SOME RESULTS ON COMPLEX SPACES. 1. Introduction. This paper deals with some recent results obtained in collaboration

Transcription

1 SOME RESULTS ON COMPLEX SPACES. GIUSEPPE TOMASSINI 1. Introduction This paper deals with some recent results obtained in collaboration with Viorel Vâjâitu. The first one (cfr. [22]) corcerns a new proof of the following theorem of Colţoiu [3] about the classical problem of the increasing union of Stein domains of a (reduced) complex space. Theorem 1. Let X be a normal Stein space and D X a relatively compact domain which is union of Stein domains. Then D is a domain of holomorphy. The second result proved in [23] deals with weakly 1-complete complex surfaces and extends to the singular ones a theorem of Ohsawa (cfr. [16]): Theorem 2. An irreducible weakly 1-complete complex surface X is holomorphically convex if and only if O(X) C. As an application we get 1991 Mathematics Subject Classification. Primary 32A38, 32D10, 32E20; Secondary 32T05. 1

2 2 GIUSEPPE TOMASSINI i) a variant of Simha s theorem [18] concerning the Restraumproblem for holomorphically convex surfaces ii) a characterization of holomorphically convex domains in two dimensional tori. 2. Proof of Theorem 1 As in [3] the statement of Theorem 1 is a consequence of the following facts: a) for every regular point x X lying on D and every sequence {x ν } D with x ν x, there exists a holomorphic function f O(D) such that the sequence {f(x ν )} is unbounded b) the subset D Sing(X) is everywhere dense in D. The proof of a) follows the same line as in Fornaess-Narasimhan s paper ([7]) where it is established Theorem 1 under the additional hypothesis that D is locally Stein. It is done using L 2 -estimates of Hörmander [10] and Skoda [20 ]. Part b) uses the resolution of singularities for complex spaces (cfr. [3]). Our contribution consists of a proof of b), based on an analyticity theorem for q-concave spaces stated in [ ]. It should be observed that another proof which is independent of resolution of singularities, can be derived using the techniques as in [4].

3 SOME RESULTS ON COMPLEX SPACES. 3 We recall that a subset A of a complex space Y is said to be q concave if for every point x A there exists a neighbourhood U of x such that U A is q-complete with corners i.e U A admits a continuous q-convex with corners, exhaustive function. We observe that 1-complete spaces are Stein. We have the following results: (1) Let A be an analytic subset of a Stein domain Ω in C N. Then Ω A is q-complete (with corners) (cfr. [17, Th. 5]). (2) Let D 1, D 2 be two domains in a complex space Y. If D 1 is q 1 -complete, D 2 is q 2 -complete, then D 1 D 2 is (q 1 + q 2 ) (cfr. [25, Prop. 3]). (3) An increasing union of q-complete domains in C N is q-complete (cfr. [24, Prop. 7]). The analyticity theorem that will be used for the proof of the property b) above is Theorem 3. Let X be a complex space of pure dimension n and A a q-concave subset of X, 0 < q < n. If the Hausdorff (2n 2q)-measure of A is locally finite, then A is analytic. Now assume that X that is a normal Stein space and D X is an increasing union of Stein domains D i, i = 1, 2,.... Should the property b) fail there exist x D and a connected open neighbourhood U of x such that U D Sing (X). Thus, in view of the fact that

4 4 GIUSEPPE TOMASSINI U Sing (X) is connected, we may assume that U = X and D = X A and that X is an analytic subset of a Stein domain of Ω C N. Let A i := X D i, Ω i := Ω A i. The sequence of open subset {Ω i } is increasing to Ω A. Let us show that each Ω i is (N n 1)-complete. Indeed, since D i is Stein, in view of Siu s theorem on Stein basis (cfr. [19]) there is a Stein domain D i Ω such that D i X = D i. Then Ω i = (Ω X) D i, where (Ω X) is (N n)-complete (property (1)) and D i is 1-complete. Therefore, owing to property (2), Ω i is (N n 1)-complete. From the property (3), then, it follows that Ω A is (N n 1)-complete too and consequently that A is (N n 1)- concave. Since A Sing (X) and Sing (X) has complex dimension at most n 2, the Hausdorff (2n 2q)-measure of A is 0. On the other hand, the identity 2n 2 = 2N 2(N n+1) and Theorem 3 imply that A is an analytic subset of pure dimension n 1. That is a contradiction. 3. Weak 1-completeness and holomorphic convexity We recall that a complex space X is said to be weakly 1-complete if it carries an (upper semicontinuous) psh exhaustion function ϕ : X [, + ). X is said to be 1-convex if ϕ can be chosen strictly psh outside of a compact subset of X. In this case X carries an exhaustive strictly psh function ψ : X [, + ) and X is obtained as a modification of a Stein space at a finite number of points and viceversa (cfr. [2]). In particular, a 1-convex space is holomorphically convex, hence

5 SOME RESULTS ON COMPLEX SPACES. 5 weakly 1-complete. Grauert,s example (cfr. [15]) shows that there exist (connected) weakly 1-complete spaces X which are not holomorphically convex (for which O(X) = C). A class of non holomorphically convex, weakly 1-complete manifolds is provided by toroidal groups (cfr. [9], [14], [21]). Ohsawa proved that, for a regular weakly 1-complete complex surface X, holomorphic convexity is equivalent to O(X) C (cfr. [16]). Theorem 2 which is announced in Introduction, extends this result to singular surfaces. Observe that we cannot use Ohsawa s result in dealing with the singular case because, as proved by Markoe (cfr. [11]), there exist locally irreducible non holomorphically convex complex spaces whose normalization is holomorphically convex. Let us now outline the proof of Theorem 2. We use the following facts (α) Let X be a complex space and ϕ : X [, + ] a psh function. If for an unbounded sequence {c ν } of real numbers the sublevels {ϕ < c ν } are holomorphically convex then X is holomorphically convex. (β) Let X be a complex space, K a compact subset of X and f a holomorphic, non-constant function on X. Let Z 1,..., Z m be the irreducible components of Z(f) := {f = 0} which intersect K and an open set Ω such that Ω Z j, 1 j m.

6 6 GIUSEPPE TOMASSINI Then, for every holomorphic, non-constant function g on X, sufficiently close to f, the irreducible components of Z(g), which meet K, meet also Ω (cfr. [6] ). Assume first that X is normal and let ϕ : X [, + ] a psh exhaustion function. Let f be a fixed holomorphic, non-constant function on X. Denote Sing(f) the analytic set of the singular points of f. Then f is constant on every irreducible component of Sing(f), hence f (K Sing(f)) is finite for every compact K X. Thus, in view of (α), we may assume that Λ 0 := f (Sing(f)) is finite. For x X, we denote N x (f) the connected component of f 1 (f(x) containing x and we study the set B of those points x X such that N x (f) is compact. Suppose B =. In this case we choose c R such that, for c close to c, the set {ϕ = c } contains no compact irreducible component of f 1 (Λ 0 ). Then, for any z C, f 1 (z) {ϕ = c} is contained in a Stein space, namely the union of the noncompact irreducible component of f 1 (z). Using Siu s theorem on Stein neighbourhoods we produce a smooth Φ which is strictly psh near {ϕ = c}. It follows that the sublevel {ϕ < c} is 1-convex. Thus there exists a sequence {x ν }, c ν + } such that the sublevels {ϕ < c ν } are holomorphically convex, and consequently, by virtue of (α), X is holomorphically convex. The case B is more involved. Using in an essential way (β), we show that, in fact, B = X, that is to say f has compact levels.

7 SOME RESULTS ON COMPLEX SPACES. 7 Finally we consider the Stein factorization σ : X X. f defines a holomorphic function f : X C with discrete fibres. In particular, X is at most of dimension one and contains no compact analytic curves, hence X is Stein. It follows that X is holomorphically convex. In general, we apply what we have already proved to the normalization X nor of X. 4. Applications The first application of Theorem 2 concerns the Restraumproblem for holomorphically convex complex surfaces. Originated from the 14 th Hilbert problem, the Restraumproblem has been studied in the algebraic and analytic contest (see e.g. [1], [8]). Simha proved the following complex analytic analogue of a result of Nagata (cfr. [18]): Theorem 4. Let X be a Stein surface and π : X nor X its normalization. Then, for any purely 1-dimensional analytic set A, X A is a Stein space if and only if π 1 (A) has no isolated points in X nor. In particular, if X is locally irreducible, X A is Stein if and only if has no isolated point. Simha s statement is false in dimension larger than two (cfr. [13]). Theorem 2 allows us to extended Simha s theorem to holomorphically convex surfaces.

8 8 GIUSEPPE TOMASSINI Theorem 5. Let X be a locally irreducible holomorphically convex complex surface and A a complex curve in X with no compact connected component. The X A is holomorphically convex. We first prove that if is a weakly 1-complete surface and A a (pure dimension) Stein curve in X. Then X A is weakly 1-complete. Remark 1. A similar result holds in higher dimension. Precisely, if X is a weakly 1-complete manifold and A a Stein hypersurface, then X A is weakly 1-complete. Let now {A λ } λ Λ be the decomposition of A in irreducible components and write Λ as a special increasing union of subsets {Λ n } as follows: Λ 0 := {λ Λ : A λ is non compact} Λ n+1 := L n {λ Λ : µ Λ n such that A λ A µ }. Set Σ n = λ Λn A λ, n = 0, 1, 2,... it follows that Σ 0 and Σ n+1 Σ n are Stein curves n = 0, 1, 2,.... Applying the previous result we deduce that, for each n, X Σ n is weakly 1-complete hence holomorphically by Theorem 2. Moreover, X A = n (X Σ n ). Thus, in order to conclude the proof we have to show that, for any point a A and any sequence {x ν } ν X A converging to a, there exists f O(X) which is unbounded on {x ν } ν. This is obvious because {A λ } λ Λ is locally finite.

9 SOME RESULTS ON COMPLEX SPACES. 9 As a second application of Theorem 2 we get the following: Theorem 6. Let T 2 be a two dimensional complex torus and D a locally Stein domain. Then D is holomorphically convex if and only if O(X) C. Consider the boundary distance function δ. Then the function log δ is psh (cfr. [12]) and exhaustive. Thus D is weakly 1-complete. References [1] J. Bingener-U. Storch, Resträume zu analytischen Mengen in Steinschen Räumen, Math. Ann. 210 (1974), [2] M. Colţoiu, A note on Levi s problem with discontinuous functions, Enseign. Math. 31 (1985), [3] M. Colţoiu, Remarques sur les réunions croissantes d ouverts de Stein, C. R. Acad. Sci. Paris 307 (1988), [4] M. Colţoiu-K. Diederich, On Levi s problem on complex spaces and envelopes of holomorphy, Math. Ann. 316 (2000), [5] M. Colţoiu-N. Mihalache, Strongly plurisubharmonic exhaustion functions on 1-convex spaces, Math. Ann. 270 (1985), [6] J. P. D ly, Cohomology of q-convex spaces in top degrees, Math. Z. 204 (1990), [7] J. E. Fornæss-R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), [8] J. E. Goodman-A. Landman, Varieties Proper over Affine Schemes, Invent. math. 20 (1973), [9] H. Kazama, On pseudoconvexity of complex Lie groups, Mem. Fac. Sci. Kyushu Univ. 27 (1973), [10] L. Hörmander, L 2 estimates and existence theorems for the operator, Acta math. 113 (1965), [11] A. Markoe, Invariance of holomorphic convexity under proper mappings, Math. Ann. 217 (1975),

10 10 GIUSEPPE TOMASSINI [12] K. Matsumoto, Boundary distance functions and q-convexity of pseudoconvex domains of general order in Kähler manifolds, J. Math. Soc. Japan 48 (1996), [13] T. Meis, Schrift. Math. Inst. Univ. Münster 16 (1996), [14] A. Morimoto, Non-compact complex Lie groups without non-constant holomorphic functions, Proc. Conf. Complex Analysis, (Minneapolis 1964) Springer-Berlin 1965, [15] R. Narasimhan, The Levi problem in the theory of functions of several complex variables, Proc. Internat. Congr. mathemaicians. (Stockolm 1962), Inst. Mittag-Leffler, Djursholm (1963), [16] T. Ohsawa, Weakly 1-complete manifold and Levi problem, Publ. Res. Inst. Math. Sci. 17 (1981), [17] M. Peternell, Continuous q-convex exhaustion functions, Invent. Math. 85 (1986), [18] R. R. Simha, On the complement of a curve on a Stein space of dimension two, Math. Z. 82 (1963), [19] Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), [20] H. Skoda, Applications de techniques L 2 à la théorie des idéaux d une algèbre de fonctions holomorphes avec poids, Ann. Sci. École Norm. Sup. 5 (1972), [21] S. Takeuchi, On completeness of holomorphic principal bundles Nagoya Math. J. 56 (1975), [22] G. Tomassini-V. Vâjâitu, An application of q-concave sets to domains of holomorphy, (to appear in Arkiv Math. ) [23] G. Tomassini-V. Vâjâitu, On completeness of holomorphic principal bundles, (preprint). [24] V. Vâjâitu, Pseudoconvex domains over q-complete manifolds, Ann. Scuola Norm. Sup. Pisa 29 (2000), [25] V. Vâjâitu, The analyticity of q-concave sets of locally finite Hausdorff (2n 2q)-measure, Ann. Inst. Fourier (Grenoble) 50 (2000), Giuseppe Tomassini, Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italia. tomassini@sns.it