Singular Points of Complex Hypersurfaces (AM-61), Volume 61 9781400881819

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Annals of Mathematics Studies Number 61

SINGULAR POINTS OF COMPLEX HYPERSURFACES BY

John Milnor

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1968

Copyright © 1968, by Princeton University Press ALL RIGHTS RESERVED

L.C. Card: 69-17408

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

TO MY MOTHER

Preface T he topology a s s o c ia te d with a singular point of a com plex curve h as fa scin a te d a number of g eom eters, ever s in c e K . B r a

u n er

* showed in

1928 th at each such singular point can be d escrib ed in term s of an a ss o ­ cia te d knotted curve in the 3 -sp h e re . R e ce n tly E . B R IE S K O R N h as brought new in terest to the su b je ct by d isco v erin g sim ilar exam ples in higher dimen­ sio n s, thus relatin g alg eb raic geom etry to higher dim ensional knot theory and the study of e x o tic sp h eres. T h is m anuscript will study singular points of com plex h y p ersu rfaces by introducing a fibration which is a s s o c ia te d with each singular point. As p rereq u isites the reader should have some knowledge of b a sic al­ gebra and topology, as presen ted for exam ple in LANG, A lgeb ra or VAN DER WAERDEN, Modern A lg eb ra , and in SPA N IER , A lg eb ra ic T opology . I want to thank E . B riesk orn , W. C asselm an , H. Hironaka, and J . Nash for helpful d iscu ss io n s ; and E . Turner for preparing n otes on an e arlier version of th is m aterial. dation for support.

A lso I want to thank the N ational S cie n ce Foun­

Work on th is m anuscript was carried out at P rin ceton

U n iversity, the In stitu te for Advanced Study, The U n iversity of C alifornia at L o s A n geles, and the U n iv ersity of Nevada.

See the Bibliography.

Proper names in c a p ita l le tte rs w ill always in d icate

a reference to the Bibliography.

CONTENTS §1.

Introduction.............................................................................................................

3

§2.

E lem entary fa c ts about real or com plex algeb raic s e t s ................

9

§3.

T he curve se le c tio n lem m a.............................................................................

25

§4.

T he fibration th e o r e m .......................................................................................

33

§5.

T he topology of the fiber and of K ...........................................................

45

§6.

T he c a s e of an iso la te d c ritic a l p o in t......................................................

55

§7.

T he middle B e tti number of the fib e r.........................................................

59

§8 .

Is K a top o lo g ical s p h e r e ? .............................................................................

65

§9.

B rieskorn v a rie tie s and weighted homogeneous polynom ials. . . .

§10.

The c l a s s i c a l c a s e :

§1 1 .

A fibration theorem for re a l s in g u la ritie s ...................................................97

Appendix A.

cu rv es in C .............................................................

71 81

Whitney’s fin iten ess theorem for algeb raic s e t s ................. 105

Appendix B . The m ultiplicity of an iso la te d solution of an aly tic eq u a tio n s............................................................................ I l l B ib liograp h y.........................................................................................................................

117

Annals of Mathematics Studies Number 61

§1.

INTRODUCTION

L e t f ( z j , z n + 1) ke a non-constant polynomial in n + 1 com plex v a ria b le s, and let V be the algeb raic s e t co n sistin g of all (n+ 1 ) -tu p les z = ( z l f . . . , z n + 1) of com plex numbers with f(z) = 0 . s u r fa c e .)

(Such a s e t is ca lle d a co m p lex hyper-

We want to study the topology of V in the neighborhood of some

point z ° . We will u se the following co n stru ctio n , due to BRA UN ER.

In te rse ct

the h ypersurface V with a sm all sphere S£ cen tered at the given point z ° . Then the topology of V within the disk bounded by S£ is c lo s e ly related to the topology of the s e t

k

= v n sg .

(Compare § 2 .1 0 and § 2 .1 1 .) As an exam ple, if z ° is a regu la r point of f (that is if some partial d erivative d i/d z j does not vanish at z ° ) then V is a smooth manifold of real dimension 2n near z ° .

The in te rse ctio n K is then a smooth ( 2 n — 1 )-

dim ensional manifold, diffeomorphic to the (2n — l)-s p h e r e , and K is em­ bedded in an unknotted manner in the (2n + l)-s p h e r e S£ . (See § 2 .1 2 .) B y way of co n tra st, con sid er the polynomial f(z l , z 2 ) = z P + z 2q in two v a ria b le s, with a c ritica l point (d i/d z ^ = d i/d z 2 = 0 ) at the origin. A ssum e that the in tegers p, q are re la tiv e ly prime and > 2. 3

4

SINGULAR POINTS OF CO M PLEX H Y PE R SU R FA C E S

ASSERTION ( Brauner).

T h e in tersectio n o f V = f“ 1(0 ) with a s p h e re

Sg c e n te r e d at the origin is a knotted c ir c le of the type known as a “ torus k n o £ of type (p, q )” in the 3 -sp h e re Sg . [P ro o f: It is e a s ily verified that the in te rse ctio n K lie s in the torus co n sistin g of all (z^, z 2 ) with | z j = p ositive co n sta n ts.

\z^\ = 77 where £ and 77 are

In fa ct, K c o n s is ts of all p airs

rj

77l//(^)

as the param eter 6 ranges from 0 to 277 : Thus K sw eep s around the torus q tim es in one coordinate direction and p tim es in the other.] F o r exam ple the torus knot of type (2 , 3 ) is illu strated in F ig u re 1.

Figu re 1. The torus knot of type (2, 3). *•*-> /%/\m A / i t%/\ I t rn /\m i o l o n n ✓'vr r»/\n A o t r n o f mn/>n (B y using more com plicated polynom ials one r* can of co u rse arrive at much a

more com p licated knots.

Compare § 1 0 .1 1 .)

B R IE S K O R N has studied higher dim ensional analogues of th ese torus

knots.

F o r exam ple let V (3, 2, 2, . . . , 2) be the lo cu s of z ero s of the poly­

nomial f ( z i , . . . , z n + 1) =

+ z 22 + -

+ z n + 12 .

§ 1 . INTRODUCTION

5

F o r all odd v alu es of n th is h ypersurface in te rs e c ts Sg in a smooth mani­ fold K which is homeomorphic to the sphere S2 n ~ 1 . In som e c a s e s (for exam ple when n = 3 ) K is diffeomorphic to the standard ( 2 n — l)-s p h e r e , while in other c a s e s (for exam ple n = 5 ) K is an “ e x o ti c ” sphere. But in all c a s e s K is embedded in a knotted manner in the (2n + 1 )-sp h ere S£ . T h e se B rieskorn sp h eres will be studied in more d etail in §9. T he o b ject of th is paper is to introduce a fibration which is useful in d escrib in g the topology of such in te rse ctio n s k =

v n sg c

Sg .

Here are some of the main re s u lts , which will be proved in S ectio n s 4 through 7 . FIBRATION T h e o r e m . If z ° is any point o f the co m p lex h y p ersu r­ fa c e V = f- 1(0 ) a nd if Sg is a s u ffic ie n tly sm all s p h e re c e n t e r e d at z ° , then the mapping 0 ( z ) = f(z)/|f(z)| from Sg — K to the unit c ir c le is the p ro jectio n map of a smooth fib er b u n d le*.

E a ch fib er Fd =

C S£ - K

is a smooth p a ra lle liz a b le 2n -d im ensional manifold. If the polynomial f h a s no c ritic a l points near z ° , e x ce p t for z ° its e lf, then we can give a much more p re c is e d escrip tion. THEOREM.

If z ° is an is o la te d critica l point o f f, then ea ch fib er F^

h a s the homotopy type of a bouquet Sn V ••• V Sn of n-s p h e re s , the num ber of s p h e re s in this bouquet ( i .e ., the m iddle B etti num ber of F q ), b ein g s trictly p o sitiv e.

E a ch fib e r can b e c o n s id e re d as the interior of a smooth

com pact manifold-w ith-boundary, C losure (F^) = F ^ U K , *

T he term “ fiber bundle” w ill be used as a synonym for “ lo c a lly triv ial fiber s p a c e .”

SINGULAR POINTS OF COMPLEX H YPER SU RFA C ES

6

w here the common boundary K is an (n — 2 ) - c o n n e c t e d m anifold. Thus all of the fibers F q fit around their common boundary K in the manner illu strated in F ig u re 2.

T he smooth manifold K is con nected

if

n > 2, and simply con nected if n > 3. Here is a more d etailed outline of what follow s.

Section 2 d e scrib e s

elem entary properties of real alg eb raic s e ts , following WHITNEY.

A fun­

damental lemma concerning the e x is te n c e of real an aly tic cu rv es on real algeb raic s e ts is proved in §3. lemma.

All of the subsequent proofs rely on this

The b a sic fibration theorem is proved in

§4. Fu rth er

d e ta ils on

the topology of K and F^ are obtained in §5.

Figu re 2.

N ext we introduce the additional hypothesis that the origin is an is o ­ la ted c ritic a l point of f.

Then a much more p re c ise descrip tion of the fiber

is p o ssib le ( § 6), and a p re cise formula for the middle B e tti number of the fiber is given (§7).

The topology of the in tersectio n K is then d escribed

in term s of a certain polynomial A( t ) with integer c o e fficie n ts which gen­ e ra liz e s the A lexander polynomial of a knot. (§ 8 .)

§1. INTRODUCTION

7

T he B rieskorn exam ples of sin gu lar v a rie tie s which are to p o lo g ically m anifolds are d escrib ed in § 9 , and the c l a s s i c a l theory of singular points of com plex cu rv es is d escrib ed in § 10.

The la s t se ctio n proves a general­

ization of the fibration theorem to certain sy stem s of real polynom ials. As an exam ple, a polynomial d escrip tion of the Hopf fibrations is given. Two appendices con clu d e the p resen tatio n .

§2.

E L E M E N T A R Y F A C T S ABOUT R E A L OR C O M PLEX A LG EB R A IC SETS

Let $

be any infinite field, and let 3>m be the coordinate s p a ce con­

sistin g of all m -tu ples x = ( x 1 , .

x ) of elem ents of

p ally in terested in the c a s e where 0

(We are princi­

is the field R of real numbers or the

field C of com plex numbers.) D EFIN ITION. A su b set V C $ m is ca lle d an a lg eb ra ic s e t * if V is the lo cu s of common zero s of some co lle ctio n of polynomial functions on 0 m. The ring of all polynomial functions from the conventional symbol ^ [ x p . . . , x m].

to will be denoted by

Let

I ( V ) C [xr . . . , x m] be the id eal co n sistin g of th o se polynom ials which vanish throughout V. The Hilbert *‘b a s is ’ ’ theorem a s s e r ts that every ideal is spanned (a s 0 [ x j , . . . , x m]-m od u le) by some finite co lle ctio n of polynom ials.

It follow s

th at every algeb raic s e t V can be defined by some finite c o lle ctio n of polynomial equations. An important co n seq u en ce of the Hilbert b a s is theorem is the following: 2 .1 D esc e n d in g chain condition.

Any n ested seq u en ce V j 3 V2 3 V3

D ••• of alg eb raic s e ts must term inate or s ta b iliz e (V- = V *+1 = Vj+2 = '*') after a finite number of ste p s. ~ It is custom ary in algebraic geometry to allow as “ p oin ts” of V a lso m -tu ples of elem ents belonging to some fixed alg e b ra ica lly clo se d exten ­ sion field of $ ; but I do not want to allow th is ..

9

SINGULAR POINTS OF COM PLEX H YPER SU RFA C ES

10

Note that the union V U V ' of any two alg eb raic s e ts V and V 7 in Om is again an algeb raic set. A non-vacuous algeb raic s e t V is ca lle d a variety or an irre d u c ib le a lg eb ra ic s e t if it cannot be e x p re sse d as the union of two proper alg eb raic su b s e ts.

Note that V is irreducible if and only if I(V ) is a prime id eal.

If V is irreducible, then the field of quotients f / g with f and g in the in tegral domain [x1 . . . . , x m] / I ( V ) is called the fie ld o f rational fu n ctio n s on V. over

Its tran scen d en ce degree

is called the alg eb raic dim ension of V over 0 .

If W is a proper su bvariety of V, note that the dimension of W is le s s

than the dimension of V.

(See for exam ple L A N G , A lg eb ra ic G eom etry,

p. 2 9 .) Now let V C

be any non-vacuous alg eb raic se t.

C hoose fin itely

many polynom ials f ^ .. . , f ^ which span the ideal I(V ) and, for each x e V, con sid er the k x m m atrix (d F /d x j) evaluated at x.

L e t p be the la rg e s t

rank which th is m atrix a ttain s at any point of V. DEFINITION. m atrix

A point x is the fie ld o f rea l (or co m p lex) num­

b ers, then the s e t V —S (V ) of n on-singular p o in ts of V forms a smooth, non-vacuous m anifold.

In fa ct this m anifold is rea l (or co m p lex) analytic,

and h a s dim ension m —p o ver T he reader is referred to WHITNEY, E lem en ta ry Structure of R ea l A l­ g e b ra ic V a rieties, for the elegan t proof of 2 .3 . In the c a s e of an irreducible V, Whitney shows th at the dim ension of the analytic m anifold V —S (V ) ov er $

is p r e c is e ly equal to the a lg eb ra ic

dim ension o f V o v er Here is another b a s ic resu lt. THEOREM 2 .4 (Whitney).

F o r any pair V D W of a lg eb ra ic s e t s in a

rea l or co m p lex coordinate s p a c e , the d iffe re n c e V — W ha s at most a fi­ n ite num ber of top ological com ponents. F o r exam ple, V its e lf h as only fin itely many com ponents; and the smooth manifold V —X (V ) h as only fin itely many com ponents. A proof of 2 .4 , only slig h tly different from W HITNEY’ S proof, will be given in Appendix A. Here are three exam p les.

(Compare F ig u re 3 .)

E a ch exam ple will be

a curve in the real plane having the origin a s unique singular point. EX A M P L E A.

The v ariety co n sistin g of all (x , y) in R 2 with y2 - x 2 ( l - x 2 ) = 0

illu stra te s the most well behaved and e a s ily understood type of singular point, a “ double p o in t” at which two real a n a ly tic b ran ches with d istin ct tan gen ts (nam ely y = x \ fl — x 2 and

y = - x > / l —x 2 ) c ro s s each o th e r.*

T h is can a lso be seen from the param etric representation x = sin 6, 2y = sin 2d (which shows that the curve is a “ L is s a jo u s figu re” ).

12

SINGULAR POINTS OF COMPLEX H YPER SU RFA C ES

(F ig u re 3°A.

F o r a definition of the term “ b ran ch ” s e e § 3 .3 .)

E x a m p l e B.

The cubic curve y2 - x 2 (x - 1) = 0

of Fig u re 3 - B h as an iso lated point at the origin; yet th is curve is a lso irreducible. Over the field of com plex numbers, exam ples of th is type

(R E M A R K .

cannot occu r.

In fa ct a theorem of R l T T im plies th at the manifold of sim-

pie points of a com plex v ariety V is everyw here d ense in V.

Compare

VAN D E R W A ERD EN Z u r a lg e b ra is c h e G eom etrie III, or A lg e b ra is c h e G eo-

m etrie, p. 1 3 4 .)

Figu re 3 =B .

The curve

y = + xyx-1

Figu re 3 =A.

The curve

y = t x\ / 1 —x 2

Figure 3 -C . The curve x 2 = y (1 + \J 1 + y )

E X A M P L E C.

The equation y3 = x 100 can be solved for y as a 33-

tim es differentiable function of x, yet th is equation defines a v ariety V C R which h as a singular point at the origin.

The equation y3 + 2 x 2 y — x 4 = 0 ,

which is illu strated in F ig u re 3 - C , can actu a lly be solved for y a s a real

§2.

13

ELEM EN TA R Y FA C TS ABOUT ALGEBRAIC SETS

an aly tic function* of x, but th is equation a lso d efines a v ariety having a sin­ gular point at the origin. If we allow x and y to vary over the com plex numbers, then the phenom­ enon b ecom es e a s ie r to understand.

In fa c t, the com plex curve y3 = x 100

is “ knotted” near the origin (com pare §1), and the com plex curve y2 + 2 x 2 y = x 4 h as three d istin ct non-singular bran ches p a ssin g through the origin. REM ARK.

A com plex v ariety can never be a smooth manifold throughout

a neighborhood of a singular point. P ro o f: Suppose that the com plex v ariety V w ere a differentiable mani­ fold of c l a s s C 1 throughout a neighborhood U of the origin in C m. The tangent s p a c e of th is smooth manifold U D V at any sim ple point is c le a r­ ly a v ecto r sp a c e over the com plex numbers.

Since the sim ple points are

d ense (s e e the Remark above), it follow s by con tin u ity that the (re a l) tan­ gent sp a ce T z C C m of U fl V at an arbitrary point z is a ctu a lly a com­ p lex vecto r sp a ce .

(T h at is , T z = iT z .) Now rep lacing U by a sm aller

neighborhood U a n d renumbering the co ord in ates if n e c e s s a ry , the im plicit function theorem show s th at U ' H V can be con sid ered as the graph of a (^ -sm o o th mapping F from an open su b set of the (z^, . . . , z s p a ce into the (z n + 1> . . . , z m) coord in ate s p a ce .

) coordinate

The d eriv ativ e of F at

each point is com plex linear, h en ce the Cauchy-Riem ann equations are s a t­ isfied , and F is com plex an aly tic. manifold.

T his proves that U 'H V is a com plex

N ext let h (z ) be any com plex an aly tic function, defined in a

neighborhood of 0, which v an ish es on V, and let f j , . . . , f ^ be polynom ials which span the prime id eal I(V ) C C [ z v . . . , z ].

The lo c a l a n aly tic

Null-

s te lle n s a tz (s e e for exam ple, GUNNING and R O S S I, p. 9 0 ) a s s e rts that some power hs can be ex p re sse d as a linear combination a ^ where a 1, . . . , a^ are germs of a n aly tic functions.

+•••+ a^f^

P a s s in g to the larger

ring C [[z]] co n sis tin g of all formal power s e rie s at the origin, it follow s a fortiori th at hs belongs to the corresponding id eal C [[z ]] I(V ).

But th is

P ro o f: Solve for x 2 as a function x 2 = 0 for all x e Ua . Now ch oose a smooth partition of unity* {Aa i on Dg —x ° , with Support (Aa) C Ua . Then the v ecto r field

v (x ) = X

A ^O v^x)

T h at is ch o o s e sm ooth re a l-v a lu e d fu n ctio n s Xa on De — X ° so that

Aa ( x ) > 0 ,

S a Aa ( x )

= i,

and so that e a c h point of DE — X ° h as a neighborhood w ithin w hich only fin itely many of the Xa are n on -zero .

See for exam ple DE RHAM, V a r i e t e s d i f f e r e n t i a b l e s

or L A N G , D i f f e r e n t i a b l e m an ifold s.

SINGULAR POINTS OF COM PLEX H YPER SU RFAC ES

20

on Dg — x ° c le a rly h as the required p rop erties. Norm alize by settin g w (x) = v ( x ) / < 2( x - x ° ) , v (x) > and con sid er the differential equation d x /d t = w(x) . T h at is , look for smooth cu rves x = p (t) , defined s a y for a < t < /3, which s a tis fy d p (t) / dt = w (p (t)) . Given any solution p (t), note that the d erivative of the com position r(p (t)) is given by d r/d t = X

(