Entire Functions of Several Complex Variables (Grundlehren Der Mathematischen Wissenschaften 282)

Proof We first consider the case n = 1. The Riesz Representation Theorem tells us that we can write
Letting z=O, we obtain

or alternatively 1

flog -

1~1<1

I~I

1

2n

2n

0

L1
f (1-
§4. Construction of a System F of Entire Functions such that Y=F- 1 (O)

127

from which it follows that

I

1

1~1<1

log-I"I LlC/>(~)dr2m~l, ..

and hence

_1 2I" (1_1~12) C/>(ei6 ) dOl :54

127t 0

Iz-e1612

for Izl < 1/3.

-

a~l

Let a=_I_ I LlC/>(z) so that if RR> 1/3 and

1 -1 <1. Since e<3, we can ogR

I J IOg(,~Z-!~,)LlC/>(~)dr2(dR

for Izl
If a=O, 1C/>(z)1 < C1+4 for Izl < 1/3. If a=t=O, by the convexity of the exponential function

exp (-1/27t

I

I~I
log (liZ - ~~I) 1-z

Llc/>(~) dr2m)

Iz-~I Llc/>(~»)

~exp ( I~I~R -a log ( II-z~1 27ta dr2(~) ~

I

I~I
(IZ-~I

II-z~1

)-a Llc/>(~) dr2(~)' 27ta

and since a < 1, we have

I

exp-C/>(z)dr(z)~e-Cl-4

Izl
)

I

I

(IZ-~I

Izl
C 2 e- CI -

)-a Llc/>(~) dr (z) 2

27ta

4,

which establishes the result for n = 1. For n> 1, we use polar coordinates to d () obtain I exp( - C/>(z» dr(z)= I dw 2n (rx) I ItI 2,,- 2 exp( - C/>(t rx» r22 t , IIzll <1/3 lIall ~ 1 III <1/3 7t where W 2n is the measure on the unit sphere. 0 Corollary 5.11. If C/> is a plurisubharmonic function in a connected set Q c: ccn , each ZE Q not in the pluripolar set C/> = - 00 has a neighborhood Uz c: Q in which exp( -C/» is integrable. Proof Suppose C/>(zo)=t= - 00. Then by the upper semi-continuity of C/>, C/>(z)(zo)+ 1 for Ilz-zoll (z) - C/>(zo). o

Theorem 5.12. Let C/>E PSH(CCn ) and ZoECC" such that exp ( - C/» is integrable in a neighborhood of zoo Then for 8>0, there exists fEJft'(CC") such that If (z)1 2 exp - C/>(z) f(zo)= 1, (1 + IIzIl 2 ),,+£ dr(z)< + 00.

J

128

5. Holomorphic Mappings from (['" to

Proof Let Wo be a neighborhood of

(['on

Zo

such that I exp [ - tP(z)]dr(z) < + 00; Wo

such a neighborhood exists by Corollary 5.11. Let XECC;;"(CC n) such that O~X~I, X=1 on B(zo,b)~wo' Set I/I(z) = tP(z)+ 2n log liz -zoll +dog (1 + IlzI12).

Then I/IEPSH(CC n) and

L 021/1(Z) WjWk~ L a2dog(1 + Ilz112) WjWk~ j,k OZjOZk

- j,k

EIIWI1 222 -(1 + Ilzll )

OZjOZk

. . 1 I/Wexp-I/I(z) for WECC n. Let /3=OX. Then - I 2 <+2 dr< +00, sInce supp/3 IS compact and zortsupp /3. E (1 + Ilzll ) Thus (cf. Lemma IIU1), we can find u such that au = /3 = a x and 2

exp-tP(z) d J Ilz-luIzoI12n(I+llzI12)' r(z)<+oo, Since /3=0 in B(zo, b), u is holomorphic in B(zo,b) and since Ilz-zoll-2n has a non-integrable singularity, u(zo) = 0, Define f(z)=x(z)-u(z)E.tf(CC n); f(zo)= 1, and exp-tP(z) JIf(z)1 2 (1+llzI12)n+edr(z)<+00,

0

Note that if V(z) is plurisubharmonic, then dO' y(w) HR(z)= V(Z)+C 2n _ 2

I

IlzlI
I

W

_ 11 2n-2 Z

is harmonic, hence CC in Ilwll
dO'y(w) RI -2n+2 U(Z)=C 2n _ 2 I Ilw-zI12n-2 >C2n _ 2 0 (r+t) dO'y(t) R

=C2n_2[O'y(t)(r+t)-2n-2]~+C2n_2

~ Then

I (r+t)-2n+! O'y(t)dt o

I

(r+t)-I

Vy(t)dt~ vy(O) log (1 + II~II)' and

-V=-U-H R

exp(-U)~(I+~)' Ilzll

for rx=Vy(O). Thus if

Vy(z) ~ 2n, the function exp ( - V) is not integrable in a neighborhood of z, Theorem 5.13. Let V E PSH (ce n ) with Co = 1 as a number of complete left stability. Then for every E > 0 and all rx> 0, there exists C(n, e, rx) independent

§4. Construction of a System F of Entire Functions such that Y=F- 1 (O)

129

of rand (n+ 1) entire functions F=(fl' ... , f"+I) such that E(I, V)=F-1(0) and (5,17)

sup log IIFII Ilzll

~

~nMy(r+ (X) +(n +e)

log (1 +r)+ C(n, e, (X) + CF'

r

a-

Proof We shall use the L2-estimates with weight for the solution of the equation (cf. Appendix III). Let e>O and cpEPSH(CC"), Zo a point such that e-'" is integrable in a neighborhood of Zo (Corollary 5.11). Then, by Theorem 5.12, there exists fE.tt'(CC") such that f(zo)=l and (5,18)

IlfII;=

J If(zWe-"'(Z)(l + IlzI1 2 )-n-£dt(z)< + 00. C'

Wr; let H", be the Hilbert space of all fEJt'(CC") such that iifli",<w. The closed set 1'/ C (C" of points z for which e-'" is non integrable in a neigh-

borhood of z is an analytic variety, the set of common zeros of the elements in H",. We recall that

E(2n, cp)cI'/.

(5,19)

On H"" a point ZoECC" defines a linear functional Zo given by zo(f)=f(zo), which is zero for ZoEI'/. To see that Zo is continuous, we use the CauchySchwarz Inequality and set t/I(z)=cp(z)+(n+e) 10g(1 + Ilz112) for e>O: If(zo)1 ~(t2"r2n)-1 J If(z)1 dt(z) by subharmonicity and 8(zo,r)

8(zo,r)

8(zo,r)

so Izo(f)I=lf(zo)I~Cllfll", with C=(r2n)-1/2exptM",(1+llzoll)=C(zo) and C(z) is bounded on every compact subsets of (Cn independantly of f EH",. The linear form Zo thus belongs to the dual space H~ and A",: z-+zEH~ is a mapping of CC n into H~. Furthermore, I'/cI'/'={z: cp(z)= -oo}, so 1'/ is of measure zero in CC". Let zortl'/' and set cp = 2n V. The function e-'" is integrable in a neighborhood of Zo by Corollary 5.11. Thus, we can find fl EH", such that fdzo) = 1, and from (5,18) and Lemma 3.47, we obtain the estimate sup log IfI (z)1 ~nMv(r+(X) +(n+e) 10g(1 + r) + C(n, e, (X)+ C I Ilzll

~r

with C(n, (x, e) =(n + e) log (I + (X) -n log (X -1/210g t 2n' What is more, E(l, V)cfl-I(O), since if vy(z);;;I, for cp=2nV, we have v",(z);;;2n and zEE(2n, cp),

(5,20)

E(l, V)=E(2n, cp)

with r'" non integrable for every point of E(l, V). This shows that fl(z)=O for zEE(l, V). Let Xj be the irreducible branches of /1-1 (0) which are not contained in E(I, V). For every j, we choose a point ZjEXjnC E(I, V). Since co = 1 is a

130

5. Holomorphic Mappings from

cr"

to

crm

number of complete stability for V, we have vy(Zj)=O and v",(Zj)=O with e-'" integrable in a neighborhood of Zj. We can thus find fEH", such that f(Zj) = 1. Then Zj(f)=O defines a proper closed subspace of H",. Since a countable union of closed subspaces is of first category in H "" there exists f2EH", such that f2(Z):f:0 for every j. Then E(1, V)cfl- I (0)nf2- 1 (0)=X 2. We continue in this way by considering the countable family Xj2) of the irreducible branches of X 2 not contained in E(1, V). We choose zjEXy), zj¢E(1, V). As before, there exists f3EH", such that f3(zj):f: 0 and E(1, V)cX 3 = {z: /,.=0, n= 1, 2, 3}. By iteration, we obtain fk' k= 1, ... , n such that

E(1, V)cZ = {z: /,.(Z) =0, k= 1, ... , n} and the set of points in Z n CE(1, V) is discrete. Thus, as before, we find

!,.+ 1 EH", such that fn+ 1 (z):f:O for ZEZ n CE(1, V). Then

n/,.-1(0)

n+1

E(1, V)=

k= 1

and IIFII satisfies (5,17), since each fk does.

D

Theorem 5.14. Let Y be an analytic variety in (Cn of pure dimension p with indicator v(r). Then Y={Z:fk(Z)=O, k= 1, ... , n+ 1}

where the /,. satisfy one of the following estimates:

I

Mk(r) ~ C(t;) log2 rv(r+u),

(5,21)

1+,

Mk(r)~

C(t;, cx)(1 +r)d [ v(t+M)t-d-1dt,

Mk(r)~

C(t;)r 2 u(r + t;).

Remark 1. Theorems 5.13 and 5.14 show that if tEf,,~p(Cn), Acsuppt, and v/(z)~ 1 for ZEA and v/(z)=O for zr/:A, then A is an analytic subset of (CR, and we can obtain A as F-1(0) where log IIFII satisfies one of the estimates in (5,21). Remark 2. The entire functions fj are zero on Y and have no common zero outside Y, but the theorem does not give at ZE Y the value of the integer n+ 1

)

vw(z)~ 1, where W=t log ( j~1 Ifjl2 .

§ 5. The Case of Slow Growth The use of a partition of unity in the construction of U(z) means that there is a certain degree of arbitrariness in the behavior of U(z). It is perhaps worth the effort to try to extend the method of canonical potentials, which

§5. The Case of Slow Growth

131

permit a constructive solution for a Cousin data of finite order, to the case of general analytic varieties. Even in the case of co-dimension 1, the use of canonical potentials loses much of its precision when one treats Cousin data of infinite order. Thus, it is more reasonable to treat only the case of finite order, and we give below an extension of the canonical potential to analytic varieties Y of dimension p such that v1y1(r) is of finite order. We shall use kernels g~(a, z)= -cplla-zll-2P, 1~p~n-1 g~(a,

z)=log Ila-zll,

p=O

and construct as in Chapter 3 the kernels ep(a, z, q), for q an integer,

q~O.

Theorem 5.15. Let t be a positive closed current of degree n - p and U t = t /\ f3 p the trace of t. Suppose that the indicator vt(r) satisjies 00

J vt(r)r-

(5,22)

3 dr<

+ 00.

1

Then for every n, the canonical potential I(z)= -c p Jep(a, z, 1)dut(z) is plurisubharmonic. We shall need the following lemma:

Lemma 5.16. If q = 0 or q = 1, the kernel ep(a, z, q) differs from pluriharmonic function. Proof For q=1" ep(a,z,1)=:~~(a,z)+llall-2P q=O,ep(a,z,O)=go(a,z)+llall p.

g~(a,

Ilal~~+2ReLaizi

z) by a

and for

D

Proof of Theorem 5.15. Suppose first that O¢supp t so that the potential Iq(z) converges. Let X(z) be in CC; such that X(z)= 1 for Ilzll ~ 1 and X(z)=O for Ilzll ~2,_and let Xj(z) = X(z/j). "!:here exists ~ constant M>O such that Mllzll-l~lax) and Mllzll-2~laaxJ Since aXj=o for Ilzll~j and for Ilzll ~2j, there exists a constant C(p, n, X) such that, if Ij(z)= -cpJ Ila-zll-2Pxia)dut(a), we have by Proposition 5.5: (5,23) with Icpj(z)l~

J Iiall

[lIz-all-llaXj(a)I+laaxia)IJllz-all-2Pdut(a), by (5,11).

>j

Let Ilzll ~R andj>2R Then for lIall =r andj>2R, ac

(5,24)

Icpiz)1 ~ C'

J vt(r)r- 3 dr+ C", j

where C' and C" depend only on t, Rand n but not on zEB(O, R). Thus, cP j(z) tends uniformly to zero on every compact subset of ([n. Furthermore, by (5,23), if q=O or q= 1, ep(a,z,q)= -cplla-zll-2p+ Iq(a,z)

132

5. Holomorphic Mappings from

crn to cr m

where lq(q, z) is pluriharmonic (by Lemma 5.16). Thus L(1i' }.) = L(1j, ;,).

Since L(1, }.)= lim L(1i' }.);;;;O by (5,22) and (5,23), the theorem is proved.

0

j-cc OC;

Remark 1. The hypothesis

J v,(r)r- 3 dr < + 00

implies that the terms J I' J 2

I

and J 3 disappear in (5,7), the representation for ioaI. We thus have i) ioaI(z)=q~[t*(nIXP+ I) A q*t], O~p ~n -2, ii) iaaI(z)=nt, p=n-l. Remark 2. Theorem 5.15 was proved under the hypothesis that O~supp t. If in fact OESUPP t, we subtract from I(z) a pluriharmonic function

II (z)= -c p J [llz-all- 2p + 2(1 + IlaI1 2P )-I]dO",(a) (5,25)

(;"

J [llz-all-2p+2(1+llaI12P)-1

II(z)= -cp

(;"

+ 2p(1 + lial1 2p + 2)-I1Re(a, z) ] dO", (a), oc

if

J v,(r)r-

3 dr

<+

00.

Then Theorem 5.15 remains valid.

I

Remark 3. For p =0, we replace -llzll- 2p by log Ilzll and in (5,23), we replace 00

00

J

(1 + liaI1 2P )-1 by 10g(1 + Iiall) if v,(r)r 2dr< + we write I

00,

and if

Jv,(r)r-

3 dr<

+

00,

I

Remark 4. By (5,25) we can calculate }.(1 1,0, r) and the indicator v8 (r) of the i

-

current ()=- aaI q , which has the same Lelong number for each ZECC" as the n given current t. For 1 ~p~n -2, we obtain for r= Ilzll >0: ve(r)=r ~2h

k p(u,r)=(2p)-1

~

/

cu cr

~

cr

}.(1I,O,r)=r

Ju2P kp(u,r)v,(u)du 0

and hp=hp(u,r) is the mean value of Ila-zll- 2p for a

given Ilall=u and ZES 2"(0,r), r=llzll>O. If p=O, we use -loglla-zll and write t instead (2p)-I. The method of the canonical representation allows us, under the hypothesis (5,22), to improve the estimates given in (5,21) for F- I (0) if [Y] is the current of integration on an analytic subvariety of dimension p. First of

§6. The Algebraic Case

133

all, we obtain the equality V(z)=Iq(z) and hence, from (3,14)

(5,26)

Mv(r)=M/(r)~A(p,q)r'I[i vr(s)s-q- 1 ds+r

I

V r (S)S-q-2 dS].

Thus, by (5,21), we obtain Y=F- 1 (0), with oc>O, e>O and M F(r) ~ nM v(r + oc) + e log (1 + r) + C(n, e, 0). We resume this result in the following theorem: Theorem 5.17. If Y is an analytic subvariety of pure dimension p with in00

Jv[Yl(t)t- 3 dt< + 00, then for every e>O, there exists

dicator v[Yl(t) such that

I

a representation Y = F- 1 (0) with MF(r) = sup log 11£11 ~nMv«1 +e)r)+e 10g(1 +r)+ C(n,p, e) Ilzll

~r

and Mv(r) satisfies (5,26). Thus for q=O or q=1 (5,27)

sup log IIFII Ilzll
~nA(p, q)r'I [J v y(t)t- q- 1 dt + r 0

r

Vy(t)C q- 2 dt]

r

+e 10g(1 +r)+ C(n, p, e). Proof As in the proof of Lemma 3.47, by the Mean Value Property for subharmonic functions, we obtain sup IIFII i ~ C(er)-2n exp[2Mv«1 +e) r)+ 2n 10g(1 +r)] IIzll
o

if we choose the ball B(z, r) so be of radius e r. Thus we obtain a much better control of the growth in this case.

§ 6. The Algebraic Case Let Y be an algebraic variety in (Cn. It can be defined as the common zero set of (n + 1) polynomials 1j(z) and the plurisubharmonic function (5,28) defines Y as the set

Y=E(I, V)= {ZEcrn:

v[Yl(z)~

I}

and

v[Yl(z)=min(order lj at z) j

and V verifies lim Mv(r)=p with p=sup(deglj). r-oo logr

134

5, Holomorphic Mappings from

ccn to

CC..

Definition 5.18. A function VEPSH(CC·) is said to be of the minimal growth

Mv(r) 0 'f I'Im-I--=a, I Sa.a>.1 cass ogr

.-ao

Proposition 5.19. Let Y be an algebraic variety (that is. the common zero set of a family of polynomials in CC·) of pure dimension p, Then Y is the set E(I. V) for V of minimal growth class Sa for a ~m C(p). where C(p) is a constant which depends only on p and m = degree Y = max card {Y n Jl}, the maximum being taken over all (n - p) dimensional planes Jl in CC· such that Yn Jl is discrete.

Proof Y is a finite union of irreducible algebraic varieties 1'; of dimension p. A linear subspace Jl of dimension (n - p) cuts 1'; in a finite number of isolated points ni(Jl) (except perhaps for Jl belonging to an analytic set '1i c G._ p(CC·) - cf. Theorem 2.42). For each 1';. mi(Jl) is constant on G._ p(CC·) - '1i and L mi(Jl) = m. For JlEG._ p(CC·). we denote by [Jl] the current of integration on the analytic variety Jl. Let p(z)ECC;;'(B(O. 1» be such that 0 ~p(z) ~ 1. p(z)d't'(z) = 1 and set P.(z) =e- 2 ·p(z/e). e>O. 1.= [Y]*P •• where [Y] is the current of integration on Y. Suppose that Of/; Y. Let Jlo be a fixed subspace of CC· of dimension (n - pl. U(n) the space of unitary matrices. and w the Haar measure on U(n) normalized so as to have total mass equal to 1. Then from the hypotheses. we obtain the inequality

S

m~

SST." [JlO(y-1 (z))]dw(y). B(O,.) U(.)

Let ~o= S [Jlo(y-I(z))]dw(y). which is a positive closed current of type (P. pl. U(.) If IP(CC·) is the projective space and n: CC·-{O} -+IP(CC·) is the natural projection. then ~o determines a current ~o of type (p -1. p -1) on IP(CC·): if qJ is a form of type (n-p. n-p) on IP(CC·). we set (~o.qJ)=(~.qJ(n(z))). Similarily. if [Jlo] is the current of integration over Jlo. [Jlo] determines a current iio of type (p-l.p-l) on IP(CC·). Let qJ be any (n-p. n-p) form with CC oo coefficients on IP(CC·). and set '(qJ)= S qJ(y(z»dw(y). We shall show that

'(qJ)=k(qJ)~·-p for

U(.)

O(=!.- c210g Ilz112. n Since any element of IP(CC·) can be transformed into any other element by an element of U(n) and both '(qJ) and O(P are invariant with respect to elements of U(n). it suffices to show 'AqJ)=k(qJ)O(: for any arbitrary point zEIP(CC·). In particular. we let Z=(ZI' .... =._1.1). We show that the space of U(n -1) invariant forms of type (n - P. n - p) in /\ CC·- I is one dimensional. Let ~z(qJ)=LeIJel"eJ where (e l . . . . . e._ I ) is a standard orthonormal basis in CC·- I• 1=;1 < ...
§6. The Algebraic Case

135

Similarly, by considering a permutation of the coordinates, we see that ClI=CJJ • Hence (z(cp)=k(cp)L;e//\el' We then have

k(cp)=

J (J(P-l/\~= J J (J(P-l/\cp(y(z»dw(y)= J (J(p-I/\cp, 1P(C[n) U(n)

1P(C[n)

since

J

1P(C[n)

(J(n-I = 1 and (J( is invariant with respect to elements of U (11).

1P(C[n)

Furthermore, (eo, cp)=

J ([JlO(y-l(z))], cp(n(z)))dw(y) U(n)

=

J [110]' cp(n(y(z)))w(y)=(,uo, (cp» U(n)

since n(y(z»= y(n(z». Thus (~o, cp) = (,uo, (cp» = (,uo, k(cp )(J(n- P) = (,uo, (J(n- p)«(J(p-I, cp) = «(J(P- 1, cp)

and ~O=(J(P-l. Since ~o and (J(P are both constant on complex lines and both determine the same current (J(P - I in IP «Cn), we have ~ 0 = (J(P. Returning to our original equation, we thus see that m~ J J (J(P.

1'./\

8(0.r) U(n)

If we now let e go to zero, we obtain v[YI(r) is of genus O. Let

m~v[YI(r).

In particular, the indicator

l(z) = - c p J[liz - all- 2p -lia 11- 2p] dO'[YI(a) = -c p Jep(a, z, O)dO'[YI(a) be the canonical potential. Then, by (5,22) and Theorem 5.15, we see that l(z) is plurisubharmonic, and Y=E(V, 1). By Proposition 3.14, since C 2 (p,O) = 1 for all p, we obtain

(5,29)

l(z) 5, c -

i v[YI(t)t dt +c'(p) r 7v[Y(t) dt. t

Pro

r

2

Thus for Ilzll=r,l(z)~mc(p)logr, and so l(z) is of minimal growth class at most Smc(p)' D

Theorem 5.20. Let Y be an analytic variety of pure dimension p such that v[YI(t)~m. Then Y is an algebraic variety which can be defined by polynomials ~ of degree at most nmc(p) + 1/2. Proof Since l(z)= -cpJ[llz-all-2P-llall-2p]dO'y(a) satisfies (5,29), we obtain an estimate of the functions P,. by Theorem 5.17 such that for every k, deglp'.I~nmc(p)+1/2 so by Corollary 1.7, P,. is a polynomial of degree c(p)mn+1/2 at most. D

Corollary 5.21. An analytic subvariety Y is algebraic if and only if v[YI(t) is bounded, and degree Y = lim v[YI(t). t-oc

136

5. Holomorphic Mappings from (['" to

(['m

§ 7. The Pseudo Algebraic Case The study of infinite order differential operators has called attention to a class of analytic varieties Y of slow growth of genus 0 such that, if ex;

h 2 (r)=r

J v[Yj(t)t-

2 dt

r

and r

hl(r)=

Jv[Y](t) t-

I

dt,

ro

This condition is satisfied, for instance, when v[Y](t)~ C(log+ t)'. Then C I(z)~s+1 (log+ r)s+1 (1+£(/')), where £(r)~O, when r~(JJ. We can then define Y by entire functions fk(z) (i.e. Y={z: fk(z)=O, k=1, ... ,n+1}) such that lim M" (r)(log r) -;, = 0 for A> S + 1. This class of functions has the r~oo

following property: for every £ > 0, there exists R, such that if If (z)1 < 1 and Ilzll > R" the distance of z to the null set X f verifies d(z, Xf) ~ 1: Ilzll.

§ 8. Counterexamples to Uniform Upper Bounds We now turn our attention to the study of problems (i), (ii), and (iii) outlined in the introduction to Chapter 5. We begin by producing examples which show that we cannot obtain uniform upper bounds for problems (i) and (ii). We define the entire functions gk(Z) of the variable ZECC by gk(Z) = (1-z2- i ). For 1:>0, let C, be a constant such that log(1 +r)~ C.r' for

TI

i*k

r~

1. Suppose that z is such that 2P~ Izl <2 P + I . Then 00

log Igk(z)1 ~

L

log(1 + IzI2- i )

i= 1 oc

p

~

L log(l +lzI2- )+ L i

log(l +lzI2- i )

i= p+ 1 p

~C,

L

Izl'2-i'+2~C',lzl',

i=1

where C: does not depend on k. Thus for every k and Izl> 1 Igk(z)1 ~ exp (C~/2Izl'!2) ~ C! exp Izl'. Let P(w, c) =

TI c

j= 1

(1) .. mteger. . w --;- for c a positive )

§8. Counterexamples to Uniform Upper Bounds

f

137

If C 1 (, ..
2

I

f(z, w)=

2- Cm gm(z)p(w, Cm),

m=l

which converges uniformly on compact subsets of CC 2. We obtain the estimate for e> 0 00

If(z, w)1 ~ C~ exp Izl'

I

2- m2 Iw+ 11m~ C~' exp(Jzl'+lw+ II')

m=l since, if h.

00

I

2-m2Iw+1Im~

m=l

I

2- m lw+ll m +explw+lJ'.

m=l

00

Let g(z, w)=

fl (1-z2-

i)

and F(z, w)=(g(z, w), f(z, w)), F: CC 2 _CC 2 . Then

i=l

F-1(0)={ (2m,

y): mEZ,j= 1, ... , Cm}

If S(r) is any positive increasing function of r, by setting rm=2m+ 1 and .. card (F- 1 (0) n B(O, r)) Cm=S(rm), we obtam hm sup S( ) ~ 1. Thus, no upper r

r-oo

bound of card(F-1(a)nB(0,r)) by a function of Iiall, r, and log IIFII is possible (independent of F). Let Y c CC 3 be the analytic variety defined by

Y = {(Zl' Z2' Z3): f(ZI' Z2)=g(Z3)=0}. If we choose c 1 = 1, then 00

2-Cm

f(0,0)=-1+m~2 cm!2 (-l)'m, and

Im~2 2-Cm(cm!-l)'m I~(eI/2_1), 00

2

so eI/2~lf(0,0)1~2-el/2. For fixed r, there are at most (log2)-llogr values of Z3 for which g(Z3)=0, IZ31
n

y](r) ~ C~Y)r,

and so r- 2 (1

If]

1 (log 2)

(r)~--

e(Y)

,

logr·r'

where the constants depend only on e but not on the choice of the cm' m~2, as long as C 1 = 1. On the other hand, Y n {z: ZI = Z3} = {(2m, j, 2m): mEZ, j= 1, ... , cm} and thus for any positive increasing function S(r), by a proper

138

5. Holomorphic Mappings from CC" to CC oo

. .. card(Yn{z:z =Z }nB(O,r» choIce of cm ' we obtam lIm sup 3

S()r

r-oo

::::

1. Hence, we

cannot find an upper bound for Y n L n B(O, r) by a function of r depending only on u[y\(r) and valid for all hyperplanes L in G::"; for the variety Y, there exists no upper bound for the growth of n( Y, L, r)= card Y n L n B(O, r) by u[y](r). This leads us to look for estimates valid everywhere outside a small exceptional set. It is in this context that we shall study problems (i), (ii), and (iii).

§ 9. An Upper Bound for the Area of F- 1 (a)

for a Holomorphic Map We begin by proving some lemmas which will provide the foundations of our calculations. In what follows, if D is a domain given by a fCX! function p (i.e. D={z: p(z)
and gradp~O on bdDr. Let () be a positive closed (n-p,n-p) current in G::n with fCoo coefficients. Then i) () /\ (3P-l /\ iap defines a positive measure on bd Dr' ii) if V EfC2 (D), then (5,30)

J

V()/\(3P-l/\iap=J V()/\(3P-l/\ioap

bdQr

Qr

+ J (r-p)ioaV /\()/\(3P-l. Qr

Proof For (1), we have to prove that for zOEbdDr' there exists a neighborhood U of Zo such that I(h)=

J h()/\pp_l/\iap~O bdQr

for hEfCg(U nbdD r), h~O. Given a continuous function qJ(xl' ... ,xm ) in 1Rm , we can calculate Rest. qJlxm=o as a limit Rest. qJlxm=o=lim c £-0

1

JqJ(XI' ... ,Xm)1X (Xm)dXm' e

F~ '(a)

§9. An Upper Bound for the Area of

where

lX(t)E~OC(-I,

+1), SIX(t)dt=1 and

for a Holomorphic Map

e-IIX(~)

139

is an approximation to

the Dirac measure <5(0). Similarily, for a form 'P,

By hypothesis, grad p =F 0 on bd Qr, so we can choose U such that p is a coordinate in U. Then if h is a continuation of h to all of U,h~O, we have

-r)

-

-

p - dp/\h8/\!3p_I/\iap J(h)=limc l S IX (-

e

£-0

by the definition of a positive form, since ii IX ~ O. Note that by this procedure, we have given an orientation to bdQ r which is consistent with Stokes' Theorem S IjJ = S dljJ. bdDr

Dr

To prove (ii), we first apply Stokes' Theorem to the left hand side of (5,30). We obtain, since d(e/\j3P-I)=O,

S ve/\j3P-l /\iap= S dV /\e/\j3P-1 /\iap+ S ve/\{3p-1 /\iaap. bdQ r

Dr

Qr

Furthermore, since (p-r)=O on bdQr' we obtain by an integration by parts:

= -

a

S (p - r) i a v /\ e /\ j3P -I. Dr

o

Lemma 5.23. Let Y be an analytic variety of pure dimension p in cr n and let be the positive closed current of integration over (the regular points of) Y. Let VE~cc(crn)nPSH(crn) with V~O, and set Mv(r)= sup V(z). Then for

elY]

liz II

~r

every y> 1, there exists a constant C depending on y and on n, p, q, such that (5,31)

S

elY] /\ j3P -q /\ (i aav)q ~ C r- 2q aly](r) [M v(r)]q.

B(O. yOr)

Proof Let e,. be a sequence of positive closed currents with '{jX coefficients such that v ~ elY] for the weak topology (cf. Proposition 2.11). Set

e

T.=(ic~V)'

and

a;(t)=

J B(O,t)

T./\e v /\j3P-s.

140

5. Holomorphic Mappings from

cr' to cr m

Then by Stokes' Theorem Mv(r)

J

J J

()v A T.A(3P-s=

M v (r)T.A(3P-s-I A iallzII 2 A()v

bdB(O. ySr)

B(O, ySr)

~

VT.A(3P-s-I AiallzII 2 A()v

bdB(O. ySr)

by Lemma 5.22 (i)

J

~

(r2-llzI12)T.+IAPP-s-IA()v

B(O. ySr)

by Lemma 5.22 (ii), since V ~ 0 y5 r

J 2ta;+1 (t)dt

~

o

after an integration by parts yS,

J1, 2tdt

~a;+I(yS+lr)

)IS +

We thus obtain by iteration:

J

()vAPp-qA(iaaV)q~Cr-2q[Mv(r)Jq

B(O, yOr)

J

()v A

(3P.

B(O, r)

If we now let v tend to infinity, we get

J

()IY)

A (3P-q A (iaaV)q~ Cr- 2q [M v (r)Jq ~ ()IY) A (3P.

~~~M

~~~

Since aly)(r) is an increasing function of r, it is continuous outside a countable set E. If we apply the above inequality for a sequence rm,EC E such that rm , increases to r, we obtain the conclusion of the Lemma. 0 Lemma 5.24. Let Y be an irreducible analytic variety of dimension p contained in a domain Q c cr and let Y' and Y be its singular and regular

points respectively. Let F = (fl' ... , fm): Y -+ crm be a holomorphic map and let f = {ZE Y: rank (efl' ... , afm) < sup rank (afl' ... , afm) = k}. Then for every zef

ZE(Y - f), there exist neighborhoods Uz of Z in Yand Vz of F(z) in that i) F(Uz) is a complex analytic manifold of dimension k in Vz, ii) dim(UznF-I(a»=p-k for aEF(Uz). Proof Let ZE Y - f. Then there exists a neighborhood biholomorphic map ~: »-:-+B(O, l)cCC P, ~(z)=O. Let F(z)=F(~-I(z»,

ZEB(O, 1),

F(z) = eft (z), ... ,fm(z»: B(O, 1) -+ cr m,

»-:

cr

m

such

of Z and a

§9. An Upper Bound for the Area of F-1(a) for a Holomorphic Map

141

and rank (aJ,., ... , aIm) = k. We suppose for simplicity that det

[;~. (O)r=I, ... ,k =t=0 .=I. ... ,k

J

(otherwise, we permute the functions and variables). Thus, by the Inverse Mapping Theorem, there exist neighborhoods T and T' of 0 in B(O, 1) such that the map 1t: T -+ T' given by z' = (J,. (z), ... , J,.(z), Zk+ I' ... , zp) is a biholomorphic homeomorphism. Set f/(z)= ~o1t-I(Z). Then afj* =0 in T' for j k, for otherwise rank [afl*, ... ,afm*]>k for at least one point of T', since fj*(z)=Zj' j=I, ... ,k. Thus for j>k, fj* depends only on the variables (ZI' ""Zk)' and if Uz=~-l(T), F(Uz)=F(T)=F*(T')= {WEce m : w j = fj*(Wl' ... , wk),j=k+ 1, ... , m} ~

of F(z). We can

dim (Uz I1F- 1 (a»=dim (T' I1F* - I (a» = p -k.

o

is a complex manifold of dimension k in a neighborhood choose fl * , ... , fk * as local coordinates on ~, and hence

Lemma 5.25. Let Y be an analytic variety of pure dimension p in a domain Q c ce n and let F: Y -+ ce m be a holomorphic map. Let Y' be the singular points of Yand Y={ZEY-Y': rank[afp ... ,afm]<m}. Then there exists an Fa-set E of Lebesgue measure zero in cem such that for a¢E, if YI1F- I (a)=t=0, dim(YI1F- I (a»=p-m and no irreducible branch of (YI1F- 1 (a» is contained in Y' u Y. Proof Let lj be the irreducible branches of Y and Y; be the singular points of lj. Set k j = max ra,nk [afl' ... , afm] and Yj- Yj

Yj= {ZE lj- Y;: rank [afl' ... , afm]
We use induction on the dimension p. If p= 1 and kj=O, then the 1;, i = 1, ... , m are constant on lj and so ~(lj) = aj • If kj = 1, then ljl1 Y; is a countable set. For a¢ F(lj) F(lju Yj), dim(F-l(a)11 Y)

f~nctions

U

U

kj=O

kj=1

~

1 and no branch of F- 1 (a) 11 Y is contained in Yu Y'. We now suppose that p is arbitrary. For any aEce m such that F- 1 (a) 11 Y =t=0, dim(F-l(a)11 Y)~p-m. If kj<m, it follows from Lemma 5.24 that Ej=F(lj-(Yju 1]'» is an F,,-set of Lebesgue measure zero, since a proper complex manifold is an Fa-set of Lebesgue measure zero. We can find a countable number of open subs~ts Qij of cem and analytic varieties Jt;j in Qij such that dim Jt;j ~ (p -1) and lj u lj = U Jt;j. It follows from the induction =

i

hypothesis that we can find Fa-sets Ej of Lebesgue measure zero in cern such that if F- I(a)l1W; j =t=0, dim(F-I(a)n_W;j)~p-l-m for a¢Eij' If kj=m, set Ej = E;j and if kj<m, set Ej=Ej E;j' Then for a¢ E j , if

U 1

U i

U j

142

5. Holomorphic Mappings from

cr" to cr

m

F-'(a)fl Y*0, dim(F-'(a)fl Y)=p-m and no branch of F-'(a)fl Y is contained in UOJ U l'j). 0

Theorem 5.26. Let Y be an analytic variety of pure dimension p in CC n and let F: Y --+CC m be a holomorphic map, m <po Set MF(r)= sup IIFII. For aECC m, let

°

liz II ~r

a be the positive closed current of integration over (the regular points Oa/\{3p_m' Let e>O, {3>1 be given. of) F-'(a)flY and set a[Y](a;r)= Then the set B(O. r)

J

s={aEcc m: dimF-'(a)*p-m and G"[y](a; r) ~O} lim sup ---c.-:::-:::-----:-;;.----:-:-~.:--,-.,__':_-___,__...,.,__-.,_____-_r_ r-oo r 2m(logrla[y]«(1 +e)r)(logMF«1 +e)r»m is of Lebesgue measure zero. Proof Let V(z)=log(1 + 11F112). Then for Ilzll ~r, O~ V(z)~log(1 +MF(r)2). It follows from Lemma 5.23 (5,31) that for y< 1, there exists a constant C such that C[log+ MF(r)ra[y](r)'r-2m~ J (ioaV)m /\ O[y] /\ {3p-m' B(O, ymr)

It follows from Lemma 5.24 that if suprankF*m, then F-'(CC m ) is of y Lebesgue measure zero. Let '1.E~;: (B(O, r)) be a sequence of functions such that '1. increases to the characteristic function of B(O, ymr) -(Y' u f). For a fixed v, if ZESUPP '1 v' we can find a neighborhood Uz~B(O,ymr)-(Y'uf) and coordinates (g" .. ·,gm' .. ·,gm+" ... ,gp) in Uz such that

{z: gm+' = ... =gp=O, Igil
Lcx =

J i '1v(iaaV)m /\ O[y](z) /\ Pp-m CX

= Hic w 8w log (1 + Ilw 112)]m JCX i'1AY](w) /\ {3p_m(z(w» and summing over i, we obtain

J'1v(icaV)m /\ 0IY) /\ Pp-m = J [iowa log (l + Ilwl12)]m J'1vOIY](w) /\ Pp_m(z(w». w

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes

Thus

J

143

(io8v)m /\ elY] /\ f3 p -m

B(O. ym,)

J

= lim v~ 00

= lim

'1v(io8v)m /\ elY] /\ f3 p -m

B(O. ym,)

J[io

w

8w log (1 + Ilwl1 2)]m

J J

'1AY](w) /\ f3 p_m(z(w»

B{O,ym,)

v~oo

= Hio w 8w log (1 + Ilwl12)]m lim \' -+ 00

'1AY](w) /\ f3 p_m(z(w»

B(O, )lmr)

J

= O'[Y](w; ym r )[io w 8w log(1 + Ilwl12)]m by the Monotone Convergence Theorem. Set "m=(l +::)-1/2, r t =(1 +£)'/2 and Ft = {aE(Cm: O'[Y](a; rt ) ~(logrt)'J' CO'[Y]((l + c)I/2rt)rt-2m [log+ MF«l +c)I/2rt)]m}

where

f3'=C;f3).

Set I1(W)=[io w 8w log(1+llwf)]m, which is a positive

measure. Then by Lemma 5.23,

JO'[Y](a; rt) dl1(a) ~ C rt- 2m [log+ M F«l + C)I/2 rt)]m O'[Yj(rt (1 + C)I/2) and hence 11(~)(log rl' C[log+ M F«(1 +

~

C)I/2 rt)]m rt- 2m 0'[Y]«1 + C)I/2 rt )

J O'[Yj(W; rt ) dl1(w) Fe

and so 11(~)~(logrt)-P'. Let E t =

UFj.

00

Then I1(Et)~

j~t

!~M(c,b), I1(Et)~b.

For a¢Et and

I

(logrj)-P' and for

j=t

!~M(c,b)

O'[Y](a; rt')~ C(logrtl'[log+ MF«l +c)I/2rt·)]mrt-;-2mO'[Y]«(1 + C)I/ 2rt') and thus for rE[rt ._ 1,rt .)

O'[Y](a; r) ~ C(log rtl' [log+ M F«(1 + C)I/2 rt.]m r; 2m O'[Yj«l + £)1/2rt') ~ C(logr)ll'[log+ MF«l + c)r)r O'[Yj «(1 +c)r)·r- 2m .

nE "'-

Thus rff=

t=

t

and l1(rff) =0.

1

o

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes Let Gq«Cn) be the Grassmannian of all complex linear subspaces of (Cn of dimension q. Then Gq(CC n) can be given the structure of a compact com-

144

5. Hoiomorphic Mappings from

crn

to

cr m

plex manifold of dimension q(n -q). We can describe the local coordinate neighborhoods VI as follows: let 1= (il < ... < iq)' where i j ~ n, and let I if p= i. Ip=(/pi'" .. ,lp"), p=I, ... ,q with Ipij= { if P=l=i~ for ijEI and Ipij=cpj for

°

ij¢I and identify with cECI:q("-q) the linear space spanned by Ip, P = I" ... , q.

To see that Gq(CC") is a compact manifold, we embed it as a submanifold of a complex projective space. Let CI:s(q) = /\ CI:" be the linear subspace spanned q

by the exterior products of degree q. If II"'" Iq are elements in CI:", we associate the subspace spanned by II' ... , Iq with n(/I/\'" /\ lq), where n(CCS(q) - {o})-> IP (CI:S(q)) is the projection of CI:s(q) into its projective space. Note that if 11' ... , Iq and I;, ... , I~ span the same subspace, then 11/\ ... /\ Iq = C(I; /\ ... /\ I~), when: C is the Jacobian of the transformation (II' ... , lq) -> (I; , ... , I~), and so n(/I /\ ... /\ Iq) is well defined and furthermore is holomorphic on VI' (For a more complete discussion of Gq(CI:"), see [H]). Although the notion of the Lebesgue measure of a set in not well defined on a complex manifold, since it changes with a change in the choice of local coordinates, the notion of a Lebesgue measurable set and a set of measure zero are invariant with respect to a local change of variables and so preserve their sense on complex manifolds. If O, we set P(z) =

C; I JIog I~~: Zi Wi + z"1
Lemma 5.27. The junction V'P(z) is plurisubharmonic in CC" and C(ioo in CI:" -{O}. Proof It is enough to prove that V'P(z) is locally in C(iOO for ZECC" - {O},

and what is more, it is sufficient to show that .1 VEC(ioo in CI:" - {O}, since if zoECI:"-{O} and aEC(ioOO(CI:"-{O}) is such that :l=1 in a neighborhood of Zo, then it follows from Green's Theorem that for C2 "_ 2 = [(2n- 2) 0)2"]-1 _ -1 a(z)V(z)= C 2 "-2

J Ilz-z'112"

2.1(a(z')V(z'))dr(z')

and

G GZ P(a(z) V(z)) =

_

-1

G

C2"- 2JI z _ Z' 112"- 2( -1)1111 Cz'P (.1 a(z') V(z'))dr(z'),

which is continuous for every p. Suppose zo=l=O. If (ZO)I = ... =(ZO)"_I =0 and z.=I=O, then zo¢supp.1 V and hence .1 V(z')=O in a neighborhood of Z00 Suppose then that (ZO)k=l=O for k =1= n. Set "- I s=

L Zi w;+z"'

;=1

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes

145

and let ~k(W, z)-+(w, z) be the map described above and Jdw, z) the Jacobian of ~;: I. Since

AzIOgl~t: Zi Wi+ Znl=2nb Ct: ZiWj+Zn) =2nb(s), where b is the Dirac measure at zero and b(s) represents the current of integration over the hyperplane s = 0, we have

J

A V(z) = 2n b(s)IP(~;: 1 (w, z)) IJk(w, zWdr(w) =

Jb(s)ip(w, z)dr(w),

where ip(w, z)= 2nIP(~;: 1 (w, z)) IJk(w, ZWEct'oo in a neighborhood of zo, since

o

ozil A V (z) =

0

Jb(s) ozil ip(w, z)dr(w).

o

Suppose now that K c:
Lemma 5.28. There exist constants C I' C 2 and v0 such that CI+logllzll~V'Pv(z)~C2+logllzll

for v~vo'

V'Pv(uz) = V'Pv(z) + log lui, it is enough to show that CI~V'Pv(Z)~C2 for Ilzll=1 and v~vo' Let A be such that Kc:B(O, A). Then V'Pv(z)~(A+ 1)+log Ilzll. Let IXEct';' (B(O, 3A)) such that O~IX~1 and IX= 1 on B(O,2A). Then if Ilzll=1 and Vo is so large that supplXvc:B(O,2A) for v~vo' we have n-I ZjWj+z n l V'Pv(z)=C;v JIog i~t2A+l) IPv(w)dr(w)+log(2A+I) Proof

Since

L

n-I

~ C;vl

L ZjWj+zn Jlog l-j~-I---IIX(w)dr(w)+log(2A+ 1)

(2A+l) n-I ZjWj+z. ~[m(K)]-1 JIog i~1 :x(w)dr(w)+log(2A+l) (2A+ 1)

L

~

C1

by Lemma 5.27.

o

Lemma 5.29. Let Y c:
146

5. HoIomorphic Mappings from

cr" to cr m

Proof Let lj be the irreducible branches of Y, each of which is of dimension p. Let WECC n. Then if ljn{z: it ZiWi=O} is non-empty and is not of pure

dimension (p-l), we have ljC{Z: it, ZiWi=O} The set Aj={WECC n: ljc{z:

L ZiWi=O}} Yc

is

a

linear

subspace,

and

Aj=FCC n,

furthermore

since

n {Z: ±ZiWi=O} and 0= w'eb n {Z: ±ZiWi=O}. If n is the projection

weA

i= 1

i= 1

of CC n into IP(CC n), then n(A j) is of measure zero in IP(CC"), and if Y~ U n(A), dim(Ln Y)=p-l. j D

Lemma 5.30. Let Y c

cc

be an analytic variety of pure dimension p and O[YI the closed positive current of integration over the regular points Y of Y. Then the set of WECC n- 1 such that the simple extension of n

(2n)-' iaalog

to

cc

n

I~t: ZiWi+Znl

A

O[Y]

as a positive closed current of degree (p -1) is not the current of

integration on yn{z:

~t: ZiWi+Zn=O} is of Lebesgue measure zero.

Proof We begin by remarking that if fEJf(B(O, 1», then f is constant on every connected component of the analytic variety

~f = ... =~f =o}. uZ

Z={zEB(O,I):

uZ,

n

Indeed, for every ZEZ, the set of regular points of Z, there exists a neighborhood Uz of Z and a biholomorphic map, n z: Uz-+B(O, l)cCC" where s depends on z. Set J(u)=f(n;l(u»

.

aJ(u)

In

B(O, 1). Then -~-= oU j

J is

of oz. L -;-~=O so n

i=1 uZi uU j

constant in B(O, 1), and hence f is constant in Uz and thus on every component of Z. Let Y' be the analytic variety of singular points of Y, which is of dimension at most (p -1). Set Al ={WECC n- 1 : Y' n{z:

dim(Y')n{z:

and

~t: Zi Wi+ Zn=0}>(P-2)}

A2 ={WECC n- 1 : yn{z: dim
~t: Zi Wi+ Zn=0}=F0

J:

~t: Zi Wi+ Zn=0}=F0

and

ZiWi+Zn=O}>(P-l)}

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes

147

It follows from Lemma 5.29 that Al and A2 are of Lebesgue measure zero in cr n - I . Let ZE Y-0. For simplicity, we assume ZI =1=0. Then there exists an open neighborhood Uz of Z in Ysuch that inf{z~: Z'EUz}>O and a biholomorphic map ~z: Uz--+B(O, l)ccr p • For W fixed, set

n-I

{

}

Yw= uEB(O, 1): ;~I z;(u)w;+zn=O

(n-I

)

}

~ {UEB(O,I):-. a .I Z;(U)W;+Zn =O,j=I, ... ,p, Yw= au) 1 1=

and A={WEcr n - l : YwcYw}' Then A is an Fa-set (and hence measurable), since Fv={w: Yw nB(O,I-I/v)cYw nB(O,I--=-i/v)} is closed and A=UFv' Let - I Z;(U)W; - Zn(U) f(u)

n-I

in B(O, 1). Suppose for w'=(w 2 ,

... ,

wn) fixed, w=(w I , w')EA. Then for uEYw

af(u) -I ( n~1 az;w; OZn) --=ZI(U) - L. - - - CUj ;=2 aU j OUj +ZI(U)

-2 (n-I az; .I Z;(U)W;+Zn(U) ) -a .=0; 1=

But we have seen that

2

j=I, ... ,p.

Z)

f == constant on each connected component of

{ UEB(O,I): of =O,j=I, ... ,P}, and thus for every w', there are at most a oU j countable number of WI = f(u) such that (WI' w')EA, so A is of measure zero in cr n - I . We now choose a countable dense set Z;E Y-0 and neighborhoods 00

UZi and sets A; as defined above. Then A 3 =

UA; is of measure zero. ;=1

Suppose that w¢AI uA 2 uA 3 . Let Zw= yn{z:

~t>;W;+Zn=O} with Zw as

regular points and Z~ as singular points. Then, since w¢A z , It follows from above discussion that since w¢A3'

dim(Z~)~p-2.

1 _ In_1 1 -icc log I Z;W;+Zn /\8, 2n ;= 1

is the closed positive current of integration over Zw in cr n - Y' - Z~. Its simple extension t is a closed positive current of degree (p - 1) in cr n - Y', since dim(Zw)~p-2, and the simple extension of t to cr n _ Y' is again a closed positive current of degree (P-l), since dim(Zwn Y')~p-2 for w¢A 2 (cf. Chapter 2). Lemma 5.31. Let Y c cr n be an analytic variety of dimension p ~ 1 and let elY] be the closed current of integration on Y. Let Kccr n - I be a compact set and

148

5. Holomorphic Mappings from q:. to q:m

IP.E~; (CC n - l ) a sequence of functions such that IP. decreases to XK' the characteristic function of K. Then for r> I, if VIP. is defined as in Lemma 5.27, lim J__ ioaVIP'(z)"OIYl"PP_1 .-OOB(O,r)-B(O,I)

2n [uIYl(w; r)-ulyl(w, 1)]XK(w)dr(w), m(K) 1["-1

J

=--

where Ow is the current of integration over the set yn{z:

~r.l ZiWi+Zn=O} 1=

and

uIyl(w;r)=

1

J

0w"Pp-l'

B(O,r)

Proof Let Y' be the singular points of Y and let 1/1 "E~; (B(O, r)) be a sequence such that the 1/1" increase to X(B(O,r)-Y'-B(O,I))" An integration by parts gives us

J1/1"ioaVIP. "Oly] " Pp-l = JVIP'ioal/l" " 0IYl" Pp-l' and it follows from Fubini's Theorem that

JVIP'ioal/l""olY]"P P_1 =C;;}

I["t

[JiOgl:t: ZiWi+Znlioal/l."OIY]"PP_l]IPV(W)dr(W)

l

A second integration by parts gives us

C;;.1

I:t:

)_1 [J 1/1"ioalog =C;;}

Zi Wi + Znl " 0ly]" Pp-l ] IP.(w)dr(w)

+

)_1 [JiOgl~t: Zi Wi+ z

oal/l""OIYl"PP_l]IP.(W)dr(W).

It follows from Lemma 5.30 and the Monotone Convergence Theorem that lim

,,-00

J1/1"ioaVIP." 0ly]" Pp-l = C;;.1 )_

1U~~ J1/1" ioalog

I:t:

Zi Wi + Znl "OIYl" Pp-l ] IP.(w)dr(w).

or

J__ ioavlP." 0IYl" Pp-l B(O,r)-B(O,I)

=2n C;;.1

I["J- [UIyl(W; r) -Uly](W'; I)] IP.(w)dr(w). 1

Finally, it follows from the Lebesgue Dominated Convergence Theorem that lim Jic2VIP'"O[Yl"Pp_1

v-x

=2nm(K)-1

I["J- [u[Yj(w;r)-u[Yj(w; l)]XK(w)dr(w). 1

0

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes

149

Lemma 5.32. Let YcCC n be an analytic variety of pure dimension p ~ 1 and let K c CC n - 1 be a compact set and q>.E~.f (CC n - l ) a sequence of functions such that q>. decreases to XK' Then if k> 1 is a constant, there exist constants 1'1,1'2,1'3 and 1'4 which depend only on K and k such that for r> 1

J

r- 21'1 0"[Y](Y2r) ~ 1'3 O"[yp)+

iaaV
8(O,r)-B(O,I)

and hence

J

.

.

21t [O"[Yl(w,r)-O"[y](w,I)]XK(w)dr(w) r -2 1'1 0"[Yl(Y2r)~Y30"[Y](2)+ m(K) ~ 1'3 0"[Y](2) + Y~O"[Yl(kr)r- 2.

Proof Let A; = {ZECC n : V
Since by Lemma 5.27, V
bdA~

By Lemma 5.22, iaV'I'v A 0,.. A{3P-I is a positive measure on bdA; and hence by Lemma 5.22 (ii) t.(r) J iaV
bdA;

~

J Il z I1 2iaaV'I' vA{)/lA{3P_I A~

+

J (logr-V
By grouping these inequalities, we obtain for s < 1 t,(r)J iaaV
J O,.A{3p'

AVsr

By Lemma 5.28, for v~vo' CI+logllzll~V'I'v(z)~C2+logllzll, and so A;cB(0,reC2 ) and B(O,sre-c')cA;r' Thus r2

J

iaaV
8(0, r)

Let

"'E~'" (B(O,

J (),. A{3p. 8(0, ,2r)

2» such that", == 1 on B(O, 1). Then

J",ioaV
J

8(0,2)

(J,..

A f3 p

(cf. Chapter 2),

ISO

5. Hoiomorphic Mappings from

cr" to cr

m

since lV'I'vl is bounded on supp iaaf/ independantly of v for v ~ yo' We now let Jl.-H~) and obtain

__ J__ iaaV'l'v "Oly) " Pp -1 -Y3ux(2)~ Y1 UIY)(Y2 r)'r- 2. B(O,r)-B(O,1)

Since this is true for a dense set of r for which the function

__ J__ iaaV'l'v" °IY] "

Pp -1

B(O,r) - B(O,1)

is continuous, we can choose a sequence rt increasing to r for which the inequality is valid. This establishes the left hand inequality for all r, To establish the right hand inequality, we note that by subtracting a constant, we may assume that logllzll-C3~V'l'v(z)~loglizll for v~vo and C 3 ~o. We apply Lemma 5.22 (ii) to V'I'v - log Ilzl! and obtain

J

(Pr 2 -ll z I1 2 )iaaV'I'v"O,,"PP _l

B(O,kr)

J (Pr 2 -llzI1 2)iaalog Ilzll" 0"" P -1 J (V'l'v -log Ilzll)O,," P -1" iallzl1 2 - J (V'I'v-log IlzlI)O,,"Pp ' p

B(O, kr)

p

bdB(O,kr)

B(O,kr)

By Lemma 5.22 (i), the first term on the right hand side is negative, and so

(k 2 _I)r2

J__ iaaV'I'v"O,,"PP_l B(O,r)-B(0.1)

~C3

J O""p +Pr 2 B(O,J O,,"iaalog Ilzll "P J 0,," Pp (cf. Chapter 2). p

B(O, kr)

~ C4

p

-1

kr)

B(O,kr)

We now let Jl. tend to infinity and obtain

J__ iaaV'I'v" °IY] " Pp -1 ~Y4r-2 B(O,r)-B(O, 1)

J

°IY] " pp •

B(O,kr)

Finally, we choose an increasing sequence of r for which uIy)(kr) is con0 tinuous and apply the above inequality.

Theorem 5.33. Let Y cCC· be an analytic set of pure dimension p. Then for O~q~p, e>O, and P> 1 · { LEG._q(CC): I Imsup r-ex.

r

-2 q (1

uILn y)(r) ogr )pq uIY) «(I +e»9= r

o}

is of Lebesgue measure zero in G._q(CC). Proof We shall show by induction on q that L) [G (..... )]q. I' {L -(L l ' ... , q E .-1 \l.- • Imsup r-x r

2q

U[YnL, ... nLq)(r) ..... O} (1 ogr )pq u[Y] «1 +e )q/P);r

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes

151

is of Lebesgue measure zero for the product measure on [G n _\ (CC)]q. Let q = 1 and suppose that measure (E={LEGn_dCC): lim sup '-+00

r

-2

I CT~nLl(r)

(ogr) CT[Yl«(1 +e)

lip

r)

=l=0}>0).

Then we can find a compact set K c E c U0 of strictly positive measure, where

Let rt=(1 +e)t/2p, k=(1 +e)1/2p. By Lemma 5.32, we obtain

J [O"[Yl(w; r)- u[Y](w; I)JxK(W) d r(w);;:;; C•. Ku[Y]((1 + ::)1 /2Pr)· r- 2.

a:n - ,

By reasoning as in Theorem 5.26, we conclude that . CT[Yl(W; r)-CT[Yl(W; I) } { WEK: hmsup -2(1 )fl «I )l lp )=1=0 '-00 r ogr CT[YI +e r is of Lebesgue measure zero, which is a contradiction. Thus measure (E)=O. We now assume the induction hypothesis for t;;:;;q -1. Suppose now that L¢E as defined above. By the induction hypothesis, the set

E = {i(L p ... , L q_ 1)E [G n- 1 (CC)]q-l: lim sup '-+00

CT[YnL,n ... nLq_d(r) q-l r- 2(q-l)(logrf(q-1l CT[Yl«1 + e)2P r)

=l=0}

is of measure zero. This establishes the induction hypothesis for all q. To terminate the proof, we show that if '1: [G n_ 1 (CC)]q is given by

f/(L 1 ,

...

,Lq)=

n

L i , then for EcGn_q(CC) such that the 2q(n-q) Lebesgue

i=l

measure of E is positive, the product measure of '1-1 (E) is positive in [G n_ 1 (CC)]q. Let ccs(q) = A cc n as a linear space. For 1= (i\ < ... < iq), we let -

q -

I=U\< ... <jn-q:ji¢I} and sign (I, I) the sign of the permutation (I, ... ,q)-+(l,l). If e\, ... ,en is the standard basis in CC n, we set el =ei , /\ ... /\ eiq , and we ~efine the map ~q: CCs(q)-+ccs(n-q) to be the linear map such that eq(e l ) = sign (I, l)el' Let 1tq: CCS(q) - to} -+ IP(CCS(q). Then the map ~q=1tn_qeq1t;;\: Gq(CC)-+Gn_q(CC)

is holomorphic and ~q(L)=H, the space orthogonal to L. To see this, we calculate in local coordinates. Let UI be the coordinate patch defined at the

152

5. Holomorphic Mappings from

cr"

to

cr

m

beginning of Section 9. For simplicity, we assume that 1= (1, ... , q). Then to

we associa te

(-CI(n_q)' ... , -cq(n_q)' 0, ... , 1) which is clearly a holomorphic homeomorphism of VI onto VI' We define the holomorphic map WI: [Gn _ 1 (
wJlL 1, L 2 ,

... ,

Lq) = ~ 1 n;;.! 1 (L 1) /\ ... /\ ~ 1 n;;_\ (Lq).

Then w11(0) is a proper analytic subset of [G n _ 1(
Theorem 5.34. Let Y c
is of Lebesgue measure zero in Gn_q(
Proof We begin by showing by induction on q that {L=(L I ,

... ,

Lq)E[G n _ 1 (
lim (J [YnLtn ... nLqj (k rm )

m~

xc

(J[Yj(rm).r,; 2

°for all k>O

}

is of Lebesgue measure zero for the product measure. For q = I, we suppose that

Then we can find a compact set KcEnVo, such that measure (K»O, where

§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes

153

(f (tr ) . Let f~I)(L)= ~YtLl m, tEZ. Then for every t, hm f~')(L)=O for LEK. It r m (f[Yl(rm)

m_ 00

follows from Egorov's Theorem that we can find K' c K compact with measure (K') >0 such that lim f~')(L)=O uniformly on K' (for t fixed). By m-oo

Lemma 5.32, we can find constants C l , C 2 and C 3 which depend only on K' such that

r;;2 C l (f[Y](rm)~ C 2(f[y](2)+ J[(f[Yl(w; C 3 r m)-(f[Yl(w; l)]xdw)d-r(w), which contradicts the fact that for t~ C 3 , lim f~')(L)=O uniformly on K' m-oo

Thus measure (E)=O. We now assume the induction hypothesis for s~q-l. Suppose that L¢E as defined above. Then there exists ko>O and a sub. . (f[y Ll(kor ,) sequence rm, (which depend on L) such that hm ()~2 *0. By the induction hypothesis, the set m-oo rm , (f[Yl rm

t,)

EL =

{L = (Ll' ... , Lq_1)E[G lim

m'-oo

n_

1(CC)]q-l:

(f[LnYnL,n ... nL.l(kkorm,) -2(q-ll

r m,

(f[YnLl

(k ) orm'

=0 for all k>O}

is of Lebesgue measure zero. The conclusion now follows in the same manner as that of Theorem 5.33. 0 Of course, for the purpose of applications, one wants to choose the sequence rm so that (f[y](rm) has "maximal growth". As an example of an illustration, we present the following corollary:

Corollary 5.35. Let Y c ccn be an analytic variety of pure dimension p such that r- 2p (fy(r) is of finite order p. Then the set of LEGn _ q, O
r 2(p-Q)(f[YnLl(r) is of order bigger than p is of Lebesgue measure zero. Suppose that rm is an increasing sequence such that (f[Yl(rm)~i.r!(rm)+2P, A. > O. Then there exists a set E c Gn _ q of Lebesgue measure zero such that for L¢E, we can find a subsequence rm , for which lim (r ,)-2(p-q) (f[YnLl(rm,) >0. m (rm,)p(r m ,)

m'-x,

It then follows from Theorem 1.18 that

lim sup r-oo

(f

(r)'r- 2(p-q) [YnLJ

rP(r)

>0

.

D

154

5. Holomorphic Mappings from (C" to (Cm

Historical Notes The representation with growth conditions of an analytic subvariety X of (Cn of co-dimension greater than one poses certain delicate problems which are only partially resolved. First of all, X is not in general a complete intersection, and secondly even when it is, the Lelong- Poincare equation Ox=iaaloglFI (studied in Chapter 3) for X a subvariety of co-dimension one (which reduces the problem to a linear equation) is replaced by a nonlinear equation of complex Monge-Ampere type: 8x =(iaalog 11F11)q for X a complete intersection of pure dimension q. Two results which were published the same year (1972) have enriched the theory of holomorphic mappings. One is the counter-example of Cornalba and Shiffman [lJ, which shows that there is no control (even asymptotically) as a function of I F I for the growth of the analytic subvariety F- 1 (a) of co-dimension two for F an entire holomorphic mapping of (C2 into (C2. An average estimate has been obtained by Carlson [2J and Gruman [8, 12]. The latter shows that the set of a for which no such estimate is possible is pI uri polar. Another result, due to Skoda [2, 3J was thus surprising. He showed that for X an analytic subvariety of CC n of pure dimension q, 0 ~ q ~ n -1, one can express X as X = F- 1 (0) for F an entire holomorphic mapping F: (Cn->cc n + 1 where IIFII has an asymptotic growth controlled by that of vx(r), the indicator of X. A comparaison of the results shows the "statistical" nature of the control (which is the point of view adopted by Stoll [8J in his study of the transcendental Bezout problem). The study of the trace of an analytic subvariety X on linear subspaces (undertaken in § 9) was also motivated by the counter-example of Cornalba and Shiffman [1]. An upper bound outside exceptional sets was first given by Carlson [2J using the Crofton Formula and general properties of positive monotonic functions. Much more surprising was the fact, first shown by Gruman [8, 11, 12J, that one could obtain a lower estimate of the trace of X on linear subspaces outside a small exceptional set. In fact, one can obtain estimates outside sets much smaller than those of Lebesgue mesure zero. For results in this direction, we refer the reader to the articles of Molzon, Shiffman and Sibony [IJ, Alexander [2J, Gruman [8, 11, 12J, and Molzon [4].

Chapter 6. Application of Entire Functions in N urn ber Theory

As has already been noted, one of the basic motivations for studying entire functions of finite order is that all of the familiar transcendental functions fall into this category (and what is more, for meromorphic functions of finite order, one arrives at the familiar elliptic functions). Thus, by studying the algebraic or arithmetic properties of entire functions of finite order, one implicity studies the algebraic or arithmetic properties of the fundamental transcentental functions. This in turn has a wide variety of applications in transcendental number theory. We give here an illustration where the methods of holomorphic functions of several complex variables were used to solve a problem which contains several classical problems of number theory. The following method uses an idea which goes back to c.L. Siegel and consists of constructing an entire fonction with "many zeros" in the class of those whose values have given algebraic properties. We present here the solution of E. Bombieri in CC· given in connection with his joint paper with S. Lang. The solution relies heavily on the technics of positive closed currents and the resolution of the operator aa.

§ 1. Preliminaries from Number Theory A complex number (X is said to be algebraic of degree p if there exists an irreducible polynomial P(x) = a o x P + ... + a p with ak a real integer and ao =1= 0 such that P(oc)=O. If we suppose that the ak have no common divisor, then P is unique and is called the minimal polynomial for (X. If a o = 1, we say that IX is an algebraic integer. A complex number which is not algebraic is said to be transcendental. We denote by Q the field of rational numbers. If K is a subfield of CC, we say that K is an extension of finite type if there exist Yl, ... ,YmEK such that K=Q(Yl, ... ,Ym). If K is a vector space of finite dimension over Q, we denote this dimension by [K: Q]. If K, an extension of finite type, is a field of algebraic numbers, then it follows from the Theorem of the Primitive Element that K is then simple, that is K = Q(oc) for some algebraic oc, and in addition we then have Q(oc) = Q [oc], that is the field generated by adding (X to the rationals is the same as the ring

156

6. Application of Entire Functions in Number Theory

generated by adding IX to the rationals. We then say that K is a number

field. Proposition 6.1. Let IX be an algebraic number. Then the following two

properties are equivalent: i) there exists a non-trivial unitary polynomial Q(X)EZ [x] such that Q(IX)=O; ii) there exists a non-trivial Z-module M generated by a finite number of algebraic elements such that IXM c M.

Proof If p is the degree of Q in i), then M=Z[IX, ... ,IX P] is a non trivial Z-module of finite dimension such that IX M c M, so i) implies ii). To see m

that ii) implies i), we let v l '

..• ,

vm be a basis for M. Then IX Vi =

L aij Vj

j= 1 with aijEZ. Let A=[aij]{:;::::::::' Then det(A-IXI) annihilates the module M, so this determinant must be zero. Then P(x)=det(A-xl) is a polynomial with integral coefficients such that P(IX)=O. 0

It follows from ii) above that the set of algebraic integers is a ring, since (IX + f3)MN =IXMN +f3MN cMN and IXf3MN cMN. It also follows from ii) that AIX is an algebraic integer for IX an algebraic integer and },EZ, since AIXM=IX(},M)=IXMcM. For IX an algebraic number, we let Da be the ideal in Z defined by Da={A.EZ: AIX is an algebraic integer}. n

This ideal is non-zero, since if Q(x) = L CiX i is the minimal polynomial of IX, then CiIXED a, for i=O n

Q(x)=xn+c n_ l x n- l +c n_ 2Cnxn-2 + ... + COC:- 2 =

L CjC:- j -

l Xj

j=O

satisfies Q(CnlX)=O, and hence (ii) of Proposition 6.1 is fulfilled. A positive element of Da is called a denominator for IX, and the positive generator d(lX) of Dais called the denominator of IX. Let IX be an algebraic number and P(x) its irreducible polynomial. Denote by 1X1' ••• , IXm the complex roots of P (which are all different, since P

n (X-IX). The IXj are j= n

is an irreducible polynomial in Q[x]). Thus P(x)=

1

said to be the conjugates of IX, and we set fi1 = max IlXjl. We then define the size of IX, S(IX), by S(IX) = max (log r;]. log d(IX». 1 ~j~m Proposition 6.2. Let K be a number field. Then for IX EK, IX =F 0, -2[K: Q]S(IX);;;; - [K: Q] 10gd(IX)-([K: Q] -l)1og~ ;;;;log IIXI.

§ 1. Preliminaries from Number Theory

157

Proof Let m be the degree of the irreducible polynomial for tx, so that Q]. Notice that the irreducible polynomial for d(tx)·tx is

m~[K:

?(x)= xm +a m_ 1d(tx)x m- ' +a m_ 2d(tx)2 xm- 2 + ... + aod(txt

(where xm + am_ 1x m- I + am_ 2 xm- 2 + ... + ao is the irreducible polynomial for m

tx). Hence ?(x) = m

we have

TI j~

(x - d(tx)tx). Since the coefficients of ?(x) are integral,

1

TI d(tx)'ltxjl~l, from which the conclusion follows. j~

0

1

We shall need the following technical lemma from linear algebra. Lemma 6.3. Let A be a IIOIl-zero subgroup oflRn. Assume that in any bounded region of lR" there exists only a finite number of elements of A. Let m be the maximal number of elements of A which are linearly independent over lR. Then we can select m elements of A which are linearly independent over lR

which form a 7l-basis for A. Proof Let {WI' ... , wm} be a maximal set of elements of A linearly independent over lR. We proceed by induction on m~n. Suppose that m= 1. Let w=tw l , where ItI is the smallest possible non-zero value. Suppose that wEA, and let t be such that w=tw i . Choose qE71 such that qt~t«q+ l)t. Then w-qw=(t-qt)w1EA and O~t-qt«q+l)t--qt-=~ which contradicts the choice of t unless t = q ~ in which case w is a 7l-basis for A. Suppose now that m> 1. Let V be the vector space over lR generated by {WI' ... , wm } and let Vm _ 1 be the space generated by {WI' ... , wm _ I }. Set Am_I = An Vm_ l . Then in any bounded region of Vm_ l , there exists only a finite number of elements of Am - I ' By the induction hypothesis, we can find {w~, ... , w~_ I} which form a 7l-basis for Am_I' Let S be the set of elements of A which can be written in the form m

L tiw; with O
I

certainly bounded and hence contains only a finite number of elements (including wm). We select an element vm in the set whose last coordinate tm is the smallest possible non-zero value. We shall show that {w~, ... , W~_I' vm } is a 7l-basis for A. We write vm=c1w'I+ ... +Cmwm, O~cm~l, cjElR. Suppose that VEA and m

write V= Lxjwj, xjElR. Let qm be the integer such that qmcm~xm< j~1

(qm + 1) C m· Then the last coordinate of v - qm Vm with respect to {w~, ... , w~} is equal to xm-qmcm and O~xm-qmcm«qm+ l)cm-qmcm~cm~ 1. Let qi be integers such that qi~xi
158

6. Application of Entire Functions in Number Theory

induction hypothesis, it can be written as a linear combination of {w~ , ... , w~ _ I' vm } with integral coefficients. Furthermore, it is clear that w;, ... , wm_ l , vm are linearly independent over 1R. 0 Lemma 6.4. Let K be a number field with [K: Q] = n. Then we can find a lL-basis for I K' the algebraic integers in K, of dimension n.

Proof By the Theorem of the Primitive Element, there exist exactly n embed dings 0"1, ... ,0" n of K into <e. We map I K into <en by r: rx ...... (0" I (rx), ... , O"n(rx)), which is an additive embedding and hence its image is an additive group. In any bounded region of <en = 1R 2n, there are only a finite number of elements of r(I K)' for in any bounded region, the 0" ;(rx) are bounded for rxEI K. Each such rx is the root of a polynomial (X-O"I(rx)) ... (x-O"n(rx)) whose coefficients are integral, since on the one hand they are algebraic integers and on the other hand, being symmetric in the conjugates, they are rational. By Lemma 6.3, if m is the number of elements of I K which are independent over 1R, then we can find a lL-basis for I K of dimension m. Since [K: Q] = n, m ~ n because a basis (wi"'" wn) for Kover Q is a basis for the Q-vector space generated by IK (i.e. for rxEI K, nne.

rx=

L CjW j = i~L -'-d(wj)wi' i~ d(wj) I

CjEQ,

I

and d(w;)wiEI K)' On the other hand, m ~ n since a lL-basis for I K is a Q-basis for K. 0 Lemma 6.5. Let Yj=

L aijxi' j = 1, ... , m, have i~

suppose m < nand in lL n of Yj=O, j

=

integral coefficients aijElL and

I

la i ) < A. Then there exists a non-trivial solution m 1, ... , m, with IXil < 1 + (nA)n-m.

XI"'" Xn

Proof As the Xi take on the (2M + 1) integral values between - M and M. we obtain (2M + l)n points in 1R m contained in the cube -nAM ~Yj~nAM. Since there are exactly (2nAM + different points with integral coordinates in this cube, it follows that there exist two different points (XI' ... , xn) and (x; •... ,x~) having the same image as soon as (2nAM + Ir«2M + I)n. Then x;' = Xj - x; gives a non-trivial solution with Ix;'1 ~ 2M. We choose

lr

m

m

for 2M the even integer in the interval (nA)n-m -1 ~2M~(nA)n-m that (2nAM + l)m «nAr(2M + l)m ~(2M + l)n-m(2M +

+ 1 so

lr

=(2M + l)n.

0 n

Lemma 6.6. Let K be a number field with [K: Q]=s. Let Yj=

L

aijxj'

i=l

j = 1, ... , m with n > ms and suppose that the a ij are algebraic integers in K

*I. Preliminaries from Number Theory

159

n

with IcGl
j = 1, ... , m and IXil < 1+ (en A)n-ms, where e depends only on K.

i=1

Proof Let WI"'" Ws be a basis for I Kover Z, which exists by Lemma 6.4. Let 0'1' ... , O's be the s embeddings of K into ce. For aeI K , let r(a) =(0'1 (a), ... , 0' sea»~, which embeds I K as an additive subgroup of ce s = 1R 2s. Then Sk = r(w k ), k = 1, ... , s, are linearly independent over the real numbers. Since s

a=

L

s

L

,P'wk, r(a)=

k=1

a(k l sk and so there exists a constant c depending

k=1

only on K (or rather on the basis

Iul=

(WI' ••• ,

ws» such that

sup 100i(a)l~c-1 sup lce(ill. 1 ~i~s

1

~i~s

s

In particular, we can write aij=

L

al~'wk with al~)eZ and laijl
k=1

apply Lemma 6.5 to the ms equations n

L al~)xi=O,

j=l, ... ,m, k=l, ... ,s.

i=1

If P(XI' ... ,xm )=

L

CIlX Il

o

is a polynomial with complex coefficients, we

11l1~d

set IPI = max Icili. If the coefficients are algebraic numbers, we set IIPII = max [CJ, and we define the size of P by s(P)= max(deg P, log IIPII). Let P(x l , ... , xm) = L CIlX be a polynomial with complex coefficients c and' Q(x l , ... , xm) = Laaxil be a polynomial with real non-negative coefIl

ll

ficients. We say that Q dominates P if Ical ~aa for all a, and we denote this by P
ii) PI P2
aQI. a-<-cJ =l, ... ,m.

... ) aPl

III

Xj

Xj

Furthermore P
Lemma 6.7. Let K be a number field. Let fl' ... , fm be functions of n complex variables holomorphic in a neighborhood V of qEce· such that /;(q)EK for all i and such that the ring K[fl' ... ,fm] is mapped into itself by the deriv-

e

c

atives D I =:;--, ... , D. = -~-. Then there exists a constant Co such that for LZ]

CZ.

every polynomial Q(x], ... ,xm) with coefficients in IK and of degree at most d, IID~'

... D!"(Q(fl' ···,fm»(q)11 ~ IIQII (d+lkl)lkl C~I+d.

Furthermore, there is a denominator for D~' ... D:"(QU], ... ,fm)(q» bounded by C~I+d.

158

6. Application of Entire Functions in Number Theory

induction hypothesis, it can be written as a linear combination of {w~, ... , w~ _ I ' vm } with integral coefficients. Furthermore, it is clear that w~, ... , wm _ l , Vm are linearly independent over JR.. 0 Lemma 6.4. Let K be a number field with [K: Q] = n. Then we can find a Z-basis for I K' the algebraic integers in K, of dimension n.

Proof By the Theorem of the Primitive Element, there exist exactly n embeddings 0"1, ••. ,0". of K into
IX= i~1

n

CiWi =

c.

i~l d(~i) d(wi)wi,

and d(wJwiEI K). On the other hand,

m~n

CiEQ,

since a Z-basis for IK is a Q-basis

~K

0 n

Lemma 6.5. Let Yj =

L aijxi' j = 1, ... , m, have integral coefficients aijEZ and i=l

suppose m < nand laijl < A. Then there exists a non-trivial solution XI"'" Xn m

in

zn of Yj=O, j= 1, ... , m, with IXil < 1 +(nA).-m.

Proof As the Xi take on the (2M + 1) integral values between - M and M, we obtain (2M + 1)" points in JRm contained in the cube -nAM ;£Yj;£nAM. Since there are exactly (2nAM + It different points with integral coordinates in this cube, it follows that there exist two different points (XI' ..• , xn) and (x~, ... ,x~) having the same image as soon as (2nAM+l)m«2M+l)n. Then x;' = Xi - X; gives a non-trivial solution with Ix;l~ 2 M. We choose m

m

for 2M the even integer in the interval (nA)n-m-l~2M;£(nA)·-m+l so that (2nAM + «nA)m(2M + l)m;£(2M + l)n-m(2M + l)m

1r

0

=(2M+l)n. n

Lemma 6.6. Let K be a number field with [K: Q] =s. Let Yj=

L aijxj'

i=1

j

= 1, ... , m with n > ms and suppose that the aij are algebraic integers in

K

*I. Preliminaries from Number Theory

159

n

fG;jl < A. Then there exists XiEZ not all zero such that

with

m

j= 1, ... ,m and IXil< 1 +(cnA)n-ms, where c depends only on K.

l:aijx = 0, i

i=1

Proo}: Let WI"'" Ws be a basis for I Kover Z, which exists by Lemma 6.4. Let 0'1' ... , as be the s embeddings of K into ce. For aEI K' let rea) =(0' I (a), ... , 0'. (a», which embeds I K as an additive subgroup of ce s =1R 2s. Then sk=r(w k ), k= 1, ... ,s, are linearly independent over the real numbers. Since s

a=

L

•

lX(k)W k ,

L

r(a)=

k= I

lX(k)Sk

and so there exists a constant c depending

k= I

only on K (or rather on the basis (WI' ... , ws» such that

Ia1 =

sup lUi(a)1 ~ c- I sup lac( i1 I· 1 ~i~s

1

~i~s

s

In particular, we can write aij=

L

1X1~)Wk with IXI~)EZ and IlXijl
k=1

apply Lemma 6.5 to the ms equations n

L IXI~) Xi = 0,

j = 1, ... , m, k = 1, ... , S.

i=1

If P(x l , ... , xm)=

L

CIlX Il

o

is a polynomial with complex coefficients, we

llll~d

set IPI = max !clli. If the coefficients are algebraic numbers, we set IIPII=max[CJ, and we define the size of P by s(P)=max(degp,logIIPII). Let P(x l , ... , xm) = cllx· be a polynomial with complex coefficients Cll and' Q(x l , ... , xm) = a.xll be a polynomial with real non-negative coefficients. We say that Q dominates P if Iclll ~all for all IX, and we denote this by P
L

L

... ) aPI

ll1

aQI. a-<-aJ =l, ... ,m. Xj Xj

Furthermore P
Lemma 6.7. Let K be a number field. Let fl' ... , fm be functions of n complex

variables holomorphic in a neighborhood U of qEce n such that J;(q)EK for all i and such that the ring K [fl , ... ,fm] is mapped into itself by the deriv-

.

atrves D I

a

c

= -~-, ... , Dn = -~-. Then there exists a constant Co such that for eZ I

every polynomial Q(x 1 ,

cZ n

... ,xm )

with coefficients in IK and of degree at most d,

IID~' ... D!"(Q(fl' .. ·.!m» (q)11 ~ IIQII (d+lkl)lkl C~l+d.

Furthermore, there is a denominator for D~' ... D:"(Q(j;, ... .!m)(q» bounded by C~l+d.

160

6. Application of Entire Functions in Number Theory

Proof There exist polynomials F:j(X I , ... , xm) with coefficients in K such that DJi=F:j(fI, ... ,fm)' Let c5 be the maximum of the degrees of F: j , i=l, ... ,m, j = 1, ... , n. For any polynomial PEK [xl' ... , xm], we set _

ap L -;-(Xl' ... ,xm)F:j(x m

DjP(x l ,

... ,xm)=

i~1 uX j

l , ...

,xm)·

Then P< IIPII(1 +XI + ... +xm)d and F:j
so

D- j P
By iteration, one then obtains a dominating polynomial J5~1

... J5~n P <

IIPII C~I(d + Ikl)lkl(1 +XI + .. , +

Xm)d+

Ikl~.

We now substitute the values h(q) for Xi to obtain the first estimate. To prove the result on the denominator, we proceed by induction on Ikl. For Ikl=O, the result is trivial. Thus, we assume the result for all Ikl up to jo -1. Let a be the common denominator of fl (q), ... ,fm(q). Then ad+ Ikl~ is . -k-k a common denommator for Dll ... Dnn(Q(fp .. ·,fm)(q»· 0 If F is a field and A is a ring containing F, we say that the elements {xl' ... , xm} are algebraically dependent over F if there is a polynomial P with coefficients in K such that P(x l , ... , xn) = 0; otherwise, we say that {Xl' ... , Xm} are algebraically independent. A subset E of A is algebraically independent if every finite subset of E is algebraically independent. If L is an extension of a field F, a subset B is a transcendence basis for L if B is a maximal set of algebraically independent elements of Lover K. The transcendence degree of L is the number of elements in this basis (perhaps

infinite).

§ 2. A Schwarz Lemma Lemma 6.8. Let F(z) be holomorphic in a neighborhood of {z: Ilzll ~R} in CC n• For O~r~R set M(r)=sup log IF(z)l. Then IIzll~r

M(r)

~M(R)-

R2 +r2 v(r) log--:----::-(4n-2)Rr

i where! =- co log IFI and v(r) is the indicator of

!.

1t

Proof By the Riesz representation of the R?n-subharmonic function log IFI, for zEB(O, R), and g(z, a) the Green function of the ball B(O, R), we have

log IF(z)1 = H R(Z) - winl Jd(J(a) g(a, z);

§2. A Schwarz Lemma

161

HR is the harmonic function which takes on the values loglFI for Ilzll =R;

HR(Z)~M(R),

and a=21n L1loglFI is the trace-measure of

T.

Let a'ECC n be

on the line Oa such that Ilalllla'11 = R2. Then for Iiall = t and Ilzll = r, ila

ll g(z,a)= Ila-zI1 2- 2n - (R(llz-a'ID

)2-2n

from which we obtain r )2-2n

O~g(z,a)~g'(r,t)=(r+t)2-2n- ( R+/it

and R

M(r)~M(R)-wlnl

Jg'(r, t)da(t).

o

Since g' (r, t) = 0 for t = R, we obtain after an integration by parts: R og' M(r) ~ M(R)- w 1nl a(t) -;- (r, t)dt. o (It

J

Thus if we replace a(t)T2n_2t2n-2 by v(O,t), we obtain M(r)sM(R)-

j v(t)t [(_t_)2n-1 _~ (~)2n-l] dt. t+r R rt+R2

0

For the first integral, we obtain A(r)=

R v (t) J o t

_t_

~

~v(r)

( t )2n-1 R( t -dt"?v (r) t+r - t r t+r u2n- 2 --du

J-

)2n-1 -dt t

1/2J l-u

and -! < (J( =~ < 1. To calculate the integral we write R+r

In

=[ -IOg(l-U)-(U+u; + ... +~:n~~)I/2 R+r

In~log--+

2r

2n-2 L k-I[2-k_l]. hi

1/2 1 J t-Idt>-, we obtain

k+

For n~3, using

k-I/2 4

In~Lk-I(2-k-l)+ I

or A(r)~log(4

k 2n- 3/2

J

9/2

R+r 3' n- )r

t-Idt~ -~i+log9-log(4n-3)

162

6. Application or Entire Functions in Number Theory

The formula is proved to be true for n = 2 by direct calculation. For B(r) the second integral, we obtain a bound R)20-

B(r)~v(r) (r

2"J--du u20 - 2 l

l-u

r2 r O<,A,=-2--2<11=-R <1 R +r +r

so that R)20-2 1-). ( R 112o-210g--=v(r) - r 1-11 R+r

B(r)~v(r) (-

)20-2 log R(R+r) 2 2' R +r

If we replace fl 20 - 2 by 1, we obtain for n~2 A(r)-B(r)~v(r)log

R2+r2 R2+r2 ) ~v(r)log(4 2 (4n-3 Rr n- )Rr

To obtain a bound which is valid for n= 1, it is sufficient to replace 4n -3 by 4n -2; moreover, we note that for the large values of n, a bound R

0

for M(r) is obtained only for large values of -. r

§3. Statement and Proof of the Main Theorem With these preliminaries out of the way, we are now in a posItIon to undertake the proof of the main result. We will say that a merom orphic function is of finite order at most p is it is the quotient of two entire functions of finite order at most p. Theorem 6.9. Let K be a number field and let f = (fl' ... , fm) be meromorphic functions in CC O of finite order p. Assume furthermore that i) the transcendence degree of K [f] is at least (n + 1) (that is, there are at least (n + 1) of the J;'s which are algebraically independent over K); ii) the partial derivatives

~, i = 1, ... , n, map the ring K [f]

into itself. iJz i Then the set of points ~ECCo where f(~) is finite and in Km is contained in an algebraic hypersurface of degree at most n(n+ 1) p[K: Q]. Remark 1. Condition ii) can be relaxed so as to include the case where the . ., c. _ C'f CJj _ IUf) . field K(f) IS mapped mto Itself by -;;-, 1-1" ... ,n, lor 1 -;;--Q(f) WIth c~

~j[x],

Q[x]EK[x], we set fm+l =Q(f)

cj. ~EK[

J],

c~

-1

-

.

. Then f=(fl' .. ·,fm+l) satIsfies

i=I" ... ,m+l, and we can apply Theorem 6.9 with the ad-

CZ i

ditional restriction that Q(fW)*O.

§3. Statement and Proof of the Main Theorem

163

Remark 2. If K c L eM are three fields, then dimK M = dim KL + dim L M. In particular, K(fI,f2)=K(fI)(f2)' etc. Thus, we can find ~,j=il' ... ,in+1 such that h" ... ,hn + 1 are algebraically independent. We assume without loss of generality that these are fl' ... , fn+ I' Let S be a finite set of points ~i' i= 1, ... , t, in (Cn for which f(~) is finite and in Km. Let j=Ul' ... ,jm+l) be a multi-index and consider the function aJ/' ... fj~j'. We use Lemma 6.6 to choose integers a j in K F(z)= I o ;fij).;;:J

such that DAF(~)=O for A={}'I' ... ,A..}, sup}'i
11,1 DAft' ... fj~l'(~)11 «(n + I)J + L)L Ci(n+ 1)J+ 2L, by Lemma 6.6, we can find integers aj not all zero with r

log lajl <-(- ) (L 10g(L+J)+J)+o(L) IJ-r

. h r=t (L+n) and IJ =]"+ I. If we choose ]"+ 1= [K: QJ{t ~ log L} then wIt n r

-(-~(logL)-I.

D

IJ -r)

For every L, we let y(L) be the integer such that D<1 F(~)=O for maxaj
~'ES

such that

-([K: Q]-l)ylogy+O(y)

where O(y) depends only on f and S.

164

6. Application of Entire Functions in Number Theory

Proof DU ' G.. (~')=g(~')(n+I)J DU ' F(~') since DU F(~')=O for max aj<s. But U' a' and some ~', so

~ = D F(~')E'K and is not zero for some

log ID'" Gy(OI = (n + I)J log Ig(~')1 + log I~I ~ O(J) -([K:

Q] -1 )s(~)+ O(log d(~))

by Proposition 6.2, The result now follows from Lemmas 6.7 and 6.10.

Proof of Theorem 6.9. Let

T.=~ 10gIG .. (z)l. ny ,

Suppose that

I

{~: I~il

r~ II~'II +n.

0 Then

< 1, i= 1, ... , n} cB(O, r) and by Cauchy'S Inequality ID u G,,(O;;;;)" max IG,,(z)l, liz II

~r

or cquivalently log /D u ' G/O;;;;y logy +max log /G;(zJ/. liz II

~r

Since v yT ., = ,'v T.. , by Lemma 6.8, we have , , R log /D u ' G/O;;;;y log y+max log IG .. (z)/-v T (0, r)'Y logIlzll~R y 4nr for R~KII +n. Since g and each fj are of order p, on I z I = R I

log /G ,,(z)/;;;; C J RPH +n log (J + L). Let R=y" with IX<{(n+l)p}-I. Then RPH
8

small enough,

and so maxlog/Gy(z)/=o(ylog}') for R=y", IX<{(n+l)p}-l, Ilzll

~R

If r is fixed, r> hence S), we obtain

II~' I + nand

y is sufficiently large (depending on rand

log /D u , Gy(~')/;;;; y log y + o(}' log y) - YVT,,(O, r) [y log y + C]. By Lemma 6.11, ~ satisfies i) vdO,r);;;;[(n+l)p+o(l)][K:Q] for every rand y sufficiently large dependin'g on r; ii) vT,.(~',r)~1 for ~'ES, By i), the total mass of VTs is uniformly bounded in every compact subset of <en as y --+ co, and so the sequence T; has a subsequence ~ which converges weakly to a positive closed current T. For K a compact subset of <en, da T= lim daTI-" In particular, this implies that VT(~')~ 1 for ~'ES and

J

K

J

I'-x

K

vT(a,r);;;;(n+l)p[K: Q] for r large. By translating if necessary, we may assume that r- I vT(O, r) is integrable in a neighborhood of the origin. We then use the result of Chapter 3 and set (n-2)! V(z)= 2nn - I

)4:"

(

1

W1 2n -

2

Historical Notes

where

(J T

=T

1\

~, (n -I)!

165

so that V (z) is a plurisubharmonic function (cf.

Theorems 3.17 and 3.26) with V(z)~(n+l)p[K:QJ logllzll when Ilzll---+oo and vv(~)~ 1 for ~ES. In particular, this implies that exp-2nV (z) is non integrable in a neighborhood of ~ES. Thus, by Theorem 5.12, there exists a non-trivial entire function F(z) such that

J IF(z)12exp-2nV(z)d r(z ) < + 00. «;n

(1 + IIzl12t H

Since exp-2n V(z) is non-integrable for For large I z II,

~ES

we obtain,

IF(z)12(l + IlzI1 2 )-,,-< exp-2n V(zgjF(Z)j2 1Iz jj

2n(n

I

F(~)=O

l)p[KQl

for

~ES.

2n- 2,

for Ilzll >RE' so

J IF(z)1 2(1 + IlzI1 2)-n(n+l)p[K:Ql-n-

E

dr(z)<

+ 00.

«;n

Thus, F(z) is a polynomial of degree at most n(n+ l)p[K: QJ, for if F(z) = L C,z', since z', -ZfJ are orthogonal on the boundary of the ball of radius R for a {3, OC't IC 12R 21.IR 2n-l IF(zW(1 + IlzI1 2)-Qdr(z)= ~! '(1 +R2)Q dR

*

·

In

J

for t, =

Iz'I 2 dr, and this last sum is finite only if C. = 0 for a> q - n.

bdB(O.l)

Thus, F is a polynomial P of bounded degree such that P vanishes on S. By multiplying by a constant, we may assume IPI = 1. We now choose an increasing sequence SI such that U SI is dense in Sand 1; the correI

sponding polynomials as constructed above. Then deg 1; ~ n(n + l)p [K: Q]. We can then find a subsequence which converges to a polynomial P$O and P(S)=O.

D

Historical Notes For n = 1, Theorem 6.9 is known as the Schneider-Lang criterion, and one can deduce many famous results on transcendence from it: that e' is transcendental for a algebraic (Hermite-Lindemann), that fJ.P is transcendental for fJ. algebraic, {3 algebraic and not rational (Gelford-Schneider) (cf. the book by Waldschmidt [IJ). Lang [IJ proved an n dimensional version when the set S was a product of one dimensional sets. Since for n = 1, card S is finite, Nagata conjectured that for n ~ 2, the set S was contained in an algebraic hypersurface. This was proved by Bombieri [1 J in 1970 using the

166

6. Application of Entire Functions in Number Theory

properties of a closed positive current () and the representation of the solution V of the equation ioaV=O. The proof used the Schwarz Lemma for n variables given in a joint paper of Bombieri and Lang [1], which was written a short time before (cf. Lemma 6.8). The paper of Bombieri indicated implicitly the way to prove that the sets of density v,(z) ~ c > 0 for t a positive closed current were analytic varieties (as obtained later by Siu [1] in 1974 using the L2 -estimates of Hormander). For more details, see Lelong [12b]. The estimate for the degree of the algebraic hypersurface S was improved by Skoda [6] and Demailly [1]; Bertrand and Masser used the n dimensional result to obtain results on transcendence and algebraic independence. For a very thorough treatment of the above as well as many related topics, we refer the reader to the book by Waldschmidt [1].

Chapter 7: The Indicator of Growth Theorem

We have seen in Chapter 1 that the radial indicator h1 of an entire function f(z) of normal type with respect to a proximate order p(r) satisfies i) h1(z)=tPh1(z) for t~O; ii) h1(z) is plurisubharmonic in CC n• We now show the converse for strong proximate orders; that is, suppose that h(z) is a function which satisfies i) and ii) above. Then for every strong proximate order p(r) there exists an entire function f(z) such that h1(z) = h(z). Let k be an integer, (1 ~ k ~ n), and", a function plurisubharmonic in CC n • Suppose that o(E~~(B(O, 1», 0(~1, O(z) dr(z)=l, 0( a function of Ilzll. We define by induction a sequence {"'~} of plurisubharmonic functions as follows:

J

i) "'~ = ... = "'~ = '" ii) "'~=O(*"'jkforj>kwith

"'t=[ sup "'~_l(Zl,· .. ,Zj_l' Zj+~j' Zj+l,· .. ,Zn)]*' I~jl< 2

For 1 ~j~n, we identify CCj with the subspace of cc n defined by {ZECC n: zp=O for p>j} and z(j) will be the projection of z on CC j , that is if z = (Zl' ... , zn), z(j) = (Zl' ... , Zj' 0, ... ,0). We will let drj be the Lebesgue measure on CC j . We shall use this to prove the following result on the extension of entire functions with growth conditions.

Theorem 7.1. Let ",EPSH(CCn) and f holomorphic on CCk (k
g(Z)E.Jf'(CCn) such that exp - ",:(z) ! Ig(z)12 (1+llzI1 2)3(n-k) drn(z) < +00.

Proof We shall prove the result by induction on n by showing that if fE.Jf'(CC j) and

exp - ",:(z(j) Jj If (1 + IIz(j)112)3(j-k) drj(z (z(j)W

(j) _

)-MJ <

00

168

7. The Indicator of Growth Theorem

there exists gE£(['j+t) such that glc[;i=f and

(;t

Ig(z(j+ 1')12 exp -I/i (z(j+ I)) (1+llz(j+I'112)3(/+\-kl drj+l(z)
Since I/Ij~ I is plurisubharmonic, we have 1/1;+ I ;;;: I/I~~ I and for IZj+ II < 1,

1/1/+ I (z(j+ I');;;: I/I~(z(j)),

so (7,1 )

s

ej+1

l=j+tI<1

< =

1)

If (z(j))1 2 exp -I/I~(z(j))dr/z(j)) (1 + Ilz(j)11 2 )3(j-k)

Let WECC;' (Izl <1) such that w(z)=1 if Izl~I/2, w(z)=O if Izl;;;:l, and let

C=supl~~I. We shall find g(Z(j+I)) so that (7,2)

°

and such that Zg = as a distribution. In fact, if fJ is the (0, 1) form f(z(j)) - fJ=--GW then fJ has CC x coefficients, cfJ=O, and IfJl~2C1fl so Zj+ I

By Appendix III, we can find X such that Ixl 2 exp-I/Ik

aX = fJ and

(z(j+l))

_

dr. <4C 2 M S (1 + Ilz(j+I)11 21+1 )3(j+2/3-k) J+I = J'

(;j+l

Then g defined by (7,2) satisfies Zg=O as a distribution. If oc,(z) 1

= 2(j+l) OC(Z/e) and g,(z)=g*:x,(z), then g,ECC% and (1g,=O, so g, is in fact e a holomorphic function. Since g, = g, * OC,. = g" * OC, = go' by the Mean Value Property for harmonic functions, g, is independent of e. Furthermore, g, --> g when e-->O in L2(B(0,r)) for every r so g,=g almost everywhere, and hence g is (equivalent to) a holomorphic function. Finally, from (7,1) and (7,2), since Ilzj+1112~1+llz(j+llf, it follows that

S

(:j+l

Ig(Z(j+l»)1 2 exp-I/Ik (ZU+II) . (1 + Ilz(j+ 1'112)3(/+\ - kl dr j+ d zU + 1»

_

~ (1 + 4 C 2 )MJ .

o

Let PSHp(t)(CC n ) be the set of plurisubharmonic functions of order p and normal type with respect to the proximate order p(t).

7. The Indicator of Growth Theorem

169

Proposition 7.2. Let I/IEPSHp(I)(CCn ) and Zo *0. Then given e>O, there exists R(e, zo) and J(e, zo) such that for Ilz-zoll <15 and t>R

I/I(tz)~ (h:(Zo) +e) tP(I). -

if. Let Proo.

'

Zo Zo =~.

IlzollP

Since h:(z) is upper semi-continuous, there exists a

neighborhood 2z~...of -z~ such that h:(z) ~h:(z~)+e/2 for ZE Uz~' We apply Hartog's Lemma (Theorem 1.31) to the family V,(z)=t-P(I)I/I(tz) on Uz ' (which we suppose compact by passing to a smaller set if necessary). D Corollary 7.3. Let I/IEPSHp(I)(CC n) and let A be a continuous positively homogeneous function of degree p with A ~ h:. Then given e > 0, there exists r > such that l/I(z)~(A(z)+ellzIIP)llzIIP(lIzll)-p for IIzll ~r.

°

Proof Since the sphere of radius 1 is compact, for t> t I' we have by D Corollary 1.32: l/I(tz)~(A(tz)+etP)tP(')-P for IIzll=l and t>r l ·

Theorem 7.4. Let 1/1 EPSHp(,)(CC n). Then the functions t/I~ introduced in Theorem 7.1 all have the same radial indicator function h:. Proof There exists a>O such that I/I~I/I~~I/I' for I/I'(z) = [ sup I/I(z+m*. For zo*O and e>O, there exist 15 and r such that II~II ~a

I/I(tz)~ (h:(Zo) +e) tP(I) - IlzollP and

for

t~r

for

t~r'.

Ilz-zoll ~J, so that I/I'(tz )< (h:(zo) +e) tP(I)

o = IlzollP

But this implies that

h",,(z)~h:(z)

for all z and hence

h:,(z)~h:(z).

Theorem 7.5. Let I/IEPSHp(,)(CC n ) and f an entire function such that

JIfl2 exp-I/Idr< 00. 4:n

Then f is of normal type with respect to pet) and

h!(z)~

1/2h:(z).

Proof By the Cauchy Integral Formula, we have f(z)2= (2 ~)n Jf (rl ei61 + zl ' ... , rnei6n + zn)2 dlJ I so

...

dlJ.,

170

7. The Indicator of Growth Theorem

Thus If(z)12~ Cn[sup exp I/I(z+ I~jl ~

2

m

J

If(z+ ¢)12

1~I~jl~2

·exp-( sup I/I(z+mdr I~JI ~ 2

~Cn[sup expl/l(z+m l~jl~2

J

If(z+¢Wexp-l/I(z+¢)dr

J

If(z+~)2Iexp-l/I(z+~)dr)

1~I~jl~2

~C~[supexpl/l(z+m ( l~jl~2

1~I~jl~2

so (7,3) loglf(z)I~C~+1/2 sup I/I(z+~) and hence f(z) is of finite order II~II ~2n

with respect to p(t) and h1(z)~!h:(z) by Theorem 7.4.

D

Theorem 7.6. Let I/IEPSH(Cn) be positively homogeneous of order p. Then we can find a decreasing sequence of plurisubharmonic functions {I/I q} each positively homogeneous of order p and ~oc on (Cn - {O} such that lim 1/1 q(z) = 1/1 (z).

q~oc

J

Proof Let IX(Z)E~oX (B(O, 1)), 0 ~ IX(Z) ~ 1, IX(Z) dr(z) = 1 and IX depending only 1 on Ilzll. Let IX,(Z) = e2n IX(Z/e). Then I/I,(z) = Jl/I(z')IX,(z-z')dr(z') is ~c>C, plurisubharmonic, and decreases to 1/1 (z), but it is not in general positively homogeneous of order p, so we must change the construction slightly. Let

~,(z)= Ilzll- 2nJ1/1 (Z')IX, el~ln dr(z')

for Ilzll =FO or equivalently

~,(z)

= JI/I(z -II zll w)IX,(w)dr(w) for all z. Then ~ ,(z) is ct x on (Cn - {O}, is positively homogeneous of order p (since 1/1 is), and since ~,=I/I, for Ilzll=l, ~, decreases to 1/1 when e tends to zero. It remains to show that 1/1, is plurisubharmonic in (Cn. Since IX,(W) depends only on Ilwll, there exists a positive continuous function A(r) such that

,

I/I'(z) =

JA(r)T,.(z)

o

with

1 T,.(z)=w2n

J

Ilwll ~r

l/I(z-llzII w)dwzn(w).

Thus, it is sufficient to prove T,.(z) plurisubharmonic in (Cn. Let i be the unitary group on (Cn, which is compact, and dy the normalized Haar measure on 1. Let Zo be a fixed point of (Cn of norm rand l/I(y)=I/I(z-llzlly(zo))· Then T,.(z) = I/I(}')d~'. Furthermore, there exists '1Ei

J

such that r'1(z)= Ilzllzo. so

T,.(z) =

r 1/I(~')=I/I(z-r~''1(z)),

and if ¢(";')=I/I(z-r~'(z)),

J¢(i''1)d"'/= J¢(r')d,'= JI/I(z-r,'(z))d)'.

r

r

r

Since for every YEi, z-r·),(z) is a holomorphic function of z, I/I(z-ry(z)) is plurisubharmonic in z and hence so is T,.(z). D

7. The Indicator of Growth Theorem

171

By a Lipschitz continuous function, we will mean II/I(z)-I/I(z')1 ~ C Ilz-z'll for z, Z' ES 2n - I.

Proposition 7.7. Let p(r) be a strong proximate order and I/I(z) a Lipschitz continuous plurisubharmonic function positively homogeneous of order p. Then there exists a plurisubharmonic function t/i(z) such that 1/1 (z)t P(liz 11)- P~ t/i(z) and h~(z) ~ I/I(z) (where the indicator is taken with respect to p(r)). Proof Suppose that e(r) is a continuous decreasing function of r such that lim e(r)=O. Then there exists an increasing convex function ~ such that r- oc

i)

~'(logr)~e(r)rp(r)

.. ) l'

Wogr)

and

We

r~ro,

s

~(s)

define

for

0

Im~=. r-oo r

II

e"(logr)~e(r)rP(r)

by

I

~(s)= S ~'(t)dt,

~'(t) =

CpS e(er)erp(er) dr

with

o

o I

C p =2sup(p, 1). Then ~'(t)~e(el)Cp S erp(e'")dr~e(el)e'P(et) for t large enough, 0 1 and since e(s)~- for s~sn' we have n C I 1 ~'(t)~

Cn+---.£. S erp(er)dr~ Cn+- C~elp(et). n s n Thus ~(t) ~ Cnt + C~ e'p(et), whic~ shows ii). By adding a multiple of n log(l + r2), we may assume that i) holds for all r. We note that if a = (aI' ... , an) is a complex vector, then ,,2 ,,2 02!,() [111121L..ajrjl] lL.,ajr) (7,4) L ~ ajiik=e'(logr) -;- j 4 +~"(logr) j 4 j.k OZjOZk r r r ~!e(r)rp(r)-21IaI12.

Since I/I(z) is Lipschitz continuous and positively homogeneous of order

p,

10~(Z)I~CIIZIIP-I;

that is, as a distribution °ol/l(z) is equivalent to a

c~

~

function with the above bound. Let a be a complex vector. Then as a distribution, setting Ilzll =r, we have

L j.k

02(I/I(z)rP(r)-p) _ _ p(r)-p 021/1(Z) _ ~ 0ajak- L r ~;:)- ajak cZj Zk j,k CZj{/Zk crP(r)- pol/l(z) _

+L-;:)--~ajak

j,k

(/Zj cZ k orP(r)- p ol/l(z)

_ + L ---;=--~-ajak j,k L Zk cZj c2 r P(rl- p +1/I(z)I ~ ~ aiik j,k czjczk ~

-e(r)rP(rl-21IaI1 2

172

7. The Indicator of Growth Theorem

for some e(r) such that lim e(r)=O. Thus we can find Wogr) plurisubhar. ~(logr) monic with 11m ----;>ir)=0, such that rfr(z)=Wogr)+t/J(z)rP(r) is plurisubharmonic. r~ x r 0

Theorem 7.8. Let t/J be a subharmonic function in CC positively homogeneous of order p. Then for 0 ~ e~ 2n, there exists an entire function f (z) of order p(depending perhaps on 0) such that . log If(te i6 ) 11m sup p I~'X: t

. t/J(e I6 ) and

If (z)1 ~ C exp rfr(z)

where rfr(z)= [sup t/J(z+ ~)]* + C 1 10g(1 + IzI2)+ C 2(log(1 + IzW. I~I ~3

n (l-2- i z), which defines an entire function. 00

Proof. Let h(z)=

i= 1

Suppose that for somej, 1/4~lz-2il~t, so that

and

k
independent of j. Let cpErt';-(B(O, 1/2)) such that O~cp~1 and cp=1 for Izl~i, and set g(z) x

=

L

cp(z-2iei6)expt/J(2iei6) (for every z, there is at most one summand).

i= 1

We shall write f(z)=g(z)-h(ze- i6 )v(z) where v(z) is chosen so that f(z) is holomorphic. We must have 2f =0 or 8g=h8v. Since 8g=0 for

Iz-2iei61~i if p=

8:,

then

IPI~cl:~1 and

JIpl2 exp-2~(z)dr(z)< + ex; C[

where ~(z)=[ sup t/J(z+m*+log(1+lzI 2). I~I ~1:2

By Appendix III, there exists r such that h = p and

J Irl2 exp-(2~(z)+ 10g(1 + IzI2))dr(z)< + "x. C[

Then g - h r defines a holomorphic function.

7. The Indicator of Growth Theorem

173

Suppose that 2j~lzl~2i+l. Then, since log (I +x)~x for x~O, x (IZI) i+ I 1 loglh(z)l~k~,log 1+2k ~k~,log(I+2i+'-k)\~,2k 'Yo

i+ I

~k~1 U+2-k)log2+1=

U 1)U 2) + 2 +

log2+1

~ C 1 (log(lzl + 1))2 + 1.

Let l/i(z)=I/i(z)+log(I+lzI 2)+C, (log(I+lzI))2, which is subharmonic. Then h,(eiO)=IjJ(eiO ) and by Theorem 7.5,

!fl ~ C exp ([ sup I/I(Z + ~)J* + 2log (I + !Z!2) + C~ (log (1 + Iz!2))2.

0

I~I~P

Theorem 7.9. Let ljJ(z) be a plurisubharmonic function positively homogeneous of order p in ccn. Then for ZoECC n, there exists an entire function g(z) (depending perhaps on zo) such that h:(z)~IjJ(z) and h:(zo)=IjJ(zo) (where the indicator is with respect to rP). Proof By a rotation, we may assume that Zo = (zp 0, ... ,0). Let ljJ(u) = ljJ(u, 0, ... ,0). Then by Theorem 7.8, we can find an entire function in CC feu) such that h;(zo)= ljJ(zo) and log If (u)1 ~ [sup ljJ(u + I~I ~3

m* + C

1

log (1 + lu1 2)2

+ C 2 log(1 +luI 2)+log c. By Theorem 7.1, there exists an entire function g(z) such that g(u,O, ... ,0) = feu) and Slgl2 exp-~(z)dT(z)< + rx; where

~(z)=2[ sup ljJ(z+ m* + Cn [log(1 + Ilz112)]2 + C~ log(1 + Ilz112) II~II ~a

for some a>O. Then h:(z)~h$(z) by Theorem 7.5 and h$(z)=h~(z)=IjJ(z) by Proposition 7.2. 0

Corollary 7.10. Let ljJ(z) be a Lipschitz continuous plurisubharmonic function positively homogeneous of order p in ccn and p(t) a strong proximate order. Then for ZoECC n, there exists an entire function g(z) (depending perhaps on zo) such that h:(z) ~ ljJ(z) and h: (:0) = ljJ(zo) (where the indicator is with respect to r PI ,)).

Proof Let I/i(z) be the plurisubharmonic majorant of ljJ(z)tPlt)-P constructed in Proposition 7.7. Then, as in Theorem 7.8, we construct an entire function f(uz o) of the variable u such that f(2 i z o)=1/i(2i z o) and If(z)1 ~ [sup I/i(z+ ()]* + C 1 log(l + Ilz112)+ C 2(log(1 + Ilzll))2 I~I ~3

174

7. The Indicator of Growth Theorem

(we choose g(UZO)=jtl

7.1, to extend

f

cp(UZo-2jzo)exp~(2jzo)).

We then use Theorem

to (Cn, as in the proof of Theorem 7.9.

0

If cp is a continuous positively homogeneous function of degree p in (Cn, we let B~(t) be the Banach space of entire functions f such that

lim If (z) exp - cp(z) IlzlIP( liz 11)- PI = 0 Ilzll~'"

with supremum norm. Let E~(I)=

n

B:~lllzIIP' which is a Frechet space. If

q

q

I/IEPSHp(t)(Cn), let m(l/I) be the set of continuous plurisubharmonic functions q; positively homogeneous of order p such that h~(z);£qJ(z). Theorem 7.6 shows that h~ (z) = inf cp and that m(l/I) is an ordered filtered set with a ",Em(I/!)

countable basis. We set E~(t) =

n E:(I), which is also a Frechet space.

",Em(I/!)

Theorem 7.11. For I/IEPSHp(t)(Cn), let f be an entire function of normal type with respect to p(t). Then fEE~(t) if and only if h1(1/I);£h~(z). Proof If fEEl/!' then fEE", for every cpEm(l/I) and hence 1

If (z) exp ( - cp(z) -- Ilzll P) IlzlIP( liz I )-PI q

is bounded for every q. Thus

JIf(z)1

2

exp - (2cp(z) + e liz liP) liz IIP(lIzll)- P dr(z) < +

e for every e>O. By Theorem 7.5, h1(z);£h:(z)+21IzIIP, so

OC>

h1(z);£h~(z).

On the other hand, for qJEm(I/I), 10glf(z)I;£(cP(z)+ellzIIP)llzIIP(llzll)-p for Ilzll ~rt by Corollary 7.3, so fEE", for every cp and hence fEEl/!' 0

Theorem 7.12. Let 1/1 be a plurisubharmonic function positively homogeneous of order p. There exists an entire function f(z) in (Cn whose indicator function h1(z) with respect to rP is 1/1 (z). Proof Suppose that 8EPSH p(
7. The Indicator or Growth Theorem

175

Let {wd be a countable basis for the open sets of CC"+ 1 and {w s } those elements such that wnCD",=I=q,. Set u.=D",uw s' Then if D",,*D,,,, us~D", for some s (depending on qJ). Let ms= {qJ: qJEPSH(CC n), qJ positively homogeneous of order p, uscD",} and let V. = interior (

n D",). Then if 1/1. = [sup qJ]*, 1/1

~ems

s

is plurisubharmonic

~Ems

and positively homogeneous of order p and V.=D",s' Thus I/Is~I/I, 1/1 ..$1/1. Suppose that fEE", and h1(z)=I=I/I. Then uscD h* for some s, and hence h1Ems' Thus, h1(z)~I/Is(z) and fEE~~) by Theorem' 7.11. Since UE",s is of s

first category in E"" we see by the Baire Category Theorem that there exists fEEl/! n C(U EobJ, and for this f, h1(z) = 1/1 (z). 0 s

Corollary 7.13. Let I/I(z) be a Lipschitz continuous plurisubharmonic function positively homogeneous of order p. Then there exists an entire function f of normal type with respect to the strong proximate order p(r) such that h1(z) = 1/1 (z). The proof is the same as that of Theorem 7.12 based on Corollary 7.10. We note that for n = 1, any positively homogeneous subharmonic function is Lipschitz continuous [D]. It seems unreasonable to hope for such a strong result for proximate orders in general. Nontheless, for the circled indicator function, we have the following result.

Theorem 7.14. Let I/I(z) be a plurisubharmonic function complex homogeneous of order p. Then there exists an entire function f of normal type with respect to the proximate order p(r) such that hL(z) = 1/1 (z). Proof By Theorem 1.23, there exist non-zero constants qJ{q) such that the type (J of the entire function feu) of the complex variable u with respect to the proximate order p{r) is given by «(Jpe)l/p= lim sup qJ(q)C!/q, where q-oc·

f(u)=

L cqUq is the Taylor series expansion at the origin. q

Since I/I(z) is plurisubharmonic in CC n, D= {z: I/I(z) < I} is a domain of holomorphy (cf. [A, B]), and so there exists a function fez) such that fez) cannot be extended as a holomorphic function to a neighborhood of any boundary point of D. Let I(z)= ~(z) be the Taylor series expansion of

L q

I (z) in terms of homogeneous polynomials. Then the entire function (ep)l/p]q _ f(z)= ~ [ qJ(q) ~(z) satisfies hL(z) = 1/1 (z).

D

176

7. The Indicator of Growth Theorem

Historical Notes The indicator Theorem was first proved by e.O. Kiselman [2] for p = 1 (using aka's Theorem on domains of holomorphy) and independently for all p by A. Martineau [4,5] along a line already suggested by the work of Kiselman. The method of the proof relies on the resolution of the 8equation and the L2 -estimates of Hormander (see Appendix III). For n> 1, the indicator h,(z) is not necessarily continuous (for an example see P. Lelong [13]) and an approximation process (see Theorem 7.6) and delicate arguments were needed to circumvent the possibility that the indicator is irregular. For Corollary 7.13 and related results, see Agranovic [1].

Chapter 8. Analytic Functionals

Let M be a complex submanifold of cr" of complex dimension m. Let .tt'(M) be the space of holomorphic functions on M equipped with the topology of uniform convergence on compact subsets of M, and let .Yf(M)' be its dual space of continuous linear functionals. We shall call the elements of .tt'(M)' the analytic functionals on M. Obviously, .tt'(M) is contained in C6'(M), the space of continuous functions on M equipped with the compact-open topology (which is a Frechet space topology): suppose K; an exhaustive sequence of compact sets in 00

M, that is K;cK;+l'

U K;=M

and for each given compact KcM, there

;=1

exists m such Kj:::::JK for i>m; then if K; is an exhaustive sequence in M of compact subsets, the topology on C6'(M) is defined by the semi-norms p;(f)=sup If(z)1 zeKi

and it is independent of the exhaustive sequence K;. With this topology, .tt'(M) is a closed subspace of C6'(M). By the Hahn-Banach Theorem, each element Jl.E.tt'(M)' is the restriction to .tt'(M) of j'lEC6'(M)' . The elements of C6'(M), are just the complex measures with compact support in M. Thus, each analytic functional on M can be represented by a measure Jl. (not unique!) with compact support in M. Since the choice is not unique, this gives rise to an equivalence class Y,,= {Jl.'EC6'(M)': Jl.'(f) = Jl.(f) VfE.tt'(M)}, and we study}'" to find certain representatives with extremal properties. Definition 8.1. A carrier of an analytic functional Jl.E.tt'(M)' is a compact subset K of M such that for every open neighborhood w of K, there exists a constant COl such that 1Jl.(f)1 ~ COl sup If(z)l, fE.tt'(M). Ol

One would like to have some intrinsic idea of the support of an analytic functional similar to the notion of the support of a measure or distribution. In contrast to the preceeding examples, no such smallest support exists in general for analytic functionals (see the examples below). Basically, this is because a holomorphic function which vanishes on an open set is identically zero (in contrast to a C6'" function, which can vanish on an open set without being identically zero). However, sometimes we shall be able to find a

178

8. Analytic Functionals

smallest carrier in a class of carriers. This is the problem we shall study here. We shall first study the problem in ccn and then reduce the general case to the problem in ccn.

§ 1. Convex Sets and the Fourier-Borel Transform Let KcCC n be a convex compact subset and set hK(u)=sup1R.e/ n


zeK

L UiZi · Then

i) hK(u) is positively homogeneous of order 1; ii) hK(U) is subadditive, that is hK(u l + u 2 ) ~ hK(u l ) + hK(U 2 ). Conversely, suppose that h(u) is a positively homogeneous subadditivt. tion, and define K by K={ZECC n :

1R.e
for all UECC n }.

Then K is convex and compact and h(u) = hK(u). A real valued function which satisfies i) and ii) will be called a support function, and if K is a compact convex subset of ccn, hK(U) will be called the support function of K.

Definition 8.2. Let JLE£(CC n)'. Then we define ~(u), the Fourier-Borel transform of JL by ~(u)= JL(exp
any compact subset of ccn. Thus JL is uniquely determined by its FourierBorel transform. Furthermore, if JL is carried by the compact convex set K. it follows from Definition 8.1 that (8,1)

1~(u)I~C.exp(hk(U)+&llull)

for every &>0.

In the following sections, we shall prove the converse to this inequality: that is, if (8,1) holds, then JL is carried by the convex compact subset K.

§ 2. The Projective Indicator If K is a convex compact subset of ccn, £(K) will be the linear space of functions defined and holomorphic in an open neighborhood of K. If {.oj} is a family of open convex sets with -:x:

=

.oj +

n .oj, then :x:

1 ~.Qj

and K =

j~

1

Jff(K)

U Jff(.Q). and we equip £(K) with the inductive limit topology induced

j~1

§2. The Projective Indicator

by thf" {0

181

(1- t)~(2)

<m

I,

~(2»'

in a unique way as z = t m1 (z) + (i-t) <~2)(Z), z)

<mp ~(2)(Z»

Proo}.

n rIC \

K=

.t.,

j= 1

Jff(K)=

". ro

U"

j=l

Theorem, sincc" tinuous function, compact subsets, t~ J1(f)= f dJ1j for Jj j>N~. Then

J

o

CPI'(~)\\ is holomorphic in ~o' """' ~n' si~

.

Lemma 8.6. Suppose that K is cm•

W0,\

function in K which is zero at infinil" for J in an open neighborhood

178

8. Analytic Functionals

smallest carrier in a class of carriers. This is the problem we shall study here. We shall first study the problem in ([n and then reduce the general case to the problem in ([n.

§ 1. Convex Sets and the Fourier-Borel Transform Let K c([n be a convex compact subset and set hK(u)=supIRe
n


L

u;z;. Then

i) hK(u) is positively homogeneous of order 1; ii) hK(U) is subadditive, that is hK(U l +u2)~hK(Ul)+hK(U2)' Conversely, suppose that h(u) is a positively homogeneous subadditive function, and define K by K = {ZE([n: IRe
Then K is convex and com pact and h (u) = hK (u). A real val ued function which satisfies i) and ii) will be called a support function, and if K is a compact convex subset of ([n, hK(U) will be called the support function of K.

Definition 8.2. Let J.lEJf([n)'. Then we define ~(u), the Fourier-Borel transform of J.l by ~(u)= J.l(exp
any compact subset of ([n. Thus J.l is uniquely determined by its FourierBorel transform. Furthermore, if J.l is carried by the compact convex set K, it follows from Definition 8.l that (8,1 )

1~(u)1 ~

C, exp(hk(u)+t:llull)

for every 6>0.

In the following sections, we shall prove the converse to this inequality: that is, if (8,1) holds, then J.l is carried by the convex compact subset K.

§ 2. The Projective Indicator If K is a convex compact subset of ([n, Jf(K) will be the linear space of functions defined and holomorphic in an open neighborhood of K. If {Q) is

n x.

a family of open convex sets with x.

=

U Jf(Q), and j~

1

Q j + 1 ~ Qj

and K

=

Qj'

then Jf(K)

j~l

we equip Jf(K) with the inductive limit topology induced

§2. The Projective Indicator

179

by the topologies on .Yf(Q). This is independent of the choice of the family {QJ. Since .Yf(CCn) is dense in .tf(Q j) (cf. [A, B]), .tf(CC n) is dense in .tf(K) and it follows from Definition 8.1 that J1E.Yf(K)' if and only if J1E.tf(CC n)' and J1 is carried by K. Let JP(CC n+ 1) be the complex projective space of dimension n. We will use coordinates ~ =(~o,~) with ~Eccn and ~oECC. For K a compact convex set we *

let K be the open set of JP(CCn+l) formed by the hyperplanes ~ such that ~nK=0 (~nK=0 if and only if G,z>+~o is never zero for zEK). If J1E.tf(K)" we consider the value of J1 on the holomorphic function

~o


. Since this is homogeneous of degree zero, it depends only on

l.

Definition 8.3. For J1E.tf(K)" we define the projective indicator of J1 by Q)/L(e)

=J1 « z, ~~>o+ ~

J.

This is defined on

K.

Theorem 8.4. The map J1-+ Q)/L is a bijection of the space .tf(K)' onto the linear space .tfo(K) of functions holomorphic in l on (for the points ~ with ~ 0 = 0).

k

and zero at infinity

We prove the theorem in several steps. Lemma 8.5. Q)/L is holomorphic and zero at infinity, that is for

Proof~ By K=

n

linearity,

Q)i~)=~oJ1 «z, ~~+~J,

so

Q)/L(~)=O

~o=O.

when

~o=O.

Let

Qj where Qj is an open convex neighborhood of K. Then since

j=l

00

.tf(K)=

U .Yf(Q), if J1E.Yf(K),

for every j, J1E.Yf(Q/, By the Hahn-Banach

j=l

Theorem, since .tf(Q j) is a closed tinuous functions on Q j with the compact subsets, there is a measure J1(f) = Jf dJ1j for JjE~(Q). Suppose j>N~. Then

subspace of the space ~(Q), the contopology of uniform convergence on J1j with compact support in Qj such that ~ n OJ = 0, which for every ~ is true for

is holomorphic in ~o' ... , ~n' since lnsuppJ1j=0.

o

Lemma 8.6. Suppose that K is convex and compact. Let t/J be a holomorphic * function in K which is zero at infinity. If J E.Yf(K), let f be a representation for J in an open neighborhood (J) of K and let K={z: p(z)~O} for a ~2

180

8. Analytic Functionals

strictly convex function p such that

K c w.

We let

0(0+ I)

-(-1)-2 20 - 1 (1 -) (2ni)" JKf(z)o~~-1 ~o I/J(~) Q(z,~(z))

_

(8,1)

T",(f)

where

?P)

.". ( ~ cp op ~= -.L.o-Zj-~-'-~-' ... '-:;-:::)=1

cZj CZ 1

-

.

~

1-

_

II

Q=L.o-(-I)1+ ~j/\ c~k/\dzl· j=1 k*j 1=1

and

('"11

The map thus defined depends only on J and not on the choice of f or and the map J -+ T",(j) is a continuous linear functional on Jr(K).

K

Proof. We begin by noting thai the expression has a meaning. Indeed,

t/I(~). outSI·deo f"\;0 = 0, -. - IS

0

f d egree - 1·;; a"-I (t/I(~)). In '>0 an d so ~ -;;- IS o~o

\;0

degree (- n) in

~.

0

f

'>0

Since II

.Q(z,~(z))=L(-ly+l~j(z) /\ a~k(Z) /\ dZ I

k*j

is of degree

+n

in

- 2

~,

0

-

1

"' "11-1

(t/I(~))

0\;0

-;;-

1= 1

-

Q(z, ~(z)) is of degree zero. In a neigh-

'>0

borhood of infinity (~o=O), we can choose local coordinates in which, for • 1 an d hence -. t/I(~). I h · ./,. . f· . . Instance, \;j= - IS hoomorp IC· SInce 'I' IS zero at In Inlty. 20 - 1 (t/I(~)) \;0 Thus O~~-I ~ is everywhere defined and holomorphic and the integration is well defined. We show now that the value is independent of the choice of K. It suffices to show that for K 2 ~ K 1 ~ w the value remains unchanged, for if K and K' are two compact subsets of w with nonempty intersection, we need only choose K 2 c K n K' and if the integral has the same value for K and K 2 and for K' and K 2' then it has the same value for K and K'. Thus, we verify

t f(z)~.o_1 ('11-1

A=

bdK,

8 - 1 ~ -11-1 ( ~o 0

__

-.- Q(z,~(z))

C\;o

- J f(z) bdK,

(t/I(~)) \;0

(t/I(~)) -.~o

- ""

Q(z, ~(z))=O.

Suppose that Pj defines K j , that is Kj=[z'Pj(z)~O, Pj strictly convex]. Let X=([,ll x IP((['I1) and let L j be the manifold II

C

L -Zk (fj,

~~)=

(z,~(j)(z)),

where

~V)=~~j, (

':'k

which we identify in a natural way with bdK j • Choose a

cZk (to remain fixed). For zEK I -K2' we let miz) be the point on bdK j where the half line from Zo to infinity passing through z intersects k=1

point

ZOEKl

§2. The Projective Indicator

181

bdKj' If tE[O, 1] set

Since for zEK I -K 2, we can write z in a unique way as z=tm 1(z)+ (l-t)m 2 (z), we have (1_t)(~2)(Z),Z>

(~I)() >=t(~I)(Z),Z> z , z (m 2, ~1)(Z»

(mp ~2)(Z» since (mi , ~j)(z» =0. We show that Y,={z': (Z',~(I)(Z»=O} does not intersect K for zEK I -K2 so that l/IWI)(z» is well defined. Let

. { I

Y.= z': t

1Re(7' ~1(7» 1Re(z', ~(2)(Z» ,-, -(I-t) 1Re(m 2, ~1 (z» 1Re(m 1, ~(2)(Z»

} 0 .

Then Y,c ~ (we note that since we can multiply ~Ul(z) by any complex number, we may assume without loss of generality that (m2' ~1)(z» and (ml' ~2)(Z» are both real, so the inclusion is trivial). For zEK, 1Re(z, ~(1» • ~ ~ ~1) 1Re(z, ~1)(Z» <0 and SInce m 2EK 2 cKp 1Re(m2' ~ (z» <0 so t 1R ( t{1)( » >0. On the other hand 1Re(z, ~2)(Z» <0 but e m 2, '> z

1Re(ml,~(2)(z»>0, so -(I-t)1Re(Z,~(2)(z»>0 1Re(m1, ~(2)(Z»

.

The manifold 1'12=(Z,~I(Z» has boundary 1'1-1'2' We apply Stokes' Theorem on this manifold: A=

an-l J d ( f(z) a~n-l 1'12

'>0

(l/I(~») ) -~- Q(z,~) '>0

an- 1 (l/I(~») _ = J ozf(z) o~n-l -~- Q(z, ~)+ 1'12

'>0

'>0

n

1 (l/I(~») _ -~- Q(z, ~).

a J f(z)d~ o~n-l -

'>0

1'12

'>0

The first term is zero since f(z) is hoi om orphic. n

On 1'12' (z,~>= -1 and

L

(dZk~k+Zkd~k)=O so, since

k=1

o

this last factor is also zero and the Lemma is proved.

Proof of Theorem 8.4. Let C(2)= J.l(z") for J.lEJt'(K') and IX a multi-index of positive numbers. Then there exists C(K) such that for sup l~i~O 11 < C(K) j

the series converges uniformly on K and if (~Z)=(~1 ZI' ... , ~nzn):

~o

L (_1)121 ~ (~Z)2

~O+~IZ1+"'+~nZn"

IX!

~O

•

182

8. Analytic Functionals

of the origin

In

.

Let t/lEPo(K). In order to calculate the ClaP we consider the closed ball BR with center at the origin and radius R > Ro so that K c: Bw For all ~EjjR' the Taylor series at the origin of t/I converges uniformly on BR . Let t/I(u) = L a(a)u a. Then (a)

(l)n-1 on-I (t/I(~») ""(I r:x 1+ 1) ... (I r:x 1+n- l)a(a)~a o):n-I -):- =L... ):a+n'

(8,2)

<'0

<'0

<'0

(a)

We shall show that for 11", = T", as defined by Lemma 8.6, «PI'" = t/I. The hyperplane tangent to the sphere of radius R at the point n

z=(zl> ... , zn) has the equation _R2 +

L (zj-zp'j=O with projective coor-

dinates ~o= _R2, ~j=Zj' Thus

j=1

(_I)n(n;I)+n (2ni)n

C(Il)

x

za

(

oi/

Il

)

~(lllcl+l) ... (lal+n-l)a(a)(_R2)lal+n

(±j=1 (-I)i+ zj /\.dz k=1AdZ 1

k

k ).

k*J

Since the series converges uniformly on BR , we can evaluate term by term, so we calculate _

}'(a)'(Il)-

J (I a 1+ 1) ... (I a1+ n -1) Il:a ( ~ .+1 _ =- n 2lal+n Z Z L... (-1)1 Zj /\ d_ k / \ (-R ) j=1 Hj k=1

dZ k

) •

bdBR

We first apply Stokes' Theorem to obtain _J(lal+l) ... (lal+n-l)-(fJ:a~ j+l_ _ n ~) 2 (1.I+n) 0 Z Z L... (-1) Zj /\ dz k /\ d" k BR (-R ) j=1 Hj k=1

}'(a),(p)-

=(_I)n(n;1) J (lal+1) .. ·(lal+n) fJ:a /\n (d d-) 2 (Ial+n) ZZ Zk" Zk BR (-R) k=1

so that A.(a).(fJ)=O for (a)=I=(p) (we consider the integral on circles in every 2n

complex line and note that

J ein8·e-im8d(J=O for m=l=n) and

o

, A

-( (P),(fJ)- -

1)

n(n+I) (2i)"(lf3l+1) ... (lf3l+n)nn f3! 2 (_I)IPI+n (lf3l+n)!

so C

(8,3)

Hence
(P)

=(-I)IPI~ 1131 ! alP)'

L a(a)ua in a neighborhood of u =0 so
o

§3. The Projective Laplace Transform

183

§ 3. The Projective Laplace Transform Let K be a compact convex set in <en and hK(z) the support function of K. Suppose that f(z) is an entire function of exponential type such that h,(z);£hK(z) (here indicators are calculated with respect to p= 1). Consider the integral Of:)

r~,A(~)=~O

S f(-O,t)exp-(~o;,t)(;,dt) o

fixed, which converges absolutely for A(~,)')={~o:IRe(~oA.» and defines a holomorphic function of ~o in this set. In fact, this value is independent of I"~ for if )'1 and 1.2 are two distinct values such that A=A(~';'1)nA(~')'2)=I=0 then rl(~o)=Tc.Al(~O) and r2«(0)=Tc.A,(~0) are both holomorphic in A and if 1(01 is small enough, it follows from Cauchy's Theorem that r l (~o)- r2(~0)=O, since we are integrating around a closed curve (two half lines emanating from the origin). Thus, by the uniqueness of analytic continuati<2n, r l (~0)=r2(~0) in A, and the value r~j~o) depends only upon «(0' ~)=~, the projection of ~ into P(<e n+ I ). We set

for

h*( -

~E<en ~ ).)}

oc

EJ(~)= ~o

S f( -~A.t) exp -(~o),t)),dt o

for any value of ), for which the integral converges absolutely; EJ(¢) called the projective Laplace transform of f

IS

Definition 8.7. The natural domain of convergence for E J(~) is the interior in

JP(<en+ I) of the set of points for which the integral converges absolutely. Lemma 8.8. Let f(z) be an entire function of exponential type and K a compact convex set and suppose that h7(:::);£h K(z). Then the natural domain of .

.

.

convergence for EJ(~) contains K and l! J(~) is holomorphic in K and zero at infinity. Proof Suppose ~ =1= 00, ~EK. By choosing local coordinates, we can let be the equation of the plane determined by ~. We consider the map of <en into <e defined by f(z)=(z,u). Then f(K)=G is a convex

¢ = {z: (z, u) = I}

*

compact set in <e which does not contain the point 1, since ¢EK. Let hG(J;) be the support function of G and let v be a value for which hG(v)
S f(uvt)exp(-vt)vdt o

is absolutely convergent, since

If(uvt)l;£ C£exp(hK(uvt)+e Iluvll),

184

8. Analytic Functionals

£>0, and hK(uvt)=hG(vt). Hence If(uvt)exp-vtl~exp-'1ltl for some '1>0, which implies that the integral converges absolutely. Suppose If(z)I~C£exp(hK(z)+£llzll) for £>0. By the Cauchy Integral Formula, if (i = (0, ... , (, 0, ... ,0) (( in the /h place), then

so

l I~ aaf Zj

~

sup If(u 1, ... , uj +(, ... , un)1

1~1=1

c£ exp( sup hK(())(exp £) exp(hK(z)+ £ Ilzll) 1,1 :5 1

since

hdz+U~hK(Z)+hK(()'

Thus we also have the absolute convergence

of each of the integrals Saf (- u vt) exp - (v t)(v dt), and we can differentiate aZ j _ under the integral sign, so £.If(~) is holomorphic. 0

Theorem 8.9. Let feu) be an entire function of exponential type. We slIppose for some compact convex set K and for every £>0, If(u)1 ~ C£ exp (hK(u) +£ilull). Then f is the Fourier-Borel transform of an element JiEK(K)' and if £.I f(~) is the projective Laplace transform of f then n(n+ 1) -(-1)-2-

f(u)= where

(hon

a

n- 1

JK exp
R is any bounded strictly convex

--t-

(£.I

(~))

__ Q(z, ~(z))

neighborhood of K with ~2 boundary.

Proof It follows from Lemma 8.6 that £.I f(~) determines a continuous linear functional on K(K) which we denote by Jif" Then from (8,1) we have n(n+l) f~f(u)=Jif(exp
-( -1)-2- S

(2nW

bdK

a

n

_

-

1

exp
m)Q(z, - ~(z)). ---z;;-

(£.If

a u' I -(-,)so that 'XC

1'1=0

IX!

I

au' ). ~(-l)m(i.t)m m 1,I=m IX.

f(-ui.t)=I (

Thus, if Ilull is small enough so that have

If(-ui.t)I~CexpAli,lt

for A
§4. The Case of M a Complex Submanifold of cr"

185

Letting A= 1, we obtain .S! (u)=" (-1)1«la f

~

!:1! u~

W a!

.S!f(~)= ~o L, (-1)1«l a ~ ()!~ )'. a. (<<)

'>0

For fixed u o , we have exp (u o , z) =

00

L

u' z« _0_,

1~1=1 a!

the convergence being uni-

form on compact sets. Thus by (8,2) and (8,3)

§ 4. The Case of M a Complex Submanifold of <en We begin by noting that a Stein manifold is biholomorphically equivalent to a complex submanifold of <en (cf. [A, B]), so by transposition the following discussion applies to Stein manifolds.

Definition 8.10. Let K be a compact subset of M. We define the supporting function H K of K by H K(cp)=sup1Recp(z)

for cpEJft'(M).

zeK

(The supremum over the empty set is defined as - 00.) Then H K is positively homogeneous of order 1 and convex, that is HK(tcp)=tHK(cp), t>O;

HK(CP+t/I)~HK(CP)+HK(t/I).

The restriction of HK(CP) to the linear functions .S!={(z, 0; ~E<en} is just the usual supporting function for a compact set.

Definition 8.11. Let fF be an arbitrary subset of Jft'(M). We define the fF -hull, ofa compact set KcM by

K§"

K§'={ZEM: 1Recp(z)~HK(CP) 'v'cpEfF}.

We shall say that K is fF -convex if K§' = K and M will be fF -convex if K §i is compact whenever K is compact.

186

8. Analytic Functionals

K.? is the largest subset of M such that H K.~ I.~ = H KI.? We note the following properties: i) if.~ c~E.tf(crn) and Kl cK2~M, then K1,,§cK 2 ,.?; ~ -:::-ii) (K'§),ji =K.? = (K,ffi),§ if ffc~, so K.? is ~-convex;

-

-

n

iii) (()K),ji c (K i ).?, so the intersection of a family of ff-convex sets is ff-convex. If ff =~, the family of linear functions, then Kf! is just the convex hull of K intersected with M; if ff = IP, the polynomials, then Kn' is just the polynomially-convex hull of K intersected with M; if ff = .tf(M), then K.1E(M) is just the holomorphically-convex hull of K. To see this, it suffices to consider tht: family eiR!p, !pE§, O~O;'£2n;. Definition 8.12 A compact set K is called an ff-support of J1E.tf'(M) if K is an ff-convex carrier of J1 and for every carrier LcK, L.?=K.?

By Zorn's Lemma, J1 has an ff -support if and only if J1 is carried by some ff -convex set. In general, an ff -support is not unique. For n = 1, consider the 1

linear functional J1(f)= Jf(z)dz. This has a unique convex support, namely o

the set {z: Imz=O, O~lRez;'£ I}, but it does not have a unique polynomialIy-convex support, since any simple arc connecting and 1 is polynomially convex (cf. [B, Theorem 1.3.1.]), Later on, we shall see that there does not always exists a unique convex support for an analytic functional.

°

§ 5. The Generalized Laplace Transform and Indicator Function Definition 8.13. Let J1E.tf(M)'. The generalized Laplace transform A of II is defined by A(qJ) = II (e'l'), qJE.tf(M) and the indicator of J1 is defined by . log IA(tqJ)1 p (qJ ) = I1m sup . t-oc t

The restriction of A to l,) is the Fourier-Borel transform of J1 (when we identify l,) with ern by duality) and is an entire function of exponential type (order 1 and finite type); the restriction of p to l,) is just the radial indicator of the Fourier-Borel transform of J1. Suppose that J1 is carried by K. Then for any open neighborhood L of K, there exists a constant CL such that IA(tqJ)1 = 1J1(te'l')1 ~ C L supexp lRe(tcp), L

§5. The Generalized Laplace Transrorm and Indicator Function

187

and so for t>O,

Hence P(CP)~HL(CP). Since this holds for all L, we have in fact P(CP)~HK(CP). Let E be a complex linear topological space and Q an open subset of E. A function p defined in Q with values in [- 00, + 00) is plurisubharmonic (PEPSH(Q)) if P=l= - 00 is upper semi-continuous (i.e. {OEQ: p(O) < C} is open for topology of E for every real C) and if for every compact disc in Q defined by Z=OI+A0 2cQ for IAI~r 1 p(OI)~-2

n

2"

r P(OI +rei
<>

Another way to describe this situation is to say that p =1= - 00 is upper semicontinuous for the topology on E and its restriction to the intersection of Q with any finite dimensional subspace M is plurisubharmonic or the constant - 00 on any component of Q II M. If J.1.EJl'(M)" then P is analytic in Jl'(M), since it is continuous and P(c/>+AI/I)=J1(e
Theorem 8.14. Let E be a complex linear topological space, separated and having a countable basis for the neighborhoods of the origin. Let Q be an open subset of E and {PiLEI a family of plurisubharmonic functions in Q indexed by I, either the integers or the real numbers, with their natural order. Suppose that {P;}iEI is uniformly bounded above on every compact set of Q. Then p*(O) = lim sup p(O') is plurisubharmonic in Q, where p = lim sup Pi (p* is 0'-8 iEI the upper regularization of pl. Proof Let Oland O2 be fixed elements of E and r such that cp + ).1/1 EQ for 1.1.1 ~r. Let (J be any element in a neighborhood U of the origin for which 0t+).02+UcQ for 1;.I~r. Since {p;} is bounded above on 01+),02+(Jfor 1.1.1 ~r (this is a compact set), by Fatou's Lemma applied to the functions P(A)=Pi(OI +A0 2+(J),

12n

P(OI +(J)~limsupiEI 2n

J Pi(OI +(J+rei
0

1 2n

~-2 n

1 ~-2 2"

(where

n

J P(OI +(J+re
0*

2"

J P*(OI + (Jrei
0

J denotes the lower Lebesgue integral over the interval [0,2nJ). By

0*

our assumption on the topology of E, there exists a sequence {fJ) f= 1 which

188

8. Analytic Functionals

tends to 0 in E such that limsupp(lJ1+U}=p*(lJ 1). Furthermore, p* is j-oo

bounded above on {lJ 1 +).lJ 2: li.l~r}, since the Pi are uniformly bounded above on U{lJ 1+Uj +iclJ 2: 1A.I~r}u{lJl+A.lJ2: li.l~r}, which is j

compact. Thus, applying Fatou's Lemma to this sequence, we obtain 1 p*(lJ 1)= lim sup p(lJ 1+ Uj)~-2 ~-2

.

J lim sup p*(lJ 1+ (Jj+re"'P lJ 2) dcp

7t 0

j-oo

1

2"

j .... oo

2"

J p*(lJ 1+ rei'PlJ2)dcp.

0

7t 0

As far as our applications are concerned, we shall be intersected in the case where!#' is a linear subspace of .tt'(M). Then the upper regularization of the restriction of p to !F, p the indicator function of an analytic functional Jl., is plurisubharmonic and positively homogeneous of order 1 in IF. The upper regularization depends in general upon the subspace considered and even if !#'cr§, we may have p}ip~.

§ 6. Support for Analytic Functionals Theorem 8.15. Let Jl.E.tt'(CC")'. Then for every ~E i?, we have Pi!(e) =inf(HK(~); K carries Jl.) where Pi! is the upper regularization of the rest ricK

tion of p to i? Proof Since Pi!(e) ~ H K(e) for every convex subset K which carries Jl., clearly To prove the converse, suppose that ~o is fixed, 11~01l = 1,

pi!(~)~inf HK(~). K

and let IX be any real number such that IX>Pi!(~o). We show that HK(~O)~IX for some carrier K of Jl.. Set qs(~)=IX1R.eL~j~oj+s(II~II-1R.eL~j~oj) and j

j

set Ns={~Ei?: WI=1, qs(~)~pi!(m, which is compact since Pi!(~) is upper Ns= {0}, for qs(~o) semi-continuous and WI = 1 is compact. Then s>O = IX > Pi!(~o) and for ~ =Ho, WI = 1, lim qs(~)= + 00. By the Finite Intersec-

n

s-oo

tion Property for compact sets, Nso = {0} for some so> 0 and so qso (e) ~ Pi!(~) for all ~. Let Kso be the compact convex set with supporting function qso. Then by Theorem 8.9, Ks carries Jl. and HK (~0)=1X so infHK(~)~pi!(~). 0 o

so

K

Corollary 8.16. Let M be a complex submanifold of CC". If Jl.E.tt"(M), then Jl. has a unique i?-support if and only if Pi! is convex.

Proof Suppose that Pi! is convex and let Ko be the convex compact subset of M such that HKo(~)=Pi!(~)' By Theorem 8.9, a convex compact subset K carries Jl. if and only if Koc;;K, so Ko is an i?-support. 0

§6. Support for Analytic Functionals

189

We shall use this specific case to study a more general situation. With some rather mild assumptions on the family ff, we shall be able to reduce the study of ff -convexity to that of linear convexity. Let Q be an open subset of M and 0(: Q -+ Q' c ([S a holomorphic map. We shall say that 0( is regular if its rank is everywhere equal to m-that is, the • aO( j . h matrIX - , k = 1, ... , n, ) = 1, ... , s has rank m everyw ere. aZk

The map 0( is proper if 0(-1 (K) is compact on Q when K is compact on Q'. Finally if 0( is one-to-one, proper, and regular, we shall say that 0( is an embedding. We shall let 0(*: £(Q')-+£(Q) be defined by 0(*(1/1)=1/100( and we shall denote its transpose by O(t

Theorem 8.17. Let ff be a linear subspace of £(M) which contains elements 0(1' ... , O(s which embed Minto ([s. Then IlE£(M)' is carried by some ff -convex set K if and only if P(1/I)~HK(1/I), 1/IEfF.

Proof If 11 is carried by K, then the inequality is obvious, so we prove the converse. Let L be an open neighborhood of K and L~ M. For every point zEbdL, there exists 1/IzEff such that 11/I(z)l>supI1/l1, since ff is a complex linear K

space, and by continuity, there exists a neighborhood Nz of z in which this inequality continues to hold. Since bdL is compact, there exists a finite number Nj = Nz ,' i = 1, ... , q, which cover bdL. Then K is a compact subset of A= {z: 11/Iz.(z)r~11/Iz.(zj)1 =a j , 1 ~i~q}. Suppose that IB is some finite dimensional subset of 3 spanned by (P1' ... , Pt)· Then

ii=(ii 1, ... ,ii s ')=(0(1""'O(S,1/I z l, •.. ,1/IZq'Pl' ... ,Pt), s'=s+q+t is an embedding of Minto ([S' (we assume without loss of generality that the coordinates ii j are linearly independent, for otherwise, we extract a maximal linearly independent subset). Then M' = ii(M) is an m-dimensional manifold in ([S'. Let bj=sup ii j and let bj be so small that A

Kc{z: liX j (z)l
Let D={WE([S': Iwjl
-

2b j ) I~J

j

Thus by Theorem 8.9, 11* is carried by some compact subset K of D',

190

8. Analytic Functionals

Suppose fEYf([,s') such that fIM'=O. Then f.1*(f) = f.1(fo &) =0, Now if fEY{(D') such that fIM'=O, we can find a sequence f"EJ'l([,s,) with f"IM,=O such that fv --+ f uniformly on compact subsets of D'. Thus f.1*(f) =0 in this case also, so that f.1* extends to a linear functional on )f(D')IM',.,D' (cf. [B, Theorem 7.2.7]). Furthermore, for every holomorphic function f defined on M', there exists a holomorphic function defined on M" = M' n D' such that 11M" = f (cf. [A, B]). Thus, the map of Frechet spaces w: .ff(D) --+ .ff(M") defined by w(F) =FI M " is surjective, and by the Open Mapping Theorem, for K, there exists a compact K c M" such that sup III ~ B sup If I for some constant B > 0,

I

K K Then for tf;E.ff(M') and its extension ~ (to .ff(D')):

1f.1(tf;)1 = 1f.1* (~)I ~ C sup Il/fl ~ CB sup 1tf;1 ~ C B sup 1tf;1 K

K

M"

~CBsupltf;l· L

Thus tf; is carried by K, since L was any compact neighborhood of K.

D

Theorem 8.18. Let ff be a linear subspace of .ff(M) which contains elements iX l ' ... , iXs which embed Minto (['S, Then f.1E.ff(M)' has a unique ff -support if and only if p% is convex.

We shall break up the proof of the theorem into two steps. Proposition 8.19. An analytic functional f.1E.ff(M)' has a unique ff -support if and only if f.1 has a unique ff -support for every finite-dimensional subspace IB of ff such that iXjEIB, j= I, ... ,s. Proof Suppose that f.1 has a unique ff -support K and let ff' be any subset of ff such that K%, is compact. Then the ff'-hull of K is contained in every ff'convex carrier of f.1, since every ff'-convex set is ff-convex. Thus f.1 has a unique ff' -support. Suppose that f.1 has two different ff -supports K, and K 2' Let L, and L2 be compact neighborhoods of Kl and K2 respectively such that K l -L 2 =1=0 and K 2 -L, =1=0. In addition, we choose L, and L2 sufficiently small so that f.1 is not carried by L, n L 2 . Since f.1 is not carried by K, n K 2' the same is true for some compact neighborhood L of K, nK2 and then we take L, and L2 so that L, n L2 c L. Now as in the proof of Theorem 8.17, we choose finite dimensional subspaces Y;; and ffz such that KU,~)cLj,j=I,2. Then if ff'=:F; +ff2' KU,ff')cL., so that f.1 has two ff'-convex carriers, ~ ~ J Kl.% and K 2.%, but f.1 is not carried by their intersection. Thus f.1 has no D unique ff' -support. Proof of Theorem 8.18. Clearly p% is convex if and only if its restriction to every finite dimensional subspace IB is convex. Thus, by Proposition 8.19, it

§7. Unique Supports for Domains in

cr"

191

is sufficient to prove the theorem for every finite dimensional subspace lB such that 0(1"'" O(s belong to lB. Let 0(1"'" O(s' 0(.+ I' ... ,0(•. , span lB and define 0(=(0(1' ""O(s') as an embedding of Minto cc s·. Let J1*=0(~(J1). Then as in Theorem 8.17, J1* defines a linear functional on Jf(O(M», and if K is an i?-support for J1*, O(-I(KIlO(M» is a lB-support for J1. Let P*W=PIB«~'O(»' Then by Corollary 8.16, J1* has a unique i?-support if and only if p*W is 0 convex, so J1 has a unique lB-support if and only if p(t/I)1 1B is convex.

§ 7. Unique Supports for Domains in

(Cn

We have already seen an example of functionals which do not admit unique supports. For n= 1, an analytic functional always has a unique convex support. This is because p(~) is always convex for n= 1 (cf. [D]). For n> 1, however, in general there is no unique support even in the class of convex supports. Let J1 be the analytic functional whose Fourier-Borel transform is COS(~I ~2)1/2 for n=2. Since the circled indicator of this function is I~I ~211/2 and

~ 11/2)2 ~O, (It~111/2_t I 1-

polydisc K t ={(ZI,Z2):

1 tl~ll+ 4t 1~21~1~1~211/2, J1 is carried by the

IZII~t, IZ21~L} by Theorem 8.9. Let a = (t, t

Then atEK s if and only if s=t. If

.

(1

K is

)

any convex carrier contained in K"

then atEK~ SlDce 2=p -, -4t =sup(1R.e
zEK" z=t=a t where

~t= (~,

:J

and 1R.e
-4t). Thus K t is a convex support. The problem

is that the convex supports have corners. We shaH see below that this is a feature of convex supports which are not unique (as well as other types of non-uniques supports).

Proposition 8.20. Let Q be a domain of holomorphy in CCn and K a compact holomorphically convex subset of Q. Then for every open neighborhood W of K, there exists a constant C", su~h that if f E~to.I)(Q) and af =0, then given e>O, there exists UE~<X)(Q) with ou=f and supu~C",suplfl+e. K

'"

Proof Let WI be an open neighborhood of K, WI ~ Q and WI holomorphically convex in Q. Then -log du(z) is plurisubharmonic in Q, and so we can find a plurisubharmonic function cp (depending only on WI' C and f) such that cp == 0 on WI and

[f Ifl2 exp-cpd,r /2 =[J lil2 exp-cpd,+ u

J U-WI

(01

~ Co(W I) sup Iii +e/C "'l

lil2 exp-cpd,r/ 2

192

8. Analytic Functionals

where

C is

a constant to be fixed later (if c=sup( -Iogda(z», we choose y(t)

=

to be a sufficiently rapidly increasing function"" of t such that y 0 for t;£ c and set cp(z)=y(sup( -Iogda(z), c)). Then there exists a solution uEICOO(Q) of the equation = f such that

au

JluI Z [exp-cp](1 + IIzII Z )-3'dr;£ JIfl z exp-cpdr

a

(cf. Appendix III).

a

(n-2) , Let t/J(iICOO(Q) such that 1/1= 1 on K. Then if C,=~, for zEK, -1 i u(z)= C.lllz_allz,-z LJ (I/Iu)dr ="2 Cn

J

-1

-

Ilz-allz,-z aa(l/Iu) "

13.-1

i [ ( 1 )="2 C, lua Ilz-aII Z' - Z "al/l"p._1

+ l I/Ia (1Iz-a\z.-z) "f "13.-1]. Since for zEK, aEsuppal/l, liz-ail >15>0, it follows from the Schwarz Inequality that for zEK, lu(z)l;£ C(w I)[

J

lulzr 'z + C' sup If I,

l.L)l-K

Cl)t

where C(w I) depends only on WI

where C'(w l )

lu(z)l;£ C'(wl)[J Ifl z exp- cpdrr 1z + C' sup Ifl, a "" depends only on WI' since cp=O on WI' Thus G lu(z)l;£ C'(w l ) [Co(W I ) sup If I+"c ]

+ C' sup If I,

WI

Wt

D Theorem8.21. Let Ko and KI be compact sets in a domain of holomorphy Qc{:' and let L be the holomorphically convex hull of Ko u K I . Suppose that K

is such that L- K is a disjoint union of two sets M 0 and M I closed in L - K such that Kj-KcM j , j=O, 1. Then every analytic functional J1.EJff(Q)' carried by Ko and KI is carried by K. Proof Let W be any open neighborhood of K. We begin by constructing a function I/IEctt(Q), such that i) 0;£1/1;£1; ii) I/I=j on wj-w for some open neighborhoods

Wj

of K j

;

§7. Unique Supports for Domains in

cr n

193

iii) IjJ is constant on every component of U - ro for some open neighborhood U of the holomorphically convex hull of Wo u W1. Let mj=Mj-w. Then L-w=mouM 1 and mOnm1 =0. Furthermore, the mj are closed in L, hence compact. Let

m1= {ZEQ:

inf Ilz-wll
Then for some I> > 0, the sets ~. are disjoint and contained in Q. Let cxE~;(B(O, 1») be such that Jcx(z) dr(z) = 1 and set IjJ = Xm, *cx (Xm, is the CI:"

characteristic function of md. Then IjJ = j in mj and 0 ~ IjJ ~ 1. Furthermore, m~ u m~ u w is a neighborhood of L, so we can find two open neighborhoods U and V of L such that V' c U c(m~ um~ uw), where V' is the holomorphically convex hull of V. Set wj=(mjuw)n V so that WOuw 1= V; hence the holomorphically convex hull of Wo u W1 is contained in U. Since U -roc(m~um~), IjJ is constant on every component of this set. By Theorem 8.20, we can find a constant C' such that for every fEJff(Q) and every 1»0 there exists UE~;:'(Q) with au=faljJ and sup lui ~ C' sup If aljJl + I> ~ C' sup If aljJl + 1>, U

roOuCOl

co

since aljJ=o on U -ro. Now J1.(f)=J1.(ljJf-u)+J1.«l-ljJ)f+u), and since J1. is carried by Ko and K 1, we obtain 1J1.(f)1 ~ Co sup IljJf -ul + C1 sup 1(1-IjJ)f+ul "'I

"'0

~

Co sup IIjJI + Co sup lui + C1 sup 1(1-IjJ)fl + C1 sup lui co

CIJ

CJJ

Wl

because ljJ=j in wj-ro. Hence 1J1.(f)I~(Co+ C1)(sup Ifl+ C' sup IfaljJl+l»

'" and since

I>

'"

o

was arbitrary, the proof is complete.

Corollary 8.22. Let Q be a domain of holomorphy in ern and Ko and K1 carriers of J1.EJff(Q)'. Then J1. is carried by K=Kon«L-Ko)uK 1 ), where L is the holomorphically convex hull of Ko u K 1. If Ko U K1 is holomorphically convex, then J1. is carried by K 0 n K 1 . Proof Set S=(L-Ko)uK 1 ,

Mo=Ko-K=Ko-S=L-S,

M1 =(L-K)-Mo·

Then MonM1 =0, MouM1 =L-K and Mo is closed in L-K, for

(L-K)nMo=LnC (KonC Mo)nMo =Ln«C KonMo)uMo)=LnMo=Mo.

194

8. Analytic Functionals

On the other hand, Mo= Mo -K =(L-S)-K =(L- K)nC S is open since Cs is open in L-K. Finally Ko-KcMo and KI-KcCKocCM o, so KI -Kc(L-K)nC Mo=MI' We can then apply Theorem 8.21. 0 Theorem 8.23. Let Q be a domain of holomorphy in (Cn and J-LE.tt'(Q)'. If Ko is an .tt'(Q)-convex support of J-L whose boundary is twice continuously differentiable, then Ko is the unique .tt'(Q)-support of J-L. Proof We show that every convex carrier KI of J-L contains Ko. To show this, it suffices to construct for every Jf'(Q)-convex compact set KI with Ko - KI =1=0 two plurisubharmonic functions F and G continuous in Q such that i) supF;;:;O, supF>O; K,

Ko

ii) sup G;;:;O, hence supG;;:;O (where L is the holomorphically convex KouK,

L

hull of KouK I) and zrtK o, G(z);;:;O implies F(z);;:;O. If zEL -Ko then F(z);;:;O by (ii) and so by (i) sup F;;:;O. Thus sup F;;:;O, where (L-Ko)c:K,

K

K=KonI.; and I.; is the hull of(L-Ko)uKI' Hence K is a holomorphically convex proper subset of K o, since F>O somewhere in Ko. Then Corollary 8.22 shows that K carries J-L, which is a contradiction. Hence Ko is the unique holomorphically convex support of J-L. We now carry cut the construction of F and G. An essential ingredient in the proof will be the fact that the hull of a compact set K c Q with respect to the holomorphic functions, the plurisubharmonic functions, and the continuous plurisubharmonic functions in Q, is the same if Q is a domain of holomorphy. Since KI is supposed holomorphically convex, there exists a CC oo plurisubharmonic fonction G in Q which satisfies sup G
Ko

Theorem 2.6.11 ]). Then for () sufficiently small G= G+ () II z 112 also has these properties. Let H=G-supG and let aEbdK o such that H(a)=O. Ko

We now choose b on the interior normal to bd K ° at a, so that z =1= a, liz -bll ;;:; Iia -bll implies zEKo (it is at this point that we make use the differentiability conditions) and set Hj(z)=H(z) -(3 - j) e(llz _b11 2 -ila _bI1 2 ),j =0,1,2 where O<e<{) is so small that supHj
monic function. Furthermore H j(z);;:;H 3 (z);;:;0 on bdK o with equality only at a. If Hiz) =0, z=l=a, then zEKo and so by the maximum principle, Hj:=O in K o , which is impossible since it is strictly plurisubharmonic. We now construct a function fECC"'(Q) such that Ko= {ZEQ: f(z);;:;O} and f ~ H 2 in Q (we construct the function locally and then piece together the local functions via a CC x partition of unity; of course, f is not plurisubharmonic). Then f - HI is convex in some neighborhood WI of a since (H _H . f -HI ~H2 -HI and f =H I only at a. The matrIX 2 0 2 0 I»)2n IS

. (0

Xj

Xk

j,k=1

§8. Unique Convex Supports

195

positive definite at a (where the Xj are the underlying real coordinates in (Cn =1R2n). Thus, in (1)1' f -HI =sup(A; A ~f -HI in WI) where the supremum is taken over all functions A(z)=1Re0 we let liz II

~I

Gq=H I +Ao+sup {A; Ao+A~f -HI in WI and IIAII
where Ao is defined by

H 2(Z) - HI (z)=H I (z) - Ho(z)=Ao(z) +o(z -a) (Ao is the first order approximation in the Taylor series development of (1Iz-bI1 2 -lla-bI1 2». Then Gq is continuous and plurisubharmonic in Q, and G~ = f in some open neighborhood w~ of a, since Ao is the best atIine approximation to f -HI at a. Furthermore, G~~f in WI by construction, and HI +Ao~Gq in Q, Ho+Ao~HI in Q and HI +Ao~H2 in Q. Since Gq decreases to HI+Ao and HI+Ao~H2<0 in KO-wl' by Dini's Theorem, for r[
and

G(z)~O)

~sup(Ho(z)+Ao(z);

z¢K Ouw 2 and HI(z)+Ao(z)~O) ~sup( -c;( -c;(llz-bI1 2 -lia-bI1 2»; z¢Kouw)
-sup(Ho+Ao))~inf( K,

-q, -supH 2»0. K,

Then (i) and (ii) are satisfied, which completes the proof.

o

§ 8. Unique Convex Supports The condition of differentiability imposed on the boundary in Theorem 8.23 is somewhat artificial, and we would like to replace it with some weaker condition. We do this below for the case of convex supports.

Definition 8.24. Let K I and K 2 be two compact convex subsets of (Cn. A convex set L linearly separates KI and K2 if for every ~ a complex linear functional, ~(KI)(')~(K2)c~(L). In particular, for every A, O~A~ 1, AKI +(l-X)K 2 separates KI and K2 linearly.

196

8. Analytic Functionals

Lemma 8.25. If an analytic functional JlEJf"(
•

.

Proof If (fJ/l is the indicator of Jl then (fJ/l extends holomorphically to CKI

.

and CK 2· Suppose that L linearly separates KI and K2 and let t be a hyperplane of CL (which is not the element at infinity). Let D(t) = {(l, ;'~I' ... , ),~"), ),E
as a function of A: (fJ/l«(l,A~»=

L

Sn(A,~)(A'-),)"

which has radius of con-

n=O

vergence R(I., ~). Furthermore, if R*(I., ~)= liminf R(X,O then ():.n-().. ~)

R(A,~)

=R*(A,~) except perhaps on a set without interior. Thus if we can show that (fJ/l extends to a holomorphic function of the complex variable A on

CLnD(t) for every t, the series for (fJ/l will converge in CL, which will prove • the lemma, since CL is open.



The hyperplane t is defined by an equation 1 + 0 =0 or 0 =0. Let ~/l be the analytic functional on Jf"(
-~

if '10=1=0 or 0 if

if '10=1=0 or (fJ~)0)=~/l(0)=0=(fJ/l(0). Then by the unicity of the convex support for n = 1,

.C

..

(~(KI)n~(K2»=C (~(KI»uC (~(K2»

and

(fJ~,"

is carried by the intersection

•

~(KI)n~(K2)'

morphically to eLand Jl is carried by L.

Thus (fJ/l extends holo0

Corollary8.26. If Jl=l=O and KI and K2 are two convex carriers of Jl, then KlnK2=1=0.

.C

.C

Proof Suppose KlnK2=0. If (fJ/l is the indicator of Jl then, (fJ/l extends to K I and

K 2' so it is defined and holomorphic everywhere hence it is constant, thus zero since it is zero at infinity. 0

§8. Unique Convex Supports

197

An open set Q c IP( CCn+ I) is starlike with respect to the origin if for every eEQ, teEQ for O~t~l. If Q« is a family of starlike domains with respect to the origin, then Q« is also, and it is simply connected. Thus there exists a

U

«EA

largest set which is starlike with respect to the origin in which cP II is holomorphic for J.LEX (CC n)'. We will denote this set by Oil and call it the starlike domain of cP Il' If K is a convex carrier of J.L containing the origin, *

*

*

then CK is starlike with respect to origin in IP(CC n + I) and hence CK c QIl' Thus a convex set L containing the origin will be a convex support of J.L (relative to the family of all convex sets containing the origin) if and only if * * CL is maximal among the open sets CM where M runs through the family of * convex compact sets which contain the origin and are contained in QIl'

Definition 8.27. A complex hyperplane will be said to be a supporting hyperplane at xoEbdK, K a compact set, if it is contained in a real supporting hyperplane at xo' Lemma 8.28. Let K be a compact convex set. Then for xEK, the convex hull of (K n CB(x, e» does not contain x for every e > 0 if and only if x is extremal. Proof Suppose that K is contained in some p dimensional subspace of 1R. 2n = CC n• If there exists eo> 0 such that the convex hull of K n C(B(x, contains x, there exist (p + 1) points of K at a distance at least eo from x such that x is in the convex hull of these points. Thus the point x is not extremal in this simplex and thus not extremal in K. On the other hand, if x is not extremal, x is contained in the interior of some line segment in K, say x=tx o +(1-t)x 1 O
eo»

Lemma 8.29. Let K be a convex support of J.L which contains the origin. For every extremal point xoEK, there exists a complex supporting hyperplane at Xo which is a boundary point of Oil' If every complex supporting hyperplane at Xo which contains the origin also contains K, then there exists a complex supporting hyperplane at Xo which does not contain the origin and is a *

boundary point of QIl' Proof If all the complex supporting hyperplanes at Xo contain the origin, *

then they are by the very definition of Q Il boundary points of the domain (they correspond to the point at infinity and CPIl is defined in a neighborhood of the points at infinity). Thus, we assume the contrary. Let V be the smallest complex subspace of cc n containing K and let U be a complementary subspace such that Un V = {OJ and V x U =CC n• We

198

8. Analytic Functionals

.

a:: q. Let CK be the complex • y • hyperplanes in 1P(a::q+ 1 ) which do not meet K. Then CK =(C K) x a:: n- q . • y Suppose that ({Jp. is holomorphic in CK x a:: n - q and that it can be extend-

assume without loss of generality that V =

y

•

ed to a neighborhood w of a boundary point (v~, u') of CK. Since 1P(a::q+ ') is a complex manifold of dimension q, we can assume without loss of generality that w = LI' x LI" where LI' is a poly disc in a:: qand LI" is a polydisc in a:: n - q• Let 1(z', z")= L c(~)(z")z" be the Taylor series expansion of ({Jp. in z'ELI' with (~)

coefficients in £(LI"). If R(z") is such that this series converges absolutely for sup Iz;I~R(z") for z" fixed, then 1

~i~n-q

1/1 (z") = -log R(z") = lim sup lal- 1 log Ic(~)(z')I, I~I ~oo

and hence R(z") is either identically + 00 in LI" or tjJ(z")* is a plurisubharmonic function. But tjJ(z")* is identically - 00 on an open subset of LI". Thus, we conclude that R(z") is identically + 00 in LI", so ({Jp. can be extended to a neighborhood Wy(v~) x a:: n - q, where Wy(v~) is a convex neighborhood of v~ in V The trace of a complex hyperplane on V is either all of V or a complex hyperplane of V. In particular, the trace on V of a complex hyperplane with support at Xo which does not contain the origin is of the form (v~, u') where v~ is a boundary point of

.C y

K, u' Ea:: n- q.

Let 1!y(xo) be the set of these boundary points. Then 1!y(xo) is compact

U

and (

Wy(v~)

•

x a::n-q)uC K contains a starlike set which is thus simply "*

O

V E7r V(XO)

.

connected and of the form (6J y(xo) xU') u CK, where wy(xo) is a neighborhood of 1!y(xo). Thus, Let

({Jp.

is holomorphic in (wy(:X: o) xU') u CK = Q.

Ln be the convex hull of

(K(lB(Xo,~)).

Then for n>n o' CLncQ.

tLn (t K) . .

u Wy (x o) for n > no' If this

This will be true if we can show that

c

last statement does not hold, since CK ~ CL n , for every n there exists a

. .

•

complex hyperplane i) ~n ( l

~n

y

of V such that

Ln =0,

ii) ~n(lK*0,

.

iii) ~n¢Wy(xO)'

.

.

If ~o is a limit point of ~n then from ii) ~o ( l K =1=0 and ~o is a supporting plane at Xo by i), which contradicts iii). Thus, Lno is a carrier of J1 which contains the origin. This is the final contradiction which proves the lemma. D

§8. Unique Convex Supports

199

Corollary 8.30. Suppose that K is strictly convex and that for each point of bdK there is only one complex tangent plane. Then if K is the support of an *

analytic functional /1, it is the unique convex support and CK is the domain of definition of CPl" Proof By translating K, we may assume without loss of generality that K contains the origin in its interior (a strictly convex set always has non*

empty interior). Then CPI' is holomorphic in CK and cannot be extended to a neighborhood of any boundary point by Lemma 8.29. Thus, if L is a convex *

*

carrier CLee K so K c L.

D

We say that a compact set K is linearly convex if its complement is a **

union of complex hyperplanes or equivalently if K = CCK. Let IP be the family of linearly convex sets. A IP-support will be said to be linearly *

convex. If /1 is carried by K, then CPI' extends to CK and if the hypotheses of Corollary 8.30 are fulfilled, K is the unique linearly convex support. Suppose that K is a convex set and V the smallest linear subspace of (Cn containing V. We shall say that K has the property (u) if ( ) {for every extremal point x of bd K, there exists at most u one complex hyperplane which has support xo' Lemma 8.31. If K is a convex carrier of /1 and /1 is carried by a complex subspace V, then /1 is carried by K n V. Proof We first show that K u V is holomorphically convex if V is a convex subset of V. Let V be defined by fp ... ,fn-q (i.e. V={z:fl= ... =fn_q=O}). Suppose KnV=I=K. For x¢KuV there exists CPx such that CPx(x)=l and cp(y) = 0 for YEV. Let M =sup ICPx(x)l. Since K is convex, there exists I/Ix such that 1/1 x(x) = 1 K

and

supll/lx(z)I<~, so I/IxCPx(x) = 1 and ZEK

M

supll/lxcpx
morphically convex. Let W be a neighborhood of K n V. Then there exist open holomorphically convex neighborhoods of K, V and K n V which we denote respectively by WI' W z and W3 such that (WI nwz)cw and W3 C(WI uW z ). Then /1 is carried by w Z nw 3 and w 1 nw 3, and since w3=(w2nw3)u(OJln0J3) is holomorphically convex, by Corollary 8.22, /1 is carried by (WI nw z ) c OJ. D Theorem 8.32 Let K be a compact convex set which satisfies condition (u). Then if K is the support of an analytic functional /1, it is the unique support.

200

S. Analytic Functionals

Proof If K is a point then the condition (u) is satisfied and furthermore K is the unique support of 11, since by Lemma 8.25 every convex support contains K. Suppose now that K is not a point. By Corollary 8.26, if K and L are two convex supports, then K nL*0 so we can suppose OEK nL. Since L is a support, there exists a point of K which is not in L. Suppose that K c V for V some linear subspace of cr". We first show that L lies in V. Since 11 is carried by a compact subset of V and by L, it follows from Lemma 8.31 that 11 is carried by V n L. Since L is a support, we must have L = V n L. Suppose that I is a complex hyperplane supporting XoE K such that K is not contained in I. Let be the trace of a real supporting hyperplane at Xo such that 1c and let be defined by the equation Re(z, 1]*) = 1 chosen so that supRe
r

r

r

ZE~K+(1- ~)L

ZEL

means that 1 does not meet AK +(l-A)L for O~),~ 1. Since the trace of AK +(l-),)L on V is ~~, 1 does not intersect ~~ either. *

Let ~ be a complex supporting hyperplane at Xo such that ~nV=~v' By Lemma 8.30, 11 is carried by },K +(1 -/.) L and hence