Linear complementary inequalities for orders of germs ... - Springer Link

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of Germs of Analytic Functions. Shuzo Izumi. Department of Mathematics, Faculty of Science and Technology, Kinki University,. Higashi-Osaka, Japan.
Invent. math. 65, 459-471 (1982)

mathematicae 9 Springer-Verlag 1982

Linear Complementary Inequalities for Orders of Germs of Analytic Functions Shuzo Izumi Department of Mathematics, Faculty of Science and Technology, Kinki University, Higashi-Osaka, Japan

wO. Introduction Let (X, ~) be a germ of a complex space. Then a germ feCx, ~ has two kinds of orders: the algebraic order v~(f) and the analytic order I~A,r along a subset A a IX]. It is easy to see the following. (i) vr (ii) For inf

r/~K

a morphism

(f, geOx,r 4,: Y--,X

and

a subset

K=~-1(r

we have

v,(fo cb)> v~ (f) (f~(gx, r

(iii) For a subset A a If we impose some plementary inequality [0, ~ ) ~ [0, ~ ) such that

(2, v)=(vr

IXl, we have #A,~(f) > re(f) (f~ OX,r condition on (X,~), 4, K or A, we have a com(CI): there exists a nondecreasing function T: 2 < T(v), where

vr162

(infv,(fo~), risK

v~(f))

or

(#A,r

re(f)).*

In some cases, we can put T(v)=av+b (a> 1).** In this paper we show some conditions for the existence of such a linear complementary inequality (LCI).*** In our main theorem (1.2) we prove that the existences of the following three LCI are equivalent for a fixed reduced and irreducible (X, ~). (1) LCI of (i). (2) LCI of (ii) for all morphisms ~ and q e Y such that grnk, q~= dim (X, ~). (3) LCI of (iii) for all complex wedges (A, 4) on X such that rnk(A,~) = dim (X, 4). If these hold, we may say that (X, ~) is " g o o d " (the smaller a, the "better" (X, 4)). Next, we show that (X, ~) is " g o o d " if it is quasi-homogeneous or if the exceptional fiber of its normalized blowing-up is irreducible. The author does not know whether all reduced and irreducible (X, ~) are "good".

0020-9910/82/0065/0459/$02.60

460

s. Izumi

In w we treat a sufficient condition on 9 for the existence of LCI of (ii) and that on A for the existence of LC! of (iii), for general (X, 3). These are generalizations of the results in [I]. In the final section we give some remarks of algebraic nature. In order to obtain these results we use the observation of order by Lejeune and Teissier. They have shown equalities (rather than inequalities) of certain orders taking canonical and favourable morphisms, etc.**** We shall often use a certain rank condition considered by Gabrielov and Hironaka's desingularization and sub-analytic sets. Throughout this study the author was stimulated by Tougeron's papers. Finally, the author wishes to express thanks to Prof. Fujiki who kindly answered to my questions.

Notes *" CI appear in many of our references, although some may be implicit. We give nonlinear CI in (1.4). As for the case (ii) with K = {~/}, existence of CI is equivalent to the condition that the induced morphism ~p: ( S x , ~ C r , . is injective and open with respect to the Krull topology and to the condition that the completion ~b is injective (cf. [B], [Mi], [B-Z]). **" If one such T exists, we can always put T(v)=a'v. But we do not always adopt this form since the infimum of a is important: in the LCI of (i), it seems to indicate complexity of (X, 3) (cf. the proof of (1.5), (4)). In the LCI of (ii), it is a generalization of the reduced order of function germ in Cy.. (consider the case X = C). ***" LCI are found in I-T3], (5.6); IS], (1.4); [-R2] , (2.2). Our (1.5) generalizes the 1st and the 3rd. Note, however, that they have quantified the constants a and b (=0), which we can not afford. Tougeron has posed a problem concerning the existence of LCI. Our results on morphisms can be viewed as partial answers to his problem. ****' The real case is treated in [-R~] (cf. [-Bo-R-]). A similar result is stated in [B-R], p. 271 (perhaps their proof supposes that the tangent cone is reduced).

Terms and Notation m A" the maximal ideal of a local ring A. vt(f)=sup {p:f~IP}: the order o f f ~ A with respect to ideal I of A. ~-l(f) = klim 1 v1(fk): the reduced order o f f ~ A . (cf. [L-T]) ~ k" I: the integral closure of ideal I of ring A in A; If A is a complex analytic algebra, _f= {f~A: 7,(f)> 1}. (cf. [L-T], (1.4), (4.3.3)) complex space X=(JX[,(gx); In this paper we always assume that the underlying topological space IX] is Hausdorff and paracompact. (cf. IF])

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real analytic space XR=(IXRI,(gx~); In this paper we always assume that X R has local models defined by coherent sheaves of ideals on smooth ambient spaces. We also assume that IXR] is Hausdorffand paracompact. (cf. [H2] ) Cgx,F: the algebra of germs of sections of (5x over neighbourhoods of closed set FcX.

@x,~=(gx, l~: the fiber of the structure sheaf (2x over 4 i.e. the local algebra of germs of holomorphic (real-anlytic) functions at 4 e X. m~=mcgx,: the maximal ideal of (~x,r ((5',,o, m,.o)=((9c=,o , me,,o ) or ((gR,, o , nt~=,o). dim~ X = dim (X, 4) = dim (~x,r the Krull dimension of (5~x,~. dim X = sup dime X. CeX

dim A: topological dimension. dim~A=inf{dimAc~ U: U is a neighbourhood of 4}.

v~(f)=v,,,~(f)=sup {p:femf}: the algebraic order o f f e (~x,r fr

= fmr (f) = 2im ~1 vr k) the reduced order of f e (gx,~.

l~x,a,t~(f)=sup{p: 3c~>0, 3 neighbourhood U=IXI of F, ~ representative f(y) o f f over U such that ]f(y)l 1 such that v,(fo ~) < a. vr for an)' f e (gx, ~. Proof There exists an irreducible component (YI,t/) of (Y,t/) such that grnk, r/'[rl=dim(X,~ ), By the desingularization theorem of Hironaka, there exist an open neighbourhood U of OeC"(R") and a morphism 7~: U ~ ( Y I , t l ) such that 7~(O)=r/ and grnko~=dim(Y~,q). Then ~o ~- U--*(X,~) satisfies q~o tP(O) = ~, grnk o 4~o 7j = dim(X, 4) and v,(fo ~') 1. Then the following conditions (1), (2), (3) are equivalent. (1) There exist a 1 > 1 and b I > 0 such that vr < a 1(v r + vr + b, for any f g~Cx, r (2) For any morphism r (Y, 17)---,(X, 4) of germs of complex spaces such that grnk, q~= dim (X, ~), there exist a 2 > 1 and b 2 > 0 such that v , ( f o q~) = 1 and b3>_-0 such that # a, ~(f) < a3 " v r

+ b3

for any f s ( g x , ~.

(1.3) Remark. The inequality in (2) implies that the induced homomorphism ~0: (gx,~--*(gy,, is injective. Its completion ~b is also injective by (4.4). We can also show that dim (X, ~) < dim ( Y,q) using the Hilbert-Samuel characteristic functions of (gx.~and (gr,,. On the other hand Osgood has given an example of rp such that ~b is injective and dim (gx,~>dim (gy,, (cf. [Mi]). Thus there exists a CI but no LCI for the orders of pullbacks in this example. See the note (*) in the introduction.

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(1.4) Remark. If we replace the linear functions aiv+b i by nondecreasing functions T~(v), (1), (2) and (3) hold for any reduced and irreducible (X,r Indeed Tz(v) exists by [Mi], Th. 2. Then we have only to put Tl(v)=2Tz(v)/k , T3(v)= Tz(v) (k: in the proof below).

Proof ( 1 ) ~ ( 2 ) : (We imitate the proof of IT2], (1.8)). It is well-known that there exists a finite morphism H: (X, ~) --, (C", O) such that g r n k r and n: (9,,o~ Cx, ~ is injective. By (1.1) there exists c > 1 such that % ( g o l l o ~ ) < c . vo(g ) for any g ~ C,,o. Let us define a sequence of linear functions eo(2), e 1(2) .... by e0(2) = c2,

%(2)=a~c(2+%_l(2))+blc

(p= 1,2, ...)

(al,b 1" in the condition (1)). We claim the following. (.) If f~(~x,r gi~(gn,O, vn(f~

and if

vr p + (gl ~ H ) f p- 1 +... + (gp o H)) > ep_ 1(2), we have vr > 2. Suppose that we have proved the cases p = 1, 2, ..., q. If v,(fo ~)>eq(2) and if vr q+ l + (gl ~ 11)fq +... + (g, +1 o 11)) =>eq(2), we have c. vr 1 ~ 11) >=c. Vo(gq + 1) ~ V,(gq+ 1 o 11 o t~) ~ eq(2). Hence, if q = 0, v~ (f) =>2. If q =>1,

v ~ ( f [ f q + (gl o Fl)f q-1 + . . . + (gq o 11)]) __>eq(2)/c. Then, by the condition (1) and the definition of eq, we have v~(f)__>2 or

vr q + (gl ~ 11)fq- 1 + . . . + (gq o H)) > eq_l(2). Even in the latter case we have v~(f)>)~ by our inductive hypothesis. Thus we have proved (.) for p = l , 2 ..... Since z is finite, there exists an integer k > 0 such that any f c Cx,~ satisfies a polynomial relation

f k +(gl ~ 11) f k - 1 +... +(gk o I])=0 for some gz~ (9., o. Hence we have only to put a 2 2 + b z = e k _ l ( 2 ). ( 2 ) ~ ( 1 ) : By the desingularization theorem of Hironaka, there exists a morphism cb: (C", O) -~ (X, 3) such that g r n k o c b = n and ~o(m~). C., o is generated by the multiple yk of the n-th coordinate. If f, g ~ (gx,r k. v~(fg) < Vo((fg ) o ~) = Vo(fo cb) + Vo(g o qb) = 1 follows from the fact that v~(f g)>v~(f)+ v~(g) and that v~(f) is unbounded. ( 2 ) ~ ( 3 ) : Let (A,r be a wedge in X such that rnk (A, r = dim (X, r = n. Then there exist an open neighbourhood U of O e C " and a morphism q~:

464

S. Izumi

U ~ X such that q)(O)=4, ~ ( U ) c A any g ~ C., o, we have

and grnko~b=n. Since pv, o(g)=vo(g) for

#x, A,r (f) < #v, o ( f ~ q)) = Vo(f~ 4)) < a 2 - re(f)

+ b2

for any f e (9x, r ( 3 ) ~ ( 2 ) : By the desingularization theorem, there exist an open subset U - { z e e m : Izl0) and a morphism 7': U--+Y such that grnk z ~ = m

for any z ~ U , __

l

0(m,). ( g u m - z,.. (gv, H

~o ~~

(Or,. = z~k r

(k=>l>0),

where H = {z c U" z m = 0}. Then there exist 0 < g < e and ~ > 0 such that ~ . d ( ~ o T(z),{)>=lz,,I k for any z6V,, where V = { z e C m : [z[=0 are

(1.5) Theorem. Let (X, 4) be a reduced and irreducible germ of a complex space of dimension n>= 1. I f this satisfies one of the following conditions, it saitsfies the three equivalent conditions in (1.2) also. (4) (X, 4) is quasi-homogeneous. (5) Let Ho: X - + X be the blowing-up of X with center me and let A" X'--+ X be the normalization morphism. Here the condition is that the exceptional fiber C = (H o o A)-*({) is irreducible. (In this case we may put a l = 1.) (1.6) Remark. A germ of a complex space is called quasi-homogeneous if there exist natural numbers n, %, ~2, ..., ~, and polynomials F1, ...,Fp on C ~ such that (X, 4) is isomorphic to the germ of the complex space at O e C" defined by the ideal a = ( F 1..... Fp)(9,, o and F1 o q~.... ,Fpor are all homogeneous, where ~: (C", O)--+(C ", O) denotes the morphism defined by x i = y ~' (i= 1.... , n). (1.7) Remark. If (X, 3) is reduced, we can choose a reduced representative X. Then X is also reduced and has a normalization. Proof. (4) We use the notation in (1.6) and identify (X, {) with the germ of the complex space defined by a. Let (Y,t/) denote the germ of the complex space defined by b = ( F 1 o ~, ..., Fp o ~b) (9., o" Since the induced morphism q~: (9 o ~ (5'~,o is finite and free, it is faithfully flat. Then q0 is injective and bc~ep((9.,o) = q~(a)(-9.,ore q~((.0.,o = q~(a) (cf. [M]). This proves that (*) f o q ~ e b if and only if l e a . Since an element of (_9.,o belongs to q)((_9.,o) if and only if all the monomials appearing in it belong to q~((9.,o), we can decompose fo 4, into homogeneous

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465

terms which belong to ~0((9,,o): fo ~ = }~ J~o q~ i>_o

(degf o 9 = i or f~o 45= 0).

Let ~: ( Y, t/) ~ (X, r denote the morphism induced by 4~ and suppose that v,(fo ~P)=p. Then any representative f e (9 o of f satisfies fo 4~e m~ o + b. Since b is homogeneous, J ~ o ~ e b (i 1.

Proof Since the complex case follows from the real case, we prove the latter. By the rectilinearization theorem of subanalytic sets ([H2], (7.1)) there exists a m o r p h i s m 4)1: R O a R ~ such that 4 ' 1 ( O ) = O , q~I(R+)~A, grnko4) ~ =n, where R+ = {(x 1.... , x,)eR"" X l > 0 ..... x, >0}. Let 4'2: R " ~ R " be the m o r p h i s m defined by

xi=Y~+...+y~_1+y2+y~+,+...+y4,

(i=1 ..... n)

and put 4' = 4)~ o q~2- 4~' R " ~ R " satisfies 45(0) = O, ~b(R") c A w {O}, grnk 0 q~ = n. Then, by (1.1), there exists a > 1 such that PA,O(f) = 1 and b>O such that

PA ~(f) < a. 7~(f) < a. v~(f) + b for any f e(gx, ~. Proof Since A ' = 1 7 - ~ ( A ) - I C [ is open, there exists a~ > 1 for each x e C a - S (S: in ('~)) such that pA,,~(g) X R '

YR

*' , X ,

Definition. Let X be a complex space with an autoconjugation Z, X n its real part, Z its complex subspace and put ZR=ZC~X' R. A subset A ~ ] Z e l is called pe-thick in Z if A c~]Zi[ is not (n i - 1)-sigmafinite for all the irreducible components Z i of Z (n~=dimZi). If A is /re-thick, IZ[ has no proper complex analytic subset which includes A. (3.3) Remark. If there exists a #R-thick subset A in Z, the reduction Zred is defined by a real sheaf of ideals. (3.4) Proposition. Let q)n" Y e ~ X n be the real part of q~: Y ~ X such that X is reduced and d i m ( X , ~ ) > l . I f ~'R(H) is #R-thick in C for some H~[DC~XR[ there exist a >_1 and b >_0 such that inf

v,(fo~)-0 such that #Xe,A,r

v~(f)