameter completely integrable if there exists a smooth function germ p : (R2 x W,O) (R,O) such that pt is a submersion germ 0 ) , where pt(u, w) = p(u,v,t ) . The pair and dpt A f,*e = 0 for each t E (R, ( p ,f) : (R2 x R,O) R x J1(R,R)is called a one-parameter family of equations with complete integral. If 7r o f t l k l ( C ) is a non-singular map for each t , c E (R, 0), then f is called a one-parameter family of general Clairaut type equations (cf. Sec. 6). Let ( p , g ) be a pair of germs of maps g : (R2 x R,O) (R2,0) and p : (W2 x R,0) (R, 0) such that pt is a submersion for each t E (R, 0 ) .

-

-

-

-

-

-


Then the divergence diagram

(R, 0 ) 2-(R2 x

w,0 ) -% (R2,O)

-

or, briefly, ( p , g ) is called a one-parameter family of integral diagrams if there exists a one-parameter family of equations f : (R2 x R, 0 ) J 1(R, R) such that ( p , f) is a one-parameter family of equations with complete integral and 7r o f = g. Define F : (R2x R,O) J1(R,R)x R by F ( u , v , t ) = ( f ( u , w , t )t, ) for a one-parameter family of equations f. In this case, F is called a oneparameter unfolding of equations associated to f . We also define G : (W2 x R,O) (R x R,O) by p(u,w,t ) = (p(u,w,t),t).The pair F ) or ( p ,F ) is called a one-parameter unfolding of equations with complete integral. If f is a one-parameter family of general Clairaut type equations, then we call (6, F ) or ( p , F ) a one-parameter unfolding of general Clairaut type equations. Let (G,G) be a pair of germs of maps G : (R2 x R,O) (R2 x R,O) and G : (R2 x R, 0 ) (R x R,0 ) such that pt is a submersion for each t E (R,O). Then the diagram

-

-

(c,

-

-

(R x R,O)

z

(R2

x R,O) % (a2 x R,O)

or, briefly, (j2,G) is called a one-parameter unfolding of integral diagrams if there exists a one-parameter family of equations f such that (j2,F ) is a oneparameter unfolding of equations with complete integral and (7rxid)oF = G where F is the one-parameter unfolding of equations associated to f. We introduce the equivalence relations among one-parameter unfoldings of equations and one-parameter unfoldings of integral diagrams by the same definitions as those in Sec. 5. Then we can show that two one-parameter unfoldings of equations F and F’ are equivalent if and only if the induced one-parameter unfoldings of integral diagrams (G, (7r x id)o F ) and (7,(7r x id) o F’) are equivalent for generic ( p , F ) and ( p ’ ,F’) (cf. Theorem 5.1). The notion of the genericity and the genericity of a property of oneparameter unfolding of general Clairaut type equations are also defined in the previous sections. The main result in this section is to give a generic classification of oneparameter unfoldings of general Clairaut type equations.

Theorem 7.1 ( [32]). For a generic one-parameter unfolding of general Clairaut type equations ( p ,F ) : (R2 x R, 0 )

-

R x J y R , R) x R,

414 M . Takahashi

the one-parameter unfolding of integral diagrams ( p , G = (r x i d ) o F ) is strictly equivalent to one of the germs in the following list: ( 1 ) p = V , G = (u,V,t). ( 2 ) p = v - ( 1 / 2 ) ~G , = (u,v 2 ,t). (3)* p = v f ( 1 / 2 ) u 2- ( 1 / 2 ) t , G = (u,v2,t). ( 4 )= ~ v Q 0 G, G = (u,v3 U V , ~ ) where , Q E 9X(z,y,t). ( 5 ) p = v Q o G, G = (u, v3 (t fu2)v,t ) , where a E 9X(z,y,t). ( 6 ) p = v + a o G, G = (u,v4 uv +p(u,t)v',t), where 0 E r n ( z , y , t ) and P E f M ( U , t ) > (aP/at)(o)# 0.

+

+

+ + +

The normal forms ( l ) (, 2 ) ,(4) and ( 6 ) are one-parameter families of Clairaut type equations which have already appeared in Theorem 5.2 (cf. [30]).The normal form (3)* is equivalent to ( p , g ) = (v - ( 1 / 2 ) u 2 , u , w 2 , t ) .Also this type has appeared as a generic general Clairaut type equation with the cross-cap singularity in Theorem 6.1 (cf. [13,31]).The normal form (5)* is a new one. It contains many interesting new equations as Example 7.2 shows.

Example 7.2 (One-parameter families of Lagrangian equations). W e consider classical equations which are more general than first order classical Clairaut equations (cf. Example 2.1). The equation of the following f o r m is called a Lagrangian equation (cf. (7J:

where ~ ( p and ) + ( p ) are smooth functions and p = d y / d x . W e consider one-parameter family of Lagrangian equations given by

-

where cp(t,p) and $ ( t , p ) are smooth functions. In this case, we consider a Iw x J1(Iw,R) defined by map germ ( p , f ) : (R2x Iw,0) ( p , f x u , v,t ) = (v,u,( P t ( V b

+ $9 (v),'Pt(v)),

and also call it a one-parameter family of Lagrangian equations. Then we can show that ( p , f ) is a one-parameter family of general Clairaut type equations. I f we put p t ( p ) = p2 and 1c,t(p) = p3 +tp, then the one-parameter family of Lagrangian equations is strictly equivalent to the normal f o r m (5)where a = - ( l / & ) x in Theorem 7.1. I n this case, the equation is not a one-parameter family of Clairaut type equations. W e can draw the pictures


of bifurcations of the equation surfaces (Image ft) and the phase portraits ({no ft(p;’)(c)},EW). See Figure 9 and Figure 10.

Fig. 9. Bifurcation of equation surfaces of one-parameter family of Lagrangian equations where pt(w) = v 2 , $J~(v)= w 3 tv.

+

Fig. 10. Bifurcation of phase portraits of Fig. 9.

Moreover, we can also draw the pictures of bifurcations of the equation surfaces about (5)* where a = 0 in Theorem 7.1 (Fig. 11 and Fig. 13) and the corresponding phase portraits (Fig. 12 and Fig. 14). These examples describe how two cross-caps meet and how the corresponding web structures bifurcate. Finally, we remark that in the generic classification of general (single) Clairaut type equations, the Lagrangian equations, except for the immersive equations, does not appear (cf. Theorem 6.1). But our theorem assert that in the generic classification of one-parameter unfoldings of general Clairaut type equations, one-parameter family of Lagrangian equations do appear (cf. Theorem 7.1).

416

M. Takahashi

Fig. 11. Bifurcation of equation surfaces of the normal form (5)+ where a = 0.

Fig. 12.

Fig. 13.

Bifurcation of phase portraits of Fig. 11.

Bifurcation of equation surfaces of the normal form ( 5 ) - where a = 0.

Acknowledgments The author would like to thank the referee of this paper for helpful comments. This work was partially supported by Research Fellowship of Japan


Fig. 14. Bifurcation of phase portraits of Fig. 13. Society for the Promotion of Science for Young Scientists.

References 1. V.I. Arnol’d, Ordinary differential equations, Springer Textbook (SpringerVerlag, Berlin, 1992). 2. V.I. Arnol’d, S.M. Gusein-Zade and A.N Varchenko, Singularities of Differentiable Maps, Vol. I (Birkhauser, 1985). 3. M. Bhupal, On singular solutions of implicit second-order ordinary differential equations, Hokkaido Math. J. 32 (2003), 623-641. 4. J.W. Bruce, A note on first order differential equations of degree greater than one and wavefront evolution, Bull. London Math. SOC.16 (1984), 139-144. 5. C. CarathBodry, Calculus of Variations and Partial Differential Equations of First Order, Part I. Partial Differential Equations of the First Order (HoldenDay, San Francisco-London-Amsterdam, 1965). 6. A. C. Clairaut, Solution de plusieurs problems, Histoire de 1’Academie Royale de Sciences, Paris (1734), 196-215. 7. R. Courant, Diflerential and integral calculus. Vol. I1 (Wiley Classics Library, 1936). 8. R. Courant and D. Hilbert, Methods of Mathematical Physics II (Wiley, New York, 1962). 9. J. Damon, The unfolding and determinacy theorems for subgroups of A and K , Memoirs AMS. 50, number 306 (1984). 10. L. Dara, Sindularitbs gbnbriques des Bquations differentielles multiformes, Bol. SOC.B r a d Mat. 6 (1975), 95-128. 11. A.A. Davydov, Qualitative theory of control systems, American Math. SOC., Translations of Mathematical Monographs, 141 (1994). 12. A.A. Davydov, Singularities of limiting directions of generic higher order implicit ODES, Proc. Stekov Inst. Math. 236 (2002), 124-131. 13. A.A. Davydov, G. Ishikawa, S. Izumiya and W.-Z. Sun, Generic singularities of implicit systems of first order differential equations of the plane, preprint. 14. S.B. Engelsman, Lagrange’s early contributions to the theory of first order partial differential equations, Historia Math. 7 (1980), 7-23.

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15. M. Fukuda and T. Fukuda, Singular solutions of ordinary differential equations, Yokohama Math. J. 25 (1977), 41-58. 16. C.G. Gibson, Singular Points of Smooth Mappings (Pitman, London, 1979). 17. E. Goursat, A course in mathematical analysis, Vol. 1 (Ginn, Boston, 1917). 18. A. Hayakawa, G. Ishikawa, S. Izumiya and K. Yamaguchi, Classification of generic integral diagrams and first order ordinary differential equations, Int. J. Math. 5 (1994), 447-489. 19. S. Izumiya, Generic properties of first order partial differential equations, Topology Hawaii, World Sci. Publ. (1992), 91-100. 20. S. Izumiya, The theory of Legendrian unfoldings and first-order differential equations, Proc. Roy. SOC.Edinburgh Sect. A 123 (1993), 517-532. 21. S. Izumiya, Singular solutions of first-order differential equations, Bull. London Math. SOC.26 (1994), 69-74. 22. S. Izumiya, Completely integrable holonomic systems of first order differential equations, Proc. Roy. SOC.Edinburgh Sect. A 125 (1995), 567-586. 23. S. Izumiya, On Clairaut-type equations, Publ. Math. Debrecen 45 (1995), 159-166. 24. S. Izumiya and Y. Kurokawa, Holonomic systems of Clairaut type, Diff. Geometry and App. 5 (1995), 219-235. 25. S. Izumiya and J. Yu, How to define singular solutions, Kodaa Math. J. 16 (1993), 227-234. 26. M. Kossowski, Fiber completions, contact singularities and single valued solutions for Cm-second order ODE, Can. J. Math. 48 (1996), 849-870. 27. M. Lemasurier, Singularities of second-order implicit differential equations: A geometrical approach, J. Dynam. Control Systems 7 (2001), 277-298. 28. V.V. Lychagin, Local classification of non-linear first order partial differential equations. Russian Math. Surveys 30 (1975), 105-175. 29. J. Martinet, Singularities of Smooth Functions and Maps, London Math. SOC.,Lecture Note Series 58 (1982). 30. M. Takahashi, Bifurcations of ordinary differential equations of Clairaut type, J. Diff. Equations 190 (2003), 579-599. 31. M. Takahashi, Holonomic systems of general Clairaut type, Hokkaido Math. J. 34 (2005), 247-263. 32. M. Takahashi, Bifurcations of holonomic systems of general Clairaut type, to appear in Hokkaido Math. J. 33. M. Takahashi, Bifurcations of completely integrable first order ordinary differential equations, (in Russian) Sovrem. Mat. Prilozh. No.33 (2005), 110125. 34. M. Takahashi, On implicit second order ordinary differential equations: Completely integrable and Clairaut type, to appear in J. Dyn. Control Syst. 35. R. Thom, Sur les equations Diffkrentielles Multiformes et leurs Intkgrales Singulikres, Boletim da Sociedade Brasileira de Matemhtica 3 (1972), 1-11. 36. V.M. Zakalyukin, Reconstructions of fronts and caustics depending on a parameter and versality of mappings, Journal of Soviet Math. 27 (1983), 27132735.

419

Cyclides as surfaces which contain many circles Nobuko Takeuchi Department of Mathematics Tokyo Gakugei University Koganei-shi Tokyo, 184-8501 Japan E-mail: nobuko@u-gakugei. a c j p

1. Introduction

A sphere in 3-dimensional Euclidean space E3 is characterized as a closed surface which contains an infinite number of circles through each point. But we do not know a surface other than a sphere or a plane, which contains many circles through each point. In the following figures, (n)is the number of circles through a point P on the surface.

a circular cylinder

an ellipsoid(not revolution) Fig. 1. Circles on the surfaces

Therefore when we judge whether the surface is a sphere or a plane or not, using numbers of circles on it, it is natural we would like to reduce the quantity of information for numbers of circles. In section 2, we introduce

420 N . Takeuchi

a standard torus Fig. 2.

Circles on the torus

conjectures concerning the above problem and give some results toward the conjectures. Moreover, in section 3 we study the cyclides which contain many extrinsic circles [2].

2. Conjectures and theorems

Let M be a closed surface of genus one. Then M is topologically obtained from a square ABCD by identifying A> with 02 and B?7 with A%. Let a and /3 be the closed curves on M corresponding to AB and BC, respectively. Then the homotopy classes [a]and [PI defined by a and /3 are generators of the fundamental group T ~ ( Mof) M (see the diagram).

A

B

D

a

B Fig. 3. Torus

Cyclides as surfaces which contain many circles

421

In 1980, Richard Blum [ l ]found a closed C" surface of genus one in E3 which contains six circles through each point. The Blum surface is defined by a quartic equation of the form:

(x; + x; + x i ) 2- 2 a 1 4 - 2 a 2 4 - 2a3xi + a = 0 , (a1 L a2 > 0 , a3 < -6) The surface contains exactly four (if a1 = a2,a3 = -&), five (if a1 = a2, a3 # -&or if a1 # a2, a3 = -&) or six (if a1 # a2, a3 # -,hi) circles through each point. The one circle of 6 circles on the surface through each point p of M belongs to a homotopy class [a] [PI and another circle belongs to a homotopy class [a]- [PI. Moreover 2 circles belong to a homotopy class [a]and the other 2 circles belong to a homotopy class [PI. In case of the surface containing 5 circles, the number of the circle belongs to [a]is one, or the number of circle belongs to [P] is one. Similarly, in case of the surface containing 4 circles, that is a standard torus, the number of the circle belongs to [a]is one and the number of circle belongs to [PI is one (see Fig.4). Then he gave a conjecture:

+

a standard torus Fig. 4.

Circles on the standard torus

Conjecture 2.1 (R.Blum). A closed C" surface in E3 which contains seven circles through each point is a sphere. In 1984, Koichi Ogiue and Ryoichi Takagi [6]have given the theorem:

422

N . Takeuchi

Fig. 5 .

Blum's surface

Theorem 2.2 (K.Ogiue and R.Takagi). A C" surface in E3 is (a part of) a plane or a sphere, if it contains two circles through each point, which are tangent to each other. Additionally, drawing from the fact that an ellipsoid contains two circles through each point except only at four points, they gave a conjecture such as

Conjecture 2.3 (K.Ogiue and R.Takagi). A simply connected complete C" surface in E3 is a plane or a sphere, if it contains two circles through each point. We had the following partial affirmative results toward conjectures 2.1 and 2.3 :

Theorem 2.4 ( [7]). A simply connected complete C" surface in E3 is a plane or a sphere, i f it contains three circles through each point. Theorem 2.5 ( [8]).A closed C" surface of genus one in E3 cannot contain seven circles through each point. Moreover, we got some theorems ( [7] , [9] , [3] ). These results are theoretically beautiful but not necessarily practical in the sense that it is difficult to verify the sphericity of a given physical solid by applying these results. On the other hand, it is well-known that a compact smooth surface in E3 which contains a circle of (arbitrarily) given radius in an arbitrary direction at an arbitrary point must be a sphere. This is a practical characterization in the sense that it is possible to verify the sphericity of

Cyclides as surfaces which contain m a n y circles

423

a given physical solid by pressing an accurate circular ring. Therefore we got the following theorem by reducing the quantity of information which is large because of its condition in an arbitrary direction at an arbitrary point.

Theorem 2.6 ( [ 5 ] ) . A smooth ovaloid in E3 is a sphere, i f the surface contains a circle of an arbitrary but fixed radius through each point. From practical point of view, it is sufficient to consider an ovaloid, because a surface which is not convex is far from being a sphere.

3. Cyclides A cyclide is a surface in E 3 defined by a quartic equation of the form 3

3

3

[a].

We pay attention to the fact that cyclides contain many extrinsic circles It is known that such a surface corresponds to a complete intersection of two quadrics in a 4-dimensional real projective space via pentaspherical representation. In the paper [lo] , we studied geometric properties of cyclides and gave a classification of cyclides from conformal point of view because we are interested in circles contained in a surface, so it is natural to apply conformal geometry.

Theorem 3.1 ( [lo]). A non-singular cyclide is conformally equivalent to a cyclide of the f o r m which is topologically a torus, a sphere or two spheres. A cyclide with singularities is conformally equivalent to a quadratic surface.

Theorem 3.2 ( [lo]). A cyclide contains n circles through each nonumbilic point and n-1 circles through each isolated umbilic point unless it is a sphere or a pair of two spheres, where n=1,2,.3,4,5 or 6. According to Theorem 3.2, a cyclide is covered by at least one family of circles. Therefore it is natural to regard a cyclide as a surface enveloped by a family of some spheres (which are called Meusnier spheres) determined by a family of circles. G.Darboux ( [2] ) took an interest in such a property of cyclides and gave concrete expressions for families of such spheres that envelope the given cyclide. We reviewed such a property of cyclides and got the following.

424 N . Takeuchi

Proposition 3.3 ( [lo]). A cyclide which is topologically a torus is a surface enveloped by three distinct families of Meusnier spheres determined by circles on the cyclide. Each sphere contains one or two circles o n the cyclide and it is tangent to the cyclide along the circle or at two points, respectively.

Moreover, we got examples of mechanical constructions of cyclides. Example 3.4 (Hulahoop surfaces). A hulahoop surface is defined in 141 to be a smooth surface obtained by revolving a circle around a suitable axis. Let r ( a ,b , r ) , r > 0, be a circle o n the xlx2-plane defined by ( X I - a ) 2 ( X Z - b)2 = r2 and let r ( a ,b, r, a ) be the circle obtained by tilting ?(a, b, r ) n7r around the diameter parallel to the XI-axis by the angle a, -- < a 5 -. 2 2 Let H ( a , b, r, a ) be the surface obtained by rotating r ( a ,b, r, a ) around the x3-axis. Then the hulahoop surface H ( a , b,r, a ) is given by

+

(xf

+ x; +

x:>2

- =4 b( XcIo s a

2

+ x; + x3.3

+ + r2)(xf + x;) - 2 ( a 2 + b2 + r2 - 2az+2b2 sinZcosz

-2(a2 b2 +=(a2 4b cos Q

Q

+ b2 + r2)x3+ (a2+ b2 + r2)2- 4a2r2 = 0.

Fig. 6.

) x3

Hulahoop surface

I t is easily seen that H ( a , b, r, a ) is a smooth surface if and only i f a = b = 0 7r 7r and a = or a # 0 and (a2 - r 2 )cos2 a + b2 # 0 . W e see that H ( 0 ,0 , r, -) 2 2 is a sphere and otherwise H ( a , b , r , a ) is topologically a torus. Then, we can see a hulahoop surface which is not a sphere and contains exactly four or five circles through each point. The one circle of 5 circles o n the surface through each point p of M belongs to a homotopy class [a] [PI and another circle belongs to a homotopy class [a]- [PI. Moreover 2 circles belong to a

+

Cyclides as surfaces which contain many circles

425

homotopy class [a]and the other one circles belongs to a homotopy class [PI. The surface containing 4 circles is a standard torus. Example 3.5. Let y be a curve given as an intersection of a sphere xf xg xz = r2 x:: x; and an elliptic cylinder - - = 1 ( r > a > b). Then y is represented as a2 b2

+ +

+

{

x1 = acos8 x2 = bsin8 23

=

dr2- a2 cos28 - b2 sin2 8.

Consider a family of circles c(8) with the following properties: (a) c(8) is vertical (b) the center of c(8) is on the ~1x2-plane (c) c(8) is tangent to Oy(8) at y(8), where 0 denotes the origin. Then c(8) is a circle with center at

br2 sin 8 or2 cos 8 a2cos2 8 b2 sin2 8 ' a2 cos28 b2 sin2 8 '

+

+

and of radius

r2 - a2 cos28 - b2 sin28 a2cos2 8 + b2 sin2 8 Therefore the surface M ( a ,b, r ) generated by the family of circles is represented as I

21

22

23 L

ar cos 8(r + Jr2 - a2 cos2 8 - b2 sin2 8 cos 9 ) a2 cos2 8 b2 sin2 8 br sin 8(r + - a2 cos28 - b2 sin28 cos cp) = a2cos28 b2 sin2 8 =

+

dr2

=

+

r dr2- a2cos28 - b2 sin28 sin cp da2cos28 b2 sin2 8

+

By eliminating 8 and cp from these equations, we obtain

r2(2r2- b2) x; 2r22i r4 = 0. b2 The surface is topologically a torus and contains exactly five circles through each point. The one circle of 5 circles on the surface through each point p of M belongs to a homotopy class [a] [P] and another circle belongs to a homotopy class [a]- [PI. Moreover one circle belongs to a homotopy class (x?

- a2) XI - 2 + x; + X i ) 2 - 2 r2(2r2 a2

+

+

+

426

N . Takeucha

[a]a n d the other 2 circles belong to a homotopy class containing 4 circles is a standard torus.

[PI . The surface

Fig. 7.

References 1. R. Blum, Circles on surfaces in the Euclidean 3-space1 Lecture Note in Math. 792, Springer, 1980 , 213-221. 2. G.Darboux, Principes des Geometrie Analytique, Garthier-Villars, 1917. 3. R.Miyaoka and N.Takeuchi, A note on Ogiue-Takagi conjecture on a characterization of Euclidean 2-Spheres,Memoirs of the Faculty of Science, Kyushu University Series A Mathematics vol.XLVI,Nol March, 1992 , 129-135. 4. K. Ogiue and N. Takeuchi, Hulahoop Surfaces , Journal of Geometry 46 (1993) 127-132. 5. K. Ogiue and N. Takeuchi, A sphere can be characterized as a smooth ovaloid which contains one circle through each point , Journal of Geometry 49 ,1994, 163-165. 6. K.Ogiue and R.Takagi, A submanifold which contains many extrinsic circles, Tsukuba J. Math. 8 , 1984 , 171-182. 7. N.Takeuchi, A sphere as a surface which contains many circles, Journal of Geometry 24 , 1985 , 123-130. 8. N.Takeuchi, A closed surface of genus one in E3 cannot contain seven circles through each point, Proc.AMS. 100 , 1987 , 145-147. 9. N.Takeuchi, A sphere as a surface which contains many circles 11, Journal of Geometry 34 , 1989 , 195-200. 10. N. TakeLChi, kyclides, Hokkaido Mathematical Journal ,29(1) ,2000, pp 119148.

427

Euler number formulas in terms of singular fibers of stable maps Takahiro Yamamoto

Faculty of Mathematics, Kyushu University, Hakozaki, Pukuoka 812-8581, Japan E-mai1:taku-chanomath. kyushu-u.ac. j p The author [24] have shown that under certain conditions, the Euler number of the source 4-manifold has the same parity as the total number of certain singular fibres of stable maps. In this paper, we show that there is no Euler number formula of the same type if we have no condition on the stable maps.

Keywords: Stable map, singular fibre, Euler number.

1. Introduction In [22] , Vassilyev constructed the theory of characteristic classes for real Lagrange and Legendre singularities. He introduced the universal complex of singular classes for which the Poincar6 dual cohomology class is well defined. The Vassilyev universal complex had been generalized by Kazarian to the characteristic spectral sequence which contains all cohomological information about adjacencies of singularities. Secondary, Saeki [17] developed the theory of universal complexes of singular fibers of differentiable maps of negative codimension. Here, the codimension of a map f : M + N between manifolds is defined to be k = dim N - dim M . For differentiable map f : M + N , the terminology singular fiber over q E N refers to a map germ along the inverse image

f : ( M ,f

+

( N ,41,

not just inverse image f - I ( q ) . For the positive codimension case, the inverse image f - l ( q ) is a discrete set of points, as long as the map is generic enough, and we can study the topology of such maps by using the well-developed theory of multi-jet spaces. However, in the negative codimension case, the fiber over a point is no longer a discrete set, and is a complex of positive dimension in general. This means that the theory of multi-jet spaces is not sufficient any more in

428

T . Yamamoto

the negative codimension case. In the theory of universal complex of singular fibers of differentiable maps, Saeki constructed the cochain complex of singular fibers and he showed that the cohomology classes induce cobordism invariants of differentiable maps. If we consider the target manifold as the Euclidean space, then cobordism invariants of maps induce cobordism invariants of the source manifolds. In this way, for the study of the topology of manifolds, universal complexes are very powerful tools. As an explicit and important example in this theory, stable maps of closed orientable 4-manifolds into 3-manifolds were studied and singular fibers were completely classified (for a precise definition of the equivalence relation, refer to Definition 2.1). Furthermore, Saeki obtained the following Euler number formula: For any stable map of an orientable closed 4-12 manifold into a connected 3-manifold, the number of singular fiber of I11 type as depicted in Figure 1 and the Euler number of the source 4-manifold have the same parity, (In the book [17], the symbol ‘?IIs” is used instead of We note that modulo two Euler number of 4-manifolds corresponds to the generator of unoriented cobordism groups of 4-manifolds (for details, see [12]). On the other hand, Saeki’s Euler number formula was generalized by the author for possibly non-orient able 4-manifolds: Under certain homological conditions, for a stable map f : M -+ N of a closed 4-manifold M into a connected %manifold N , the total number of certain singular fibers and the Euler number x ( M ) of the source 4-manifold M have the same parity,

“z12”).

where I.F(f)l denotes the number of singular fibers of f of type F ,F, and Fo are subclasses of the equivalence class F (for details, see [24]). For the notation of singular fibers in the formula, see Figure 1. In this paper, we calculate the cohomology groups of several universal complex of singular fibers of stable maps and we consider the geometrical meaning of the cohomology classes. In $2, we review the theory of the universal complex of Thom maps. In $3, we consider the universal complex of proper stable maps of 3-manifolds into 3-manifolds and we consider the geometrical meaning of the cohomology classes. Furthermore we consider the universal complex of proper stable maps of 5-manifolds into 4-manifolds. In $4, we review the characterization of a cocycle of the universal complex of the singular fibers. Then we consider the geometrical meaning of the

Euler number formulas in terms of singular fibers of stable maps 429

Fig. 1. The singular fibers of the formula

cohomology groups of the universal complex of proper stable maps of 5manifolds into 4-manifolds. Throughout this paper, all manifolds and maps are of class C”. We call p E M the singular point of C” map f : M + N if the rank of Jacobi matrix of f at p is strictly less than the dimension of the target manifold. For a finite set P , we denote by /PI the number of its elements. For a topological space X the symbol “idx” denotes the identity map of X . and the symbol x( X) denote the Euler number of X. For a smooth manifold M , its interior is denoted by IntM. The symbol “E’denotes an appropriate isomorphism between algebraic objects. The author would like to express his thanks to adviser Go-o Ishikawa for helpful advice and encouragement. The author also would like to express his thanks to Osamu Saeki for helpful comments and discussions. Finally, the author would like to his thanks to Akiko Neriugawa for useful comments on English and to the referee for helpful comments. 2. Preparation

First, we introduce the equivalence relation among the singular fibers of differentiable maps. Definition 2.1. Let Mi be smooth manifolds and Ai subsets of Mi, (i = 0 , l ) . A continuous map g : A0 -+ A1 is said to be smooth if for every point q E Ao, there exists a smooth map l j : V -+ MI defined on a neighbourhood V of q in MOsuch that ljIv,-,~~ = g IVnAo . A smooth map g : A0 -+ A1 is a difeomorphism if it is a homeomorphism and its inverse is also smooth. When there exists a diffeomorphism between A0 and A l l we say that they are diffeomorphic. Let fi : Mi + Ni be differentiable maps and qi E N , i = 0 , l . For

430 T. Yamamoto

qi E Ni, we call that the fibers over qo and q1 are Cw equivalent (or Co equivalent) if there exist open neighborhoods Ui of qi E Ni, i = 0,1, and

diffeomorphisms (resp. homeomorphisms) : ((fo)-l(U0), (fo)-l(qo)) + ((fi)-'(Ui), (fi)-l(qi)) and cp : (Uo,qo) + (U1,ql) which makes following diagram commutative ,

((fo )- (UO)7 (fo 1fo

1

(40))

A

((fl ) - (Ul) 7 (fl )- (41 ))

'p

If1

(UO,9.0) (Ul, Q1). If q E N is a regular value of the differentiable map f : M 3 N , then we call the equivalence class of the map germ f : ( M , f - l ( q ) ) -+ ( N , q ) a regular fiber, otherwise singular fiber. In [17], Saeki constructed the theory of universal complexes of singular fibers. In the following, we review the theory of the singular fibers of proper l) Thom maps 2, developed in [17].For Propositions and Corollaries of this sections, the proofs are in [17, 57 and 81, we omit here. In order to construct a universal complex of singular fibers, we need to prepare the following two: (1) a class of singular fibers T , and (2) an equivalence relation p among the singular fibers in

T.

For (l),we consider a certain class of singular fibers of proper Thom maps. Furthermore, 7 should be closed under the adjacency relation: namely, if a singular fiber is in 7, then any nearby singular fibers should also be in 7. For (2), we consider an equivalence relation which is weaker than the Co equivalence, we denote it by po: namely, each equivalence class with respect to p is a union of Co equivalence classes. This implies that for every proper Thom map f and for every equivalence class F with respect to p, the subset F ( f ) consisting of the points in the target which have a singular fiber of type F is a Co submanifold of constant codimension, which we denote by

43).

Furthermore, p should satisfy the following (we call this condition (*).):

l ) A continuous map is said to be proper if the inverse image of a compact set is always compact. 2)A Thom map f : M + N is a stratified map with respect to Whitney regular stratification of M and N such that it is a submersion on each stratum and satisfies a certain regularity conditions. (for more details, refer to [ 5 ] )

Euler number formulas an terms of singular fibers of stable maps 431

For any two proper Thom maps fi : Mi -+ Ni and any points qi E Ni, i = 0,1, such that the fibers over qi contained in T and are equivalent with respect to p, there exists neighborhoods Ui of qi in Ni, i = 0,1, and a homeomorphism 'p : UO+ U1 such that ' p ( q 0 ) = q1 and ' ~ ( UnO 3(fo)) = U1 n3(f 1) for every equivalence class 3 of fibers with respect to p. In the following, a proper Thom map f : M -+ N is called a 7-map if its fibers all lie in 7. The Universal complex of singular fibers C*(T, p) is constructed as follows. For K E Z, let CK(7,p)be the Zz-vector space spanned by the equiv= K), which may possibly alence classes 3 of singular fibers in 7 with ~ ( 3 contain infinitely terms. Then we define &-linear map 6, : Cn(7,p) -+ Cn+l(7,p) by

lC(G)=tC+l

where 3 is the equivalence class of fibers of elements of r and n F ( G ) E Zz is the number modulo two of the components 3(f ) which are locally adjacent to the component G(f). We note that the definition of the coefficient n F ( G ) E 22 and the map 6, is well-defined by virtue of the above condition (*) of equivalence relation p. Furthermore, we can prove that 6,+1 o 6, = 0. Thus we obtain a cochain complex C*(7, P> = (C"(7,P),6fc)w

We call the resulting cochain complex the universal complex of singular fibers for proper 7-maps of n-manifolds into p-manifolds with respect to the equivalence relation p, and we denote its cohomology group of dimension K by H" (7,PI. Definition 2.2. Let n(F)=n

be a K-dimensional cochain of the complex C * ( T , ~ )where n3 E ZZ. For a 7-map f : M" -+ NP, we define c( f) to be the set of points q E N such that the fiber over q belongs to some 3 with n F # 0. If c is a cocycle, then c ( f ) is a Zz-cycle of closed support of codimension K of the target manifold N . In addition, if M is closed and K > 0 , then c ( f ) is a Zz-cycle in the usual sense.

432

T. Yamamoto

Lemma 2.3. Suppose that c and c' are K-dimensional cocycles of the complex C*(T, p) which are cohomologous. Then c( f ) and c'( f) are homologous in N for every r-map f : M" + NP. Definition 2.4. Let [c] be a &-dimensionalcohomology class of the cochain complex C * ( r , p ) . For a proper -r-map f : M" + Np, we define [c(f)] E H,C_,(Np,Zz) to be the homology class represented by the cycle c(f) of closed support. By virtue of lemma 2.3, this is well-defined. When M" is closed and K > 0, we can also regard [c(f)]as an element of Hp-,(NP, Z2). Then we can define the map 'Pf : H K ( T p) ,

--$

H"(NP,Z 2 )

by ( P ~ ( [ c ] )= [c(f)]*, where [c(f)]* E H n ( N p , Z 2 ) is the Poincar6 dual to [c( f)] E H,C-,(Np, Z 2 ) . This is clearly a homomorphism induced by -r-map f.When M" is closed and K > 0, we can also regard 'Pf as a homomorphism into the cohomology group H,"(Np,Z 2 ) of compact support. Let us introduce the suspension of Thom maps.

Definition 2.5. Let f : M n

+ NP

be a proper Thom map. Then we call

the map

f xid~:M"xR-+NPxR the suspension o f f . Furthermore, to the fiber off over q E NP, we associate the fiber of f x idw over ( q , O ) E N x R. We say that the latter fiber is obtained from the original fiber by suspension. We note that the suspension of a proper Thom map is again a proper Thom map. In the following, we assume that a class of singular fibers in r consists of certain singular fibers of proper Thom maps of an n-dimensional manifold into a pdimensional manifold for a fixed dimension pair (n,p). In this case, we often write r = r(n,p).Let us consider two classes of singular fibers r ( n , p ) and r ( n 1 , p 1) and their associated equivalence relations and p " + ~ , ~ +respectively. l In addition to the conditions (*), let us impose the following condition:

+

+

(1) the suspension of any classes of ~ ( n p) ,is also an element of r(n+l , p + l),and (2) if two singular fibers are equivalent with respect to p n , p , then so are their suspensions with respect to ~ ~ + l , ~ + l .

Euler number formulas an terms of singular fibers of stable maps

433

Then, the suspension induces a natural map

+

sK : C"(r(n+ 1 , ~11,~ ~ + i , ~ + Ci K) . ( r ( n , p )P, ~ , ~ ) , +

+

+

for an equivalence class 3 E C"(T(n 1,p 1 ) , ~ ~ + 1 , ~ + we1 )define , s,(3) E C K ( r ( n , p ) , p n , p )to be the (possibly infinite) sum of all those equivalence classes of fibers of codimension K. with respect to pn,p whose suspensions are contained in 3,for each K . We note that s, is a well-defined &-linear map. Lemma 2.6. The system of &-linear maps {sK} defines a cochain map {SKI

:C ( r ( n

+ 1,P + 1),Pn+l,p+l)

--+

c ( T ( n , P ) ,P n , p ) .

We introduce equivalence relation among r-maps. Definition 2.7. For a smooth manifold N , two r-maps fi : Mi N of closed manifolds Mi ,i = 0,1, into N are said to be r-cobordant if there exist a compact manifold W with boundary the disjoint union of MO and MI and a r-map F, : W + N x [0,1] such that fi = F l ~ 4: Mi ~ -+ N x {i}, i = 0 , l . We call F a r-cobordism between fo and fl. --f

Then, we obtain the following. Proposition 2.8. Let fi : Mi -+ N , i = 0,1, be r-maps, where we assume that Mi are closed. If they are r-cobordant, then f o r every K. we have p f o IIrns,,

+

= pfl IIrns,,

: ImsK,

-+

H " ( N ;2 2 1 ,

+

where sK* : H n ( r ( n 1 , p l),Pn+l,p+l) + H K ( T ( n , p ) , p n , p )is the homomorphism induced by suspension.

+

Thus, every element of H " ( r ( n 1,p cobordism invariant among r(n,p)-maps.

+ 1 ) , ~ , + 1 , ~ + 1induce )

a r-

Remark 2.9. If r is big enough then for every smooth map f : M + N of a closed manifold, we can defin pf to be pj, where ?is an approximation of f which is a r-map. Then, we can show that this is well-defined, and that it defines a bordism 3, invariant of smooth maps into N . In paticular, if N is contractible, it define a cobordism invariant of the source manifold. 3 ) T ~ smooth o maps fo : MO N and f1 : MI + N of closed manifolds MO and M I are said t o be bordant if there exist a compact manifold W with boundary the disjoint union A40 LI Mi,and a smooth map F : W + N x [0,1] such that f i = : Mi -+ N x {i), i=O,l for details, see [2]) -+

434

T. Yamamoto

3. Stable maps of n-manifolds into p-manifolds, p = n, n - 1 In this section, we consider a more specific situation, the case of proper C" stable maps 4, . Before we move on, let us introduce an equivalence relation weaker than Coequivalence. We note that for proper smooth maps of n-manifolds into pmanifolds, n > p > 0, the regular fiber is the disjoint union of the trivial Sn-P bundles. We also note that for proper smooth maps of pmanifolds into pmanifolds, p > 0, the regular fiber is the disjoint union of the trivial covering space. Definition 3.1. We say that two fibers of proper Thom maps of nmanifolds into pmanifolds, n > p > 0 (or n = p > 0 ) are Co equivalent modulo m regular fiber components if one of them is Co equivalent to the disjoint union of the other one and lm copies of a fiber of the trivial Sn-P bundle (resp. trivial covering space) for some nonnegative integer 1. We denote this equivalence relation by pi,p(m).We shall use the same notation p;,,(m) for the equivalence relation among singular fibers of subclasses of proper Thom maps. 3.1. Stable Maps of 3-Manifolds into 3-Manifolds

Since ( 3 , 3 ) is a nice dimension pair in the sense of Mather [ll],if dim M = 3 and dim N = 3 then the set of all C" stable maps is open and dense in C"(M, N) with respect to the Whitney C" topology as long as M is compact. The following characterization of proper C" stable maps from 3manifolds into 3-manifolds is well-known. Proposition 3.2. A proper smooth map f : M .+ N of a 3-manifold into a 3-manifold is C" stable i f and only if the following conditions are satisfied. ( i ) (Local condition) For every p E M , there exist local coordinates ( a ,b, x) and ( X ,Y,2 ) about p E M and f ( p ) E N respectively such that one of the following holds:

4)A smooth map f : M -+ N is said t o be Cm stable (or stable) if the A-orbit o f f is open in Cm(M,N) with respect to the Whitney Cm topology, where C w ( M , N ) denotes the set of smooth maps of manifold M into manifold N. Here the A-orbit of f E Cm(M,N) is defined as follows. Let Diff(N) denote the group of self-diffeomorphisms of N.Then the group Diff(M) x Diff(N) acts on C m ( M , N ) by (0,W)f = r k o f o @ - l , where (0,W) E Diff(M) x Diff(N) and f E C m ( M , N ) . Then the A-orbit of f E C m ( M , N ) is the orbit through f with respect t o this action. We note that the proper Coostable map is also Thom map.


( a ,b, x ) ( a ,b, x 2 ) ( a ,b, x3 a x ) ( a ,b, x4 + ax2 + bx)

+

p p p p

: Regular

point : Fold point : Cusp point : Swallow-tail point

(ii) (Global condition) The set S ( f ) = { p E M I rankdf, < 3) is a cZosed2-dimension submanifold of M . Then for every q E f (S(f)), f -l(q)n S(f ) consists of at most three points and the multi-germ

(f,f-l(q)r l S ( f ) ) is smoothly right-left equivalent to one of the six multi-germs as follows: (1) single immersion germ which corresponds to a fold point, we denote

corresponding singular fiber by F , (2) normal crossing of two immersion germs, each of which corresponds to a fold, we denote corresponding singular fiber by F F (5’) normal crossing of three immersion germs, each of which corresponds to a fold, we denote corresponding singular fiber by F F F (4) map germ corresponding to a cusp point, we denote corresponding singular fiber by C (5) transverse crossing of a cusp germ and an immersion germ which corresponds to a fold germ, we denote corresponding singular fiber by C F (6) map germ corresponding to a swallow-tail point, we denote corresponding singular fiber by s. We note that the map germs (1) - (6) in Proposition 3.2 correspond to the map germs in Figure 2. Proposition 3.6 can be proved by using the transversality theorem and the multi-transversality theorem, since the dimensions pair (3,3) is in the nice range in the sense of Mather [ll](for details, see [5], [lo] or [6]).

Remark 3.3. According to du Plessis and Wall [3], if ( n , p ) is in the nice range in the sense of Mather [ll],a smooth map between manifolds of dimension n and p is C” stable if and only if it is Co stable. Hence, the above proposition gives a characterization of Co stable maps of 3-manifolds into 3-manifolds as well, since (3,3) is in the nice range.

436

T. Yamamoto

(5)

(4)

Fig. 2. Multi-germs of fls(f)

In this dimension pair (3,3), the inverse image of any point is a discrete set of points if the map is generic enough. Let us consider the class 7 = S$(3,3) which consists of all singular fibers of proper stable maps of 3-manifolds into 3-manifolds. As the top notations of this subsection, this class is big enough. The Co equivalence P ; , ~among the singular fibers in the class S p ( 3 , 3 ) is so strong, we may choose Co equivalence modulo two regular fiber components ~ : , ~ ( 2 )(see, Definition 3.1). We note that for smooth map of 3manifold into 3-manifold, the regular fiber is the disjoint union of the trivial covering space, we denote corresponding singular fiber by R. Thus we obtain the cochain complex (CIE(S33,317 P$,3(2)), sIE>fc* Then the coboundary operations of this cochain complex are given as follows;

Co 3 R,, Re ++ 0 E C1,

Euler number formulas in t e r m s of singular fibers of stable m a p s 0

437

C1 3 F,, Fe +-+ 0 E C 2 , C2 3 C,, C, H 0 E C 3 , C2 3 FF,(or FF,),H Se(resp. So) E C3,

where the symbols 3, (or 3,)denote the Coequivalence class which consist a singular fiber of type .F and a some copies of a fiber of the trivial covering space such that the total number of trivial covering space is odd (resp. even) . Then, we obtain the following.

Proposition 3.4. The cohomology groups of universal complex of the singular fibers for proper Coostable maps of 3-manifolds into 3-manifolds with respect to Co equivalence modulo two regular fiber components

( ( C K ( s $ ( 331, , &3(2)))1

4c)tc

are given as follows; 0

0 0

H o g Z2 @ Z2 generated by [R,] and [Re], H1 Z2 @ Z2 generated by [F,] and [F,], H 2 g Z2 @ Z2 generated by [C,] and [Ce],

where [*]denote the cohomology class represented by the cocycle

*.

Let us consider geometrical meaning of the second cohomology classes. The generator of H 2 induces the following formula.

Theorem 3.5. Let f : M -+ N be a stable map of a closed 2-manifold into a 2-manifold. Then we have

X ( M ) = C(f) (mod 21,

where C (f ) denote the number of cusp points o f f . Proof. If N = R XN', then the set of all bordism class of closed l-manifolds into N , denote it by n l ( N ) ,forms an abelian group. Since the dimension pair ( 2 , 2 ) is in the nice range in the sense of Mather, we can choose a stable map as the representative of the bordism group. O n the other hand, we have n2(R2 ) 2 n2 E< w2 >zZ,

where 1x2 is the unoriented cobordism group of 2-manifolds and w2 is the second Stiefel-Whitney class (for details, see [2]). For [C*]E H 2 (* = o or e), we can define a homomorphism [C*]: n2(R2)4 HZ(IR2,Z2) Z2 b y [C*]([f])= [ C * ( f ) ]This . homomorphism is well-defined by virtue of

438

T . Yamamoto

Proposition 2.8. W e have generator K = [K : RP2 + R2] = 1 E Z2 n2(R2)constructed b y Kobayashi 181 which has three cusp points. Thus we = 1 and [Ce](K) = 0. So the homomorphism induced by [Co] have [Co](K) is a just projection and [Ce] is a 0-map. 0 3.2. Stable Maps of &Manifolds into 4-Manifolds

Since (5,4) is a nice dimension pair in the sense of Mather [ll],if dimM = 5 and dimN = 4, then the set of all C" stable maps of M into N is open and dense in C"(M, N ) with respect to Whitney C" topology as long as M is compact. The following characterization of proper C" stable maps of 5-manifolds into 4-manifolds is well-known. Proposition 3.6. A proper smooth map f : M + N of a 5-manifold into a 4-manifold is C" stable i f and only i f the following conditions are satisfied. 0

( i ) (Local condition) For every p E M , there exist local coordinates ( a ,b, c, x,y ) and ( X ,Y,2,W ) about p E M and f ( p ) E N respectively such that one of the following holds: ( X 0 f, y

0

f, 2 0 f , W 0 f) p : regular point,

( a ,b, c, x) ( a ,bl c, x 2 + Y 2 ) ( a ,b, c, z2 - Y 2 ) ( a ,b, c, x3 a x - y 2 ) ( a ,b, c, x4 ax2 + bz y 2 ) ( a ,b, c, x4 + ax2 + bx - y 2 ) ( a ,b, c, x5 + ux4 + bx3 + cx2 - y 2 )

+ +

p : cusp, p : definite swallow-tail, p : indefinite swallow-tail,

+

+ +

+

( a ,b, c, 3x2y y3 a ( x 2 y 2 ) ( a ,b, c, 3 z 2 y - y3 4- a ( x 2 + y 2 ) 0

p : definite fold, p : indefinite fold,

p : butterfly, bz cy) p : definite D4, bx + c y ) p : indefinite D4,

+ + +

(ii) (Global condition) The set S(f) = { p E M I rankdf, < 4) is a closed 3-dimension submanifold of M . Then for every q E f(S(f)), f-l(q) n S( f ) consists at most four points and the m u l t i - g e m

(fIS(f), f -l(q)

fl

S(f1)

is smoothly right-left equivalent to one of the thirteen multi-germs as follows:


Fig. 3.

Map germ corresponding to a butterfly point

single immersion germ which corresponds to a fold point, normal crossing of two immersion germs, each of which corresponds to a fold point, normal crossing of three immersion germs, each of which corresponds to a fold point, map germ corresponds to a cusp point, transverse crossing of a cusp germ and an immersion germ corresponds to a fold point, map germ corresponds to a swallow-tail point, normal crossing of four immersion germs, each of which corresponds to a fold point, transverse crossing of cusp germ and a normal crossing of two immersion germs which corresponds to a fold point, transverse crossing of a swallow-tail germ and an immersion germ corresponds to a fold point, normal crossing of two cusp germs, map germ corresponds to a butterfly point, map germ corresponds to a definite Dq point, map germ corresponds to an indefinite D4 point. We note that the map germs (1) - (6) in Proposition 3.6 correspond to the suspension of the map germs (1) - (6) in Figure 2. The map germs (11)- (13) are described in Figures 3 , 4 and 5 respectively, in order to draw 3-dimensional objects in a 4-dimensional space, we have depicted three “sections” by 3-dimensional spaces for each object. In Figures 3, 4 and 5, around 1-dimensional complexes are the neighborhood of singular point in

440

T. Yamamoto

4

Fig. 4.

Fig. 5.

Map germ corresponding to a definite

D4

point

Map germ corresponding to an indefinite D4 point

each singular fibers. Proposition 3.6 can be proved by using the transversality theorem and the multi-transversality theorem, since the dimensions pair (5,4) is in the nice range in the sense of Mather [ll](for details, see [ 5 ] , [lo] or [6]). We call a D4 point a C 2 ~ 2 point ~ o as well. We note that the normal forms for D4 points are slightly different from the usual one (see, for example , [l]), for details, see [18, Remark 4.21

Remark 3.7. The above proposition gives a characterization of Co stable maps of 5-manifolds into 4-manifolds, since (5,4) is in the nice range in the sense of Mather [ll]. Let f : M

-+ N

be a stable map of a closed 5-manifold M into a 4-

Euler number formulas in t e n s of singular fibers of stable maps

441

Fig. 6. Neighborhoods of a singular point in a singular fibers of proper stable maps of 5-manifolds into Cmanifolds

manifold N . For each regular point x E M of f , the fiber through x is a 1-dimensional submanifold near the point (see Figure 6 (0)). For each singular point p E M of f , based on the local condition of Proposition 3.1 ( i ) ,it is easy to determine the diffeomorphism type of a neighbourhood of p in f -'( f ( p ) ) as follows. Lemma 3.8. Every point p of a stable map f : M -+ N of a closed 5manifold M into a 4-manifold N has one of the following neighborhood in its corresponding singular fiber f -'( f ( p ) ) (see Figure 6):

+

((2,y ) E R2 I x2 y 2 = 0 } , if p is a definite fold point, (2) union of two transverse arcs diffeomorphic to { (x, y) E R2 I x2 - y2 = 0 } , i f p is an indefinite fold point, (3) ( 2 ,S)-cuspidal arc diffeomorphic to ((2, y) E R2 I x3 - y2 = 0}, if p is a cusp point, (4) isolated point diffeomorphic to {(x, y ) E R2 1 x4 y2 = 0}, if p is a definite swallowtail, (5) union of two tangent arcs diffeomorphic to {(x, y) E R2 I x4-y2 = 0 } , if p is a n indefinite swallowtail, (6) (2,5)-cuspidal arc diffeomorphic to {(x, y) E R2 I x5 - y 2 = 0 } , i f p is a butterfly point, (7) union of a n arc and a point diffeomorhic to { (x, y) E R2 I 3x2y y3 = 0 } , if p is a definite 0 4 point, (8) union of three arcs meeting at a point diffeomorphic to {(x,y) E R2 I 3x2 - y3 = 0 } , i f p is a n indefinite Dq point.

(1) isolated point diffeomorphic to

+

+

442

T. Yamamoto

In Figure 6, both the black dot (1) and the black square ( 4 ) represent an isolated point, although the corresponding map germs are not C” equivalent to each other; we use distinct symbols in order to distinguish them. We note that each singular point p E M , except for a definite fold point and a definite swallow-tail point, is incident to some edges in its neighbourhood in f -Yf (PI). We also note that an inverse image of regular value of f is a closed 1dimensional submanifold of M , that is, the inverse image is a disjoint union of finite number of circles. Thus, for a regular value q of f , the fiber of f over q is C” equivalent to the disjoint union of a finite number of copies of a fiber of a trivial circle bundle. For the singular fibers of f , we have the following. Theorem 3.9. Let f : M --+ N be a stable map of a closed 5-manifold M into a 4-manifold N . Then, every singular fiber o f f is C” equivalent to the disjoint union of one of the fibers in the following list and a finite number of copies of a fiber of a trivial circle bundle: (1) one of the fibers as depicted in Figure 7, (2) a disconnected fiber -o,o,o

-0,0,1

-O,l,l

-l,l,l

-0,0,2

-0,2,2

-1,1,2

-1,2,2

-0,1,2

I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , -2,2,2 -0,3 -0,4 -0,5 -0,6 -0,7 -1,3 -1,4 -1,5 -1,6 I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , -1,7 -2,3 -2,4 -2,5 -2,6 -2,7 -0,a -1,a -2,a -0000 I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , I11 , IV ’ ’ ’ , -000 1 -O,O,l,l -0 1 1 1, $ , l , l , l , $ 4 0 , 0 ’ 2 , 1 ~ 0 , 0 , 2 , 2 , $2’2’2 I V ” ’ , IV , IV”’ , -1 1 1 2 -1 1 2 2 -1 2 2 2 -0,0,1,2 -0,1,1,2 -0,1,2,2 -22 22 I V ” ’ , I V ” ’ , I V ” ’ , IV , IV , IV , IV”’, -0,0,3 -0,0,4 -0,0,5 -0,0,6 -0,0,7 -0,1,3 -0,1,4 -0,1,5 -0,1,6 , IV , IV , IV , IV , IV , IV , IV , IV , IV -0,1,7 -1,1,3 -1,1,4 -1,1,5 -1,1,6 -1,1,7 -0,2,3 -0,2,4 -0,2,5 IV , IV , IV , IV , IV , IV , IV , IV , IV , -0,2,6 -0,2,7 -1,2,3 -1,2,4 -1,2,5 -1,2,6 -1,2,7 -2,2,3 -2,2,4 IV , IV , IV , IV , IV , IV , IV , IV , IV , -2,2,5 -2,2,6 -2,2,7 -0,8 -0,9 -0,lO -0,ll -0,12 -0,13 IV , IV , IV , IV , IV , IV , IV , IV , IV , -0,14 -0,15 -0,16 -0,17 -0,18 -0,19 -0,20 -0,21 -0,22 IV , IV , IV , IV , IV , IV , IV , IV , IV , -0,23 -0,24 -0,25 -0,26 -1,8 -1,9 -1,lO -1,ll -1,12 -1,13 IV ,IV , IV , IV , IV ,IV ,IV ,IV , IV , IV , -1,14 -1,15 -1,16 -1,17 -1,18 -1,19, ~ 1 , 2 0 f, i 1 , 2 1 , $22 IV , IV , IV , IV , IV , IV -1,23 -1,24 -1,25 -1,26 -2,8 -2,9 -2,lO -2,ll -2,12 -2,13 IV , IV ,IV , IV ,IV , IV , IV , IV , IV , IV , -2,14 -2,15 -2,16 -2,17 -2,18 -2,19 -2,20 -2,21 -2,22 IV , IV , IV , IV , IV , IV , IV , IV , IV , -2,23 -2,24 -2,25 -2,26, $$,a -O,l,a -l,l,a -0,2,a -1,2,a IV , IV , IV , IV , IV , IV , IV , IV , -2,2,a -3,3 -3,4 -3,5 -3,6 -3,7 -4,4 -4,5 -4,6 -4,7 IV , IV , IV , IV , IV , IV , IV , IV , IV , IV , -6,6 -6,7 -7,7 -3,a -4,a -5,a -6,a -7,a -5,5 -5,6 -5,7 IV , IV , IV , IV , IV , IV , IV , IV ,IV , IV , IV , 9

Euler number formulas in terms of singular fibers of stable maps

Fig. 7. -0,b

-1,b

-2,b

-0,c

List of -1,c

K

443

= 1 , 2 singular fibers -2,c

-0,d

-0,e

-0,f

-0,g

-1,d

IV , IV , IV , IV , IV , IV IV , IV , IV , IV , IV E 1 , e -1,f -1,g -2,d -2,e -2,f -2,g , IV , IV , IV IV IV , IV a n d f i a l a (3) one of the connected fibers as depicted in Figures 8 , 9, 10 and 11.

,

The figure corresponding to each fiber listed in Theorem 3.9 (2) can be obtained by taking the disjoint union of the fibers in Figure 7 or 8 corresponding to the numbers or letters appearing in the superscript. For -0,0,2 example, the figure of the fiber 111 consists of two dots and a figure of I' type, see Figure. 12. In Figures 7, 8, 9, 10 and 11 K denotes the codimension of the set of points in N whose corresponding fibers are C" equivalent to the relevant -* -* one. Furthermore, I1 , I11 and E*mean the names of the corresponding singular fibers, and "/" is used only for separating the figures. We note that the conclusion of Theorem 3.9 holds if f is proper even if M is not closed.

-

I.,

444

T. Yamamoto

Fig. 8.

List of

K

= 3 singular fibers

The proof is very similar to that of [17, Theorem 3.51, as we omit the proof here.

Remark 3.10. Each singular fiber described in Theorem 3.9 can be realized as a component (or as a union of some components) of a singular

Fig. 9. List of

ri

= 4 singular fibers; 1

446

T. Yamamoto

n=4

Fig. 10. List of

K



Fig. 11. List of

K.


447

448

T. Yumamoto

Fig. 12. The singular fiber of type IIIo*o>2

fiber of a stable map of a closed 5-manifold into It4. This can be seen as follows. Given a singular fiber, we can realize it as a singular fiber of a Morse function parameterized on D3, ft : S + [-l711,t E D3, of a compact surface with boundary S into [-1,1], where D3 denotes the unit disk in R3. We note that F : S x D3 -+ [-1,1] x D3, defined by F ( z , t ) = (ft(z),t), is a smooth map and that F has the given singular fiber over (0,O). We call S a transverse surface correspond to the singular fiber (for details, see [9]). In this way we obtain a proper smooth map FlIntSxIntD3 : IntS x IntD3 + (-171) x IntD3. Then we can extend the map to a smooth map of a closed 5-manifold containing IntSx IntD3 into R4. Perturbing the extended map slightly, we obtain a desired stable map. If the source 5-manifold is orientable, then any transverse surfaces for any singular fibers are orientable. If the source 5-manifold is non-orientable, then there may exist a non-orientable transverse surface. The transverse surface which corresponds to the singular fiber of I2 type is a punctured Mobius band. We note that there exists a stable map of a non-orientable 4-manifold into a 3-manifold such that the transverse surface is orientable for any fibers. We note that for a stable map f : M -+ N of an orientable closed 5manifold M into a 4-manifold N , the singular fibers of the following types never appear, since they have non-orientable transverse surfaces: ~ 0 , 2 5, 1 , 2 -2,2 -5 -6 -7 -0,0,2 -0,2,2 -1,1,2 -1,2,2 -0,1,2 , I1 I1 I1 I1 , I11 I11 I11 I11 , I11 , -2,2,2 -0,5 -0,6 -0,7 -1,5 -1,6 -1,7 -2,3 -2,4 -2,5 -2,6 I11 , 111 , I11 , I11 , I11 , I11 I11 , I11 , I11 I11 I11 -2,7 - 2 , ~ -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 I11 , I11 , I11 I11 I11 I11 7 I11 7 I11 , I11 , I11 , 111 , 111 , -23 -24 -25 -26 -C -9 -0,0,0,2 -0,0,2,2 -0,2,2,2 -1,1,1,2 I11 I11 I11 , I11 , I11 , I11 IV IV IV IV 7 -1,1,2,2 -1 2 2 2 -0,0,1,2 -0 1 1 2 -0 1 2 2 -2,2,2,2 -0,0,5 -0,0,6 IV , r v ” ’ , IV , I V ” ’ , I V ” ’ IV , IV 7 IV , -0,0,7 -0,1,5 -0,1,6 -0,1,7 -1,1,5 -1,1,6 -1,1,7 -0,2,3 -0,2,4 IV , IV , IV IV , IV , IV , IV , IV , IV ,

1,

-O,2,5

-0,2,6

-0,2,7

-1,2,3

-1,2,4

-1,295

--1,2,6

-1,277

-2,2,3

IV IV IV , IV , IV IV IV 7 IV 7 IV -2,2,4 -2,2,5 -2,2,6 -2,2,7 -3,5 -3,6 -3,7 -4,5 -4,6 -4,7 -5,s IV ,IV ,IV ,IV , I V , I V , I V 7IV , I V , I V 7IV -5,6 -5,7 -6,6 I v 6 , 7 , 1 ~ 7 , 7 -5,a -6,a, ~ 7 , a ~, 0 , 1 3-0914 -0,15 IV IV IV , IV IV , IV 7 IV -0,16 -0,17 -0,18 -0,19 -0,20 -0,21 +0,22 -0,23 -0,24 -0,25 IV IV , IV IV , IV , IV , IV IV 7 IV , IV 7

7 7

7 7

Euler number formulas in terms of singular fibers of stable maps -0,26

-1,13

-1,14

-1,15

-1,16

-1,17

-1,lS

-1,19

-1,20

449

-1,21

IV , IV , IV , IV , IV , IV IV , IV , IV , IV -1,22 -1,23 -1,24 -1,25 -1,26 -2,8 -2,9 -2,lO -2,ll -2,12 IV , IV , IV , IV , IV , IV , IV IV , IV , IV -2,13 -2,14 -2,15 -2,16 -2,17 -2,18 -2,19 -2,20 -2,21 -2,22 IV IV 7 IV , IV , IV , IV 7 IV , IV , IV , IV -2,23 -2,24 -2,25 -2,26 -2,b - 0 , ~ - 1 , ~ - 2 , ~ -41 -42 -43 IV , IV IV , IV IV , IV , IV , IV , IV , IV , IV -48 -49 -50 -51 -52 -53 -54 -55 -44 -45 -46 -47 IV , IV 7 IV , IV , IV , IV , IV 7 IV , IV , IV , IV , IV -56 -57 -58 -59 -60 -61 -62 -63 -64 -65 -66 -67 -68 IV , IV 7 IV , IV , IV , IV 7 IV , IV , IV , IV , IV 7 IV , IV -69 -70 -71 -72 -73 -74 -75 -76 -77 -78 -79 -SO -81 IV , IV , IV , IV , IV , IV , IV , IV , IV , IV , IV , IV , IV -S2 -S3 -S4 -S5 -86 -S7 -S8 -89 -90 -91 -92 -93 -94 IV , IV , IV , IV , IV , IV 7 IV , IV , IV , IV , IV , IV , IV -95 -96 -97 -98 -99 -100 -101 -102 -103 -104 -105 -106 IV , IV , IV 1v , IV , IV , IV 7 IV , IV ,lv , IV , IV -107 -108 -109 -110 -111 -112 -113 -114 --k --I -m -n IV , IV , IV , IV IV , IV , IV , IV , IV IV , IV , IV -u -7) -x -y -z -w -c IV IV IV , IV , IV , IV , IV and 7

zol3,

, , ,

, , , , , ,

EL.

We note that the list of the singular fibers of stable maps of "orientable" 5-manifold into 4-manifolds is also obtained in [18]and [25]. For stable maps of closed (possibly non-orientable) 4-manifolds into 3manifolds, C" equivalence and Co equivalence are coincide, refer to [24, Chapter 1 Corollary 2.71. Similarly, we obtain the following. Corollary 3.11. For two singular fibers of proper stable maps of closed (possibly non-orientable) 5-manifolds into 4-manifolds, the following two are equivalent. (1) They are C" equivalent. (2) They are Co equivalent.

Proof. If two fibers are C" equivalent then they are clearly Co equivalent. The converse is not trivial. Since Co equivalence preserves the numbers of connected components of nearby singular fibers, the singular fibers of I1 type and that of type are not Co equivalent. Similarly, for any two singular fibers in Theorem 3.9, they are not Co equivalent.

Let us consider the class T = Sg(5,4) which consists of all singular fibers of proper stable maps of 5-manifolds into 4-manifolds. As the top notation of this subsection, this class is big enough. The Co equivalence pE,4 among the singular fibers in the class S g ( 5 , 4 ) is so strong, we may choose Coequivalence modulo two regular fiber components ~ 8 , ~ ((see, 2 ) Dfinition 3.1). We note that for smooth map of ( p + l ) manifold into pmanifold, the regular fiber is the disjoint union of the trivial circle bundles.

450

T. Yamamoto

Thus we obtain the cochain complex

(C"(qX5,4),PE,4(2)>,S")". The generators of C " ( S g ( 5 , 4), ~ : , ~ ( 2 )are ) the subclasses of the classes in the table of Theorem 3.9. Then the generators of coboundary of this cochain complex are in the tables 1, 2 and 3. In the tables 1, 2 and 3, the symbols F, (or 3,)denote the Coequivalence modulo two regular fiber components class which consist a singular fiber of type 3 and a some copies of a fiber of trivial circle bundle such that the total number of trivial circle bundle is odd (resp. even), F represent F,+F, and the item Sn(Fe)can be obtained by interchanging Go with Q, in the item corresponding to F,. Then, we obtain the following. Proposition 3.12. The cohomology groups of the universal complex of singular fibers for proper CM stable maps of 5-manifolds into 4-manifolds with respect to the Co equivalence modulo two regular fiber components

( c " ( S g ( 5 7 4), P:,&h

JK)"

are given as follows;

+&I

0

H o "= Z2 generated by [is, H1 E Z2 @ Z2 generated by +Ti] -1,2 = -0,2 H 2 2 Z2 generated by [11, + 11,

0

H3 =

0

+yA],

+y:]

-6 -02 -12 -6 + II,] = [II,' + 11,' + II,]

(0)

where 0, (or 0,) denote the equivalence classes of regular fiber whose number of connected components is odd (resp. even), [*] denotes the cohomology class represented the cocycle *. Furthermore, we restrict our attention to the subclass S$(5,4)Ori c Sg(5,4)which consists of all singular fibers of proper stable maps of orientable 5-manifolds into 4-manifolds. Thus we obtain the cochain complex (C"(q35,4)OTi,P:,4(2)), S")"

The generators of this cochain complex are the subclasses of the classes in Theorem 3.9 except the classes in Remark 3.10. Then we obtain the following.

Proposition 3.13. The cohomology groups of the universal complex of singular fibers for proper C" stable maps of orientable 5-manifolds into 4manifolds with respect to the Co equivalence modulo two regular fiber components


451

n 1 -

2

3 -1,a -b + 111, + 111,, -O,l,l -8 -1,a = I11 + I11 + 111, + 111,- ., -0,1,2 -0,6 -2,a = I11 + 111, + iE:, + I11 -0,1,2 -1,6 -2,a = I11 + 1-141-2,6 1, + 111, , + I11 -0,2,2 -1,2,2 -20 = I11 I11 + I11 + 111, , + -3 -0,3 -1,3 -8 -9 -11 .17 -b 62(11,) = 111 + 111 + 111, + 111 + 111, + + 111, + E:, -4 -0,4 -1,4 -10 -11 -13 -21 62(II,) = I11 + I11 + I11 + 111, + 5:+ 111, + 111, , -5 -0,5 -1,5 -15 -17 -21 62(II,) = I11 + I11 + I11 + 111, + 111, + E;, -6 -0,6 -1,6 -14 62(II,) = I11 + I11 + 111, + EZ, -7 -0,7 -17 -13 -18 -19 -20 62(II,) = I11 + I11 ’ + 111, + 111 + I11 + 111, , -0,a - 62(iiZ) = I11 + I11 + TIb

-0,a

= 111,

-l,l,l

”

iii,

-1,a

4 -0,b -1,d + IV, + IV, , -O,l,l -O,l,l,l -0,8 -0 1a -l,l,a -1,b 63(III, ) = I V + IV, + IV,’ ’ + IV, + IV, , -1,l,l -O,l,l,l -1,8 -1 1 a &(III, ) = IV + IV, + IV,’ ’ -0,1,3 -03 -0,9 -0,ll -0 b --h &(III, ) = I v + IV, + IV + IV, + IV,’ + IV, + IV, -0,f -3 a -0,17 + IV, + IV,’ +IV, , -1,3 -0,1,3 -1,8 -1,9 -1,ll -29 -1,b 63(III, ) = Iv + IV, + IV + IV, + IV, + IV, -1,f -3 a -1,17 + IV, +IV,’ + IV, , -094 -0,1,4 ~ 0 ’ 1 0 -0,11 -0,e -4,a --i

-093

+IV,

~:,13

-194

-0,1.4

63(III, ) = IV -8

-9

+

-0,21

+IVo

+IV,

7

~ 1 , l O -1,ll

+IV,

-28

+IV,

-1,e

+IV,

-4,a

+IV,

-0,8

-1,8 --h + IV + IV, + E:,

-0,g

+ I v + Iv, + Iv, + Iv, + $ + @,:

&(III,) = IV 63(1110)

+ x+IV,

+

63(III, ) = IV

= Iv

-1,9

-29

-31

-46

Table 1. Generators for the coboundary groups of C(Sg(5,4),p;,,(2))

452

T. Yamamoto

-

generator(s1 \

-10

I

-0,lO

~ 1 , 1 0 -28

+

&(III, ) = IV

-31

+ IV, + IV, + IV, -55

-11

-58

-59

-31 -66 -67 + + I V +IV, +IV, -&12 + -70 -71 &(III, ) = IV + IV + IV, + IV, -0 a -O,l,a -0,b 63(1110’ ) = IV + IV + Z:, -1,a -O,l,a -h &(III, ) = IV + IV + IV, + E:, -b -0,b -1,b -h &(III,) = IV + IV + IV, + E: + I”vz, -d -0,d -1,d &(III,) = IV + IV + E,“, -1,e

-68

&(III, ) = IV -0111 -12

+iv:

+IV,

-36

-1,b

-0,e

+E‘+~~+E~+E~+$ 63(f$) = IV + IV -0+ I”v” +$ ,+ Ez $,2,a + Ez + E:, -o,o,2 -0,0,0,2 012 -0,0,6 -0,c -2,d &(III, ) = IV + IV ’ ’ ’ + IV +IV, +IV, , &(zE)=Iv

+IV

-1,f

-0,f

+

-0,2,2

63(III,

-l71,2

-0,1,2,2

+

) = IV

-0 1 1 2 -1,14 -2,8 + fi1,2,6 IV ’ ’ ’ +IV, + IV, -1,20 -2,14

-191,132

&(III, ) = IV --1,2,2 -0,1,2,2 &(III, ) = IV -0,172

&(III, ) = IV -0,1,6 -0,6 &(III, ) = IV -0,7 -0,197 63(III, ) = IV -1,5

-0,1,5

63(III, ) = IV -176 -0,196 63(III, ) = I V -1,7 -0,1,7 &(III, ) = IV

-0,14

&(III, ) = IV

,

-2,20

+IV,

-k + IV, + E:721 + IV, + IV, + IV,

-0,13 + IV, +

+

-5,a

-0,g

,

-6,a

+

-41

+IV, +

~ 0 , 1+ 8 fiO,19

~ 1 , 1 5 ~ : , 1 7 -1,14

+IV,

~ : 7 1 3

+

-O,2O

+IV,

-1,21

-43

-5

a

+

~ 1 , 1 8 ~ 1 , 1 9 +

1

~ 1 , 2 , 3 -3

+

6

-2

+ I V ’ +IV,’

-41

-48

+IV, ~ 1 , 2 , 4 -4,6 IV -63 -2,e +IV, +IV, ~ 1 , 2 , 5 -5,6 +IV -2,g

+IV,

+

-l,g

+IV, +IV, +IV,’ +IV, , -42 -44 -1 c -6,a + I V , +IV, +IV,’ + I V , ,

-85

+

-82

-1,~

+ E:”*+ IV, + IV, + $, + KT + mz,

63(III, ) = IV -2,21 +IV, -2,5 -0,2,5 &(III, ) = IV +IV,

-2,b

~ 0 , 1 5 -0,17

+ IV, + IV,

-0,2,4

-2,2,6

-0,c

-2,17

-2,4

222

-1,2,a

+ I V ” ’ +IV

+

+ IV, + IV,

-0,273

-0,2,a

-1

-84

-2,3

+IV,

+ IV, + IV, + IV, + IV, + IV,

-0,1,6

-0,1,5

-1,1,6

+IV

+ IV, + IV, + i v y ,

+

83(III, ) = I V -2,292 -0,2,2,2 &(III, ) = I V -0,s

~0,2,6

8

-2,9 + IV

-2,11

+IV0

-2 b -2,f +-2,lO IV,’ + I V 0 , + IV + i v y +

,

--2,15 -2,17 --2,21 + IV + IV, + IV,

,

Table 2. Generators for the coboundary groups of C(Sg(5,4),

p;,,(2))


-

453

aeneratods) \

-2,6

,

+ IV

-0,2,6

+

d3(111, -14

63(III, -15

63(III,

-2,14

+ IV

~ 1 , 2 , 7 -6,7

-2,20

-13

-80 + IV, +-2,13 IV, + IV, + IV,

-1,2,6

63(III, ) = IV -2J --0,2,7 ~~(111,= IV

-104

-81

+

+IV,

-2,c

fi2,18

-109

+ IV

,

-2,19

+IV, +IV, , + Iv1,13 + IV + IV , ) = IV + Iv1'14+ E4l+ IV + IV + $,+ -k ) = Eo'l5 + + IV, +IV, + IV, + I V 0 +IV, +IV, = IV

-54

--0,13 -0,14

-43

+~

= $,16 -88

+IV

=p 1 7

+ IV,

1 ' 1 6 -67 -89

+IV + K1,17

-18 63(III, ) = Eo'18 +

-59

-42

-44

-46

-55

-78

-79

-78

-81

-58

+ I V +IV, + -91 -94 +IV, +IV, + I v y , -46 -66 -67 IV + IV-" + IV- +

+

-59

-84

+ IV, + IV, + IV: + IV, +

-54 -78 -80 -85 + E1llg + IV, + IV, + IV, + IV, +

-19

~~(111, ) = IV --0,19

-80 -81 + IV + IV

-20

&(III, ) = fi0'20 +

-55

+IV -22

~~(111,) = IV -0122

+

-53

+IV

,

-58

+IV

-57

+IV

-66

+IV, -61

+IV

-67

+

-63

+

+IV, +IV,

+ IV, +IV, + IV, +IV, + IV, + E:l + IV, + E!, 23 63(III, ) = Iv0'23 + + IV, + IV, + IV, + IV, + + -34 B3(III, ) = I v + IV, +IV, +IV, + 1v- + + IV, + IV, + IV, +ivy, d3(111, ) = IV -0,25 + Iv1,25+ E%+ IV, + IV, + IV + 63(III, ) = Eo'26 + + E%" + Ep + + -65

-68

-70

-73

-75

-94

-42

'-24'

-OJ4

-55

-65

-45

-55

-66

-45

-28

-51

-75

-93

-99

-79

-25

26

$02

-2,a

b3(111, )

-0,2,a

+

= IV -0,c

6 3 ( E 3 = IV

-04

6 3 ( E 3 = IV

~ 1 , 2 , a -2,b

+ IV -1,g + IV

-6,a

+IVf,+Iv:, + 57: + Iv" + Iv" + E;, + Ivy + Iv:: + EZ+ Iv: +IV

-1,c

$05

+IV

Table 3. Generators for the coboundary groups of C(Sg(5,4), ~ : , ~ ( 2 ) )

454

T. Yamamoto

are given as follows; 0 0

0

H o g 2 2 generated by Po+Gel H1 2 Z2 generated by [I: I:] = [I: H 2 = (0) -12 -12 H 3 g 22 generated by [III, 111, ]

- +-

- +-I:]

+

where the symbols are as above. In the next section, we consider the geometrical meaning of the cohomology classes of universal complex of the singular fibers of proper stable maps of 5-manifolds (resp. orientable 5-manifolds) into 4-manifolds. 4. Characterization of the cocycle

In this section, we give a necessary and sufficient condition for a certain cochain of the universal complex to be a cocycle in terms of the homomorphisms pf induced by 7-maps f . For Lemmas and Propositions on this subsection, the proofs are very similar to that of [17, 121, we omit the proofs here.

Definition 4.1. Let f : M + N be a Thom map and g : V + N be a smooth map which is transverse to f and to all the strata of N . Set

F = { ( q y ) E M x N I f(.)

=g(y)} C M x

V

and consider the following commutative diagram;

-

V

V

-

9

M

9N ,

7

where 5and are the restrictions of the projections to the first and second factors respectively. We note that is a smooth manifold of dimension dim V + dim M - dim N and is a proper Thom map. We call the pullback of f by g and say that ?is obtained by pulling back f by 9.

7

7

Definition 4.2. Suppose that

+ + 1)

T ( n , p ) and ~ ( nl , p

are given such that the suspension of an element of ~ ( n , p belong ) to ~ ( +n1,p + 1). Let f : M -+ N be an element of ~ ( n1,p 1) and

+

+


g : IntDP -+ N a smooth map which is transverse to f and to all the strata of N . We note that the pull-back fof f by g is then a proper Thom map of an n-manifold to a pmanifold. If the fiber of always belong to r ( n , p ) , then we say that T ( n , p ) is transversely complete with respect t o T(n 1 , p 1).

f

+ +

For example, S g ( 4 , 3 ) is clearly transversely complete with respect to S3574). Lemma 4.3. I f T ( n , p ) is transversely complete with respect t o 7(n+l , p + l ) , then the natural &-linear map SIC

+

: CK(7(n 1 ,P

+ I ) , Pn+l,p+l)

+

CYT(72.7PI, P n , p )

induced by the suspension is injective f o r any K I p , where ~ ~ + l , and ~ + pn,p are equivalence relations f o r the fibers of elements of T ( n 1 , p 1 ) and 7 ( n , p ) , respectively.

+

K

Let c be an any cochain in C K ( ~ ( n , p ) , p nwith ,p) 0 < - ( n - p ) . Since we always have Cn+l(7(*,K ) , p*,,) = 0,

6, : C,(T(*, K ) , P * , K )

+

K

+

< p . Set *

=

Cn+l(7(*,61, p*,,)

is the zero map, and hence s,c E C.(T(*, K ) , p*,,) is a cocycle of the complex C ( T ( * , K ) , P*,,), where stc : C n ( ~ (Pn) , P n , p )

+

C K ( 7 ( *K ,) , P*,,)

is the homomorphism induced by the ( p - rc)-th suspension. Therefore, for a map f : M -+ N , the homology class [ ( s , c ) ( f ) ] E H,"(N;Zz)represented by (s,c)( f ) is well-defined.

Proposition 4.4. Suppose that T ( * , K ) is transversely complete with respect t o r ( n , p ) , where 0 < K < p and p - n = K - X = k. T h e n a cochain c E C K ( T ( n , p ) , p n , p is ) a cocycle of the complex C(T(n,p),p,,,) i f and only if [ ( s , c ) ( f ) ] = 0 E H o ( N ; Z z )f o r every T ( * , " ) - m a p f : M -+ N such that both M and N are closed and that f i s 7-cobordant t o a nonsingular map.

As a direct conclusion, we obtain the following corollary. Corollary 4.5. I f there exist a cochain c E C 3 ( S g ( 5 ,4 ) , ~ g , ~ ( 2such ) ) that N of closed 4-manifold [ ( s ~ c ) ( f= ) ] x ( M ) f o r every stable map f : M into 3-manifold then [c] E H 3 ( C ( S g ( 5 ,4 ) , ~ ; , ~ ( 2 ) ) . -+

Proof. Let c E C 3 ( S g ( 5 , 4 ) , p g , 4 ( 2 ) such ) that [ ( s ~ c ) ( f= ) ]x ( M ) f o r every stable map f : M -+ N of closed 4-manifolds into 3-manifolds. Since the

l

T. Yamamoto

456

Euler characteristic is a n invariant of unoriented cobordism, [(s3c)(f ) ] = x ( M ) E 0 E H o ( N ; Z 2 )f o r f which is Sm-cobordant t o a nonsingular map.

0

Saeki’s Euler number formula (for details, see Introduction of this paper) -12

implies that the cohomology class of I11

is in H 3 ( C ( S ~ ( 5 , 4 ) 0 T i , ~ ~ , 4 ( 2 ) ) .

-12

Actually I11 is in H 3 ( C ( S p ( 5 4,) O T i , p&(2)) (Proposition 3.13). Combing Corollary 4.5 and the third cohomology of the cochain complex Proposition 3.12, we obtain the following Theorem.

Theorem 4.6. There is no Euler number formula of 4-manifolds in terms of the singular fibers of stable m a p s into 3-manifolds if we consider just stable maps. References 1. Y. Ando, O n local structures of the singularities Ak, Dk,and Ek of smooth maps, Trans. Amer. Math. SOC.331 (1992), 639-651. 2. P.E. Conner and E.E. Floyd, Differentiable periodic maps, Ergebnisse der Math. und ihrer Grenzgebiete, Band 33, Springer-Verlag, Berlin, Gottingen, Heidelberg, 1964. 3. A. du Plessis and T. Wall, The geometry of topological stability, London Math. SOC.Monographs, New Series 9, Oxford Science Publ., The Clarendon Press, Oxford Univ. Press, New York, 1995. 4. C. G. Gibson, Singular points of smooth mappings, Pitman, London 1979. 5. C. G. Gibson, K. Wirthmuller, A. A. du Plessis and E. J. N. Looijenga, Topological stability of smooth mappings, Lect. Notes in Math., Vol. 552, Springer-Verlag, Berlin-New York, 1976. 6. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math. 14, Springer, New York, Heidelberg, Berlin, 1973. 7. M.E. Kazaryan, Hidden singularities and Vassiliev’s homology complex of singularity classes (in Russian), Mat. Sb. 186 (1995), 119-128; English translation in Sb. Math. 186 (1995), 1811-1820. 8. M. Kobayashi, Two nice stable maps of C 2 P into It3, Mem. College Ed. Akita Univ. Natur. Sci. 51 (1997), 5-12. 9. H. Levine, Classifying immersions into R4 over stable maps of 3-manifolds into It2, Lecture Notes in Math., Vol. 1157, Springer-Verlag, Berlin, 1985. 10. J. Martinet, Singularities of Smooth Functions and Maps, London Mathematical Society, Lecture Note Series 58 1982. 11. J. N. Mather, Stability of Cm mapping: V1,the nice dimension, Lect. Notes in Math. 192, Springer, 1971, pp.207-253. 12. J. Milnor, J Stasheff, Charactristic Classes, Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton 13. Rimafiy and Sziics [13], Pontrjagin-Thom-type construction for maps with singularities, Topology 37 (1998), 1177-1191.


14. 0. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl. 49 (1993), 265-293. 15. 0. Saeki, Topology of special generic maps in to R3,in “Workshop on Real and Complex Singularities” , MatemAtica Contempothea 5 (1993), 161-186. 16. 0. Saeki, Studying the topology of Morin singularities f r o m a global viewpoint, Math. Proc. Camb. Phil. SOC.117 (1995), 617-633. 17. 0. Saeki, Toplogy of Singular Fibres of Differentiable Maps , Springer, Lect. Notes in Math. 1854 18. 0. Saeki and T. Yamamoto, Singular fibers of stable maps and signatures of 4-manifolds, to appear in Topology and Geometry. 19. 0. Saeki and T. Yamamoto, Singular fiber and characteristic classes, in preparation. 20. 0. Saeki and T. Yamamoto, Singular fibers of differentiable maps and characteristic classes of surface bundles, in preparation. 21. A. Szucs, Surface in R3,Bull. London Math. SOC.18 (1986), 60-66. 22. V.A. Vassilyev, Lagmnge and Legendre characteristic classes, Translated from the Russian, Advanced Studies in Contemporary Mathematics, Vol. 3, Gordon and Breach Science Publishers, New York, 1988. 23. T. Yamamoto, Classification of singular fibres and its applications (in Japanese), Master Thesis, Hokkaido Univ., March 2002. 24. T. Yamamoto, Classification of singular fibres of stable maps of 4-manifolds into 3-manifolds and its applications, J. Math. SOC.Vol. 58, No. 3 (2006), 721-742 25. T. Yamamoto, Singular fibers of two colored differentiable maps and cobordism invariants , Doctor Thesis, Hokkaido Univ., March 2006


459

AUTHOR INDEX Adachi, J., 1 Bekka, K., 33 Fukui, T., 33 Hillman, J., 46 Ishikawa, G . , 56 Izumi, S., 85, 109 Izumiya, S., 124 Katzarkov, L., 176 Koike, S., 33 Kuo, T.C., 207 Leinster, T., 232 Matsui, Y., 248 Miywhi, J., 271 Miyajima, K., 279 Neeman, A., 290 Norbury, P., 299 Ohsumi, M., 323 Paunescu, L., 207 Sakuma, K., 342 Takahashi, M., 388 Takeuchi, K . , 248 Takeuchi, N., 419

Yamarnoto, T., 427

Real and Complex Singularities: Proceedings of the Australian-Japanese Workshop, University of Sydney, Australia, 5-8 September 2005

Real And Complex Singularities

Real and Complex Singularities