No. 4] Proc. Japan Acad., 45 (1969) 243
54. A Remark on the Theorem o f Bishop
By Chikara WATANABE
Department of Mathematics, Faculty of Science, Kanazawa University (Comm. by Kinjiro KUNUGI, M..1. A., April 12, 1969)
1. On normality of a family of pure-dimensional analytic sets in a domain of Cn, the following theorem of Oka [4] is well-known.
Theorem of Oka. Let F be a family o f pure-dimensional analyt- ic sets in a domain o f Cn. Then F is analytically normal i f and only if the volumes o f elements o f F are locally uniformly bounded.
This theorem was proved by T. Nishino [3] in the case of two variables. The proof of this theorem in the case of n variables was given in our former paper (Watanabe [d]).
On the other hand, the concept of geometric convergence was in- troduced by E. Bishop as follows.
Let {Sj be a sequence of closed subsets in a domain of Cn. It is said that {S} vconverges geometrically to a closed set S if for any compact set K, {S, (1 K} is a convergent sequence in Comp (K)" and S= U lim (Sv (1 K) where K ranges over the compact sets. Further
K Bishop [1] proved the following.
Theorem of Bishop. Let {S,} be a sequence of purely 2-dimen- sional analytic sets in a domain D o f Cn. Suppose that {S} vconverges geometrically to a closed set S in D. I f the volumes o f Sv are uni- formly bounded, then S is also an analytic set in D.
We shall prove that in the above theorem of Bishop, S is also purely 2-dimensional if S is not empty.
2. Let D = 4 X { w <R} be a domain of Cn+1, where 4 is a do- main of (z1, ... , z)-space Cn(z). Then the following proposition is well-known (for example, Fuj ita [2]).
Proposition. Let S be a purely 2-dimensional analytic set in D.
Assume that S is contained in 4 X { I w J <R0} for some positive number Ro < R. Then the projection o f S on 4 is also purely 2-dimensional analytic set in 4.
It follows from this :
Corollary. Let D = 4 X { wl J <R} X .. . X { w, <R} b e a domain o f C+~ and S be a purely 2-dimensional analytic set in D. I f S is contained in 4 X { 1 wl <R0} x ... X { J w, <R0} for some positive number Ro < R, then 2I = (S, ~r, 4) is an analytic cover, where 7r is a projection.
1) For a definition of Comp (K), see [5].
244 C. YY ATANABE [Vol. 45,
Now let us prove the following
Theorem. Let {S,} be a sequence o f purely A-dimensional analyt- ic sets in a domain o f Cn. I f {S} yconverges geometrically to a non- empty analytic set S, then the local dimension dimpS at each point p e S is at least 2.
Proof. Suppose that there holds dime S=k<2 for some point p e S. For simplicity we may assume that p is the origin. After a suitable change of the coordinate system, we can choose (n-k) pseudo- polynomials Pk+l(z' ; ~), Pk+2(z' ; ~) • • • , Pn(z' ; ~), whose coefficients are holomorphic functions of z'= (z1, • • • , zk), and positive numbers si (i=1, 2, • • • , n) such that
(i) Sfl UCS*={(z',zk+l, • . •,zn); P1(z'; zl)=0,1=k+1, • • •,n}for a neighbourhood U of the origin.
(ii) the roots of P1(z' ; ~) = 0 are all contained in the disc < E~
(1=k+1, • • •, n) for z'=(zl, • • •, zk) with z~ <s~ (j=1, 2, • • •, k). For sufficiently small positive numbers r, p, the polydisc Q=4X { I zk+1 I <p}
X ... X { zn I <p} is a relatively compact subset of U, where 4 = {z'; I z~ I
<r, j=1, 2, • • •, k}.
We may assume that for z' E 4, the roots of P1(z' ; ~) = 0 are all contained in the disc < P = p'. Then S* is an analytic set in Q
2
and S* fl Q C 4 X { zk+1 < p'} X • • • X { I zn <p'}. Since {S,} converges geometrically to S, it follows that lim (Sv fl Q) c: S fl Q, and it is easily
v seen that for a positive number p" (p' < p" < p) there is a positive inte- ger vo such that S, fl Q is contained in 4 X { zk+1 < p"} X • • • X { zn <p"}
for v >_ vo. On the other hand, denoting 4 X { zk+l <p} X • • • X { I Z2 I <p}
by d, we have Q = 4 X { zA+l I <p} X • • • X { zn <p} and so Sy fl Q C 4 X { z~+1 <p"} X X { zn I <p"}. But from the corollary of Proposi- tion, the projection of Sv fl Q on the (zl, z2, • • •, zA)-space is 4. This contradicts the fact that Sv fl £2 is contained in 4 X { zk + l I <p"} X
... X { I zn I <p"}. Q.E.D.
On the other hand, from the estimation of the Hausdorff measure and the relation between the volume and the Hausdorff measure, the dimension of S is at most 2 if a sequence of purely 2-dimensional an- alytic sets converges geometrically to an analytic set S and if the 22- dimensional volumes of Sv are uniformly bounded ([5]). Therefore we have from our theorem, the following
Corollary. In the theorem o f Bishop, i f the limit set S is not empty, then S is also purely 2-dimensional.
3. Here we shall give some properties of geometric and analytic convergence.
Let {S,} be a sequence of closed sets in a domain D of Cn. Sup-
No. 41 Remark on Theorem of Bishop 245
pose that {S} vconverges geometrically to a closed set S in D. Since lim (Sv (1 K~) c lim (S. f K~ 1), we have S c lira lim (S. (1 K~) for every se- quence {Ku} of compact sets such that K~c K~+1 , and D= U K~.
On the other hand, we have S lim lim (Sy (1 K~) from the very defini- tion, and hence S=lim lim (Sv (1 Ku). The converse is not necessarily true as the following example shows.
Example. Let D = {(z1, z2) e C2; z1-1 <1, (z2 ( <1} be a domain of C2 and {Sn} be a sequence of analytic sets such that S1= (z1, z2) e D ;
zl = , s2=
2\{zi z2) e D ; z1=1 + I}, ... Stn
2- {(z1, z2) e D ; z1=1 + 1 ,
2nS2n1=
+{z1,
(z2) ~ e D•
2z1= + + 1 1 ... +I}, .... Then
4 2 nsince S2n (1 K~~
,and Stn 1 fl K = 0 for the compact set K = {(z1, z2) e D ; z1- 5 ~ 1
4 4
z < 1 {S} does not converge geometrically. But it is obvious
2 that lim lim (S, fl K~) = {(z1, z2) e D ; z1=1} for every exhaustion of D by
(v
compact sets K. However if lim (Sv fl K~) exists for sufficiently large
v p, then there is a subsequence {S.} of {S} which converges geometric- ally to a closed set S* and lim (Sv (1 K~) = lira (S.d (1 K~) for sufficiently
large ~u, and hence we have S* =1im lim (Sv3 (1 K~) = lim lim (Sv fl K~) p 3 p j
for sufficiently large p.
Summing up the above result, we have
Proposition 1. 1 f {Sv} converges geometrically to a closed set S in D, then it holds that S=lim lim (Sv (1 K~) for every sequence {K} ~o f
u v
compact sets such that K,1 c K~+1 • • • , and D = U K~. Further i f S= lim lim (Sv (1 K~) exists, then S is closed and there is a subsequence o f {S,} which converges geometrically to S.
Next we consider the case of analytic convergence of a sequence of pure-dimensional analytic sets.
Suppose that a sequence {S,} of pure-dimensional analytic sets in a domain D of Cn converges analytically to S.2' we shall show that
0 S fl K= lim (S, fl K) for a compact set K such that S (1 K~ 0.
v
Let S~' = U {z; p(z, z') < E}, where p(z, z') means the Euclid dis-
z'ES
tance between z and z'. If Sv j (1 K- S~' (1 K=/= 0 for a sequence of positive integers v1 < v2 < • • • , then we can choose a sequence of points pvj E Svj (1 K-Sw (1 K. Since K is compact it may be assumed that p,3-p. From the assumption p is not contained in S. On the other
2) For a definition of analytic convergence, see [21, [61.
246 C. WATANABE [Vol. 45, hand, from the definition of analytic convergence there are a neigh- bourhood U of p and holomorphic functions f r), k=1, 2, ... ,1 in U such that Sv fl U={z e U; f kv'(z)=0, k=1, 2, ..., l}.
Moreover, since f(z) converges uniformly to a holomorphic function f °(z) (z) in U, f kvj' (z) also converges uniformly to f°(z) in U.
We have f k°' (p) _ ok > 0 since p is not contained in S. But since f kv~) (z) converges uniformly to f °(z), (z), it holds I f>(p) I f(p)
I + I f k°'(pvj)-f °''(pvj) I <ok for sufficiently large j. This is a contradiction and hence we have S.3 fl K-SAE' fl K=0 for sufficiently large j.
Thus there is a positive integer v° depending only on s such that Sv fl Kc See' fl K for v> vo. This means that {S. fl K} converges to S fl K in Comp (K). Thus we have
Proposition 2. I f a sequence {S} vo f pure-dimensional analytic sets converges analytically to S, then it holds that S=lim lim (S. fl K) for every sequence {K} ~o f compact sets such that Ku C K~+1... , and D= U K~.
Remark. Even if lim (S. fl K~) exists, the sheet numbers of Sy
v need not be bounded. Hence we can not always choose a subsequence of {S,} which converges analytically. Such an example was given in former paper ([6]).
References
[1] E. Bishop: Conditions for the analyticity of certain sets. Mich. Math. Jour., 11, 289-304 (1964).
[21 0. Fujita: Sur les families d'ensembles analytiques. J. Math. Soc. Japan, 16 (1964).
[31 T. Nishino: Sur les families de surfaces analytiques. J. Math. Kyoto Univ., 1, 357-377 (1962).
[4] K. Oka: Note sur les families de fonctions analytiques multiformes etc.
J. Hiroshima Univ., 4 (1934).
[51 G. Stolzenberg: Volumes, limits, and extensions of analytic varieties.
Lecture note in Math., No. 19, Springer-Verlag (1966).
[6] C. Watanabe: On a family of pure-dimensional analytic sets (to appear in the Sci. Rep. Kanazawa Univ.).