absolutely isolated singularities of holomorphic vector fields

Invent. math. 98, 351-369 (1989) I~l ve~ tio~es mathematicae ~3 Springer-Verlag 1989

Absolutely isolated singularities of holomorphic vector fields

C. C a m a c h o ~, F. C a n o 2, a nd P. Sad l lnstituto de Matem~.tica Pura e Aplicada, 22460 Rio de Janelro, Rj, Brazil

-~ Universidad de Valladolid, Valladolid, Spain

w Introduction

Let X be a n-dimensional complex manifold and ~ a singular analytic foliation by lines on X. By this we mean that at any point P e X the foliation is generated by the vector field

D = a~--- , a~er g.c.d.(a 1 . . . . . a.) = I . (1) i= t Oxi

The multiplicity v(@; P) of ~ at the point P e X is the minimum of the multiplicities v~,(a i) (i.e., the order of the zero of ai at P). We shall say that P is a singular point of

iff v(@; P) > 1. The set of such points will be called Sing(.CJ). A singular point P ~ X is called irreducible if the linear part of ~ at P e X has at least one nonzero eigenvalue. The resolution problem for an isolated singularity P o X of @ is to prove the existence of a proper holomorphic map 4~: M --, X of a complex manifold M of dimension n such that (i) 4, 1 (p) = Ui N_ 1 Ki is a union of codimension one compact complex submanifolds with normal crossings (ii). The pull back foliation 4,*(@ I x\{e}) extends to a singular foliation of M with singular set of codimension > 2 and such that all singular points are irreducible.

A first step towards the solution of this problem is the case when the codimension of the singular set of the lifted foliation is n. We show that in this case the foliation @ can be solved by a map 4~ which is the composit ion of a finite number of blow up's. We proceed now to give the precise statements.

Let n: X ( 1 ) - , X be a quadratic blowing-up with center at a singular point P e Sing(Y). Then there exists a unique way of extending n* (~ lx ' . {e}) to a singular analytic foliation @(1) of X(1) with singular set of codimension > 2; we say that 9(1) is the strict transform o f ~ by n. Let E c X(I ) be the exceptional divisor ofn . The blowing-up n is called non-dicritical for ~ when E is invariant for C~(l), i.e., it is the union of leaves and singularities of 9(1).

Assume @ has only one singular point P e X . We say that P e X is an absolutely isolated singularity of ~ iff the following property is verified:

352 C. Camacho, F. Cano and P. Sad

(A.I.S.). Let us denote P = P(0), X = X(0), @ = @(0), Consider an arbitrary sequence of quadratic blowing-up's

~(1) re(N)

x ( o ) , x o ) , . . . . X ( N )

where the center of each re(i) is a point P ( i - 1 ) e S i n g ( ~ ( i - 1)); here ~ ( j ) denotes the strict transform of c~( j_ 1) (1 < i,j < N).

Then re(N) is non dicritical for C~(N) and # Sing(~(N)) < ~ . This paper is devoted to the proof of the following desingularization result:

Theorem A. Assume P ~ X is an absolutely isolated singularity of c2. Denote P=P(0) , X=X(0) , ~ = ~ ( 0 ) . Then there exists a fni te sequence of quadratic blowing-ups

~(I) ~(N)

x(o), X ( l ) , . . . . , X ( N )

satisfvin 9 the Jbllowin 9 properties:

1) The center of each ~z(i) is a point P ( i - 1 ) e S i n g ( ~ ( i - 1)), where c~(j) is the strict transform of ~ ( j - 1) (1 < i,j < N).

2) ~ ( N ) is generated locally at each singular point by a vector field (1) whose linear part has at least one nonzero eigenvalue.

The main tool for proving this theorem is the use of a formula relating the multiplicity of the original singularity to the Milnor numbers of the singularities which appear after a sequence of non dicritical blowing-up's. Recall that if a singular analytic foliation ~ is given locally at PeS in g (~ ) by the vector field (1), then the Milnor number is defined as

# (~ ; P) = dim c (gx, p a i ' { ~ x , p . i

The formula we just mentioned generalizes the one used in [1] for the two- dimensional case.

Like in [4] we can describe "simple singularities" for an n-dimensional vector field which are persistent under blowing-up and which generalize those obtained by Seidenberg [5] in the two-dimensional case. Our second desingularization result is the following.

Theorem B. We can extend the sequence in Theorem A above

~(1) re(2) n ( M )

X(O), X(1), . . . , X(M), M=>N

in such a way that ~C~( M) has only simple singularities (which are, of course, absolutely isolated singularities).

The isolated simple singularities which are absolutely isolated are completely determined by their linear part. This provides a first family of examples of absolutely isolated singularities. Actually, being an absolutely isolated singularity

Absolutely isolated singularities of holomorphic vector fields 353

is a quite generic property among the singularities having fixed multiplicity. We end this work by exhibiting a family of absolutely isolated singularities which possess a more complicated tree of desingularization.

The above desingularization result when n = 2 is due to Seidenberg [5]. A strategy for the general three-dimensional case was developed by Cano [3]; however a definite result is still missing. It must be mentioned that a complete resolution theorem for singularities of algebraic varieties, in any dimensions, was proved by H. Hi ronaka in the 60's.

w 1. The formulae

We start by recalling some elementary facts about quadratic blowing-up's. Let (xl . . . . . x,) be affine local coordinates of a ne ighborhood U of P e X , such that P = ( 0 , . . . ,0). Now, E (1 )= 7r(1)-1(0) is the exceptional divisor and X(1) is covered by X(1) \E and U' = 7z(l)- I(U). For each j = 1 . . . . . n let Uj = (xj :4= 0) and U ) = re( l ) - l (Ui) . In U) we introduce coordinates (x'l . . . . . x',) and rr(1) has the following expression:

1r(1)(x' 1 . . . . . X'n) = (X 1 . . . . . Xn) t t t and Xl = XaXl, �9 �9 �9 x i = x i , . . . , x, = xjx, .

In this chart E(1) c~ U) is represented by x) = 0. The vector field D which generates 9 is expressed in U as

and

3 D = a i - - , a ieCx,e . (1)

i = 1 ~ X i

~ ai--x)aj (3 lr(1)*(D) = aj-;-w +

Ox~ z_, x) ~ ' ( . X i I = l i , j

where ai = a/on(l) for i = 1 . . . . . n. If r = v (9 ; P) is the algebraic multiplicity of D at P e X , we find that 9 (1) is

given by the vector field in Uj

1 (3 1 " 0 - . + ~ Y. ( a l - x)aj)-~x. (2) D(1)j (x'ff -1 a ' tx ) (xj) i= 1

in the nondicritical case.

Theorem 1. Let X be an n-dimensional complex manifold and 9 a complex foliation of dimension one with isolated singularity P e S i n g ( 9 ) . Let z~(l): X(1)-+ X be the quadratic blowing-up with center P. Suppose that z~(1) is non dicritical and that the strict transform 9(1) of 9 has only isolated singularities. Let r = v (9 ; P), then,

: - 1 + : - 2 + . . . + 1 = ~ /a(~(1)le{l~;Q). (3) Q e E ( 1 )


Proof. Let X 1 . . . . . X, be homogeneous coordinates for P" - 1 (C) g E(1) and let Ai be the jet of order r at P ofai. Then A~ is a homogeneous polynomial of degree i" or it is identically zero.

Since =(1) is nondicritical, an easy computa t ion shows that 9(I)IE(ll is gener-

Xi ated in the affine chart (5'1 . . . . . 1 . . . . . Y,,), Yi = ~Yi' i = 1 . . . . . n , j fixed, by

D(1)j[E(1 , = ~ ( A i ( y 1 . . . . . 1 . . . . . y , ) - - Y i A a ( Y 1 . . . . . 1 . . . . . y,))~cl . (4) t = l v y l

From this it is easy to see that in the intersection of two charts corresponding to the indices j and k one has

1 D ( 1 ) j I E ( t ) - - ( y j ) r - 1 D ( l ) k [E(1) (5)

in the variables y. Thus (A 1 . . . . . A,) defines a section of the bundle L r- 1 | TP" - 1 (C) where L is

the line bundle associated to a linear divisor and T P " - 1 (C) is the tangent bundle of p . - 1 (C).

This section has only isolated points in its zero locus and the sum of its multiplicitics which equals the right hand member of (3) must be equal to the top Chern class c._ 1 ( U - 1 @ TP=(C)). In order to compute this number it is enough to

n r consider the vector field D = ~i= 1 x~ . We have ~X i

0 D(1)jle.) = ( 3 " - Y ' ) c y ,

all singularities have Milnor number one and so # S i n g ( ~ ( 1 ) ) = r "-1 + r " -2 + . . . + 1 . ~

Theorem 2. Let us take the same assumpt ions and notat ions o f the above theorem.

Then we have that

# ( 9 ; P ) = r " - r " ' . . . . . 1 + ~ IL(~(1); Q) . (6) QeE(1)

P r o o f Let QeE(1) and let us assume that (x' 1 . . . . . x',) are local coordinates around Q in such a way that E(1) is locally given by X'l = 0. Then ~(1)Q is generated by

F _ ~ D' = b l x ' 1 + i ='~2bicxi" (7)

Let us define the "adapted Milnor number" #(@(1), E(1); Q) by

tJ (2(1) , E(1); Q) = d imc ~ x(~),e i= b~COxa)'e " (8)


Then we have that

/~(_~(1); Q) = tt(~(l), E(1); Q) +/~(r Q).

Hence, in view of Theorem 1, it is enough to show that

(9)

/~(~; P) = r" + ~ t t(~(1), E(I); Q) . (10) Q~E(1)

Let ( x l , . . . , x , ) be local coordinates a round P and let us assume that ~ p is generated by

O = a , . i= 1 ~Xi

Since 9 (1 ) has only isolated singularities one has that there is at most one ai such that vp(ai) = ri > r. (Proof: if at least two of the a i have order bigger than r, then by (2) Sing(~(1)lE~)) is locally given by n - 2 equat ions in E(1) at most, hence there are non isolated singularities, contradiction). Let us assume without loss of generality that r I > r = r 2 . . . . . r,. Let Hi be the hypersurface given by ai = 0 (Hi may not be reduced) and let s = r~ - r . For a point QcE(1) we have that

#(~(1) , E(I); Q) = i ( Q ; / 4 x + sE(1),/42 . . . . . /4,) (11)

where Hi are the strict t ransforms and i ( Q ; ' , . . . . ") denotes the intersection multiplicity (see [2]). Then we have that

g ( ~ ; P ) = i ( P ; H i , . . . , H , ) = r I " r " - 1 + QEE(I)

+ ~ i ( Q ; s E ( 1 ) , / 4 2 . . . . , H n ) - ~, i ( Q ; s E ( 1 ) , I42 . . . . . 14n). QEE(1) QEE(1)

(12)

Now, since

~, i ( Q ; s E ( 1 ) , I ~ 2 . . . . . I t , ) = s r " - ' Qeg(1)

then the theorem follows from (11) and (12). []

R e m a r k . We shall need the above formulae (3) and (6) even in the degenerate case that n = 1. We adop t the convent ion

p(~l{p}; P) = 1.

Let us assume that P is an absolutely isolated singularity of @ and let us consider a sequence of quadrat ic blowing ups

n(1) 7r X = X(0) , X ( l ) , . . . , X ( m ) (13)

verifying the propert ies of (A.I.S.). Let us take the same notat ions of (A.I.S.). For each i, 1 < i < m, let E( i ) be the total inverse image of P ( 0 ) = P in X ( i ) and let

E ( i ) = E 1 (i) w . �9 �9 w El ( i ) (14)


be the decomposi t ion of E(i) in its irreducible components . Assume that E~(i) = ~(i)- ~ ( P ( i - 1)) and that Ej(i) is the strict t ransform of Ej(j), j < i. Let us define a weight p(Ej(m)) inductively as follows

p(El(m)) = 1

p(Ej(m))= ~ p(Ei(m)) . (15) { i ; P ( j - 1)EE,(j 1)}

If QeE(m) , we define the sets H(m, 1; j;Q), 1 = 0 . . . . . n - l , j = 1 . . . . . m, as follows

H(m, 1;j;Q) = {(j, i x . . . . . i,);ik +-jVk,

i~ < . . . < it, Q~Ej (mjc~Ei~(m)~ . . . . Eil(m)}

and the numbers ,

(16)

a*(~(m), E(m), EArn); Q) n - I

= ~ ( - 1 ) ~ Z a (~(m)]~ j , , ) . . . . . E,,~,,);Q). (17) 1 = 0 (j,i~ . . . . . i l )~H(m, l;j; Q)

Lemma I. With the above notations, let Q ~ E(m) and let us take Ei~ (m) . . . . . Ei, (m) such that QE Ei~ (m) c~ . �9 �9 c~ Ei,(m ) = F. Let us assume that dim F => 2 (i.e. l =< n - 2 ) . Then we have

a) v ( @ ( m ) l F ; Q ) = v(~(m);Q). b) Q is an absolutely isolated singularity of ~ ( m ) lv.

Proof Let •: X ( m + 1 ) ~ X ( m ) be the quadrat ic blowing-up centered at Q. Let F ' = E h (m + 1) c~. �9 �9 c~ Ei,(m + 1). Then the restriction ~: F ' ~ F is the blowing-up of F with center Q and F ' c~ E,, + 1 (m + 1) is the exceptional divisor of ft. Since ~ (m + 1) has only isolated singularities, then ~ ( m + 1)Iv, has only isolated singularities and hence ~ ( m + 1)[p, is the strict t ransform of ~(m)IF by ft. Moreover , since n is a non dicritical blowing-up for ~ ( M ) it follows that ~ ( m + l ) [ F is tangent to F ' ~ E,,+ t(m + 1) and hence ff is a non-dicrit ical blowing-up for ~ (m) [v . Then b) is proved. Now a) follows from the fact that ~t and ~ are bo th non-dicrit ical blowing- ups and the fact that we divide by the same power of the equat ion of the exceptional divisor in order to obta in @(m + 1), resp. ~ (m + 1)It ' , f rom N(m), resp. ~ (m)It . []

Theorem 3. With the above notations, we have that

r " - l + r " - 2 + . . . + 1 = ~ p(Ej(m)) Z p * ( ~ ( m ) , E ( m ) , E j ( m ) ; Q ) (18) j = 1 Q~E(m)

where r = v (~ , P).

Proof Let us proceed by induction on m. I f m = 1, it is just Theorem 1. Now, let us assume that the theorem is true for m - 1. First of all, let us simplify the notat ions by making


R = P ( m - 1); # ( j ) = p ( E ] ( m ) )

~* (re;j; Q) = / t * (~(m), E(m), El(m); Q) (19)

Ej= Ej(m)

I~(m,(j, i I . . . . . it); Q ) = # ( ~ ( m ) [ E / m ) c ~ E , , ( m ) . . . . . E,,Im); Q ) '

Let E j , ( m - 1) . . . . . Eje (m- 1) be the components of E ( m - 1) which contain R. Compar ing the right hand-side of(18) for m - 1 and for m, it is enough to prove the following formula

p(J t ) ' la*(m-l ,J , ; R ) = ~ P(J,)" t = 1 t = 1 Q ~ E m

Then it is enough to prove that

(la* (m,.j, ; Q) + l~* (m, m; Q)). (20)

l a * ( m - l , j ; R ) = ~ (#*(m, j ;Q)+ la*(m,m;Q)) (21) Q ~ E J m

f o r j =J l . . . . . Je- Let us compute the left hand-side of (21). We have that

n - 1

/~* (m- 1, j ;R) = ~ ( - 1)' / = 0

l~(m -- 1, ( j , i~ . . . . . it); R ) . ( j , it . . . . . i l ) E H ( m - 1 . l , j ; R )

By applying Theorem 2 and Lemma l, we have that

p(m-- 1,(j, i 1 . . . . . it); R) = r "- t -1 - r n- l -2 . . . . . 1

Now, since we have

we obtain that

+ 2 Q 6 E . c',Ejc~E, ~ . . . ~ E , ,

p(m, (j, i x . . . . . il); Q) .

l a* (m- - l , j ;R )= ~" ( - 1 ) t z=0 1 ( r " - l - a - r " - t - 2 - . . . 1 )

e - 1 + ~ (-- 1) ~ Z la(m,(J, il . . . . . il); Q) .

/ = 0 (j,i 1 . . . . . i l )~H(m 1) , I , j ;R) Q ~ E ~ c~ E;c~ E . c~. . �9 c~ E,,

Let us remark that if we define

and

14(m, l , j ;m;Q)={( j , il . . . . . i t)EH(m,l,j;Q);ik4=m V k = 1 . . . . . l} (22)

H (m, l,j;m; Q) = H(m, 1,j; Q) -14(m, l,j;m; Q) .


Then we have that

_ = ( _ 1 ) l e - 1 z=o I (r " - l - l - r " - 1 - 2 - . . . - 1 )

e - - 1

+ ~ ( - 1 ) t Z ~ II(m(j, ia . . . . . i t ) ;Q) . (23) t = 0 Q ~ E m [j, i I . . . . . i I )~ I t ( m , l , j ; Q )

Now, let us compute the right hand-side of (21). We have that

e - - I

Z # * ( m , j ; Q ) = Z ( - 1 ) t ~ ~ p ( m , ( j , il . . . . . it);Q) Q ~ E m t = 0 Q ~ E m (j , i l , . . . , i t ) 6 l ~ ( m , l , j ; m ; Q )

n - 1

+ ~ (--1) t ~ ~ #(m,(j , i 1 . . . . . it);Q).

(24)

Then, by (23) and (24), in order to prove (21) it is enough to show that

( - 1 ) t ( r n - t - l - - r ~-~-2 . . . . . I ) = 2 # * ( m , m ; Q ) l = 0 Q e E m

n - 1

+ Z ( - 1 ) t Z Z u ( m , ( j , il . . . . . /t);Q). (25) l = 0 Q ~ E ~ ( j , i I . . . . . i ~ ) ~ H ( m , t , j ; m ; Q )

Now, applying Theorem 1, we have that

P(m, (m, i l , . . . , i t ) ; Q ) = r " -1 -1 _ t _ r n - t - 2 + . .

Q E E ~

and if ik = m for some k, we have also

+ 1 (26)

# ( m , ( j , i ~ , . . . , i t ) ; Q ) = r " - l - l + r " - t - 2 + . . + l (27) Q ~ E m

(recall Lemma 1). Now, counting the expressions (26) and (27) in the right hand-side of (25), we have that (25) is equivalent to

(_1) t e - 1 l e t=o 1 ( r " - Z - l - r " - t - 2 . . . . . 1)= ( - 1 ) + z=o 1 1 1

(r " -1 -1 + r " - t - 2 + . . . + 1)

which is easily seen to be true. This ends the proof of the theorem. []

w T h e t h e o r e m o f d e s i n g u l a r i z a t i o n

Suppose P is an absolutely isolated singularity of ~ and

n ( 1 ) n(ra}

x = x ( 0 ) , x(1)~-.. . , x(m)

a sequence of blowing ups verifying the properties of (A.I.S.).


Proposition 1. Let Q ~ E(m) be a singular point o f f ( m ) . Assume that Q ~ Ej(m) and let F = Ej(m)c~ [Ei2(m) ~ . . . w Eie(m)] where E~2(m ) . . . . . Ei~(m) are the other components of E(m) which contain Q. Then we have that

/~* (~(m), E(m), Es(m); Q) = p(~(m)[E,~m~, F; Q) (28)

(recall the definition of "adapted Milnor number" (9)). Hence p*(~(m),E(m), Ej(m); Q) > 0 and if it is equal to zero then ~(m) must have a nonzero eiyenvalue in its linear part.

Proof. Let us take local coordinates (xt . . . . . x.) at the point Q in such a way that Ej(m) is given by x 1 = 0 and that Ei,(m) are given by xt = 0, t = 2 . . . . . e. Then ~(m)Q is generated at Q by a vector field

D = a i x i - - + al �9 (29) i= ~ Oxi ~?xi i = e + l

Let us denote by H i, i = 1 . . . . . e, the hypersurface given by a~ = 0 (may be non reduced). Let us denote E 1 = Ej(m), E t = Ei,(m), t = 2 . . . . . e. Let us remark now that

P(~(m)le ,~. . .~E, ; Q) = i(Q;e(1)H1 + El, ~;(2)H2 + E2 . . . . . e(e)He +

Ee, He + 1 . . . . . H,) (30)

where i (Q; ' , . . . . ") denotes the intersection multiplicity and e ( t ) = 0 if re{t1 . . . . , tk}, e ( t ) = 1 if t~{ t l . . . . . tk}. For a subset I t { 1 . . . . . e} let us denote e~ the above function. Then (28) is equivalent to

( -1)*1-1" i (Q;e t (1)H1 + E 1 . . . . . et(e)He+ Ee, He+l . . . . H,) t c {1 ..... e}

1~1

= i(Q;E1, H2, H3 . . . . . Hn) (31)

and (31) follows easily from the multilinear property of the intersection index. Thus we have ~* > 0 and if #* = 0 there is one coefficient ai, 1 _< i < e which is a unit, hence we have a nonzero eigenvalue. []

Let P be an absolutely isolated singularity of 9 . Let us denote P(0) = P, X(0) = X, ~ ( 0 ) = @. Let us take notat ions as before. We shall say that a (finite or infinite) sequence of blowing-ups

n(1) n(2)

x(0 ) , X ( l ) , x ( 2 ) , . . . (32)

"respects the desingularization procedure" iff n(i) is centered at a point P ( i - 1) e E ( i - l) such that P ( i - l) is a singular point for ~ ( i - l) and the linear part of a generator of ~ ( i - I) at P ( i - l) is nilpotent (i.e. it is zero or non zero but with no nonzero eigenvalues).

Let us remark that there exists always sequences which respect the desingularization procedure.

Theorem 4. With the above notations, there is no infinite sequence of blowing-up's which respects the procedure of desingularization.


Proof Assume that we have an infinite sequence like (32) which respect the desingularizat ion procedure. Then, by reordering, if necessary, the blowing ups, we can assume without loss of generality that the centers P(i) of ~t(i + 1), i > 1, satisfy

P(i)~rt( i)-~(P(i- 1)). (33)

At the step m, we have the following term in the right hand-side of(18) (the others being > 0):

p (Era(m))"/~* (~(m), E(m), Era(m)); P(m)).

Notice that

p(El(1)) < p(E2(2)) <. . . < p(E,,(m)) (34)

for our choice of the sequence (32) with the proper ty (33). Put /~* = # * ( ~ ( m ) , E(m), E,, (m); P (m)). Since the left hand-side of (18) is fixed and since/~*~ N - {0} (if ~* = 0, we have a nonzero eigenvalue), the sequence (35) must stabilize. Let us fix N such that for each m > N we have that

p(Em(m)) = pN(EN(N)).

This means that for each m > N, the only componen t of E(m) containing P(m) is exactly E(m). Therefore there is a formal branch curve F(N) of X(N) at P(N) such that F(N) is nonsingular and transversal to EN(N) at P(N) and such that the successive strict t ransforms F(m) of F(N) pass through P(m); F(m) is defined inductively as follows: F(N + l) = strict t ransform o f F ( N ) by ~ ( N + 1), i fm > N + l, F(m) = strict t ransform of F(m-- l) by ~(m).

Now, we choose formal coordinates (x~(N) . . . . . x,(N)) a round P(N) such that EN(N ) is locally given by x l ( N ) = 0 and F(N) is given by x2(N) . . . . x.(N) = 0. Then we can define inductively formal coordinates (xl (m) . . . . . x,(m)) a round P(m), for m > N + 1 by

xl(m ) = xl(m-- 1)

x~(m)xl(m)= xi(m-- 1) i > 2

and E,.(m) is locally given by x a ( m ) = 0 and F(m) is given by xz(m) . . . . . x.(m) = 0. Let us fix a generator D(N) of ~ ( N ) at P ( N ) given by

0 D(N) = al (N)xa ( N ) 0 x ~ + ,=2 ~ ai(N)ax,(N) "

Then, we have inductively generators D(m) of ~ ( m ) at P(m), n > N + 1,

i=~= 2 ai(m)c3x m) D(m) = a x(m)x l ( m ) ~ + ~Txl trn)

such that

al(m) = a 1 ( m - l ) /x 1 ( m ) r (m- 1)- 1

ai(m) = ai(m -- l)/xl (m) r~m- 1 1 xi(m)al (m -- 1)/x 1 (m) rim - ) - 1 (35)


where

r(j) = v (~ ( j ) ; P(j)) = min({ v(a 1 ( j ) x l (j)} w {v(ai(j);i> 2})

for j > N. (Note r(j) > 1). F o r any ai(m ), let us denote by d~(m) the restr ic t ion of a~(m) to F(m), i.e.

a~(m) = ai(m)(xl(m), 0 . . . . . 0 ) ~ C [ [ x 1 (m)]] .

Let us consider the number

6(m) = min(v(di(m));i > 2) ,

with the assumpt ion 6(m) = ~ if d~(m) = 0 for each i > 2. Because of (35) and the fact r(rn- t) > 1 (otherwise P ( m - 1) is not a s ingular point), we have that

~ ( m ) = f ( m - - 1 ) - r ( m - 1 ) for each m > N + l .

Hence, the only possibi l i ty is 6(N) = co (hence 6(m) = oo for each m > N) and then di(m) = 0 for each i > 2 and m > N.

Let us consider now the invar iant

r(m) = v(a 1 (m))

for each m > N. Then r(m) 4= ~ (i.e. a l(rn) 4= 0), since otherwise F(m) c (al(m) . . . . . a.(m) = 0) and P(m) is not an absolute ly isolated singularity. Moreover , we have that

z ( m ) = ~ ( m - - 1 ) - r ( m - l ) + l .

Hence (since z(m) cannot decrease infinitely), there is an index M > N such that for each m > M we have r(m) = 1. Not ice that

v(al(M)xl(M)) > 2 (36)

since otherwise a 1 ( M ) is a unit and we find a nonzero eigenvalue in the l inear par t of D(M). Then, there is an index t > 2 such that

v(a,(M)) = I . (37)

Since P(M) is an absolute ly isolated singulari ty, all the coefficients of D (M), except possibly one of them, have the same mult ipl ici ty, then (36) and (37) imply that

v (az (M)) . . . . . v(a,(m)) = 1 . (38)

No te that since di(M) = 0 for i > 2, then the l inear par t of a i (M), i > 2, depends only on x2 (M) . . . . , x.(M). Now, we can make a l inear change of coord ina tes in x z ( M ) , . . . , xn(m ) in such a way that the l inear par t of D ( M ) has a Jo rdan form. This change of coord ina tes does not modify the proper t ies d~(M) = 0 for i > 2, (36), (37) and (38) above. If we assume that all the eigenvalues are zero, then looking at the Jo rdan matr ix, we see that the linear par t of a,(M) must be zero, hence

v(a,(M)) > 2 (39)

and this cont radic ts (38). This is the desired contradic t ion. []


In particular, a sequence which respects the desingularizat ion procedure must be finite, and we end up with the sequence we need to prove Theorem A.

w Simple singularities

Let X be an n-dimensional complex manifold and let E c X be a normal crossings divisor of X (we can imagine that we are at an intermediary step of the desingularization process, E being the whole exceptional divisor). Let ~ be a singular analytic foliation by lines on X such that each irreducible componen t of E is invariant for 9 . Fix a point P e S i n g ( ~ ) and denote by e = e (E ,P) the number of irreducible componen t s of E through P. We shall assume tha t e > 1 (this is always achieved after one blowing-up). Then the vector field D of (1) may be written as (28).

D = bixl + i = 1 i = e + l

where E = (I-I;= 1 xi = O) locally at P.

Definition I. Assume that e = 1. Then P is a "simple point" iff one of the following possibilities occurs:

A) b l ( 0 ) = 0, the curve (x 2 . . . . . x . = 0) is invariant for ~ (up to an adequate formal choice of the coordinates (x~ . . . . . x,)) and the linear part of DI~ is of rank n - 1.

B) b~ (0) = 2 4= 0, the multiplicity of the eigenvalue 2 is one and if ~ is ano ther eigenvalue of the linear part of D, then I~/;t4sQ+ ( = strictly positive rat ional numbers).

Assume that e > 2. Then P is a "simple corner" iff (up to a reordering of (x 1 . . . . , xe), we have b 1 (0) = 2 + 0, b2(0) =/~ and ~ / 2 r

We say that P is a simple singularity iff it is a simple point or a simple corner.

Remark. We have avoided the special case of Th. A.5. in [4] which will not appea r in the context of absolutely isolated singularities.

Proposit ion 2. Assume that P is a simple singularity, let n: X ' --* X be the blowing-up of X with center P and put E' = n - 1 (E w { P } ). Let ~ ' be the strict transform of ~ by ~. Then:

a) Each irreducible component o f E' is invariant for ~' . b) I f P ' e S i n g ( ~ ' ) n n - l ( P ) , then P' is also a simple singularity of ~ ' (with

respect to E'). More precisely: b - l ) I f P is a simple point, there is exactly one simple point

P ' e Sing ( ~ ' ) n n - 1 (p). The other points in Sing ( ~ ' ) n n - x (p) are simple corners. Moreover, P and P' have the same type A) or B) o f Definition 1.

b - 2) l f P a simple corner, then all points in S i n g ( ~ ' ) u n n - l (P) are simple corners.


Proof a) We have only to prove that n - 1 (p) is invariant for 9 ' . This fails only if the linear part of D is a multiple of the radial vector field ~7= 1 xiO/Oxi. But this is evidently not true.

b) It is straightforward from the equations. (See also Th. 3.6. and Th. A.4. of [43). []

Now, assume that we are in the situation of w We shall say that the sequence (32) respects the second procedure on desingularization iff~(i) is centered at a point P ( i - 1)~ E ( i - 1)c~ Sing ( 9 ( i - 1)) which is not a simple singularity for 9 ( i - 1).

Theorem 5. With the above notations, there is no infinite sequence of blowing-ups which respects the second procedure of desingularization. (Hence after finitely many blowing-ups we get only simple singularities, which are, of course, absolutely isolated singularities).

Proof Assume that we have an infinite sequence like (32) which respects the second procedure of desingularization. As in the proof of Theorem 4, we can assume that (33) holds. Moreover, by the Theorem 4, we can assume without loss of generality that the linear part of a generator of 9(0) at P(0) is non-nilpotent. This property is stable under blowing-up, hence the linear part of a generator of 9 ( 0 at P(i) is non- nilpotent, for all index i > 0.

First case. e(E(O), P(0)) = 1 and 9(0) is generated at P(0) by

8 bi A D = b' Xl Ox-~ + i=2 ~ Oxi '

with bl(0) = 0 (E(0) = (Xl = 0)). If e(E(1), P(1)) = 1, the situation repeats at P(I). First of all, note that the eigenvalue 0 has multiplicity one; otherwise, the linear part of D is

Xl Ox~ -~ X2 ~x3 "~- " " " -t- X k - l OxW ~- i=k+l ~ Ai(~xi '

where A~ = a~(Xk+~ . . . . . X,), for an appropriate choice of the coordinates. By considering the point in X(1) given by the equations

x~ = x'k; xi = x'kx'i, i f i r

we find two coefficients of multiplicity two in our vector field, hence we have no absolutely isolated singularity, contradiction. In particular, we can write the linear part of D as

0 y~ B,(x,,

i>2 . . . . X n ) o x i

where Bi(x~,O . . . . . 0 )= 2 i 4= 0. If e(E(1),P(1))= 2, this implies that P(1) is a simple corner. Contradiction. Hence, the only possibility is to have e(E(i), P(i)) = 1, for all i > 0. This implies that the points P(i) are the infinitely near points of a formal branch F at P(0), which is nonsingular and transversal to E(0). Reasoning as in the end of the Theorem 4, we find that the condition A) of the Definition 1 occurs. This is the desired contradiction.


Second case. e = e(E(0), P(0)) > 1 and 9(0) is generated at P(0) by

D = b i x i -4- i = 1 �9 i = e + l

with bl (0) = 2 4: 0. In this case we find a contradiction by applying Th. 4.5. of [4]. Moreover, this is the last case, since if e > 2, we have necessarily bl (0) 4:0 up to a reordering of xl . . . . . xe. []

Hence, the Theorem B stated in the Introduction is proved.

w Absolutely isolated simple singularities

We have very often to deal with the following situation: suppose a sequence of quadratic blowing-ups

~ ( 1 ) ~ ( 2 ) tt(n)

x(o), x(1), . . . . . . . , X ( n )

is given, where the center of each ~(i) is a point P ( i - 1 ) 6 S i n g ( 9 ( i ) ) . Moreover, assume that 9 (n) has a finite number of singularities, all of them with nonzero first jets. Then, under what conditions on these jets it follows that 0~X(0) is an absolutely isolated singularity? Proposition 3 above gives a sufficient condition, which turns out to be also necessary to characterize the absolutely isolated simple singularities.

Let 9 be a foliation by lines on X and fix PESing(9) . Assume that the linear part LD of a generator D of 9 at P is written in the Jordan form

= ().s+jXp + Xp+ 1 )OXp i= 1 j = 1 p = s + t l + ... + t~ - i + 1

O -t- 2 s + j X s + t l + . . . +t~ t ~ X s + t t + . . . + b '

where t i > 2 are the sizes of the Jordan blocks.

Proposition 3. In the above situation, assume that i) P is an isolated singularity of g .

ii) 2j 4= 0, i = 2 . . . . . s+k; tj = 2,j = 1 , . . . , k; 2i/2ir ,for i 4:j, i,j = l . . . . . s + k , j 4 : l .

Then P is an absolutely isolated singularity of 9 .

Proof It is enough to show that the assumptions i) and ii)persist under the blowing-up with center P. This is a tedious but elementary verification. []

Lemma 2. Assume that P is an absolutely isolated singularity of 9 and that the following property holds: (*) "There are two distinct eigenvalues 2i, )~j of LD such that ),i 4= 0 and 2j/2jq~ Q + ".


Then the property ii)of the Proposition 3 holds except for a reordering of the eigenvalues )ol . . . . . 2s.

Proof First of all note that (*) is stable under blowing-up. Let us show first that tp = 2, for p = 1 . . . . . k. Let us reason by contradiction assuming that tp > 3 for some index p. Let us distinguish the following cases.

l~ __> 1. Make the transformation T~ and let 9 ' be the corresponding strict transform. Then 9'lx,~ =0 is linearizable and it has a Jordan block of size > 3. Contradiction.

2~s = 0. Then (*) implies that k > 2. Assume that t 2 __> 3. Make the transformation T~+~. Then ~[X's.l=0 also satisfies (*) and it is given, in appropria te coordinates by

i = 1 OXi i = s + 2

k s + l l + - . . +tu- 1

U=2p=s+tl+...+t. t + l

0 +(2~+, -2~+ 1 -x~+z)x~+,, +. . +,,, t~x,+t, +... +t~

and we are done reasoning by induction over the dimension n. Thus, we have tp= 2 for all p = 1 , . . . , k . Note that only one of the 2;,

i = 1 , . . . , s + k, can be equal to zero; otherwise there are at least two coefficients of a generator of ~p which have order > 2 and hence P is not an absolutely isolated singularity. Now, let us show that 2j 4 :0 for all j = s + 1 , . . . , s + k. Let us reason by contradiction, assuming that ).~+~ = 0. Making the transformation T~+ 1, the corresponding strict transform 9 ' is given by

O ' = i = 1 (f~ix'i+~ Jc~

k + ,=2Z t;~+,x~+~-~ + x ~ + 2 , + ~+ 2~-~x'~+ ~) ax's+ 2p- 1

0 + (2s+pXs+zp + ~ts+zpXs+ 1 + D'*

p

where the order of D'* is > 2. Note that if x~ + 2 divides D(x~ + 2), then cq + 2 = 0. If ~+ 2 = 0, we are done, since we have at least two coefficients of order two and we get a contradiction. Assume that ~ + 2 + 0. Not ing that 2j # 0 i f j 4: s + 1, up to a linear change of the coordinates we have

D ' = i=a 2iYi~Y/ + p=lE ()~s+pYs+2p-1 +Ys+2P)63ys+zp_l

0 - - + D ' * . q- 2s + pYs+ Zp Oys+ zp

Moreover, Y~+2 divides D'(y~+2) (note that Y~+2 =x's+~). Then, repeating the argument, we obtain the desired contradiction.


Finally, let us show that if 2j 4 : 0 and i * j then ~.i/2~ 6 Q +. Assume the contrary. Then

z = min { p + q; p, q > 0, p, q e Z, such that there are r 4: m with 2,/2m = p/q }

is finite. If z > 2, then z decreases strictly after the t ransformation T,. So, we can assume that ~ = 2. This implies that 2r = 2,. for some r < m. Let us distinguish the following cases

l~ < s. Making the t ransformation To we have that 9 ' Ix,= o is linearizable and it has a zero eigenvalue. Contradict ion.

2~ + 1 < r. Assume that r = s + 1. Making the t ransformation T~ we see that 9 ' has a zero eigenvalue of multiplicity > 2. Contradict ion.

This ends the p roof of the Lemma. []

Theorem 6. Let E be a normal crossings divisor E of X such that each irreducible component of E is invariant by 9 . Assume that P is a simple singularity of g . Then P is an absolutely isolated singularity if and only if the conditions i) and ii) of the Proposition 3 are satisfied.

Proof For the "only if" part, it is enough to remark that the property (*) of the above Lemma is satisfied in our case.

w Some examples of absolutely isolated singularities

O u r first family of examples comes from Proposi t ion 3 (we keep the same notation).

Example. Assume D is linear in some coordinate system. Then P is an absolutely isolated singularity if and only if the following property holds

2 i :#0 , i = 1 . . . . . s + k , t j = 2 , j = 1 , . . . , k ; 2 i /2 j~Q+

for i :#j , i , j = l . . . . . s + k .

The "if" part follows from Proposi t ion 3, and the "only if" part is a straightforward verification.

Let us now consider nonlinear cases. Fix a point P ~ X such that v ( 9 , P) = r > 1. Let ~ ' be the maximal ideal of d~x, P

and let D be a generator of 9p . Then we have a C-linear map

L ~ D : ~ l / j r 2 _~ .i//r / //r , + 1

f + JC2 ~ O ( f ) + Jlr+ l

If we change the generator of 9 p , then LrD is multiplied by a scalar. Thus, we have a well defined mapping

ff~{9; v (9 ; P) = r) ~ Pro j (Homc(J l / J l [ 2, ~/r/J//~ + 1 )) __ p ( c s ) .

Proposition 4. There is a dense subset H c p(CN), such that if g ~ H , then

i) P is an absolutely isolated singularity of 9 ii) By blowing-up P we get only absolutely isolated singularities.


P r o o f It is enough to prove ii). Fix a generator

D = a i i = 1 ~ X i

of @p and put

Ai = class of a i in J t " / ~ '" + 1, i = 1 . . . . . n .

Then each A~ is a homogeneous polynomial of degree r. Write

A , = E A , . , X i ' . . . . X ' ; . l = ( i l . . . . . i n )

Thus, we can think of ( A i , , ) as being homogeneous coordinates in P ( C N ) . Note that there is a nonempty Zariski-open set W c P (C Iv) such that (Ai, t )~ W

if and only if the following properties hold:

a) (Ai = O; i = 1 . . . . . n) c P (C N) is empty.

b) ~ i= l ...... , A i ? / ? X i is not a multiple of ~ i = t ..... , X I O / O X i . In particular, P is an isolated singularity of 2 and ~ is not dicritical at P. Let n:X'--+ X be the blowing-up centered at P. Recall that E = ~ t - l (P) is

covered by open sets U~ ~ X ' , j = 1 . . . . . n, and the strict transform 9 ' is generated in U~. by

' ' ' ; 5 Dj = (Aj(x '~ . . . . . x j _ 1,1, x j+ 1 . . . . . x ' ,) + x } ( . . . ) ) x (~ X;,

0

. . . . . ))Ox + ~ ( a i ( x s . . . . . 1 . . . . . x ' , ) - - x j A a ( x , . . . . 1 . . . . . x , ) + x a ( . . .

i4: j ' 'i

where E (~ U} = (x} = 0). Let us consider homogeneous coordinates ( X i ; i = 1 . . . . . n) and ( V u , 7 " ) ,

u = 1 . . . . . n. Then there is a homogeneous polynomial

Polj . u((Ai, t ) ; ( X i ) ; ( V. , 7 . ) )

of degree 1 in (Ai, 1 ), of degree r in (Xi) and of degree n in ( V., 7"), such that

Polj,.((Ai, l); (~ ~ . . . . . 1 . . . . . a.); (l, 7"))

is the characteristic polynomial of the linear part of D~ at the singular point Q = (a~ . . . . . 0 . . . . . a . ) E E n U~. We have also a homogeneous polynomial

Pol~,.((A~. ~ );(X,);( V., 7"))

such that

eol~,.((A~,t);(ax . . . . . l . . . . . ~.);(1, 7.)) = ( d / d T . ) ( P o l j . . ( ( A ~ , ~ ) ; ( a l . . . . . 1 . . . . . a . ) ; ( l , 7"))) .

Now, let Zj ~ P ~ - ~ x P~- ~ x Pc x . , . x Pc be the Zariski closed set given by

Pol i . . ( ( Ai, l );( X i ) ; ( V, , T,)) = O .

X ~ A i - x I A ~ = O, i 4=j .


Fix an element t E Q + , t 4: 1. Let Z~,, c Zj be given by the intersection of Zj with

n-- t . . [(Pc s - l - W) • Pc 1 • Pc • . - - • Pc]u(P~ T,)) = O)

w(X~=O)w(V, ,=O;u= 1 . . . . . n)u(T,, = 0;u = 1 . . . . . n)

w ( V , T , , , - t V , , T , = O ; u , u ' = 1 . . . . . n).

Let K~, t be the projection of Z~,, in P ~ - 1. Put Hj, t = P~- 1 _ K j , , . Then Hj, t is a Zariski open set in P ~ - 1 and (Ai, ~) ~ Hj, t if and only if the following properties are satisfied:

a) P is an isolated singularity of 9 and 9 is not dicritical at P. b) The linear part at each singular point of 9 ' in U~ c~ E has n-distinct nonzero

eigenvalues. In particular each singular point of 9 ' in U'jmE is an isolated singularity of 9 ' .

c) If 21, 22, are two eigenvalues of the linear part of a generator of 9 ' at a singular point in U)c~E, then 21/22 d/= t.

Put Hj = OtHj , t. Assume that Hj 4: ~b, then H = ("]jHj is the coset of a countable family ofZariski hypersurfaces, hence a dense set and we are done. But if we consider

D ~--~- x r , ~ q t- ~~ ( /~ /x5 - 1 " - l l x i / X J ~ X ~ 7t- # i X i " j ieej i

with tr.deg.(Q({2~}, {&})/Q) = 2 ( n - 1), we find easily that H j . 4'. E5

Example. The above Proposition 4 provides examples which have a quite trivial desingularization. The following one is an example of absolutely isolated singularity having nontrivial desingularization

O~ = XTXl 8x~ + ~ j x j + ~x'~ x ~ - - , i=2 j = 2 OXi

where the ~ j and the #i are algebraically independents. In fact, if m > 2, after one blowing-up we find absolutely isolated simple singularities and exactly one singularity of the type D,,_ 1- After m blowing-ups we find only absolutely isolated simple singularities. We may represent the final result by the diagram (a linear chain)

0 - - - - - 0 - - - 0 " ' �9 �9 �9

P(0) P(I) P(2) P(m- 1) P(m) Each P(j) is connected compact complex manifold of codimension one and P ( j ) m P ( j - 1) contains 3 simple singularities for 1 < j < m. Also P(0) and P(m) contain other 4 simple singularities.

Example. We present now an example whose tree of desingularization is not a linear chain. Take

x,i, + 4 8 3 9 ( 0 ) = ~. + ~ [ x , ( A i l ( x 2 , x z ) A i 2 ( x 2 - x l , x 3 - x l )

t,,.~ 1 i=2

q_ k l i X . ~ + 2)Ai3(x2 _ x 1 ' x 3 _ x l ) q_ x 2 x m + 3 ] Ox--i


where Aij, i = 2, 3, j = 1, 2, 3 are homogeneous linear polynomials chosen in a generic way such as to allow (0, 0, 0)~ C 3 to be an isolated singularity, as well as the singularities which appear in the following desingularization procedure.

a) # S i n g ( ~ ( 1 ) ) = 19. Five of these singularities have only nonzero eigenvalues, and 12 of them are saddle-nodes (one eigenvalue is zero). All of them are simple singularities once the Aij are generically chosen. There remain 2 singularities to be desingularized (if the coordinates are Ul = x l , u2 = x2 \x l and u 3 = x 3 \ x l , these singularities are located at the points (0, 0, 0) and (0, 1, 1)). In fact, they are similar to D,, in the example above; the equat ion for ~(1) is

@(1) = u7 + 1 ~ + ~ [ui(Ail(u2, u3)Ai2(u 2 - l, u 3 - l) Oul i=2

g " [ - ] ~ i b l n ~ ) A i 3 ( U 2 - - 1, 1'/3 - - I ) ] - -

(~u i

Therefore, each of them is desingularized after a linear sequence of blowing-ups. Again it must be stressed the role of the generic choice of the linear polynomials A i2 in @(0) (and also of/~i, i = 2, 3) in order to get isolated simple singularities.

b) The d iagram of desingularization becomes

Pl(1) P1(2) P l (m-1) PJ(m)

~ 0 �9 " ' ' �9 �9

P ( O ) O . . .

Pz(I) P2(2) p2(m-1) p2(m)

R e f e r e n c e s

1. Camacho, C., Lins, A., Sad, P.: Topological invariants and equidesingularization for holomorphic vector fields. J. Differ. Geom. 20, 143~ 174 (1984)

2. Fulton, W.: Intersection Theory. Berlin, Heidelberg, New York: Springer 1984 3. Cano, F.: Desingularization Strategies for 3-dimensional Vector Fields. (Lecture Notes in

Math., Vol. (1259). Berlin, Heidelberg, New York: Springer 1987 4. Cano, F.: Final forms for a 3-dimensional vector field under blowing-up. Ann. Inst. Fourier 37,

151-193 (1987) 5. Seidenberg, A.: Reduction of singularities of the differential equation AdY= BdX. Am. J.

Math. 90, 248 269 (1968)

Oblatum 24-11-1989

absolutely isolated singularities of holomorphic vector fields

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