Every Stein subvariety admits a Stein neighborhood

Inventiones math. 38, 89-100 (1976) Inve~tio~es mathematicae �9 by Springer-Verlag 1976

Every Stein Subvariety Admits a Stein Neighborhood*

Yum-Tong Siu

Department of Mathematics, Yale University, New Haven, Connecticut 06520, USA

In this paper we prove the following result.

Main Theorem. Suppose X is a complex space and A is a subvariety of X. l f A is Stein, then there exists an open neighborhood U of A in X such that U is Stein.

Some special cases of the Main Theorem were obtained by Douady [11, Appendice] and LeBarz [3]. Douady proved the special case where X is a relatively compact open subset of a Stein space Y and A=Xc~ V for some subvariety V of Y. LeBarz proved the special case where A and X are both nonsingular with the additional property that the tangent bundle of some open neighborhood of A in X admits a holomorphic connection or is generated by global holomorphic cross sections.

Our method of proof uses Richberg's result [7] and the techniques which Grauert [2] devised to prove that a compact subvariety with a weakly negative generalized normal bundle is exceptional and uses a special plurisubharmonic function which arises from considering the envelope of holomorphy of a certain Reinhardt domain. The required Stein neighborhood is then explicitly constructed and its Steinness is proved by producing a continuous strictly plurisubharmonic exhaustion function and invoking Narasimhan's result [4].

The Main Theorem has the following three corollaries as immediate consequences.

Corollary 1. Suppose V is a complex submanijbld of a complex manifold M. I f V is Stein, then there exists a biholomorphic map from an open neighborhood W of V in M onto an open neighborhood of the zero cross section of the normal bundle of V in M such that its restriction to V agrees with the canonical map from V onto the zero cross section. As a consequence, there is a holomorphic retraction from W onto V.

Corollary 1 follows from the Main Theorem and [1, p. 162, Hilfssatz 11].

* Research partially supported by an NSF Grant and a Sloan Fellowship

90 Y.-T. Siu

Corollary 2. Suppose S is a Stein manifold, M is a complex manifold, and q: S -* M is a holomorphic immersion. Let N be the normal bundle of q. Let S be identified with the zero cross section of N. Then there is a holomorphic immersion of an open neighborhood of S in N to M which extends q.

This follows from Corollary 1, because it is well-known that we can construct a manifold l f /wi th a locally biholomorphic map p: ~ / ~ M and a proper holomorphic embedding ~: S --, ~ / s uch that q = p o q-

Corollary 3. Suppose S is a complex submanifold of a complex manifold T, xeS , M is a complex manifold, and q: S ~ M is a holomorphic map which is a local embedding at x. I f S is Stein and dim T _-< dim M, then q can be extended to a holomorphic map from an open neighborhood of S in T to M which is a local embedding a t x .

The proof of Corollary 3 goes as follows. Let S' be the image of S under the map q': S-+ T x M defined by the inclusion map S ~ T and q. By the Main Theorem, we can find a Stein open neighborhood W of S in T and a Stein open neighborhood W' of S' in T x M. There exist a proper holomorphic embedding r: W ' ~ •N and a holomorphic retraction s from an open neighborhood P of r(W') in ~N to r(W'). There exists a holomorphic map t from an open neighborhood Q of x in T to M which extends qIQc~S and is a local embedding at x. Let p': Q ~ T x M be defined by the inclusion map Q ~ T and p. We can find a holomorphic map u: W--, ~N which extends ro q' and whose components have first order partial derivatives at x agreeing with those of the components of r o p' at x. Let n: T x M --, M be the natural projection. Then ~ o s o u: u- 1 (p)__~ M is the required extension of q.

Results weaker than Corollaries 1, 2, and 3 were obtained by Royden [9] and Seabury [10]. The result obtained by Royden [9] is useful in the infinitesimal theory of the Kobayashi metric [8].

The composition of this paper is as follows. In w 1 we reduce the Main Theorem to the special case where X is finite-dimensional. In w we show that every compact subset of A admits a Stein open neighborhood in X. In w 3 we discuss a special plurisubharmonic function which arises from considering the envelope of holomorphy of a certain Reinhardt domain and which is important for the proof of the Main Theorem. In w 4 we sew up the proof of the Main Theorem.

i wish to express my gratitude to Michael Schneider who posed to me during the 1975 American Mathematical Society Summer Institute in Williamstown, Massachusetts, the question whether every Stein submanifold of a complex manifold admits a Stein open neighborhood. This paper stemmed from my attempts at answering his question.

w 1. In this section we reduce the proof of the Main Theorem to the special case where X is finite-dimensional. (This reduction is not absolutely essential. It is carried out here because it helps to simplify slightly the arguments in w 4 and it is very easy.) Suppose we have proved the Main Theorem when X is finite- dimensional. We want to prove the general case. Let Xv be the union of all v-di-

Every Stein Subvar ie ty Admi ts a Stein NelgL,,, 'q 'hood 91

mensional components of X. Let

Y~= ~)X~, i - O

A~=Ac~X,,,

B,,=AnY,..

We now construct by induction on v a Stein open ne ighborhood U,, of B,, in Y,. such that U,.c~ Y~ ~= U,._~. Suppose we have U,. and we want to construct U,,+I. There exists an open subset W' of Y,,~I such that W'c~ Y,,= U,., Let W" be an open ne ighborhood of A,. ~ - W' in X,, ~1 - Y,,. Let W - W'w W" and V = U~uA~+~. Since U~ = W ~ Y~, U~ is a Stein subvariety of W. A,. ~ is clearly a Stein subvariety of W. Hence by [5], V is a Stein subvariety of W. Since W is finite-dimensional, we can find a Stein open ne ighborhood U~+I of V in W. U,.+~ is a Stein open ne ighborhood of A in Y~+~ whose intersection with Y,,, is U~. This completes

x

our construct ion by induction. Let U = ~ U,,. Then every v-dimensional branch i 0

of U is a branch of U,, and is therefore Stein. Hence by [5], U is a Stein open ne ighborhood of A in X. In the rest of this paper we assume that X is finite- dimensional. We assume also that X is reduced.

w 2. We prove in this section that ever), compact subset K ~?fA admits a Stein open neighborhood in X.

Let J be the sheaf of germs of holomorphic functions on X which vanish identically on A. By replacing A by a relatively compact Stein open ne ighborhood A' of K in A and replacing X by an open subset X ' whose intersection with A is A', we can assume without loss of generality that or ~2 is generated by a finite number of global cross sections gl . . . . . gk-

Let L be the linear space over A which is associated to j / j 2 (see [2, p. 351, Def. 5, and p. 352, Satz 7]). L can be naturally regarded as a linear space over X. Let L be the linear space over X which is associated to J . The sheaf-epimorphism j _ . j / j 2 gives rise to an embedding L~--~L. Let ~z: L - ~ X be the natural projection. We claim that L = ~ t (A) set-theoretically. The inclusion L~--~ ~ - 1 (A) is clear. Take x~A and let

be an exact sequence for some open ne ighborhood G of x in X such that the holomorphic functions al . . . . . ~q on G defining a form a minimal set of generators for i x . Choose G so small that we have a sheaf-epimorphism

r ~--,a l(J21G). Let p be defined by the matrix (Pii)l <i<q,1 <=JSp of holomorphic functions on G and ~ be defined by the matrix (z,)~ ~ q , ~ ~<,. of ho lomorphic functions on G. Then 7t I (G) is b iholomorphic to the complex subspace of G • C q whose ideal- sheaf is generated by

q

pii(x) z i (1 <j<p) i 1

92 Y.-T. Siu

where x ~ G and (z 1 . . . . . zq)~ ff5 q. This biholomorphism maps L c~ g- I ( G) biholo- morphically onto the complex subspace of G • (E q whose ideal-sheaf is generated by

q

2 PiJ(X) Zi (1 <=j<=p), i=1

q

~ r,(x) z i ( l < / < r ) . i=1

Since al . . . . . aq form a minimal set of generators for i x , it follows standardly from Nakayama 's lemma that all zlz (1 <i<q, t< l<r ) vanish at x. Hence set- theoretically lr- 1 (x) = L. This shows that L = n ~(A) set-theoretically.

The cross section gi of j / j 2 corresponds to a holomorphic function hi on L which is linear along each fiber (1 <i<k). Let ~ be a positive C a strictly plurisubharmonic exhaustion function on A. Since gl . . . . . gk generate j / j r , it follows

k

fr~ the definiti~ ~ L that (qJ~ rt) + l~ ( ~=l lhilZ) is a strictly

function on L-OL, where 0 L is the zero cross section of L. Let k

�9

Then ~o is C ~ on L and strictly plurisubharmonic on L - 0 L and the zero-set of q0 is precisely 0 L.

By Richberg's result [7, p. 275, Satz 3.3], q~ can be extended to a C a strictly plurisubharmonic function ~5 defined on an open neighborhood D of L - 0 c in L, and 0 can be extended to a C a strictly plurisubharmonic function ~ defined on an open neighborhood Q of A in X. By replacing Q by Q c~ {~ >0}, we can assume without loss of generality that ~ is positive on Q.

Let c be the supremum of ~ on K. Let T be the set of all points of A where ~ < c and let T be the set of all points of Q where ~ __< c. Take a relatively compact open neighborhood W of 0Lc~rt-I(T) in L. There exists e > 0 such that qg>e on rt- ~ (T) ~ ~3 W. Choose a relatively compact open neighborhood W' of 0 L c~ rc ~ (T)

e in W such that q~<- on rr-I(T)c~OW '. Choose an open neighborhood V of A in Q such that 2

(i) (W-W' )c~r t - l (Vc~ 7") is a relatively compact subset of D,

(ii) ~ > e o n / r and

q3<~ on rc (iii) I ( V c 3 T ) ( - ~ W / .

There is a canonical holomorphic cross section s of L which is 0 on A and is nowhere zero on X - A (see [2, p. 353] for the construction of s). There exists p > 0 such that

Wn(p s) (7"c~ v)=r162 Now define 17 to be set of all points x of V with ~ ( x ) < c which satisfy one of the following two conditions:

(i) ps(x)~ W'; (ii) ps(x)~ W - W' and (o(ps(x))<e.

Every Stein Subvariety Admits a Stein Neighborhood 93

Then I? is a Stein open neighborhood of K in X, because, if q5 denotes the

plur isubharmonic function on l? which equals ~ on (/c~(ps) 1 ( ~ ) and equals

the maximum o f ~ and ~b o (ps) on ( / - (ps ) 1 ( ~ ) , then

1 1

is a cont inuous strictly plur isubharmonic exhaustion function on 17".

w 3. For the last step of the Main Theorem, we will need a special p lur isubharmonic function. We introduce it here and explain where it comes from. We will need only certain properties of this special plur isubharmonic function and will not need any knowledge of its origin. However, the knowledge of its origin provides a better understanding of it and at the same time gives ready proofs of some of its properties.

First we introduce the following notations. For an m-tuple g =(g~ . . . . . g,,) of numbers, IJgl[ denotes the maximum of [gil for l<_i<_m. When gl . . . . . g,, are functions, IlglL is defined in the same way and is a function. The notat ion {equalities and inequalities involving h~ . . . . . h,,} means the set of all points in the intersection of the domains of definition of hi . . . . , h,, which satisfy the given equalities and inequalities. These notat ions will be used in the rest of this paper.

Take 0 < a < b , 0 < c < d and positive integers k,l. Let z=(z~ . . . . . Zk) be the coordinate of [~k and let w=(w~ . . . . . w~) be the coordinates of C ~. Let D c l l 2 k • I~ l

be the union of {[bzll < a, Nw][ <d} and {[/zlb <b , UwH <c} and le t / ) be the envelope of ho lomorphy of D. We want to describe /5 explicitly. Since D is a complete Reinhardt domain in Ckx ~2 ~, b is the smallest logarithmically convex complete Reinhardt domain in I1~ k x ~ which contains D [6, p. 120, Prop. 5].

For l<_i<_k, l<=j<l, let ~ij: ~ k x f f j - - ~ 2 be the projection which sends (z,w) to (zi,wj) and let Dijcl~ 2 be the union of {]zi[<a, lwjl<d } and {[zi[<b, Iwjl<c}.

We observe that

(1) D= N iJ'(O0" l <=i<~k l<_j~l

The inclusion c is clear. To prove the other inclusion ~ , we take a point (z ' ,w ' ) - - ( f 1 . . . . . z~,w' 1 . . . . w'l) belonging to the right-hand side. Clearly Ilz'll<b and I[ w'[I < d. Suppose (z', w') is not in {N z I[ < b, l[ w [I < c}. Then [w)l > c for some j. Since for every i (z'i,w))6Dij, it follows that Iz'A<a for every i and (z',w') is in {[Izll < a , Ilwl[ <d}. Likewise, one proves that, if (z', w') is not in {[Izll < a , IIw]l <d}, then it must be in {l[zll <b, Ilwll <c}. Thus (1) is proved.

For 1 _< i < k and 1 =<j__< l we define the plur isubharmonic function

log ]z~l-loga log Iw j [ - log c

Z~j- l o g b - l o g a q l o g d - l o g c '

on I1~ 2. Then the smallest logarithmically convex complete Reinhardt domain /)ij in C 2 which contains Dij is {Iz;l<b, Iwj[<d, zi~<l}, as one can easily see by

94 Y.-T. Siu

looking at the image D',i of Du-((I12 x0)w(0 x C)) under the map which sends (z i, w j) to (log Izil, log Iwjj) and constructing the Euclidean convex hull olD' u in IR 2.

From (1) it follows that b is the intersection of gi~l(bu) for 1 <_i<k, 1 <=j<l, because the image and the inverse image of a logarithmically convex complete Reinhardt domain under =u is also a logarithmically convex complete Reinhardt domain. Denote by k,~ Z~:b,c,d or simply by X the maximum of Zuo n u for l< i<-k and l<=j<=l. That is,

k,t / l o g l z i l - l o g a log lwj l - logc~ Za'b'c'd= M~<-Xk \ ~g b----~og a ~ ~g d~og c 1"

1 <_j<l

This is the special plurisubharmonic [unction which we will use later./) is simply {llzll <b,/Iw[I <d, X< 1}

Denote by O~,~,c, d the set {tlzlj __<b, Ilwll =<d, Z<__l}. Let

B5 = {llzll = a , Jlwll < d } ,

B~ = {a__< llzll _-<b, ttwll = d } ,

Z = {a < IIz[I <b, Ilwll =0}.

The properties of Z which we will need later are the following.

(2) Z<I on B 1 - B 2 .

(3) Z>I on Bz-BL. (4) Z < I on Z.

(5) For f ixed a, b, d (and k, l), if K is a compact subset of {llzll >a, Itwll >0} in ~k , l Ck x C~, then there exists a positive number Co such that, for c < c o, D,,,v,~,d is

disjoint from K.

(2) and (4) hold, because B 1 - B 2 and Z are both contained in D. (3) holds, because, if x ~ B 2 - B 1 , then Izf(x)l>a for some i and Iwj(x)i=d for some j and as a result Zu(~u(x))> 1_

To prove (5), we write K as the union of compact subsets K u (l <=iNk, 1 < j< l ) so that Iz~[>~ i and Iwjl>coj on K u for some (~>a and coi>0. When both zi and w~ are nonzero, the inequality Zu< 1 is equivalent to

Iogs

Hence it suffices to choose c o such that

for all 1 _< i _< k, 1 <=j _< I. This completes the proof of (5). w 4 will be the last step of the proof of the Main Theorem. Since the main ideas

of w may be obscured by the massive necessary technical details (mainly set- theoretical), before we go into w 4, we first illustrate how the function Z is used


in the key point of the arguments of w 4 by considering here the trivial example of trying to find a Stein open neighborhood basis of C k x 0 in C k x ~ . Suppose (2 is an open neighborhood of ~kx 0 in Ckx C ~. Choose a decreasing sequence of positive numbers d~ (v>2) such that {qLz[I <v, HwH <d~} is contained in ~2 and

~k,I d,. + 1 < �89 d2 (v > 2). By using (5), we define c,. > 0 (v > 2) inductively so that D~ _ , v .... d~ is disjoint from {]]zll = v - 1, cv_t < ]lwl[ <dr}, {v-�89 <v, Nwll =d~+l}, and {llzl[=v,d~+~<llwll<d~} (where we substitute the empty set for {Hzll=v-1, c,._t <]lwl] <d~} when v=2). Then the open set W bounded by the (possibly nonsmooth) surfaces

{llzl] <2, Ilwll =�89 d2}

M~c~ {;(k~~ ~ ....... d = l } (v>2)

is a Stein open neighborhood of q2 k x 0 in ~, where

M~=-{v -~<l l z l l<v , ltwll<d~} (v>2).

The main ideas of w 4 are merely modifications of the arguments used in this trivial example. We do not have global coordinates z and w in the general case, but by w 2 we have a Stein open neighborhood U of any arbitrary compact subset of A and we can find global holomorphic functions on U defining A c~ U and also we can extend to U holomorphic functions f~ on A which map A properly into some C". These functions on U defining U ~ A will play the r61e of w and the functions on U which are extensions of f~ will play the r61e of z. We will not have globally defined plurisubharmonic functions like )(~], ....... dr, but we will have the analog of M~ and the analog of 7,~:! ~ ....... d~. [M~. By taking the supremum of the analogs of zk~'~ ........ ,~IM~ and using Richberg's result [7], we will be able to construct a continuous strictly plurisubharmonic exhaustion function for some open neighborhood of A in X which is then Stein by Narasimhan's result [4].

w 4. Now we start the construction of a Stein open neighborhood of A in X. Since A is finite-dimensional and Stein, we can find holomorphic functions fl . . . . . f , on A such that they define a proper map from A into ~?". Denote the n-tuple (fl . . . . . f,) by ./7.

Recall that from w 2 we have a positive C" strictly plurisubharmonic function on an open neighborhood Q of A in X whose restriction 0 to A is an exhaustion

function on A. For any positive integer v, let 7v be the infimum of 0 on A - {H flJ < v}. Then 7~---" or as v--~ oc.

By w 2, for any positive integer v we can find a Stein open neighborhood U~ of {HfLI <v} in x . We can assume without loss of generality that U,. is relatively compact in Q. The n-tuple f l U~c~A of holomorphic functions on Uvc~A can be extended to an n-tuple .[(~) of holomorphic functions on U~. Since U~ is relatively compact in X, the minimal number of generators of the ideal sheaf of A at x is bounded uniformly for x e Uvc~A and we can find a finite number of holomorphic functions g]V) . . . . . g ~ on U,. such that they generate the ideal sheaf of Uvc~A. Denote the mv-tuple (g~l . . . . . g~lv) by gl~l. By replacing g~Vl by the product of g~) with a large positive number and by replacing U~ by a smaller set, we can assume

96 Y.-T. Siu

without loss of generality that

{ll f(V)ll <v, IIg(~)lt-51}

is relatively compact in U~. For 2_<__v< oe we define inductively positive numbers d~ as follows. We set

d 2 = l and define 0<dr+ 1 <�89 (v>2) such that the following four conditions are satisfied.

(i) {ll/(")ll <#, Ilgl~+~)[I <d~+l} is contained in {llgt")/I <d,} for 2__</~__<v.

(ii) {11/{~)[I <v, IIg{~+a)ll <d~+x} is contained in {ll/l~+~q I <v+�89

(iii) {llf{~+l)tl < v - � 8 9 IIg~+l)ll <dr+l} is contained in {l{f(~lll <v}. (iv) {~<~,~ 1, [Jg~+~)[I <d~+l} is contained in {llf(~+l~ll <v - �89

The following two statements are direct consequences of (i), (ii), and (iii).

{llf{~)ll <v, IIg~a)]l _-<d).} is contained in {llf(")ll < p - � 8 9 for v+ l </~_-<2.

{Llf(~)lL <v - 3 , IIg(a)ll <da} is contained in {tlf(")L] <~} for v - 1 <p__<2 and v____2.

(1)

(2)

We define

R1 =~J,

R2 = {]l f(2)ll <2, II g(2)ll <d2},

/~2 = {ll f{2~ II =< 2, 11 g~2)I1 ~ d2},

R~={ v - 3 < [I f(~ll <v, II g(~)ll <d~}

/ ~ = { v - 3 < I1 f(~)ll <v, IIg(~)ll <d~}

(v>3),

(v__>3).

Because of (1),/~v is disjoint f r o m / ~ for 2+ v - 1 , v, v+ 1. A consequence of (iv), (1), and (2) is that

(3) {~<7~ 1} is disjoint from /~z for 2 > v + l .

For v_>2 we w r i t e / ~ - R v naturally as the union of three sets in the following way. Define

Bi={llf(~)ll=v -3, ILg~V)LI <d~} (v>3),

Be =~,

B~a={v-3<llft~)ll<v, I[gl~)bl =d~} (v => 3),

B~ = {11 f(2)l[ <2, Ilgl2~ll =d2},

T h e n / ~ - Rv equals B[ wB~ u B ; for v>2. From (1) and (2) we conclude that

(4) B~ is disjoint from/~v+l.

F rom (i) we conclude that

(5) B~ is disjoint from/1~+1.


F r o m (1) we conclude that

(6) B~ is disjoint from/~v 1.

For 2 __< v < ~ , let Fv: Uv--~ C " x C "v be defined by fl~) and g~). Observe that U,. ~ A equals F~- 1 (C" x 0).

For 2 < v < oo we define inductively 0 < c~ < d~ as follows. Suppose we have defined c~ 1. For 2 < # < v - 1 define

- - n , m 0 , - z , (,~ .... ~oF , (U>3),

0,, = Max(2 [I gl2~]l, Zi,""~2, ~, d~ o F2) (/~ = 2),

where Z"d ~ . u, ~,, d, (2 </~ < V -- 1) was defined in w 3. Let

E 2 =~[ ,

E ~ = { l l f tv ~ l l = v - l , l tgt~)l l<d~,/h._l>l} if v > 3 ,

n ; = {11 fl~+~ll > v - � 8 9 H f~")lq <v, Llgl=+~ll =d~+l},

H ' = {ll f ~ l l =v, II g")lt---_d~} - {llg(V+l)H <d~+l},

n~ = H; u H",

/~ = F~(H~).

Let z = (Zl . . . . . z,) be the coordinates of C" and let w = (Wl . . . . . w, ,) be the coordinates of r We claim t h a t / ~ u _0~ is a compact subset of { II z II > v - 3, II w II > 0}. C l e a r l y / ~ u/4~ is a compact subset of { II w II > 0}. To prove the claim, it suffices to show that Hft~)ll > v - 3 on E~, H'~, and H'~'. For H'~' it is clearly the case. For E~ it follows from (2). For H~ it follows from (1) and (2). Our claim is thus proved. By (5) of w 3, there exists 0 < c~ <d~ such that D~_~, ~,~.a~-""~ is disjoint f r o m / ~ u/4~. This completes the definition of c~ by induction, because the definition of c 2 does not depend on the existence of some q .

For v > 2 define

- - - 1 ~ n , m v G~-F~ (Dv_~ . . . . . . j .

Then G~ equals

{tlf<~)ll <v, IIg<~)lI __<d~, 0~<1}

and is disjoint from E~ w H~. By (2), (3), (4) of w 3 (and the definition of 02), we have for v => 2

(7) 0~<1 on B [ - B ~

(8) 0~>1 on B ~ - B ~

(9) 0~<1 on A n R ~ .

We derive now the following two statements for v > 2.

(10) 0~<1 on /~+a ~ {0~+~ < 1} c~ (/~v- R0.

(11) 0~+1<1 on / ~ n { 0 ~ < l } ~ ( / ~ + l - R ~ + l ) .

98 Y.-T. Siu

The proof of (10) is as follows. By (4) and (5) , /~ + 1 ca (/~ - R,.) equa ls /~+ 1 ~ B ~ .

Since {0 v > 1 } ca /~ + 1 ca B~ is contained in E~ + 1 which is disjoint from G~ + i and since G,,+~ contains /~+~ ca {0,.+~ < 1}, it follows that {0,,> l} ca/~,.+~ c~ B; and /~+~ ca {0~+ 1 __< 1} are disjoint. This completes the proof of (10).

The pro of of(11) is as follows. By (6),/~ ca (/~ ~ 1 - R~ + 1) equa l s /~ ca (Bf + 1 w B~ +1). By (7) it suffices to show only that 0~ + 1 < 1 on/~ ~ ca { ()~ < 1 } ca B~ + ~. Since/(,. ca B~ + is contained in Hv which is disjoint from G~ and since G~ c o n t a i n s / ~ {0~< 1}, it fo l lows/~ca {0~< 1} ca B~ is empty. This completes the proof of (11).

Let R be the union of R~ for 2 < v < o~ and define 0 to be the function on R which is the maximum of O~IR~ for 2__< v < ~ in the sense that, for xeR , O(x) is the maximum of 0~ for all v satisfying xeR~. The function 0 in general is not continuous.

We now prove the following four assertions.

0 is continuous and plurisubharmonic on some open neighborhood of {0= 1 } (i) in X.

(H)

(III)

(iv) positive

{0< 1} is an open neighborhood of A in X.

For every v > l , {~<v , 0 < l } is compact.

For any open neighborhood W of {0= 1} in X and any v> 1, there exists a number q depending on W and v such that 0 < 1 - q on {~ < v, 0 < 1 } - W.

The proof of (I) is as follows. Take x~R with 0(x)= 1. Then x~R~ for some v. Since/~, does not intersect R v unless # = v - 1, v, v + 1, 0 is obviously cont inuous and plur isubharmonic on the open subset R , . - ~ ( R , - R , ) of R~. It

/t--v 1 . v + l

suffices to consider only the case x~R~ca(R , -R~) for # = v - 1 or v + l . Then the definition of 0 gives U~(x) = 0(x) = 1. By (10) for # = v - 1 and by (1 l) for # = v + l, 0 , (x )< 1. Hence, if we define 2 by {#, 2} = { v - l , v+ 1}, then 0 equals ()~ on the open ne ighborhood {0~<0~} c~R~-/~z of x in X and is cont inuous and plurisubharmonic there. This concludes the proof of (I).

The proof of (II) is as follows. F rom (9) it follows immediately that {0<1} contains A. To prove that {0 < l} is open in X, we take a point x of {0 < 1 }. Then x~R~ for some v. We distinguish between two cases.

Case(a). x~R~caR, for # = v - 1 or v + l .

Case(b) . x e R ~ c a ( R , - R , ) for / ~ = v - l , v + l .

In Case (a), O(x)= Max (Ov(X), 0,(x)). Then {0 < 1} contains the open neighborhood R~caRuc~{O,,<l}c~{O,<l} o f x in X.

In Case (b), O(x)=O~(x). By (10) or (11), 0 , ( x ) < l . If we define 2 by {# ,2}= {v - 1, v + 1 }, then {0 < 1 } contains the open ne ighborhood R~ ca {0~ < 1, (t, < 1 } - /~x of x in X. This concludes the proof of (II).

The proof of (III) is as follows. Take a sequence {x~} in { ~ < V o , 0 < l } for some v o > 1. We want to prove that some subsequence of {xz} converges to a point of {~<Vo, 0<1}. By (3), {~<Vo, 0<1} is disjoint from ~ /~, for some

/~>vl

v~ ~2. Hence {x~} is contained in ~ R, . By replacing {x;.} by a subsequence,

we can assume without loss of generality that {x~} is contained in R~ for some


v > 2 and it converges to a point x , o f / ~ . Since Ov(xa)<O(x;)< 1, it follows that 0~(x,) < 1.

We claim that x,~R. Suppose the contrary. Then x,~B~uB~2uB~ 3. Since 0 v ( x , ) < l , it follows from (8) that x,q~B~2-B~. By (i) and (1), B ~ R , . 1. Hence x, eB~3. Since x, eG,. which is disjoint from H,., it follows that x , must belong to B~ c~ { [I g(~+ 1)ll< d~+l}, which by (1) and (2), is contained in Rv+ 1. We have thus a contradict ion. So xcR.

Let I be the set of all # such that x, eR,,. I has at most two elements. 0 ( x , ) = Max 0,(x,) . For 2 sufficiently large, xxe ~ R , and Max O,,(x,)<O(x~.)< 1. Hence

u~l /~I #~I

0(x,) < 1. It is clear that ~ ( x , ) < Vo. This completes the p roof of (III). The proof of (IV) is as follows. Suppose the contrary. Then for some v o > 1

and some open ne ighborhood W of {0 = 1} in X, there exists a sequence {x;,} in {~<Vo, O < l } - W such that the limit of 0 ( x ; ) i s 1 as 2 ~ c . Since by (III) {~ < v o, 0 < 1 } - W is compact , we can assume without loss of generality that the sequence {x;,} approaches a limit point x , in {~ < v 0, 0 < 1 } - W . Since x , is in the domain of definition of 0, x, eR,. for some v. Without loss of generality we can assume that {x; ,}cR~. If x, eR, for / ~ = v - 1 or v + l , then 0 ( x , ) = Max (0~(x,), O,(x,)) and for 2 sufficiently large xa ~ R~, and O(x;.) = Max (0~(x~), 0,(x~). Hence 0 ( x , ) = l , contradict ing x , r W. So we can assume that x,r ~voR~+~. It follows that 0 ( x , ) = O,.(x,). We distinguish between two cases.

Case (a). R~-(R~_ 1 voR,.+l ) contains a subsequence {xx, } of {x~.}.

Case (b). R , contains a subsequence {x;,} of {xa} for # = v - 1 or v + 1.

For Case (a), we have 0(xx,)= 0,.(xa.). Hence 0v(x , )= 1 and 0 ( x , ) = 1, contra- dicting x , r W.

For Case (b), we have x , ~ / ~ and 0(x;.,)= Max (0,.(x~..), 0,(x~,)). Since 0~(x , )= 0 ( x , ) < 1, the limit of O~,(xa,) must be 1 as i -, oc. As a consequence, 0 , ( x , ) = 1. This contradicts (10) or (11), because x , belongs to R,,~{tk.<I}c~(Ru-R,). This completes the p roof of (IV).

We are now ready to sew up the p roof of the Main Theorem. By (II) {0< 1} is an open ne ighborhood of A in X. By (I) there exists an open ne ighborhood W of {0= 1} in X such that 0 is cont inuous and p lur i subharmonic on W. Since by (II) and (Ill) {~ < v, 0 = 1} is compac t for every v > 1, we can find an open ne ighborhood W' of {0=1} in W such that W'm{(9<v} is relatively compac t in W for every v > 1. By (IV) for every integer v > 1 there exists a positive number q~ such that 0 < 1-r /~ on { ~ < v , 0 < 1 } - W ' . Without loss of generali ty we can assume that r/~+l <q~ for all v > 1. Choose a positive C 2 function p on the set of all posit ive numbers such that the first and second order derivatives of p are nonnegat ive

1 1 everywhere and p(v)>tl~+ ~ for all v. Let q~ be the m a x i m u m of 1 0 and p o

1 1 on {0<1}. Since ~ < on { ~ < v + l , O<I} -W' , it follows that 4~ equals

/~v+ I po ~ on {v<~<v+ I, 0<l}- W' for every v. Since 0 is continuous and plurisubharmonic on W, it follows that 4, is continuous and plurisubharmonic on {0 < I}. The function 4)+ ~ is a continuous strictly plurisubharmonic function on {0 < I}.

100 Y.-T. Siu

By [4, p. 195, Th. II], the proof of the Main Theorem will be finished if we can show that q~+~ is an exhaustion function on {0 < 1}. Take any positive integer v. We want to show that { ~ + ~ < v } is compact. It is clearly contained in {~<v,

�9 <v}. Let 4 be the maximum of 1 - ~ and 1 p(v)" Then {q~ < v, q~ < v} is contained

in {~<v, 0<4}. The latter is a closed subset of {~<v, 0<1}, because from > 1 - try it follows that {~ < v, 4 < 0 < 1} is contained in W' where 0 is continuous.

Since by (III) {~<v, 0<4} is compact, it follows that {~b+~<v} is compact. This concludes the proof of the Main Theorem.

References

1. Forster, O , Ramspott, K.J.: Analytische Modulgarben und Endromisbundel. Inventiones math. 2, 145-170 (1966)

2. Grauert, H.: l]ber Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331-368 (1962)

3. LeBarz, E: Sur les voisinages tubulaires en g6om6trie analytique. Math. Ann. 215, 83-95 (1975) 4. Narasimhan, R.: The Levi problem for complex spaces II. Math. Ann. 146, 195-216 (1962) 5. Narasimhan, R.: A note on Stein spaces and their normalizations. Ann. Scuola Norm. Sup. Pisa

16, 327-333 (1962) 6. Narasimhan, R.: Several complex variables. University of Chicago Press, Chicago, 1971 7. Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 257-286 (1968) 8. Royden, H.L.: Remarks on the Kobayashi metric. Proc. Maryland Conf. on Several Complex

Variables (1970), Lecture Notes in Math., 185, pp. 125-137. Berlin-Heidelberg-New York: Springer 1971

9. Royden, H.L.: The extension of regular holomorphic maps. Proc. Amer. Math. Soc. 43, 306-310 (1974)

10. Seabury, C.: Some extension theorems for regular maps of Stein manifolds. Bull. Amer. Math. Soc. 80, 1223-1224 (1974)

11. Siegfried, E: Un th6or6me de finitude pour les morphismes q-convexes. Dissertation of the University of Regensburg 1972

Received February 21, 1976

Note Added in Proof The intermediate result of w 2 on the existence of a Stein open neighborhood of K in X for a compact subset K of a Stein subvariety A of a complex space X was also proved by M. Schneider in Manuscripta Math. 18, 391-397 (I 970).

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Every Stein subvariety admits a Stein neighborhood