Open Sets with Stein Hypersurface Sections in Stein Spaces

Annals of Mathematics

Open Sets with Stein Hypersurface Sections in Stein SpacesAuthor(s): Mihnea Coltoiu and Klas DiederichSource: Annals of Mathematics, Second Series, Vol. 145, No. 1 (Jan., 1997), pp. 175-182Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2951826 .

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Annals of Mathematics, 145 (1997), 175-182

Open sets with Stein hypersurface sections in Stein spaces

By MIHNEA COLTOIU and KLAs DIEDERICH*

Introduction

Let D C Cn , n > 3, be an open set such that for any linear hyperplane H C Cn the intersection H n D is Stein. According to a theorem of Lelong [13] it then follows, that D is itself Stein. Using the solution of the Levi problem by Docquier-Grauert [7], this result can easily be generalized to open subsets in Stein manifolds. Therefore, it is natural to raise the following question (see also [4]):

Problem of hypersurface sections. Let X be a Stein space of dimension n > 3 and D C X an open subset such that Hn D is Stein for every hypersurface H C X . Does it follow that D is Stein?

This question is closely related to the following classical form of the Levi problem (for a survey see [18]):

Local Steiness problem. Is a locally Stein open subset D of a Stein space X necessarily Stein?

A complete answer to this problem is not known. There are only partial results (see [4] for a discussion of this subject). In particular, for dimX = 2 the answer is known to be positive [1].

It is immediate that, therefore, a positive answer to the "Problem of hypersurface sections" would, by induction over the dimension, also provide a positive answer to the "Local Steiness problem".

Concerning the "Problem of hypersurface sections" it is known (see [2]), that D as above is indeed Stein, if one knows in addition, that H1 (D, 0) = 0 (in this case the hypothesis dimX > 3 is not necessary; a weaker statement was already proved in [9]).

*The first author thanks the University of Wuppertal and the DFG for the given support during this research.



176 MIHNEA COLTOIU AND KLAS DIEDERICH

In this note we now produce a counter-example to the "Problem of hypersurface sections". This shows that in order to get the Steiness of D, some additional hypothesis (like H1 (D, 0) = 0 as above) is necessary, if X is singular. More precisely, our main result can be stated as follows:

THEOREM 0.1. There is a normal Stein space X of pure dimension 3 with only one singular point, and a closed connected analytic subset A C X of pure dimension 2, such that D = X \ A has the following properties:

a) D is not Stein;

b) For every hypersurface H C X (i.e. closed analytic subset of X of pure codimension 1) the intersection H n D is Stein.

This shows, again, that in the case of singular Stein spaces, the situation may be drastically different from the smooth case (see also [8], [9], [11], [12], [19] for other examples concerning the Levi problem on singular Stein spaces).

1. Tools and lemmas

For the construction of an example proving Theorem 0.1 some properties of weakly 1-complete manifolds will play an essential role. In order to fix the terminology we state:

DEFINITION 1.1. A complex manifold X is called weakly 1-complete if there exists a Coo plurisubharmonic exhaustion function up: X - R.

Notice that even for complex manifolds with smooth boundary, the re- quirement of being weakly 1-complete is much stronger than Levi-pseudoconvexity of the boundary (see [6]).

Nakano [15] proved the following generalization of the Kodaira vanishing theorem to weakly 1-complete manifolds:

THEOREM 1.2 (Nakano). Let X be a weakly 1-complete manifold, Kx the canonical line bundle of X, and E a positive line bundle on X. Then

(1.1) H1(XKX0E) 0 0 if i>1

Hc (X,E*) = 0 if 0 < i < dimX

where E* denotes the dual bundle of E.

Next we want to consider a special class of weakly 1-complete manifolds. For this we start from an arbitrary complex manifold S and a holomorphic line bundle ir: L -- S . Then we denote by or: L -- S the bundle obtained from 7v: L -- S by adding the point at infinity to each fiber (i.e. L = P(Os 0 1)



STEIN HYPERSURFACE SECTIONS 177

is the projective bundle associated to Os e ? where ? is the invertible sheaf corresponding to L). We write Lo for the zero section of L and put Loo := L\L.

In the case where S is compact 1-dimensional, which is of particular interest to us, L is a ruled surface and hence algebraic [3]. We show:

LEMMA 1.3. Assume that the complex manifold S is compact, 1-dimensional, and let L be a topologically trivial line bundle on S . Then the surfaces L, L \ Lo and L \ (Lo U Loo) are weakly 1-complete.

Proof. Since dim S = 1, S is Kihler and, therefore, the map H1 (S,, R) H1 (S, OS) is surjective. Together with the fact that L is topologically trivial, it follows that with respect to a suitable open covering of S, the bundle L can be given by constant transition functions {9ij} with JgijJ = 1 for all i, j. Hence, the function o := Jw12, where w is the coordinate along the fibers, is well defined on L. Clearly, p is a plurisubharmonic exhaustion function on L. (Notice that this function (o may depend on the chosen local trivializations of

L.) Similarly, the functions 1 I IW12+ 1 are well-defined, exhaustive and

plurisubharmonic on L \ Lo, L \ (Lo U L,,), respectively. FD

This lemma has as immediate consequence:

COROLLARY 1.4. Under the assumptions of Lemma 1.3 we have: If F is any negative line bundle over L and s a section of F defined on an open connected neighbourhood of Lo (resp. L,,), then s vanishes identically.

Proof. Indeed, from the theorem of Nakano (Th. 1.2) we get the vanishing He(L \ Lo, F) = 0 (resp. H1 (L, F) = 0). Therefore, the given section s can be extended to a global section s E Ho (L, F) . But H (L, F) = 0 by Kodaira's vanishing theorem, since F is negative on L. D1

The following lemma will be crucial for the construction of our example:

LEMMA 1.5. Let X be a compact, connected complex manifold of dimension k, r: L -- X a holomorphic line bundle, and assume that there exists a compact analytic subset A c L of pure dimension k such that AnLo = 0. Then there is a positive integer A such that the bundle LA is analytically trivial.

Proof. We define A to be the sheet number of the ramified analytic covering p := rA: A - X. Let {Ui} be an open covering of X such that LIUi - Ui x C, and let {gi;} be the corresponding transition functions for L. For x E Ui, the fiber p-1(x) can be written in the trivialization LJUi - Ui x C as p-1(x) = {si()(x), . .., s(')(x)} c CA if counted with multiplicities. Then the product hi(x) := si (x).... s ) (x) defines a holomorphic function on Ui. From the hypothesis A n Lo = 0 it follows, that hi(x) :4 0 for x E Ui. Moreover,




hi(x) = gj>(x)hj(x) when x E ui n Uj. Therefore, the collection {hi} defines a nonvanishing holomorphic section in LA showing that LA is analytically trivial. FD

The following lemma is a special case of a more general result of Mat- sushima and Morimoto (see [14, Th. 5]):

LEMMA 1.6. Let X be a complex manifold, wr: L -* X a holomorphic line bundle, and assume that L \ Lo is Stein. Then X is itself Stein.

For the convenience of the reader we give here a simple proof of this special case:

Proof. If U C X is an open set such that LIU is trivial, then any holomorphic function G on 1r-'(U) \ Lo - U x C* has a Laurent expansion along the fibers. The constant term of it is a holomorphic function g on U which does not depend on the choice of the trivialization. In this way we associate to any holomorphic function G on L \ Lo a holomorphic function g on X. Let now {x>,} be an infinite discrete sequence of points in X and c, E C arbitrary complex numbers. Since L \ Lo is Stein, there is a holomorphic function G on L \ Lo such that GC1r-1(x,) \ {O} = c, for every v E N. Therefore the corresponding holomorphic function g E O(X) satisfies g(x,) = c. and so X is holomorphically convex. Similarly, for any two points xi #7 x2 in X there is a g E 0(X) with g(xi) 7/ g(x2). Consequently, X is Stein. FD

Remark: The converse of this lemma also holds in the sense, that the Steiness of X implies L and L \ Lo being Stein.

For the convenience of the reader we recall the following result of Simha [17]:

LEMMA 1.7. Let X be a normal Stein space of dimension 2 and A C X a closed analytic subset without isolated points. Then X \ A is Stein.

Remark. A more general result is proved in [5]: If X is a normal Stein space of dimension n and A c X is a closed analytic subset without isolated points, then X \ A is a union of (n -1) Stein open subsets. In particular, X \ A is (n - l)-complete.

An immediate consequence of Lemma 1.7 is:

COROLLARY 1.8. Let X be a Stein space of dimension 2 and A C X a closed analytic subset. Assume that for any point x E A and for any local irreducible component X.,i of X at x the point x is not isolated in A n Xsk. Then X \ A is Stein.




Proof. This follows immediately from Lemma 1.7 and the invariance of the Stein property under normalization (see [16]). 0

2. Construction of the example proving Theorem 0.1

In this section we now want to construct a normal Stein space of pure dimension 3 with only one singular point and a closed connected analytic subset A C X of pure dimension 2 having the properties stated in Theorem 0.1.

We start with a 1-dimensional torus S and choose wr: L -- S to be a topologically trivial holomorphic line bundle on S such that no power Lk, k = 1, 2, 3 .. ., is analytically trivial. Let or: L -- S, Lo , Lo be defined as in Section 1 and let q: F -> L be a negative line bundle on L. We shall identify the zero section Fo of F over L with L. By the results of Grauert [10], we can blow down Fo to a point xo by a contraction map p: F -- X such that p(Fo) = {xo} and X is a normal Stein space (X is, in fact, affine algebraic), p is proper, holomorphic, and induces a biholomorphism F \ Fo - X \ {xo}.

We define A p(q-1(Lo U L,,)) C X. Then A is a closed connected analytic subset of X of pure dimension 2, and we put D X \ A.

We show at first:

Claim 1. D is not Stein.

Proof of Claim 1. Clearly D is biholomorphic to q-1(L \ (Lo U Loo)) \ Fo. If D would be Stein, then, by Lemma 1.6, the manifold L \ (Lo U Loo) also would be Stein. And, therefore, again by Lemma 1.6, S would also be Stein. This is obviously not possible. Consequently, D is not Stein. D

We now come to the proof of:

Claim 2. For every hypersurface H C X the intersection H n D is Stein.

In view of Corollary 1.8 and since X\ {xo} is smooth, it suffices for proving Claim 2 to show:

Claim 2'. If Hxo is an irreducible germ at xo of a hypersurface near x0, then the point xo is not isolated in Hx0 n A.

Proof of Claim 2'. Let us assume by reductio ad absurdum that

(*) the point xo is isolated in Hx0 n A.

We represent the germ Hx0 in a small neighbourhood U of xo by an irreducible hypersurface H. Let H be the proper transform of H, i.e. H = p-I(H\{xo}),




where the closure is taken in p-I(U), and let C H n L. Then C is a curve in L (i.e. a closed analytic subset of pure dimension 1). Let

k (2.1) C= U Ci

i = 1

be the decomposition of C into irreducible components. Observe that the curve C must intersect Lo U Loo because, otherwise, by Lemma 1.5, some power LA of L, A > 0, would be analytically trivial, which is excluded by our choice of L. We now distinguish between two cases:

Case i. Lo and L,, do not occur in the decomposition 2.1.

We assume, that e.g. C n Loo 0 (the case C 0 Lo : 0 is similar). Since L,, is not an irreducible component of C, it follows that Hnq- (L,,) is a curve which is not contained in F0 and passes through some point of L,. Therefore, xo is not isolated in H n p(q-1 (L,,)). This implies that dimxo H n A > 1 in contradiction to (*).

Case ii. Lo or L,, (or both) occur as components in 2.1.

Let us, for example, assume that L,, is an irreducible component of C. From the assumption that x0 is isolated in H n A we obtain the following: if V cc F, V C p-l(U), is a sufficiently small open neighbourhood of the zero section F0, then &V n q- (L,,) does not intersect H. Therefore, if W C Fo is a small enough connected open neighbourhood of Lo, the map

(2.2) 0 := qlH n V n q-1(W) : H n V n q-1(W)- W

is a ramified analytic covering with, say, gu sheets. Let WI, . .. , WI be an open covering of W such that FjWi is trivial:

(2.3) FWj- W, x C,

and let {9ij} be the transition functions for FIW with respect to these trivializations. For x E Wi the fiber )-!1(x) may be written with respect to the chosen trivialization over Wi in the form

(2.4) V)i- (x) = { i) (x), , s ) (x)} C ci

if multiplicities are counted. Therefore, from the elementary properties of ramified analytic coverings, it follows that the product hi(x) := s (x)*.... 4(2)(x) is a well-defined holomorphic function on Wi. One has hi = g8>hj on Wi n Wj and {hi = 0} = Wi n C. Thus the functions {hi} define a section s E HO(W, F") with {s = 0} = W n C. In particular, s does not vanish identically. On the other hand FA is negative; hence it follows by Corollary 1.4 that s =0. This is a contradiction. E




Since, consequently, Claim 2' holds, x0 is not isolated in H n A. Hence Claim 2 also holds and, therefore, Theorem 0.1 is completely proved. ED

There is a small variation of the above example, which is of some additional interest. We, therefore, mention it in the following:

COROLLARY 2.1. Let 0 < a < b be arbitrary constants and put in the situation of the example constructed in this section

(2.5) Da,b :=p(ql({w E L:a < (o(w) < b}) \Fo), where p is a plurisubharmonic exhaustion function on L as constructed in the proof of Lemma 2.3. Then the domain Dab is fat, the boundary &Dab is near all points x E &Dab \ {xo} real-analytic smooth and pseudoconvex, and Dab has the hypersection property; in other words, Dab n H is Stein for all complex hypersurfaces H c X, but Dab is itself not Stein.

Proof. Most parts of the proof of Theorem 0.1 just carry over. Only the argument showing that Dab is not Stein has to be added. But, clearly, Dab is biholomorphic to q-1(La,b) \ Fo where we have put Lab :={w E L: a < (p(w) < b}. If Da,b would be Stein, then by Lemma 1.6, La,b would also have to be Stein which would imply that Hj (La~b, 0) = 0. Therefore, the function which is identically 0 on a neighborhood of {w E L: (p(w) = b} and identically 1 on a neighborhood of {w E L: (p(w) = a} would extend to a holomorphic function on La,b, which is obviously impossible. Hence Dab is not Stein. FD

Remark: The importance of topologically trivial line bundles with the property that no positive power of them is analytically trivial, for the construction of very pathological examples of pseudoconvex manifolds was pointed out for the first time by H. Grauert in [11]. Here we use these line bundles for a different purpose, namely in order to get a "very bad" isolated singularity needed for our particular problem.

INSTITUTE OF MATHEMATICS, ROMANIAN ACADEMY, 70700 BUCHAREST, ROMANIA

MATHEMATICS, UNIVERSITY OF WUPPERTAL, 42097 WUPPERTAL, GERMANY

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(Received June 21, 1995)



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Open Sets with Stein Hypersurface Sections in Stein Spaces