AN INTERPOLATION PROPERTY OF LOCALLY STEIN SETS

We prove that, if D is a normal open subset of a Stein space X of pure dimension such that D is locally Stein at every point of ∂D \ Xsg, then, for every holomorphic vector bundle E over D and every discrete subset Λ of D \Xsg whose set of accumulation points lies in ∂D \ Xsg, there is a holomorphic section of E over D with prescribed values on Λ. We apply this to the local Steinness problem and domains of holomorphy. 2010 Mathematics Subject Classification: 32F10.


Introduction
Stein spaces are the fundament of the theory of Functions of Several Complex Variables. Their cohomology groups with coefficients in coherent analytic sheaves are trivial in positive dimension (Cartan's Theorem B) so that the standard machinery of exact sequences implies that, for every discrete sequence {x ν } of X there is a holomorphic function f on X with prescribed values on {x ν }, that is Stein spaces enjoy a nice interpolation property.
In this circle of ideas an important question is the characterization of locally Stein open subsets of Stein spaces. A definitive answer is not known, so one would like to know "how far are they from being Stein".
Our main result is the following theorem. (For a complex space X, X sg denotes the singular locus of X.) Theorem 1. Let X be a Stein space of pure dimension. Let D ⊂ X be an open set and π : E −→ D a holomorphic vector bundle. Suppose that D is normal and locally Stein at every point of ∂D \ X sg .
Then, for every sequence {x ν } of D \ X sg whose set of accumulation points lies in ∂D \ X sg , and for every set of vectors ξ ν ∈ E xν = π −1 (x ν ), there is a holomorphic section σ of E over D such that σ(x ν ) = ξ ν for all ν.
Remark. Theorem 1 remains true if X is normal and "D is a domain of holomorphy in X" is substituted for "D is locally Stein at points of ∂D \ X sg "; see the subsequent Corollary 3. (For definitions see the end of this section.) A crucial ingredient to settle Theorem 1 is the following result whose proof is given in §3.
Proposition 1. Let X be a Stein space of pure dimension and let D be an open subset of X that is locally Stein at every boundary point of ∂D \ X sg . Then D \ X sg admits a complete Kähler metric. Corollary 1. Let X be a Stein space of pure dimension and D ⊂ X an open set that is normal and locally Stein at every point of ∂D \ X sg . Then, for every sequence {x ν } of points in D \ X sg converging to a point of ∂D \ X sg and every sequence {c ν } of complex numbers, there is a holomorphic function h on D with h(x ν ) = c ν for all ν.
This corollary improves Theorem 3.1 in [12] where h is asked only to be unbounded on the given sequence. Besides it allows us to remove the relative compactness hypothesis on D in several known results ([2, Theorems 3.3, 3.4, and 3.14]; [15, Propositions 5.5 and 5.8, and Corollary 5.16]), so for the benefit of the reader we restate them subsequently.
When D is relatively compact, Corollary 1 is stated without proof in [6, Theorem 3.2.1] and it is proved in [15,Proposition 5.3]. Note that part b) is false if n ≥ 3. To give an example, we let C be a compact Riemann surface and consider π : L −→ C be a holomorphic vector bundle of rank n − 1 that is negative in the sense of Grauert. Let τ : L −→ X be the blowing down of its zero section Z C. Then X is a normal Stein space of dimension n with only a singular point {x 0 } = τ (C). Take a point α ∈ C and put H = τ (π −1 (α)). Clearly H is a complex hypersurface of X and D := X \ H is biholomorphic to π −1 (C \ {α}) \ Z, which is not Stein since Z \ π −1 (α) has codimension n − 1 ≥ 2 in the Stein manifold M := π −1 (C \ {α}). Observe also that O(D) is a Stein algebra isomorphic to O(M ), and D is a domain of holomorphy in X ( [9] or Corollary 3 below).    [2].
The notion of domain of holomorphy for domains in complex manifolds adapts to normal complex spaces [7], [10]. For not normal complex spaces it is discussed at large in [15].
A topological C-algebra A is called a "Stein algebra" if A is topologically isomorphic to O(Z) for some Stein space Z.

Linear spaces
By definition (see [5, p. 50]) a complex space L (not necessarily reduced) is said to be a linear space over another complex space X if L is a unitary X ×C-module in the category of complex spaces over X, in other words is a complex space L −→ X over X together with compositions a) + : L × X L −→ X and b) · : (X × C) × X L = C × L −→ L, such that the module axioms hold. Roughly speaking, a linear space can be thought of as a "vector bundle with singularities".
Alternatively (see [8]), we have the following local description, namely there is an open covering of X by sets U such that, over each U we have a presentation: is induced by a n × m matrix whose entries f jk are holomorphic functions on U and L| U is the complex subspace of U × C m defined by the ideal generated by the holomorphic functions To every linear space L −→ X we associate in a functorial way its coherent analytic sheaf of germs of holomorphic sections in L. This correspondence establishes a contravariant functor from the category of linear spaces over X to the category of coherent analytic sheaves on X that is an antiequivalence of categories.
The subset W L of X of those points admitting an open neighborhood U such that L| U is a vector bundle, is a Zarisky dense open subset of X; hence X \ W L is a rare analytic subset of X. Now suppose that, for every point x of X we have a hermitian metric h x on L x . Then h = {h x } x∈X is a hermitian metric on L according to Grauert and Riemenschneider [8], if there is an open neighborhood U of each point x 0 of X, an embedding L| U ⊂ U × C q and a positively definite hermitian form By using partitions of unity it is easy to endow L with a hermitian metric. We say that a linear space L is Nakano semi positive (resp., Nakano semi negative) if one can endow L with a hermitian metric h such that L restricted to Ω := W L \ X sg is negative (resp., positive) with respect to h| Ω in the sense of Nakano [11]. This means that each point x * ∈ Ω admits an open neighborhood U ⊂ Ω and z = (z 1 , . . . , z n ) local coordinates centered at x * and an isomorphism L| U −→ U × C r such that for the matrix (h jk ) we have at x * : 1) (h jk (x * )) = (δ jk ) = identity matrix, 2) (dh jk (x * )) = 0, and 3) the hermitian form Besides, if u is a real-valued smooth function on Ω, then For the sake of clarity, let us mention the following correspondence between real (1, 1)-forms and hermitian forms on an open subset of a complex euclidean space with coordinates (w 1 , . . . , w q ). To a hermitian form so that ω(α, β) = Im g(α, β), and, in particular ω(α, α) = g(α, α), which relates the complex hessian (or hermitian Levi form) and (1, 1)-forms. Now, it is straightforward to construct a hermitian metric h on L by using a partition of unity subordinated to a Stein open covering {U j } of X by local embeddings of L| Uj ⊂ U j × C nj . If ϕ > 0 is a smooth, strictly plurisubharmonic, and exhaustion function on X, there is a rapidly increasing convex function χ : (0, ∞) −→ (0, ∞) such that the modified hermitian metric he −χ(ϕ) is as desired. Details of the proof are left to the interested reader.
The Stein property of the linear spaces is treated in [14] in a more general setting, namely the following result is proved.

Proposition 2.
A linear space over a q-complete base space is q-complete. In particular, if X is Stein, then E is Stein, too.
The normalisation is such that "1-complete" means Stein. (The setting of vector bundles over q-complete spaces is treated in [17].) The canonical line bundle K X on a complex space X is as in [8]. Let π :X −→ X be a resolution of singularities of X and put K X be the direct image through π of the canonical sheaf onX. From Lemma 1 we get the following result.
Corollary 6. Let X be a Stein space. Then the linear space associated to K X or its dual K X is Nakano semi-positive.

Proof of Proposition 1
First we recall a well-known criterion for Kähler completeness.
Then, if δ represents the geodesic distance with respect to ω, for arbitrary points z 1 , z 2 ∈ M , one has δ(z 1 , z 2 ) ≥ |θ(z 1 ) − θ(z 2 )|. In particular, if θ is exhaustive, then ω is complete. Then there exists a smooth function φ : D −→ R that is strictly plurisubharmonic and for every closed subset T of D whose closure in X is disjoint from X sg , the restriction of φ to T is exhaustive.
Proof: First by [1] we note that if V X is a Stein open set, then there are finitely many discrete holomorphic mappings τ j : V −→ C n and holomophic functions f j on V , j = 1, . . . , m, with the following properties.
The induced mappings τ j : V \ {f j = 0} −→ C n are locally biholomorphic, therefore (V \ {f j = 0}, τ j ) becomes a domain over C n , and the intersection of all {f j = 0} equals V ∩ X sg .
Set U = V ∩ Ω and U j = (V \ {f j = 0}) ∩ Ω. Then U j is a Stein domain over C n via τ j | Uj because V \ {f j = 0} is a Stein manifold and U j is locally Stein in V \{f j = 0}. Let δ j be the boundary distance of the domain U j over C n . Thus − log δ j is continuous and plurisubharmonic.
For  V (x) tends to infinity as x ∈ U \ X sg goes to a point of V ∩ (∂Ω \ X sg ).
One important feature of this construction is that, if we start with another Stein open set W X to get ψ (l) W , then for every compact set M ⊂ V ∩ W , there is N V W ∈ N such that, for every k, l ∈ N, k, l ≥ N V W , the difference function ψ is strictly plurisubharmonic, continuous and exhaustive, there are constants C α ≥ 0 (we take C α = 0 if V α Ω) and a strictly increasing convex function χ : [0, ∞) −→ [0, ∞) such that we can patch the continuous plurisubharmonic functions ψ α + C α µ α + χ(ψ| Vα ) defined on V α to a continuous strictly plurisubharmonic function φ : Ω −→ [0, ∞) by setting for x ∈ Ω: This has the desired properties except smoothness. But this can be achieved since by [13,Satz 4.2] we can approximate φ in the C 0 -topology by a smooth strictly plurisubharmonic function, whence the proof.
Put θ = log σ. Thus θ < −1 on X, and if {x ν } is a sequence of points in D \ X sg converging to a point x ∈ D ∩ X sg , then {−θ(x ν )} tends to infinity.
Let γ > 0 be a smooth strictly plurisubharmonic exhaustion function on X. We may assume that is semi-positive (replace γ by χ(γ) for a suitable smooth strictly increasing convex function χ from (0, ∞) into (0, ∞), if necessary). Now we claim that the following closed (1, 1)-form ω defined on D\X sg is Kähler complete, where Indeed, first it is easily checked that so that we deduce immediately that ω is positively definite.
To check the completeness of ω notice that, if δ denotes the geodesic distance induced by ω, for arbitrary points z 1 , z 2 ∈ D \ X sg the following inequality holds: |}. We then conclude since for a sequence {x j } j of points in D \ X sg converging to a point of ∂D ∪ X sg , at least one of the sequences {φ(

Proof of Theorem 1
First we recall Demailly's version of the ∂ problem with singular weights on Kähler complete manifolds [3]. Now we start with the proof of Theorem 1. Since on D \ X sg we have the natural isomorphism of holomorphic vector bundles E (E ⊗K * X )⊗ K X the idea is to use Theorem 2 to solve a ∂ problem for (0, 1) forms with coefficients in F := E ⊗ K X on D \ X sg and due to Nakano semi positivity of that bundle (see Lemma 1), to check the hypotheses of the theorem reduces to a proper choice of a smooth convex increasing function in order to satisfy the curvature inequalities and convergence of a certain integral. Since D is normal, the holomorphic section we construct on D \ X sg with values in E extends to a holomorphic section in E over D.
To proceed, let {x j } j be a sequence of points of D \ X sg whose set of accumulation points, which is a closed subset of X, and by hypothesis is contained in ∂D and disjoint from X sg . Then, select relatively compact open neighborhoods V j and U j of x j in D\X sg such that U j are mutually disjoint and contained in D \ X sg and V j ⊂ U j . We choose U j such that T := ∪ j U j is a closed subset of D, and whose closure in X is disjoint from X sg . Moreover, we may assume that, for each index j, U j is the domain of a coordinate chart centered at x j and z (j) be complex coordinates centered at x j on U j . Now, let ρ j be a smooth function whose support lies in U j and equals 1 on V j . We define a smooth section σ in E over D \ X sg by σ = ρ j ξ j . This induces a holomorphic formσ of type (n, 0) on D \ X sg with coefficients in F , where n is the complex dimension of X. Consider the smooth (n, 1)-form with coefficients in F defined by v = ∂σ. Then v has support in T := ∪ j (U j \ V j ) ⊂ T . Now solve the ∂ equation ∂u = v with singular estimates and for that we choose in the statement of Theorem 2 the constants A p = n and obtain a smooth (n, 0) form u with coefficients in F such that u(x j ) = 0 for all indices j. Thenτ =σ − u induces a holomorphic section τ in E over D \ X sg with σ(x j ) = ξ j for all indices j. Since D is normal, the extension of τ to D is as desired. Now, we go on as follows. Keeping the notations from §3 (proof of Proposition 1), the function γ is positive, smooth of class C ∞ , strictly plurisubharmonic and exhaustive for X, and we set: Let χ : [0, ∞) −→ (0, ∞) be an increasing, smooth of class C ∞ , and convex function, to be chosen later in proof. We put Consider the singular weight function where ρ j = 1 on V j , ρ j ≥ 0 on X, and its support lies in U j , where U j and V j are as above. Define Ψ = Φ 0 + Φ 2 + 3χ(Φ 1 ), and let the singular hermitian metric on F be he −Ψ .
We would like to apply Theorem 2 for λ the constant function, λ = 1, and for that we need to select χ carefully; we wish the inequality i∂∂ Ψ ≥ ω χ , and, in order to achieve this we require that i∂∂χ(Φ 1 ) ≥ i∂∂Φ 1 , a condition that is fulfilled if we impose that χ − id be positive, increasing, and convex, and the inequality that can be easily realized if χ is large enough taking into account the computations from §3. Then we need the convergence of the integral D\Xsg |v| 2 e −Ψ dV χ , which, as a matter of fact, is an integral over T and Φ 0 and Φ 2 are smooth there, and where dV χ is the volume element associated to the complete Kähler form ω χ . By straightforward computations we obtain for some positive continuous function C on D \ X sg , where dV is the volume element associated to the Kähler form ω. Therefore, the convergence will follow by using the following lemma due to Demailly [4, p. 374].