The monodromy of weighted homogeneous singularities

lnventiones math, 39. 199-211 (1977) I l l v e r l t j o r l e $

mathematicae (t~; by Springer-Verlag 1977

The Monodromy of Weighted Homogeneous Singularities

P. Orlik and R. Randelt*

University of Wisconsin, Madison, Wt.53706, USA The Institute for Advanced Study, School of Mathematics, Princeton, N.J. 08540, USA

Varieties defined by weighted homogeneous (or quasihomogeneous) polynomials are of particular interest because they admit actions of the multiplicative group of non-zero complex numbers [24] and because the topological properties of many general singularities may be computed from quasihomogeneous or semiquasihomogeneous normal forms [4]. In this paper we use hyperplane sections and relative monodromy to study such singularities, and we compute explicitly the local integral monodromy of a general class in any number of variables.

1. Introduction

Let w = (wo . . . . . w,) be an (n + l)-tupte of positive rational numbers. A polynomial f(z0 . . . . . z,,) is said to be a weighted homogeneous polynomial (whp hereafter)

, ,i, of f satisfies io/wo + . . . + i , /w, = 1. with weights w if each monomiat c~z'~' . . . . , It has an isolated critical point at 0c117 "+~ if g r a d f = ( ? ~ f / ~ z o , . . . , ~?f/(?z,) is zero at 0ct12 "+~, but gradflz4:0 for all z + 0 in a neighborhood of 0. Unless otherwise stated, f will indicate a whp with isolated critical point at 0. Thus if we set I~= ~ ( f ) = {zE~"+llf(z)-_t}, then V~ will be a hypersurface with an isolated singularity at 0, and 1~ will be nonsingutar for small t+0.

In Section 2 we use results of L0, [t 7, ! 8] to compute the integral monodromy for weighted homogeneous polynomials of the form

p(Zo . . . . . z , ) - ~ o +~o~l + . . . + z , l ~ , , _

This is the first computation known of this invariant for indecomposabte polynomials (see (2.2)) in more than 3 variables.

Recall how the integral monodromy is defined, Let S 2"+~ be the sphere of radius e centered at 0et[ ""~, and let 2,+t K = S ~ ca V o. Then there is the well- known Milnor fibration f/[[']: S~:-K--~ S ~, see [19]. The fibre F is diffeomorphic

* The authors were partially supported by NSF grants

200 P. Orlik and R. Randell

to V~. It is a real 2 n-dimensional open parallelizable manifold having the homotopy type of the wedge of/~ copies of S". In [20] it is shown that /~ = ~ (w~- 1), thus imposing an integrality condition on the rational weights. The diffeomorphism h: V1 ~ VI by which the ends of V~ x [0, 1] are identified to form S ~ - K is called the characteristic map of the bundle. It is given by [19] as follows. Write w~= u~/rz with (u i, vi) = 1, and let d = l.c.m. (u0, ..., u,). Define d i = d / w i and ~ =exp(2~ i/d). Then

(1.1) h(z o . . . . . z,) = (~a~ Zo . . . . . ~a" z,).

The induced map in homology or cohomology, say h.A: H.(V1; A ) - - . H . ( V 1 ; A), is called the local monodromy with coefficients in the ring A. It is a fundamental invariant of the singularity.

By Milnor-Orlik [20] or Brieskorn [6] one may obtain the eigenvalues of h. r but for topological reasons (see [19, 22]), the more subtle invariant h~ is quite important. Explicit computations for the Pham-Brieskorn polynomials ~z~' have been given by Pham [25], Brieskorn [5], Hirzebruch-Mayer [15], Gabrielov [8], and Hefez-Lazzeri [14]. Computations for other polynomials of essentially two or three variables have been given by A'Campo [2, 3], Gabrielov [9, 10], and Gusein-Zade [11, 12].

Our techniques differ from the above papers in that we use hyperplane sections of VI, Morse theory, and the concept of relative monodromy due to Thom and developed by L6. Polynomials of type p above are particularly amen- able to treatment by these methods, although a similar approach may work for other types. The exact result is given in Theorem (2.11).

In Section 3 we indicate some difficulties which arise in the attempt to work with other classes. We conclude with a conjecture for the Seifert matrix for the singularity of type p.

It is a pleasure to thank L6 Dfing Tr~tng for useful conversations.

2. Monodromy

In this section we use results of L~ [17, 18] and Harem and L~ [13] to compute the integral monodromy of a basic type of whp. We first note some results which reduce the number of polynomial types that must be considered. Then the computations below for the class

(2.1) p(z o . . . . . z , ) = ~ ~ ", n > l ;

together with well-known results for polynomials in fewer than 3 variables allow one to compute the integral monodromy for many new examples.

(2.2) Definition. The whp f is decomposable if after suitable renumbering of the variables f ( z o . . . . , z , ) = f t ( z o . . . . . zk)+ f z ( z~+l , . . . , z,), where ,~: C k+l ~ is a whp of type (w o . . . . , wk) with isolated critical point and J2: ~" k__,r is a whp of type (wk+ 1 . . . . . %) with isolated critical point. Call f indecomposable if it is not decomposable.

Weighted Homogeneous Singularities 201

(2.3) Remark. A result of Oka [21] says that if f is decomposable, V~(.f) is homotopy equivalent to the reduced join of V~(.~) and V~(J;). The monodromy of f is thus the tensor product of the monodromies of .1~ and f2. Therefore it suffices to consider indecomposable polynomials, such as p of (2.1).

We now give some notation, ideas, and results from [13, 17, 18] which will be used in this section. Let L denote both a non-zero linear form L(z0,. . . , z,) = a o Z o + . . . + a , z, and the hyperplane {L(z 0 . . . . ,z,)=0} defined by L. Define QSL: {Fn + 1 ~ 2 by qSL(Z)=(f(z), L(z)). Let CL be the critical locus of qS~.

Definitions [17]. Let F L be the union of irreducible components of C L which are not contained in {f=0}. FL is called the polar curve of.]" relative to the direction defined by L.

Let AL=qS(FL) and call AL the Cerf diagram o f f relative to the direction defined by L.

One generally considers functions f : U ~ ~ which are analytic in some neighborhood U of 0 in C "+I. The same definitions of course make sense, and there is the following result.

Theorem [3, Section 2]. There is an open dense subset (2 of the space of non-zero linear Jbrms so that .lor L~F2 we have:

(i) F L is an analytic curve at O.

(ii) qbL[rL is a .finite analytic map.

(iii) There is an open neighborhood V of O, so that for every point x~ (F L r V ) - {0} the hypersurJace { f = f (x)} r { L = L(x) } c { L = L(x)} has a nondegenerate singularity.

Thus (iii) means that for xe(~c~ V)-{0}, f[r=L(~) looks like f(Y0 . . . . . Y,-0 = ~ y2 for a local system of coordinates (To . . . . . Y,-1) on L = L(x).

Definition. A partition {V/} of an analytic variety V c G "+1 is a stratification (with strata Vii) if the following conditions are satisfied.

(i) Each V/is a connected subvariety of V. (ii) For each i, the closure of V/, cl(V/), and cl(V/)-V/ are analytic varieties. (iii) Every zeG "+1 is in the closure of at most finitely many V~.

(iv) V/c~cl(Vj)+O implies Vic--cl(Vj).

We will wish to stratify G~(.t')= {z~C'+ ~lf(z)=0}.

Definition [13, Def. (1.1.3)]. V/is a good stratification of Vo (f) if for every sequence {x,} in G'" z_ Vo(f) such that

(i) x . + x~ Vo(f), (ii) the tangent hyperplane T(x , , V1.(~,,)(.f)) to Vj.(x,)(.f) at x, is defined and

(iii) T(x., V:(~.)(f))~ T the hyperplane T contains the tangent plane at x to the stratum which contains x.

See [13, 1.1.4] for a stratification which is not good. The Cerf diagram A L is a plane analytic curve. If A L is square-free (every

irreducible component has multiplicity one), then L6 observes in [18, 2.2.6]


that at every point xeFL--{0}, the hypersurface f = f ( x ) , L=L(x) has a nondegenerate singularity.

For the polynomials we will examine later, L will be in (2 if it has these two properties:

(2.4) L is transverse to the strata of a good stratification, and

(2.5) AL is square-free.

Any linear form L such that (2.4) and (2.5) are satisfied will be called generic. For generic L, the results of L6 in [17] and [18] apply, so that Vl(f) is formed by adding n-cells to V1(f)c~L.

We note the following basic lemma.

(2.6) Lemma. Suppose f: ~ " + 1 ~ C has an isolated singularity at the origin. I f either

(i) L is generic, or

(ii) f is weighted homogeneous and L(z o . . . . ,z , )=cizi: then the jbllowing diagram commutes and has exact rows (where h* is the monodromy).

0 ,FI" ' ( V I ~ , L ) ~ H " ( V 1 , V~c~L)-~H"(VO---*O

"r h* "r h* T h* 0----*/q" I ( V ~ n L ) ~ H " ( V 1 , VI~L)--~H"(V~)-~ '0 .

Proof. The rows are portions of the long exact cohomology sequence of(V 1, V 1 c~ L). The right-hand 0 follows from [19] whether or not f has an isolated singularity, while the left-hand 0 ~ / ) " 1(I/1) follows from [19] because f has an isolated singularity.

In Case(i), R.Thom (see [17, 3.6]) has shown that one may construct a relative monodromy making the diagram commute.

In Case (ii), the map h given in (l.1) is a map of the pair (I/1, 1/1 c~ L) into itself, and the result is trivial.

We are now ready to study the polynomial p of (2.l), which clearly has its only singularity at 0 (or is non-singular if ao = 1). Let L(zo . . . . . z , )=z, and take ~ = {Vo(p)-{0}, {0}} as a partition of Vo(p).

(2.7) Lemma. ~ is a good stratification of Vo(p) and L is transverse to the strata oJ'~//~ (except {0}).

Proof ~ is clearly a good stratification of Vo (p).

The gradient of L is (0, ..., 0, 1). Suppose at some z'e Vo(p), grad p=(0 . . . . . (), (+0 . With pi=Sp/Szi we have

po=aoz~ ~ ~ + z~' = 0 ,

a~ - - 0 ~ P l ~"~- a l Zo Z~l' -1 jr_ z2_ -

p ._ l=a , , l z , _2Z~nn~ 1 - { - ~ n n = 0 ,

p,=a, z,_l 4 " - 1 = ( +0.

W e i g h t e d H o m o g e n e o u s S i n g u l a r i t i e s 203

With z'=(z'o,z'~ . . . . . z',), z{~=0 implies z'~ . . . . . z ' ,=0, a contradiction. Also, n

~04=0 implies zi4=O, i = 1 . . . . . n. The identity p(z o . . . . . z,,) = - ~ (zi/wi)Pi and the i O

prescribed values for pi(z') then imply p(z ' )=(z ' , /w , )p , , ( z ' )+O, a contradiction. Thus the tangent plane to Vo(p) is not parallel to L at any point of Vo(p)-{0}.

(2.8) L e m m a . FL={ZEC"+I[po(Z) . . . . . p,, I(Z)=0}.

Proqfi The differential of 4,L is

0 . . .1

so CL is the set given above. But no irreducible componen t of CL is contained in Vo(p), for if this were the case, p ( z ) = 0 would imply p , ( z ) = 0 along this component (as in (2.7)), so that z =0 . Thus FL= CL.

We now define rk = a 0 a l . . . Ok, k = 0 . . . . . l l ; F 1 = l ; I l k ~ - F k - - F k 1 -~- Fk 2 - -

�9 " + ( - 1 ) k+ l r l, and %=g.c .d .0; , , l* , 1). An easy calculation using [20] shows t h a t / ~ , = r a n k ( H , ( V I (p)), the Milnor number of the singularity of p at 0.

Recall that 4,L = (P, L) and the Cerf d iagram is 4' (F L) = A L C 1172. We parametr ize C 2 by (x, y).

(2.9) Lemma. AL is a square-fi 'ee curre with Pu iseux expansion

X = t r''

Yc; , , = 9~cr,, t u " ~.

(That is, AL has ~, branches at 0.) We may take : x j=exp (2~ i j / r ; , ) , j = 1 . . . . . or,.

P r o q f ~(FL)=(p(FL) , Z,,). But F L is given by Po . . . . . p,, 1=0, and p (z )= ~, (zi/'wi)p~. Thus for zc/-}~, i o

4,(z)=((z. /w.) (a. z~ , z."" *), z.).

By using Po . . . . . P, 1 = 0 one may solve for z, ~ in terms of z,. Substi tuting the result of this gives

4'(z) = (C z~. ' ' ' '~ ', z . ) .

The Puiseux expansion is obta ined by taking t=z~, 7~ ..... . Clearly AL is square-free, so by (2.4) and (2.5) L is generic.

(2.10) Theorem. H, (1/1 (p), I/1 (p)c~ L) is ji 'ee with basis bl . . . . . b,, . The monodromy action induced by the characteris t ic map h is h . ( b i) = b i+ ~ . . . . . . H"(I/1 (p ), I/l (p ) ~ L) has dual basis b* . . . . . b* and h * ( b * ) = b i ~ . . . . .

Proo f This follows immediate ly from (2.7), (2.9), and [18, Section 3.3].

We now use (2.6) and (2.10) to calculate the m o n o d r o m y action on H"(V~(p)).


Recall that we always use integral coefficients. Define integers c~ o, cq . . . . , ~it,, by

cS,(t)= ]~I ( t r ' - l ) ( l)"-~=O~u, tu"+'"+~l t+~o" i = -1

(2.11) Theorem. H"(V1(p)) is the abelian group with generators gl . . . . ,gr,, and

relations ~o gi + o~1 gi4 1 -{- " ' " ~ - (Xit. g i + It . : O, i = 1, 2 . . . . , I~, _ 1. The monodromy action h* is induced by h* (gi)= gi , ,_,-

Note that r , - # , 1 = / t , , and the order of h* is r , /a , .

P r o o f We use induction on n. The case n = 1 may be proved in two different but instructive ways.

First, notice that if n = l, 1/1(p) ~ {z~ ~ 2 [ ~ ~ ~- Z 0 Z~' : l }. Here is a deformation retraction of ~(p) to a 1-complex ~2t(p): Notice that for (Zo,Zl)eVl(p) , % + 0 . Thus one may deform radially to I zol = 1 in the z 0 coordinate while continuously varying z I to remain in VI(p). Then f~I(P) consists of a o vertices, corresponding to z~ ~ 1, zl =0, together with al edges between each pair of adjacent vertices. See Figure 2.1. It is now easy to compute H ~ (Vl(p)) and h* explicitly.

�9 (-10) ~ ~ / ~ (10) 321 (p)

Fig . 2.1

It would be informative to have such an explicit retraction for n > l, but in that case the situation is considerably more complicated.

We now assume that /~"-~(V~c~L) is as described in the statement of the theorem. F rom (2.6) we have the short exact sequence

0 ---~/~rn -- 1( V1 ("1 g ) ~, H" (V~, V~ c', L) ~ , H" (V~) ,0

where the middle term is described by (2.10). We will show that the action of h* limits 6 enough so that Image 6 can be determined.

Using reduced cohomology and knowledge of the m o n o d r o m y of the polynomial z~ ~ one can in this way obtain the case n - -1 which starts the induction. This is the second proof mentioned earlier for the case n = 1.

First suppose that a , ,=g .c .d . ( r , , l~ ,_0= 1. Since r k = l & + t & ~, this implies that g.c.d.(rk, # k _ 0 = 1, k = 0 . . . . . n. H"-~(V~ m L) is given inductively as a quotient C/C' , where C is the free abelian group with generators q , . . . , % .... and h * ( c l ) = c i , ,_~. Let {b l , . . . , b , , , } generate B = H " ( V 1 , VlC~L), as in (2.10). Then 5 is induced by a map 5': C - * B such that h $ 6 ' = 6 ' h * . Let M = ( m u ) be the (r, x r,_ 0 matrix of 6'. M is determined by its first column as follows: Since g.c.d.(r, 1,/~,-2) = 1, we may find ~k satisfying 1--y k P , - 2 - = k ( m o d ~;, ~1)-Then

5' (Ck) = 6' ((h~) ~ (cO) = (h~) 7k 6' (ca) rn rn rn

= (h*) '~ Z m, , b, = Z rn,l(h~) '~ (b~) = Z m,, b , . ~ u . _ , . i = 1 i = 1 i = 1

W e i g h t e d H o m o g e n e o u s S i n g u l a r i t i e s 205

The last sum may be re-indexed to yield

rn

(*) 6'(Ck)= ~ m i + ~ , . . . . 1 bi. i - 1

We note two impor tan t consequences of (*).

(i) Lett ing k = 1 we find that the coefficient of bt in 6 ' (q ) is ml_ ~ ;,,~ ~.1, where 7i is any integer congruent to 0 modulo r n 1. Since cr,=g.c.d.(r, , /~, ~)= 1, it follows that ml, ~= m~, ,+1, ~ . . . . . . m ( , _ 1)( ..... )+1, 1' Thus if we set V M = ( m l l , rrt21 , . . . . m~, ,.0, the first column vector of M is (VM, . . . , VM)', with I'~4 repeated a, times. Every other column of M arises in the same fashion from some cyclic permuta t ion of Vv.

(ii) The first row of M is V M also. Since m~2 is the coefficient of bl in 6 ' (C2) ,

the equat ion (*) yields m12 = ml + ~ , . . . . t, where 1 - I'2 .u,, 2 ~ 2 (mod J; _ 1), We set x - 1 + 72 P, t( m o d r, 1). Rear rang ing these two congruences and using the fact that /~,_2+/*, ~=r,, ~, we conclude that x - 2 , so that m~2=m2t . Similarly m i k = m k l for any k = l . . . . , r , 1.

The image of 6' is generated by the columns of M. (That is, these columns give the coefficients with respect to the basis {bt . . . . . br,,}.) Kernel 6 ' = C' is known inductively. With respect to the basis c~ . . . . . %, ,, kernel 6' is generated by

~toCi~-O~iCi+l-{-"'~-O{~*n 1Ci+t*n I ' i = 1 . . . . . ] I n 2,

where

n 1

~ ( t ~ , _ l ) ~ x),, * , = ~ , , t " " ' + - . . + ~ ' l t + c ~ ; . 1

Let ~ ' = ( c ~ , c( 0, 0, 0) be a vector of length J;, 1 and let ='(s) be the vector obta ined by shifting ='s units to the right. Then the fact that kernel 6' is as given above yields the equat ions

VM'~ ' ( s )=O, S = 0 . . . . . P,, 2--1"

We are now ready to compare image 6' with B', where B' is the subgroup of B generated by the realat ions given in the s ta tement of the theorem.

Setting G equal to either image 6' or B', we have

a) G is free of rank p, 1.

b) BIG is free.

c) h * ( G ) = G and (h*) r" ' ( g ) = g , for all g c G .

Finally, let ~(s) (respectively UM) be the vector obta ined by juxtapos ing a, copies of e(s) (respectively VM). Then clearly

UAI . = ( s ) = 0 , s = 0 , 1 . . . . . p,, 2 - 1 .

Thus we have

d) a ( s ) . g = 0 , f o r a l l s = 0 , 1 . . . . . //n 2 - 1 " a n d f o r a l l g ~ G .


But using (c) we see that B' consists of precisely those elements of B satisfying d), so that image 6 ' c B ' . Thus a) and b) imply image 6 ' = B ' . This completes the induction step if a , = 1.

The case a , > 1 may be reduced to the preceding, as follows. Let rl = exp (2 rr i/an). Then letting p = e x p ( 2 n i / d a , ) and l , ~ = { z e ~ "~ l l p ( z ) = r / } , w e have the map f : V l - . V,; given by f(z 0 . . . . . z,) = (pdo Zo . . . . . pa, Z,). There is a commutat ive diagram

0 ---~/4"-- l (VI c~ L) ~ H" (V~, V~ c~ L)

T f* Tf* O----~ ffI" - I ( V . ~ L ) ~ H"(V. , I~r L)

as in (2.6, ii). By a slight elaborat ion of the technique used to prove (2.10), it may be shown that there is a basis {b~ ~ . . . . . b~r'], )} of H(I~, V, Ic~L ) such that f * (b~ ")) = b: _ ,,, _. ~/~,.

Then the inductive step follows from arguments like those used in the case ~r,, = l. This completes the proof.

(2.12) Corollary. For a polynomial ~?f type p sati@'4ng g.c.d.(J; , , / t , ,_0=l, the monodromy matrix H is equivalent to the companion matr ix 0"[ the polynomial (~,(t). Fur thermore O,(t)= A (t), the characteristic polynomial o.I" H.

Note that 8( t )=A( t ) also follows from [20]. In the tables of normal forms of singularities given by Arnold (see [4], for

example) there are several polynomials of form p, such as D k and Ev, and there are many which have topological properties computable from a polynomial of form p, such as K I 3 , Z H , Z13 , ~]3, etc. Dk, for k odd, and Ev satisfy the hy- potheses of (2.12), so we illustrate (2.11) with D2k.2.

z 2k+~ z 2, k>_l. (2,13) Examples . D2k+2: + z 0 _

Here w o = 2 k + l , w l = ( 2 k + l ) / k , l~o=2k, / ~ = 2 k + 2 , r o = 2 k + l , q = 2 ( 2 k + l ) , and g.c.d. (r~, #o) = 2. Then ~ (t) = (t '*k + z _ 1) (t - l ) / (&k . ~ _ 1 ) = t z k+ 2 _ tz k+ 1 + t -- 1.

H 1 (1,] (p)) has generators g~ . . . . . g4k+ 2 and relations -- gi + gi+ ~ - gi + 2 k + 1 -I- gi+ 2k~ 2 =0 , i = l , . . . , 2 k , h*(gi)=gi_2k. Choosing g2k+z,g2k+~ . . . . . gl as basis we have the ( 2 k + 2 ) x ( 2 k + 2 ) m o n o d r o m y matrix (in cohomology, acting from the left):

-0 0 t t . . . . . . t- 0 0 - 1 0 . . . . . . 0

0 - 1 . . . . . . 0 0 0 . . . . . .

0 0 0 1 1

H * =

. . . . . . . m 0

0 . . . . . . 0 - 1 1 . . . . . . 1 1

Using the fact that z 2 has m o n o d r o m y ( - 1) we may then obtain the m o n o d r o m y of D2k+ 2(n): zZk+1+ Z o Z~ + Z~ + ... + Z,Z,, n > 2, as (-- 1) ' 1 H .


We then find that for D 2 k . 2 ( 2 n + l ) , 1 2 k . 2 - H ~ d i a g ( 1 2 k , O , O ) and for D e k + 2 (2 VI), [ 2 k + 2 -[- H ~ diag (I 2 k, 2, 2). Thus for the associated K = V o (p) c~ S~, we have H2n(K4"+1; ~-)~@~, and Hen I(K 4n 1; 7])=712@g2. This agrees with the results of [26].

With choice of basis similar to that above we find that the m o n o d r o m y of D2k+1 is a matr ix of type H above of size ( 2 k + l ) x ( 2 k + l ) . The matrix of E7 with this type of basis is the 7 x 7 matrix.

-0 0 0 0

1 0

H * = 0 1

0 0

0 0

0 0

0 0 0 1 1-

0 0 0 - 1 0

0 0 0 0 - 1

0 0 0 1 l

1 0 0 - 1 0

0 1 0 0 - l

0 0 1 1 1

As a final example, the m o n o d r o m y of K13 ~3 .5 28, , "~o + Zo -~ 1 + a may be computed as above from the case a = 0 , when the polynomial has form p. Al though the various values of ae([ ' give non-equivalent singularities, the monodromies are the same.

3. Further Observations

It is natural to ask whether the techniques of the preceding section will work for other classes of whp. In this section we show that partial results may often be obtained, part icularly concerning the relative m o n o d r o m y , even when (2.4) or (2.5) fail for the otherwise convenient choice of L.

(3.1) We consider the family of singularities defined by q(zo . . . . . z,)=z'~,"z I + m a n _ _ . . . "'" + ~, zo, n >_ 1. Define L(zo, z,,) = z,.

As before, let Cc be the singular set of q~L(z)={q(z), L(z)) and let F L be the union of those irreducible componen t s of CL not contained in Vo (q). CL is clearly given by {qo=ql . . . . . q, 1=0} (where qi={?q/?zi), but FL4:C L, because the branch {Zo= q . . . . . ~, l = z , , = 0 } of CL is contained in Vo(q).

Let s , = a o ... a, + ( - l )" and t , = a o . . . a . 1 - c q . . . a , l + ' - - + ( - 1 ) " . It follows from [20] t h a t / ~ , = a o ... a , , = s , + ( - I) "+1.

(3.2) Lemma. The Ce~f diagram AL is square-free and has Puiseux expansion

Yl = o~1 "ct" X ~. TY s~

where the number o f branches equals s = (s,,, t,,).

Proofi This calculat ion is rather lengthy. We illustrate the case n = 2 : qS(CL) --(z2 q2/w 2, zz )= (z : zT '+ z'~: zo, z2). (For simplicity we will suppress coefficients.)


If Zo=0 in CL, then the equation qo=Z~o ~' 1zl+z~2=0 implies that z2=0, and this branch of C L is not in F L. Similarly z2=0 and q, =0 imply Zo=0. Thus

_ _ a ~ - I) Zo=t=0 and z24=0 in 1},, except at 0. Therefore qo=0 implies Z I - - Z 2 " z o ( a ~ . Then q, =0 implies "~o'a') '-,"t a, 2-atla, 1 ) ~.~' ZO (al - l ) ( a o - 1)__.0, or

ZaoOat--at+ I ~_ Zt~la~.--az+ I

Write this as z~ ~ = z~ ~, or z 0 = z~ ~.~ '. Thus

ZI ~ Zr~ Z2-(ao .-1)~/exo

Substituting this into qS(f}) and simplifying gives

(FL) = (z'~ ::t:, z2) = (r'~, with ~t ?,) *, *r - - ~l / t?

It is easy to see that L is not transverse to the strata of the obvious stratification. Take n = 2 as example, Then on { z o = z z = O } c V o ( q ) we have q 0 = q t = 0 , q2 4=0. There is still the following result analogous to (2.10), however. Let

D . = {z~ ~i (q)l l z, l < d .

(3.3) Lemma. For all g>0 su[]~cientIy small, Hn(I,~(q),n~,) is isomorphic to the fi'ee abelian group with generators g1 . . . . . gs., and h*(g~)~g~._t, ,, Further, H i (1/1 (q), D~) ~ O, Jbr j q= n.

Proof Choose r > 0 so that no complex number z, with O<lz l<r , is a critical value of Llv,(q). For 0 < e < r , the lemma then follows from [17] and [18].

Now the exact sequence of the pair (I/1 (q), O 0 yields

O-o H " - ~ ( D ) - , It"(V~, D).~H"(VO-~H"(D)--~O

and since we know tt = rank H" (~) we easily compute rank H" (D) = rank H" 1 (D) + t - l ) "-1. If one could show rank H ' ( D ) = [ l + ( - l)" ~]/2, then the integral monodromy of H"(VI) would follow.

(3A.l) Conjecture. I f n is even, I t ' ( l~ (q)) is a .free abelian group with generators gl . . . . , g~.~ and a single relation g~ + . . . + g.~, = O. h* (gi) = gi t. induces the monodromy.

(3.4.2) I f n is odd, H"(~(q) ) is a J?ee abeIian group with basis gl,--- , g~,, c. The monodromy is git~en by h* (&) = gl -~,, h* (c) = c.

These conjectures imply that the torsion subgroup of H , _ I ( K ; Z ) is 0 if n is odd and ~ : Z,~, d = s,/g.c.d.(s,, t,), ifn is given. Thus Orlik's conjectured algorithm [22] for computing the homology torsion for weighted homogeneous polynomials would hold for type q.

(3.5) Remark. As for the type p polynomial it would be useful to have an explicit retraction of I/1(q) to an n-complex. For n= 1 this is not difficult, but the monodromy calculation is already fairly complicated.


a �9 ~ ,rbl .~b2 (3.6) Next consider the whp g(zo,z l ,z2)=Z~o"+ZoYl '+ZoZ2-T~l ~2, where a l > b l > l , b 2 > = l . For example, one might take % = 7 , a l = 1 5 , a2=10, b l=10, b2=5. Then Wo=ao, w l = a o a l / ( a o - 1 ) , w2=aoa2 / (ao -1 ) , t ,=a2oala2/(ao-1) - a o a l - a o a 2 + a o - 1, and bl/wl +ba/w2= 1.

It is natural to try ao(zo,Zl,Z2)----z 2. It turns out that L o is transverse to the strata of the usual good stratification, but L o fails to be generic. Here's how to prove the latter statement: Ignoring coefficients, we have

go = :o" 1 + ~l + : s

g l = Z o Z ~ ' l + z ~ ' I72: ,

a~ 1 bi b~ 1 ~ 2 ~ ZO Z2- q- Z1 Z2-

al bl CLo={go=gl=0}. But g l = 0 implies z b~ 1=0 O r , o z l +z~2=0. In the latter case computations much like those of (2.9) show that the corresponding branch of the Cerf diagram is (z~2/t~,z2), where c ~ = a l b 2 + a o a l a 2 - a o b l a 2 + b l a 2 and f i = a l + ( a l - b l ) ( a o - I ) . If z~' 1=0 on FL,,, then Zo=#0 and z2~0, because g(ZO,Zl, Z2)=g2z2/w2=t=O. Thus { :o~ gives a branch of FLo, and the corresponding branch of the Cerf diagram is ~[7-2 ,-~"~ 1), z2~,~ with multiplicity b l - 1 . Thus the Cerf diagram is not square-free, and so by (2.5) L o is not generic.

In fact, there are points z where the function g restricted to Lo(z)=~. has an isolated singularity with Milnor number b~ - 1. A small perturbation of L o gives a generic line L, and this degenerate singularity then splits into b~-1 Morse singularities.

For a general whp f we thus see that it may be impossible to choose a generic line L so that I/i (f)c~ L is defined by a whp.

To return to g(ZO,Zl,Z2), let us try L(Zo,ZI,Zz)=ZI--z2 . It turns out that L is a suitable generic line, and Vl(g)c~ L ~ V~ (~), with

(Zo, z 0 = z~ ~ + Zo z~ ~ + Zo z~ ~ + z~' + "~

This is a semiquasihomogeneous singularity [4], and the monodromy for V~(g) is equivalent to the monodromy of z"o~ zT' (assuming a~ <a2). But now the Cerf diagram A, becomes much more difficult to analyze by the techniques we used in Section 2. Even if this analysis can be done, there remains the problem of finding the image of 6.

The study of fi is the study of how n-cells are attached to V~(f)n L to form I:1 (f). As mentioned in [17, (3.5)] this is an interesting general question.

(3.7) Finally, we note that a major advantage of the techniques of L6 is that they work for non-isolated singularities as well. The extra difficulties that arise are basically that the left-hand zero in (2.6) disappears and that one must be more careful in choosing the good stratification. These difficulties are often not major; for example, the relative monodromy of

b n - 1 a n ~"~ + zbo" z~' + "" + z, l z, ~ 0

may readily be computed using the methods of this paper.


4. Other lnvariants

-0{ 0

( _ 1),~, ll,,z L(p)=

Any isolated hypersurface singularity has two other integral invariants, the Seifert form L and intersection form S on H,(VI(f),TZ,). In [7] it is shown that these invariants are related by the matrix equat ions

H = ( _ I ) , + 1 L ~ I L t,

S = L + ( - 1 ) " L t

where H is the matrix of the integral homology m o n o d r o m y . Fur thermore , if one takes a basis for H,(V1(f)) which is distinguished (in the sense of Gabr ie lov [8], see also [6, Appendix] , [7, 16]), Durfee shows in [7] that L is upper t r iangular with ( - 1) "l"+a)/2 on the diagonal. In this case the first of the above equat ions can be solved uniquely for L, and the second yields S.

Example (2.13) is thus interesting, since for Dk and E7 the solution for L can be obtained. More generally, choosing the basis g , . , g , , 1 . . . . . gl for H" (VI (p)), one obtains the following matr ix L(p). Since we had to assume that the basis was distinguished, we state this answer for L as a conjecture. Let 0{o, 0{1 . . . . . 0{,. be defined as in (2. l l),

"" Then the Seifert (4.1) Conjecture. Let p(z)=z~~ zl z"22+...+z, 1 z, . matrix L(p) of p is given by

0~1 0{2 0{ # n i

0{0 0{1 CZ2 0~l*. 2

0{0 0{1

0{0 0{1 0{2

0{0 0{1

~0

References

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Received May 31, 1976; Revised Version October 27, 1076

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The monodromy of weighted homogeneous singularities