Algebraic approximations of holomorphic maps from Stein domains to projective manifolds

Jean-Pierre Demailly^{1} László Lempert^{2}

Université de Grenoble I Purdue University Institut Fourier, BP 74 Department of Mathematics U.R.A. 188 du C.N.R.S. West Lafayette, IN 47907, U.S.A. 38402 Saint-Martin d’Hères, France

Bernard Shiffman^{3} Johns Hopkins University Department of Mathematics Baltimore, MD 21218, U.S.A.

Key words: affine algebraic manifold, algebraic approximation, algebraic curve, complete pluripolar set, Eisenman metric, Hörmander’s L^{2}-estimates for \overline{∂}, holomorphic map, holomorphic retraction, holomorphic vector bundle, hyperbolic space, Kobayashi pseudodistance, Kobayashi-Royden pseudometric, Nash algebraic map, Nash algebraic retraction, plurisubharmonic function, projective algebraic manifold, quasi-projective variety, Runge domain, Stein manifold.

A.M.S. Classification 1985: 32E30, 32H20, 14C30

Table of contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 2 2. Holomorphic and Nash algebraic retractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 8 3. Nash algebraic approximation on Runge domains in affine algebraic varieties. . . . . . . . . . .p. 12 4. Nash algebraic approximations omitting ample divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 18 5. Exhaustion of Stein manifolds by Runge domains of affine algebraic manifolds . . . . . . . . . p. 26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 29

1 Research partially supported by Institut Universitaire de France

2 Research partially supported by National Science Foundation Grant No. DMS-9303479

3 Research partially supported by National Science Foundation Grant No. DMS-9204037

1. Introduction

The present work, which was motivated by the study of the Kobayashi pseudodistance on algebraic manifolds, proceeds from the general philosophy that analytic objects can be approximated by algebraic objects under suitable restrictions. Such questions have been extensively studied in the case of holomorphic functions of several complex variables and can be traced back to the Oka-Weil approximation theorem (see [We35] and [Oka37]). The main approximation result of this work (Theorem 1.1) is used to show that both the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric on a quasi-projective algebraic manifold Z are computable solely in terms of the closed algebraic curves in Z (Corollaries 1.3 and 1.4).

Our general goal is to show that algebraic approximation is always possible in the cases of holomorphic maps to quasi-projective manifolds (Theorems 1.1 and 4.1) and of locally free sheaves (Theorem 1.8 and Proposition 3.2). Since we deal with algebraic approximation, a central notion is that of Runge domain: By definition, an open set Ω in a Stein space Y is said to be a Runge domain if Ω is Stein and if the restriction map O(Y) → O(Ω) has dense range. It is well known that Ω is a Runge domain in Y if and only if the holomorphic hull with respect to O(Y) of any compact subset K ⊂ Ω is contained in Ω. If Y is an affine algebraic variety, a Stein open set Ω ⊂ Y is Runge if and only if the polynomial functions on Y are dense in O(Ω).

Our first result given below concerns approximations of holomorphic maps by (complex) Nash algebraic maps. If Y, Z are quasi-projective (irreducible, reduced) algebraic varieties, a map f : Ω → Z defined on an open subset Ω ⊂ Y is said to be Nash algebraic if f is holomorphic and the graph

Γ_{f} := {(y, f(y)) ∈ Ω × Z : y ∈ Ω}

is contained in an algebraic subvariety G of Y × Z of dimension equal to dim Y. If f is Nash algebraic, then the image f(Ω) is contained in an algebraic subvariety A of Z with dim A = dim f(Ω) ≤ dim Y. (Take A = pr_{2}(G) ⊂ Z, after eliminating any unnecessary components of G.)

Theorem 1.1. — Let Ω be a Runge domain in an affine algebraic variety S, and let f : Ω → X be a holomorphic map into a quasi-projective algebraic manifold X. Then for every relatively compact domain Ω_{0} ⊂⊂ Ω, there is a sequence of Nash algebraic maps f_{ν} : Ω_{0} → X such that f_{ν} → f uniformly on Ω_{0}.

Moreover, if there is an algebraic subvariety A (not necessarily reduced) of S and an algebraic morphism α : A → X such that f_{|A∩Ω} = α_{|A∩Ω}, then the f_{ν} can be chosen so that f_{ν|A∩Ω_{0}} = f_{|A∩Ω_{0}}. (In particular, if we are given a positive integer k and a finite set of points (t_{j}) in Ω_{0}, then the f_{ν} can be taken to have the same k-jets as f at each of the points t_{j}.)

An algebraic subvariety A of S is given by a coherent sheaf O_{A} on S of the form O_{A} = O_{S}/I_{A}, where I_{A} is an ideal sheaf in O_{S} generated by a (finite) set of polynomial functions on S. (If I_{A} equals the ideal sheaf of the algebraic subset Supp O_{A} ⊂ S, then A is reduced and we can identify A with this subset.) The restriction of a holomorphic map f : Ω → X to an algebraic subvariety A is given by

f_{|A∩Ω} = f ∘ ι_{A∩Ω} : A ∩ Ω → X ,

where ι_{A∩Ω} : A ∩ Ω → Ω is the inclusion morphism. We note that the k-jet of a holomorphic map f : Ω → X at a point a ∈ Ω can be described as the restriction f_{|{a}^{k}} : {a}^{k} → X of f to the nonreduced point {a}^{k} with structure ring O_{Ω,a}/M_{Ω,a}^{k+1} (where M_{Ω,a} denotes the maximal ideal in O_{Ω,a}). The parenthetical statement in Theorem 1.1 then follows by setting A equal to the union of the nonreduced points {t_{j}}^{k}.

If in Theorem 1.1 we are given an exhausting sequence (Ω_{ν}) of relatively compact open sets in Ω, then we can construct by the Cantor diagonal process a sequence of Nash algebraic maps f_{ν} : Ω_{ν} → X converging to f uniformly on every compact subset of Ω. We are unable to extend Theorem 1.1 to the case where X is singular. Of course, for the case dim S = 1, Theorem 1.1 extends to singular X, since then f can be lifted to a desingularization of X. The case S = C of Theorem 1.1 was obtained earlier by L. van den Dries [Dr82] using different methods.

In the case where X is an affine algebraic manifold, Theorem 1.1 is easily proved as follows: One approximates f : Ω → X ⊂ C^{m} with an algebraic map g : Ω_{0} → C^{m} and then applies a Nash algebraic retraction onto X to obtain the desired approximations. (The existence of Nash algebraic retractions is standard and is given in Lemma 2.1.) In the case where f is a map into a projective manifold X ⊂ IP^{m−1}, one can reduce to the case where f lifts to a map g into the cone Y ⊂ C^{m} over X, but in general one cannot find a global Nash algebraic retraction onto Y. Instead our proof proceeds in this case (after making some reductions) by considering the map G : Ω × C^{m} → C^{m} given by G(z, w) = g(z) + w, so that G^{−1}(Y \ {0}) is a submanifold containing Ω × {0}. We then approximate G by algebraic maps G_{ν}, which we compose (on the right) with Nash algebraic maps from Ω_{0} × {0} into G_{ν}^{−1}(Y) to obtain the approximations f_{ν}.

In the case where the map f in Theorem 1.1 is an embedding of a smooth domain Ω into a projective manifold X with the property that f^{⋆}L is trivial for some ample line bundle L on X, we can approximate f by Nash algebraic embeddings f_{ν} with images contained in affine Zariski open sets of the form X^{\}D_{ν}, where the D_{ν} are divisors of powers of L. This result (Theorem 4.1) is obtained by first modifying f on a Runge domain Ω_{1} ⊂⊂ Ω in such a way as to create essential singularities on the boundary of f(Ω_{1}) so that the closure f(\overline{Ω}_{1}) becomes complete pluripolar (in the sense of plurisubharmonic function theory). The existence of an ample divisor avoiding f(Ω_{0}) is then obtained by means of an approximation theorem (Proposition 4.8) based on Hörmander’s L^{2} estimates for \overline{∂}.

One of our main applications is the study of the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal pseudometric on algebraic manifolds ([Ko67], [Ko70], [Ro71]). We use throughout the following notation:

Δ_{R}(a) = {t ∈ C : |t − a| < R}, Δ_{R} = Δ_{R}(0), Δ = Δ_{1}. For a complex manifold X: T_{X} = the holomorphic tangent bundle of X. N_{S} = N_{S/X} = the normal bundle T_{X|S}/T_{S} of a submanifold S ⊂ X. κ_{X} = the Kobayashi-Royden infinitesimal Finsler pseudometric on T_{X},

i.e., for v ∈ T_{X},

κ_{X}(v) = inf \{λ > 0 : ∃f : Δ → X holomorphic with f_{⋆}(λ\frac{∂}{∂ζ}_{|0}) = v\}.

In addition, for a complex space Z, we let d_{Z}(a, b) denote the Kobayashi pseudodistance between points a, b ∈ Z. In the case where Z is smooth, then d_{Z}(a, b) is the κ_{Z}-length of the shortest curve from a to b; see [NO90]. (A complex space Z is said to be hyperbolic if d_{Z}(a, b) > 0 whenever a, b are distinct points of Z. A compact complex manifold is hyperbolic if and only if the Kobayashi-Royden pseudometric is positive for all nonzero tangent vectors [NO90].)

As an application of Theorem 1.1 (for the case where Ω is the unit disk Δ), we describe below how both the Kobayashi-Royden pseudometric and the Kobayashi pseudodistance on projective algebraic manifolds can be given in terms of algebraic curves.

Corollary 1.2. — Let M be an open subset of a projective algebraic manifold X and let v ∈ T_{M}. Then κ_{M}(v) is the infimum of the set of r ≥ 0 such that there exist a closed algebraic curve C ⊂ X, a normalization γ : C′ → C ∩ M, and a vector u ∈ T_{C′} with κ_{C′}(u) = r and dγ(u) = v.

Proof (assuming Theorem 1.1). — Let r_{0} be the infimum given above. By the distance-decreasing property of holomorphic maps, κ_{X}(v) ≤ r_{0}. We must verify the reverse inequality. Let ε > 0 be arbitrary. Choose a holomorphic map g : Δ → M and λ ∈ [0, κ_{X}(v) + ε] such that dg(λe_{0}) = v, where e_{0} = \frac{∂}{∂ζ}_{|0}. By Theorem 1.1 with Ω = Δ, there exists a Nash algebraic map f : Δ_{1−ε} → M such that df(e_{0}) = dg(e_{0}). Let f̃: Δ → M be given by f̃(ζ) = f((1 − ε)ζ) and let λ̃ = λ/(1 − ε). Then df̃(λ̃e_{0}) = v. Since f̃ is Nash algebraic, there exists an algebraic curve C ⊂ X such that f̃(Δ) ⊂ C. Let γ : C′ → C ∩ M be the normalization of C ∩ M. We then have a commutative diagram

\begin{aligned}Δ & \underrightarrow{f̂} C′ \\ f̃ & ↘ ↓ γ \\ & C ∩ M . \end{aligned}

Let u = df̂(λ̃e_{0}). We have dγ(u) = v. Thus

r_{0} ≤ κ_{C′}(u) ≤ λ̃ = \frac{λ}{1 − ε} ≤ \frac{κ_{M}(v) + ε}{1 − ε} .

Since ε > 0 is arbitrary, we conclude that r_{0} ≤ κ_{M}(v).

Suppose that M is as in Corollary 1.2 and v ∈ T_{M,x} (x ∈ M). We note that if C, C′, γ, u are given as in the statement of the corollary, then the analytic set germ C_{x} contains a smooth irreducible component tangent to v. In this case we say that C is tangent to v. In general, (dγ)^{−1}(v) is a finite set of vectors (u_{j}) in T_{C′} (C_{x} may contain several smooth components tangent to v) and we define κ_{C∩M}(v) = min_{j} κ_{C′}(u_{j}). In particular, if we let M be Zariski open in X, we then have the following result:

Corollary 1.3. — Let Z be a quasi-projective algebraic manifold. Then the Kobayashi-Royden pseudometric κ_{Z} is given by

κ_{Z}(v) = inf_{C} κ_{C}(v), ∀v ∈ T_{Z} ,

where C runs over all (possibly singular) closed algebraic curves in Z tangent to v.

In the case where C is hyperbolic (e.g., the normalization of C is a compact curve of genus ≥ 2), the Kobayashi-Royden pseudometric κ_{C} coincides with the Poincaré metric induced by the universal covering Δ → C′ of the normalization C′; thus in some sense the computation of κ_{Z} is reduced to a problem which is more algebraic in nature. Of particular interest is the case where Z is projective; then the computation becomes one of determining the genus and Poincaré metric of curves in each component of the Hilbert scheme. This observation strongly suggests that it should be possible to characterize Kobayashi hyperbolicity of projective manifolds in purely algebraic terms; this would follow for instance from S. Lang’s conjecture [La86], [La87, IV.5.7] that a projective manifold is hyperbolic if and only if it has no rational curves and no nontrivial images of abelian varieties; see also [De94] for further results on this question.

We likewise can use algebraic curves to determine the Kobayashi pseudodistance in quasi-projective algebraic varieties:

Corollary 1.4. — Let Z be a quasi-projective algebraic variety, and let a, b ∈ Z. Then the Kobayashi pseudodistance d_{Z}(a, b) is given by

d_{Z}(a, b) = inf_{C} d_{C}(a, b) ,

where C runs over all (possibly singular and reducible) one-dimensional algebraic subvarieties of Z containing a and b.

Remark. — If C is a one-dimensional reducible complex space, then by definition d_{C}(a, b) is the infimum of the sums

∑_{j=1}^{n} d_{C_{j}}(a_{j−1}, a_{j})

where n is a positive integer, C_{1}, …, C_{n} are (not necessarily distinct) irreducible components of C, a_{0} = a ∈ C_{1}, a_{n} = b ∈ C_{n} and a_{j} ∈ C_{j} ∩ C_{j+1} for 1 ≤ j ≤ n − 1.

Proof of Corollary 1.4 (assuming Theorem 1.1). — Let ϵ > 0 be arbitrary. It suffices to find algebraic curves C_{j} ⊂ Z as in the above remark with

∑_{j=1}^{n} d_{C_{j}}(a_{j−1}, a_{j}) < d_{Z}(a, b) + ϵ .

By the definition of the Kobayashi pseudodistance, there exist points a = a_{0}, a_{1}, …, a_{n} = b in Z and holomorphic maps f_{j} : Δ → Z such that f_{j}(0) = a_{j−1}, f_{j}(t_{j}) = a_{j} (t_{j} ∈ Δ), for 1 ≤ j ≤ n, and ∑ d_{Δ}(0, t_{j}) < d_{Z}(a, b) + ϵ. Let ρ < 1 be arbitrary. By Theorem 1.1 (applied to the desingularization of Z), we can find Nash algebraic maps g_{j} : Δ_{ρ} → Z with g_{j}(0) = f_{j}(0) = a_{j−1} and g_{j}(t_{j}) = f_{j}(t_{j}) = a_{j}. Then for 1 ≤ j ≤ n, g_{j}(Δ) is contained in an algebraic curve C_{j}. Hence

∑_{j=1}^{n} d_{C_{j}}(a_{j−1}, a_{j}) ≤ ∑_{j=1}^{n} d_{Δ_{ρ}}(0, t_{j}) = ∑_{j=1}^{n} d_{Δ} (0, \frac{t_{j}}{ρ} ) .

Since ρ < 1 is arbitrary, our conclusion follows.

Eisenman [Ei70] has also introduced an invariant pseudometric on the decomposable p-vectors of an arbitrary complex space Y. Let B^{p} be the unit ball of C^{p}. At each nonsingular point of Y, the Eisenman p-metric E_{Y}^{p} is defined by

E_{Y}^{p} (v) = inf \{λ > 0 : ∃f : B^{p} → Y holomorphic with f_{⋆}(λ\frac{∂}{∂z_{1}} ∧…∧ \frac{∂}{∂z_{p}}_{|0}) = v\}

for all decomposable p-vectors v ∈ Λ^{p}T_{Y}. (The space Y is said to be strongly p-measure hyperbolic if E_{Y}^{p} (v) > 0 whenever v ≠ 0.) For the case p = dim Y, E_{Y}^{p} is called the Eisenman volume. The following generalization of Corollary 1.3 to the Eisenman p-metric is obtained by a parallel argument.

Corollary 1.5. — The Eisenman p-metric of a quasi-projective algebraic manifold Z can be computed in terms of the Eisenman volumes of its p-dimensional algebraic subvarieties. More precisely

E_{Z}^{p}(v) = inf_{Y} E_{Y}^{p} (v)

for all decomposable p-vectors v ∈ Λ^{p}T_{Z}, where Y runs over all (possibly singular) p-dimensional closed algebraic subvarieties of Z tangent to v. (A decomposable p-vector v at a singular point y ∈ Y is said to be tangent to Y if the analytic germ of Y at y contains a smooth irreducible component tangent to v; the above definition of E_{Y}^{p} (v) carries over to this case.)

If Y is compact and the canonical bundle K_{Y} of a desingularization Ỹ → Y is ample, then a form of the maximum principle tells us that the Eisenman volume Ep ; _{Y} is bounded from below by the volume form of the Kähler-Einstein metric (with

curvature −1) on Ỹ (see [Yau78], [NO90, §2.5]). It would be very interesting to know when these two intrinsic volumes are actually equal. In connection to this question, one would like to know whether the infimum appearing in Corollary 1.5 can be restricted to subvarieties Y with Ỹ satisfying this property, in virtue of some density argument. This would make the Eisenman p-metric theory completely parallel to the case of the Kobayashi-Royden pseudometric κ_{Z} = E_{Z}^{1}.

The following result of Stout [St84] on the exhaustion of Stein manifolds by Runge domains in affine algebraic manifolds allows us to extend Theorem 1.1 to maps from arbitrary Stein manifolds.

Theorem 1.6 (E.L. Stout). — Let S be a Stein manifold. For every Runge open set Ω ⊂⊂ S, there exists an affine algebraic manifold Y such that Ω is biholomorphic to a Runge open set in Y.

Theorem 1.6 has been refined into a relative version by Tancredi-Tognoli [TT89]. An immediate consequence of Theorem 1.1 and Stout’s Theorem 1.6 is the following approximation theorem for holomorphic maps from Stein manifolds into projective algebraic manifolds:

Corollary 1.7. — Let Ω be a relatively compact domain in a Stein manifold S, and let f : S → X be a holomorphic map into a quasi-projective algebraic manifold X. Then there is a sequence of holomorphic maps f_{ν} : Ω → X such that f_{ν} → f uniformly on Ω and the images f_{ν}(Ω) are contained in algebraic subvarieties A_{ν} of X with dim A_{ν} = dim f_{ν}(Ω). Moreover, if we are given a positive integer k and a finite set of points (t_{j}) in Ω_{0}, then the f_{ν} can be taken to have the same k-jets as f at each of the points t_{j}.

We must point out at this stage that Theorem 1.6 completely breaks down for singular Stein spaces. In fact, there are examples of germs of analytic sets with non-isolated singularities which are not biholomorphic to germs of algebraic sets; Whitney [Wh65] has even given an example which is not C^{1}- diffeomorphic to an algebraic germ. The technique used by Stout and Tancredi-Tognoli to prove Theorem 1.6 consists of a generalization to the complex case of the methods introduced by Nash [Nh52]. In the present work, we give a different proof of Theorem 1.6 based on some of the arguments used in the proof of the approximation theorem 1.1. Our proof starts with the observation that every Stein domain Ω ⊂⊂ S can be properly embedded in an affine algebraic manifold in such a way that the normal bundle is trivial. This key step also yields the following parallel result for holomorphic vector bundles over Stein manifolds. (A similar result has also been given by Tancredi and Tognoli [TT92, Theorem 4.1]; see Section 3.)

Theorem 1.8. — Let E be a holomorphic vector bundle on an n-dimensional Stein manifold S. For every Runge open set Ω ⊂⊂ S, there exist an n-dimensional projective algebraic manifold Z, an algebraic vector bundle Ẽ → Z, an ample divisor D in Z, and a holomorphic injection i : Ω ↪ Z ^{\} D with the following properties:

(i) i(Ω) is a Runge domain in the affine algebraic manifold Z ^{\} D;

(ii) i^{⋆}Ẽ is isomorphic to E_{|Ω}.

Theorem 1.8 is proved by observing that there is a holomorphic map from S into a Grassmannian such that the given vector bundle E on S is isomorphic to the pull-back of the universal quotient vector bundle. The desired result then follows from Theorem 1.6 applied to Ω ⊂⊂ S and from the existence of Nash algebraic approximations of the map from Ω to the Grassmannian (see Section 5).

A substantial part of the results of this work was obtained during a stay of the third author at Institut Fourier, Université de Grenoble I, and he wishes to thank Institut Fourier for its hospitality.

2. Holomorphic and Nash algebraic retractions

Let S be an n-dimensional Stein manifold. The Bishop-Narasimhan embedding theorem ([Bi61], [Na60]) implies that S can be embedded as a closed submanifold of a complex vector space C^{N} with e.g. N = 2n + 1 ; by Eliashberg and Gromov [EG92], it is in fact enough to take N = [(3n+3)/2], and this value is optimal for even n. Such optimal values will not be needed here. We first recall a well-known and elementary lemma about the existence of holomorphic retractions for Stein submanifolds.

Lemma 2.1. — Let M be a Stein (not necessarily closed) submanifold of a complex manifold Y and let n = dim Y.

(i) There exist a neighborhood U of the zero section in the normal bundle N_{M} of M, a neighborhood V of M in Y and a biholomorphism ψ : U → V, such that ψ maps the zero section of N_{M} identically onto M.

(ii) Let π : N_{M} → M be the natural projection. Then ρ = π ∘ ψ^{−1} : V → M is a holomorphic retraction, i.e. a holomorphic map such that ρ_{|M} = Id_{M}.

Moreover, if Y is affine algebraic and M is a closed algebraic submanifold, then ψ and ρ can be taken to be Nash algebraic.

Proof. — We first outline the proof of the well-known case where Y = C^{n}. (Details can be found in [GR65, p. 257].) As M is Stein, by Cartan’s Theorem B the normal bundle sequence

0 ⟶ T_{M} ⟶ T_{C^{n}|M} ⟶ N_{M} ⟶ 0  (2.1)

splits; i.e., there is a global morphism σ : N_{M} → T_{C^{n}|M} which is mapped to Id_{N_{M}} by composition with the projection onto N_{M}. Then the map ψ : N_{M} → C^{n} given by

ψ(z, ζ) = z + σ(z) · ζ , ζ ∈ N_{M,z} , z ∈ M ,  (2.2)

coincides with Id_{M} on the zero section (which we also denote by M) of N_{M}, and thus the derivative dψ satisfies the equality (dψ)_{|T_{M}} = Id_{T_{M}}. Furthermore, at each point z ∈ M, ψ has vertical derivative (d^{v}ψ)_{z} := dψ_{|N_{M,z}} = σ_{z}, and hence dψ is invertible at z. By the implicit function theorem, ψ defines a biholomorphism from a neighborhood U of the zero section of N_{M} onto a neighborhood V of M in C^{n}. Then ψ_{|U} : U → V is the required biholomorphism, and ρ = π ∘ (ψ_{|U})^{−1} is a retraction. When M is affine algebraic, N_{M} can be realized as an affine algebraic manifold such that π : N_{M} → M is an algebraic map. The splitting morphism σ can also be taken to be algebraic. Then Γ_{ψ} is an open subset of an n-dimensional subvariety G ⊂ N_{M} × Y, and Γ_{ρ} = (Id_{Y} × π)(Γ_{ψ}^{−1}) is contained in the n-dimensional algebraic variety (Id_{Y} × π)(G).

In the case of a general complex manifold Y, it is enough to consider the case where Y is Stein, since by [Siu76] every Stein subvariety of a complex space Y has arbitrarily small Stein neighborhoods. (See also [De90] for a simpler proof; this property is anyhow very easy to check in the case of nonsingular subvarieties.) Thus we can suppose Y to be a closed n-dimensional submanifold of some C^{p}, e.g. with p = 2n + 1. Observe that the normal bundle N_{M/Y} of M in Y is a subbundle of N_{M/C^{p}}. By the first case applied to the pairs M ⊂ C^{p}, Y ⊂ C^{p}, we get a splitting σ′ : N_{M} → T_{C^{p}|M} of the normal bundle sequence for N_{M/C^{p}}, a map ψ′ : N_{M/C^{p}} → C^{p} inducing a biholomorphism ψ′ : U′ → V′, and a retraction ρ′′ : V′′ → Y, where U′ is a neighborhood of the zero section of N_{M/C^{p}}, V′ a neighborhood of M in C^{p}, and V′′ a neighborhood of Y. We define ψ to be the composition

ψ : N_{M/Y} ⟶ N_{M/C^{p}} \underrightarrow{ψ′} C^{p} \underrightarrow{ρ′′} Y.

Of course, ψ is not defined on the whole of N_{M/Y}, but only on N_{M/Y} ∩(ψ′)^{−1}(V′′). Clearly ψ_{|M} = Id_{M}, and the property (d^{v}ψ)_{|M} = σ′_{|N_{M/Y}} follows from the equalities (d^{v}ψ′)_{|M} = σ′ and (dρ′′)_{|T_{Y}} = Id_{T_{Y}}. It follows as before that ψ induces a biholomorphism U → V from a neighborhood of the zero section of N_{M/Y} onto a neighborhood V of M. If Y and M are algebraic, then we can take ψ′ to be algebraic and Γ_{ρ′′} to be contained in an algebraic p-dimensional subvariety G′′ of C^{p} × Y. Then Γ_{ψ} is contained in some n-dimensional component of the algebraic set

(N_{M/Y} × Y ) ∩ (ψ′ × Id_{Y})^{−1}(G′′).

Therefore ψ is Nash algebraic, and ρ = π ∘ ψ^{−1} is also Nash algebraic as before.

In fact, the retraction ρ of Lemma 2.1 can be taken to be Nash algebraic in slightly more general circumstances, as is provided by Lemma 2.3 below. First we need the following elementary lemma due to Tancredi and Tognoli:

Lemma 2.2 [TT89, Corollary 2]. — Let A be an algebraic subvariety (not necessarily reduced) of an affine algebraic variety S, and let Ω be a Runge domain in S. Suppose that h is a holomorphic function on Ω and γ is an algebraic function on A such that h_{|A∩Ω} = γ_{|A∩Ω}. Then h can be approximated uniformly on each compact subset of Ω by algebraic functions h_{ν} on S with h_{ν|A} = γ.

Tancredi and Tognoli [TT89] consider only reduced subvarieties A, but their proof is valid for nonreduced A. We give the proof here for the reader’s convenience: To verify Lemma 2.2, one first extends γ to an algebraic function γ̃ on S. By replacing h with h − γ̃, we may assume that γ = 0. Choose global algebraic functions τ_{1}, …, τ_{p} generating the ideal sheaf I_{A} at each point of S, and consider the surjective sheaf homomorphism (of algebraic or analytic sheaves) O_{S}^{p} \underrightarrow{τ} I_{A} given by τ(f^{1}, …, f^{p}) = ∑ τ_{j}f^{j}. Now choose f = (f^{1}, …, f^{p}) ∈ H^{0}(Ω, O^{p}) such that τ(f) = h. Since Ω is Runge in S, we can approximate the f^{j} by algebraic functions f_{ν}^{j} on S. Then the functions h_{ν} = ∑ τ_{j}f_{ν}^{j} provide the desired algebraic approximations of f.

We now find it convenient to extend the usual definition of a retraction map by saying that a map ρ : V → S from a subset V ⊂ X to a subset S ⊂ X (where X is any space) is a retraction if ρ(x) = x for all x ∈ S ∩V (even if S ⊄ V). The following lemma on approximating holomorphic retractions by Nash algebraic retractions is a variation of a result of Tancredi and Tognoli [TT90, Theorem 1.5]:

Lemma 2.3. — Let V be a Runge domain in C^{n} and let S be an algebraic subset of C^{n} such that the variety M := S ∩ V is smooth. Suppose that ρ : V → S is a holomorphic retraction and V_{0} ⊂⊂ V is a relatively compact domain such that ρ(V_{0}) ⊂⊂ M. Then there is a sequence of Nash algebraic retractions ρ_{ν} : V_{0} → M such that ρ_{ν} → ρ uniformly on V_{0}.

Proof. — First we consider the special case where ρ is the holomorphic retraction constructed in Lemma 2.1. In this case, the conclusion follows from the proof of Theorem 1.5 in [TT90], which is based on an ingenious construction of Nash [Nh52]. We give a completely different short proof of this case here: Let V, S, M satisfy the hypotheses of Lemma 2.3, and suppose that σ : N_{M} → T_{C^{n}|M}, ψ : U → V, and ρ = π ∘ ψ^{−1} : V → M are given as in the proof of Lemma 2.1 (for the case Y = C^{n}). We first construct a “normal bundle sequence” for the possibly singular subvariety S. Let f_{1}, …, f_{r} be polynomials generating the ideal of S at all points of C^{n}. Let

dI_{S} ⊂ O_{S} ⊗ T_{C^{n}}^{⋆} = T_{C^{n}|S}^{⋆} ≃ O_{S}^{n}

denote the coherent sheaf on S generated by the 1-forms (1_{O_{S}} ⊗ df_{j}). (This definition is independent of the choice of generators (f_{j}) of I_{S}.) We consider the “normal sheaf”

N_{S} = Image (T_{C^{n}|S} → Hom(dI_{S}, O_{S})) ,

where we identify T_{C^{n}|S} ≃ Hom(T_{C^{n}|S}^{⋆}, O_{S}). Thus we have an exact sequence of sheaves of the form

0 ⟶ K ⟶ T_{C^{n}|S} \underrightarrow{τ} N_{S} ⟶ 0 .  (2.3)

Note that N_{S|M} = N_{M} and the restriction of (2.3) to the submanifold M is the exact sequence (2.1). Let F = Hom(N_{S}, T_{C^{n}|S}) and choose a surjective algebraic sheaf morphism O_{S}^{p} \underrightarrow{α} F. Recall that

σ ∈ Hom (N_{M}, T_{C^{n}|M}) = H^{0}(M, F) .  (2.4)

Since M is Stein, we can choose h ∈ H^{0}(M, O_{S}^{p}) = O(M)^{p} such that α(h) = σ. Since V is Runge, we can find a sequence h_{ν} of algebraic sections in H^{0}(S, O_{S}^{p}) such that h_{ν} → h uniformly on each compact subset of M. We let

σ_{ν} = α(h_{ν}) ∈ H^{0}(S, F) .  (2.5)

Let M̃ ⊃ M denote the set of regular points of S. Then N_{S|_{M}} = N_{M} and the sections σ_{ν|_{M}} ∈ Hom(N_{M}, T_{C^{n}|_{M}}) are algebraic. Recalling (2.2), we similarly define the algebraic maps ψ_{ν} : N_{M} → C^{n} by

ψ_{ν}(z, ζ) = z + σ_{ν}(z) · ζ , ζ ∈ N_{_{M},z} , z ∈ M̃ .  (2.6)

Choose open sets V_{1}, V_{2} with V_{0} ⊂⊂ V_{1} ⊂⊂ V_{2} ⊂⊂ V. Then ψ_{ν} → ψ uniformly on the set U′ = ψ^{−1}(V_{2}) ⊂⊂ U, and for ν ≫ 0, ψ_{ν|U′} is an injective map with ψ_{ν}(U′) ⊃ V_{1}. It follows that the maps ρ_{ν} = π ∘ ψ_{ν}^{−1} : V_{0} → M converge uniformly on V_{0} to ρ. (We remark that this proof gives algebraic maps ψ_{ν}, while the method of [TT90] provides only Nash algebraic ψ_{ν}.)

We now consider the general case. Choose a Runge domain V_{1} such that V_{0} ⊂⊂ V_{1} ⊂⊂ V and ρ(V_{0}) ⊂⊂ V_{1}, and let ε > 0 be arbitrary. It suffices to construct a Nash algebraic retraction ρ′ : V_{0} → M with ∥ρ′ − ρ∥ < 2ϵ on V_{0}. By Lemma 2.1 and the case already shown, there exist a neighborhood W of M ∩ V_{1} and a Nash algebraic retraction R : W → M ∩ V_{1}. After shrinking W if necessary, we can assume that ∥R(z) − z∥ < ε for all z ∈ W. By Lemma 2.2 (with A, S replaced by S, C^{n}), we can find a polynomial map P = (P_{1}, …, P_{n}) : C^{n} → C^{n} such that ∥P − ρ∥ < ε on V_{0}, P_{|S} = Id_{S}, and P (V_{0}) ⊂ W. We consider the Nash algebraic retraction

ρ′ = R ∘ P_{|V_{0}} : V_{0} → M .

Then ∥ρ′ − ρ∥_{V_{0}} ≤ ∥R ∘ P − P ∥_{V_{0}} + ∥P − ρ∥_{V_{0}} < 2ε.

Lemma 2.4. — Let M be a Stein submanifold of a complex manifold Y, and let E → Y be a holomorphic vector bundle. Fix a holomorphic retraction ρ : V → M of a neighborhood V of M onto M. Then there is a neighborhood V′ ⊂ V of M such that E is isomorphic to ρ^{⋆}(E_{|M}) on V′. In particular, if the restriction E_{|M} is trivial, then E is trivial on a neighborhood of M.

Proof. — By [Siu76] we can find a Stein neighborhood V_{0} ⊂ V. As V_{0} is Stein, the identity map from E_{|M} to (ρ^{⋆}E)_{|M} can be extended to a section of Hom(E, ρ^{⋆}E) over V_{0}. By continuity, this homomorphism is an isomorphism on a sufficiently small neighborhood V′ ⊂ V_{0} of M.

Finally, we need an elementary lemma on the existence of generating sections for holomorphic vector bundles on Stein spaces:

Lemma 2.5. — Let E be a holomorphic vector bundle of rank r on an ndimensional Stein space S. Suppose that I is a coherent sheaf of ideals in O_{S}, and let A = Supp O_{S}/I. Then we can find finitely many holomorphic sections in H^{0}(S, I ⊗ E) generating E at every point of S ^{\} A.

Lemma 2.5 follows by an elementary argument using Cartan’s Theorem B. (In fact, a careful argument shows that we can find n + r generating sections.) Note that if I = O_{S}, then A = ∅.

Corollary 2.6. — Let E be a holomorphic vector bundle over a finite dimensional Stein space S. Then there is a holomorphic map Φ : S → G into a Grassmannian such that E is isomorphic to the pull-back Φ^{⋆}Q of the universal quotient vector bundle Q on G.

Proof. — By Lemma 2.5 (with I = O_{S}), we can find finitely many holomorphic sections g_{1}, …, g_{m} spanning E at all points of S. We define the holomorphic map

Φ : S → G(m, r), x ↦ V_{x} = {(λ_{1}, …, λ_{m}) ∈ C^{m} : ∑ λ_{j}g_{j}(x) = 0}

into the Grassmannian G(m, r) of subspaces of codimension r in C^{m}. Then the generators g_{j} define an isomorphism γ : E → Φ^{⋆}Q with γ_{x} : E_{x} → Q_{Φ(x)} = C^{m}/V_{x} given by

γ_{x}(e) = {(μ_{1}, …, μ_{m}) ∈ C^{m} : ∑ μ_{j}g_{j}(x) = e} , e ∈ E_{x} ,

for x ∈ S. (See, for example, [GA74, Ch. V].) Hence E ≃ Φ^{⋆}Q.

3. Nash algebraic approximation on Runge domains in affine algebraic varieties

In this section, we prove the main approximation theorem 1.1 for holomorphic maps on a Runge domain Ω in an affine algebraic variety. We also give an alternate proof and variation of a result of Tancredi and Tognoli [TT93] on the exhaustion of holomorphic vector bundles on Ω by algebraic vector bundles (Proposition 3.2).

We begin the proof of Theorem 1.1 by first making a reduction to the case where S = C^{n}, X is projective, and f is an embedding. To accomplish this reduction, let S be an algebraic subvariety of C^{n} and suppose that Ω ⊂ S and f : Ω → X are given as in Theorem 1.1. By replacing X with a smooth completion, we can assume that X is projective. We consider the embedding

f_{1} = (i_{Ω}, f) : Ω → X_{1} = IP^{n} × X,

where i_{Ω} denotes the inclusion map Ω ↪ C^{n} ↪ IP^{n}. Approximations of f_{1} composed with the projection onto X will then give approximations of f. Since Ω is O(S)-convex, we can construct an analytic polyhedron of the form

Ω̃ = {z ∈ C^{n} : |h_{j}(z)| ≤ 1 for 1 ≤ j ≤ s} ,

where the h_{j} are in O(C^{n}), such that

Ω_{0} ⊂⊂ S ∩ Ω̃ ⊂⊂ Ω .

Then Ω̃ is a Runge domain in C^{n}. Consider the Runge domain

Ω̃_{ε} = {z ∈ Ω̃ : |P_{j}(z)| < ε for 1 ≤ j ≤ ℓ} ,

where the P_{j} are the defining polynomials for S ⊂ C^{n}. In order to extend f_{1} to Ω̃_{ε}, we apply a result of Siu [Siu76] to find a Stein neighborhood U ⊂ X′ of f_{1}(Ω). Then by using the holomorphic retraction of Lemma 2.1 (applied to U embedded in an affine space), shrinking ε if necessary, we can extend f_{1|S∩_{Ω_{ε}}} to a holomorphic map

f̃_{1} : Ω̃_{ε} → U ⊂ X_{1} .

By Nash approximating the embedding

f_{2} = (i_{Ω_{ε}}, f̃_{1}) : Ω̃_{ε} → X_{2} = IP^{n} × X_{1} ,

we also Nash approximate f.

Thus by replacing (Ω, X, f) with (Ω̃_{ε}, X_{2}, f_{2}), we may assume that Ω is a Runge domain in C^{n} and f : Ω → X is a holomorphic embedding. We make one further reduction: Choose a very ample line bundle L on X. By the proof of Lemma 2.5, we can find algebraic sections σ_{1}, …, σ_{p} of the algebraic line bundle α^{⋆}L^{−1} → A without common zeroes. We then extend the sections σ_{j|Ω} to holomorphic sections σ̃_{1}, …, σ̃_{p} of f^{⋆}L^{−1}. Again by Lemma 2.5, we choose holomorphic sections

σ̃_{p+1}, …, σ̃_{p+n+1} ∈ H^{0}(Ω, I_{A} ⊗ f^{⋆}L^{−1})

such that σ̃_{1}, …, σ̃_{p+n+1} generate f^{⋆}L^{−1} at all points of Ω. By the proof of Corollary 2.6, the sections σ̃_{1}, …, σ̃_{p+n+1} define a holomorphic map Φ : Ω → IP^{p+n} and an isomorphism of line bundles γ : f^{⋆}L^{−1} → Φ^{⋆}O(1). Similarly, the sections σ_{1}, …, σ_{p} define an algebraic morphism

Φ_{A} : A → IP^{p−1} ⊂ IP^{p+n}

and an algebraic isomorphism γ_{A} : α^{⋆}L^{−1} → Φ_{A}^{⋆}O(1) such that Φ_{|A∩Ω} = Φ_{A|A∩Ω} and γ_{|A∩Ω} = γ_{A|A∩Ω}.

We set X′ = IP^{p+n} × X and let L′ → X′ be the total tensor product L′ = O(1)×L. Then L′ is very ample on X′ and is thus isomorphic to the hyperplane section bundle of a projective embedding X′ ⊂ IP^{m−1}. Hence X′ can be identified with the projectivization of an affine algebraic cone Y ⊂ C^{m}, and the line bundle L^{′−1} = O_{X′}(−1) can be identified with the blow-up Ỹ of the cone Y at its vertex. Let τ : Ỹ → Y denote the blow-down of the zero section X′ of L^{′−1} (so that τ : Ỹ ^{\} X′ ≈ Y ^{\} {0}).

We consider the maps

f′ = (Φ, f) : Ω → X′ , α′ = (Φ_{A}, α) : A → X′ ,

and we note that f′_{|A∩Ω} = α′_{|A∩Ω}. The pull-back bundle f^{′⋆}L′ is trivial; in fact, a trivialization of f^{′⋆}L′ is given by the nonvanishing global section

γ ∈ Hom(f^{⋆}L^{−1}, Φ^{⋆}O(1)) = H^{0}(Ω, f^{⋆}L ⊗ Φ^{⋆}O(1)) = H^{0}(Ω, f^{′⋆}L′) .

We can regard the nonvanishing section 1/γ ∈ H^{0}(Ω, f^{′⋆}L^{′−1}) as a holomorphic mapping from Ω to Ỹ ^{\} X′. Therefore f′ : Ω → X′ lifts to the holomorphic embedding

g = τ ∘ (1/γ) : Ω → Y ^{\} {0} .

We likewise have an algebraic morphism

β = τ ∘ (1/γ_{A}) : A → Y ^{\} {0}

such that β is a lift of α′ and g_{|A∩Ω} = β_{|A∩Ω}. It is sufficient to Nash approximate g by maps into Y ^{\} {0} agreeing with β on A. Hence Theorem 1.1 is reduced to the following:

Lemma 3.1. — Let Ω be a Runge open set in C^{n}, let Y ⊂ C^{m} be an affine algebraic variety, and let g : Ω → Y′ be a holomorphic embedding into the set Y′ of regular points of Y. Suppose that there is an algebraic subvariety A (not necessarily reduced) of C^{n} and an algebraic morphism β : A → Y such that g_{|A∩Ω} = β_{|A∩Ω}. Then g can be approximated uniformly on each relatively compact domain Ω_{0} ⊂⊂ Ω by Nash algebraic embeddings g_{ν} : Ω_{0} → Y′ such that g_{ν|A∩Ω_{0}} = g_{|A∩Ω_{0}}.

Proof. — Consider the map G : Ω × C^{p} → C^{p} given by

G(z, w) = g(z) + w .  (3.1)

Choose Runge domains Ω̃_{1}, Ω̃_{2}, Ω̃ in C^{n+p} such that

\overline{Ω_{0}} × \{0\} ⊂ Ω̃_{1} ⊂⊂ Ω̃_{2} ⊂⊂ Ω̃ ⊂ G^{−1}(C^{p} \ Y_{sing}) ,

and let M = G^{−1}(Y) ∩ Ω̃ = G^{−1}(Y′) ∩ Ω̃. Since G is a submersion, it follows that M is a closed submanifold of Ω̃. We note that G(z, 0) = g(z) for z ∈ Ω, and thus M ⊃ Ω_{0} × {0}.

By the proof of Lemma 2.1, there is a neighborhood V ⊂ Ω̃ of M together with a biholomorphism ψ : U → V, where U ⊂ N_{M} is a neighborhood of the zero section M, such that ψ_{|M} = Id_{M}. Let

V_{i} = {z ∈ Ω̃_{i} : |h_{j}(z)| < iδ for 1 ≤ j ≤ ℓ} , i = 1, 2 ,

where we choose h_{1}, …, h_{ℓ} ∈ O(Ω̃) defining M, and we choose δ > 0 such that V_{2} ⊂⊂ V. We then have the holomorphic retraction ρ = π ∘ ψ^{−1} : V → M, where π : N_{M} → M is the projection. By shrinking δ if necessary, we can assume that ρ(V_{1}) ⊂ M ∩ V_{2} = M ∩ Ω̃_{2}.

We identify A ⊂ C^{n} with A × {0} ⊂ C^{n+p} so that we can regard A as an algebraic subvariety (not necessarily reduced) of C^{n+p}; then G_{|A∩_{Ω}} = β_{|A∩_{Ω}} .

Since Ω̃ is Runge, by Lemma 2.2 we can choose a sequence of polynomial maps G_{ν} : C^{n+p} → C^{m} such that G_{ν|A} = β and G_{ν} → G uniformly on a Runge domain D ⊂ Ω̃ containing \overline{V_{2}}. We write M_{ν} = G_{ν}^{−1}(Y) ∩ D′, where D′ is a Runge domain with V_{2} ⊂⊂ D′ ⊂⊂ D. Then for ν ≫ 0, M_{ν} is a submanifold of D′. Moreover, since π_{|M} is the identity, for large ν the restriction of π to ψ^{−1}(M_{ν})∩π^{−1}(M ∩V_{2}) is a biholomorphic map onto M ∩ V_{2}. The inverse of this latter map is a section t_{ν} ∈ H^{0}(M ∩ V_{2}, N_{M}), and t_{ν} → 0 uniformly on M ∩ V_{1}. We now consider the holomorphic retractions

ρ_{ν} = ψ ∘ t_{ν} ∘ ρ_{|V_{1}} : V_{1} → M_{ν} .  (3.5)

Then ρ_{ν} → ρ uniformly on V_{1}. Choose a domain V_{0} such that \overline{Ω_{0}}×\{0\} ⊂ V_{0} ⊂⊂ V_{1} and ρ(V_{0}) ⊂⊂ V_{1}. Since V_{1} is Runge in C^{n+p} and ρ_{ν}(V_{0}) ⊂⊂ V_{1} for ν ≫ 0, by Lemma 2.3 we can find Nash algebraic retractions ρ′_{ν} : V_{0} → M_{ν} sufficiently close to ρ_{ν} such that ρ′_{ν} → ρ uniformly on V_{0}. Then the Nash algebraic approximations g_{ν} : Ω_{0} → Y′ of g can be given by

g_{ν}(z) = G_{ν} ∘ ρ′_{ν}(z, 0) , z ∈ Ω_{0} .  (3.6)

It remains to verify that

g_{ν|A∩Ω_{0}} = β_{|A∩Ω_{0}} .  (3.7)

Since ρ′_{ν} is a retraction to M_{ν}, we have

G_{ν} ∘ ρ′_{ν|M_{ν}} = G_{ν} .  (3.8)

It suffices to show that I_{M_{ν}} ⊂ I_{A|D′} (where I_{A} now denotes the ideal sheaf on C^{n+p} with O_{A} = O_{C}^{n+p}/I_{A}), since this would give us an inclusion morphism A ∩ D′ ↪ M_{ν}, and then (3.7) would follow by restricting (3.8) to A ∩ D′ and recalling that G_{ν|A} = β. Let h ∈ I_{Y,G_{ν}(a)} be arbitrary, where a ∈ A ∩ D′. Since G_{ν|A} = G_{|A}, it follows that

h ∘ G_{ν} − h ∘ G ∈ I_{A,a} .  (3.9)

Since Ω × {0} ⊂ M, we have

h ∘ G ∈ I_{M,a} ⊂ I_{Ω×{0},a} ⊂ I_{A,a} ,

and thus it follows from (3.9) that h ∘ G_{ν} ∈ I_{A,a}. Since G_{ν}^{⋆}I_{Y,G_{ν}(a)} generates I_{M_{ν},a}, it follows that I_{M_{ν},a} ⊂ I_{A,a}.

Tancredi and Tognoli ([TT93], Theorem 4.1) showed that if E is a holomorphic vector bundle on a Runge domain Ω in an affine algebraic variety S, then for every domain Ω_{0} ⊂⊂ Ω, E_{|Ω_{0}} is isomorphic to a “Nash algebraic vector bundle” E′ → Ω_{0}. Tancredi and Tognoli demonstrate this result by applying a result of Nash [Nh52] on Nash-algebraically approximating analytic maps into the algebraic manifold of n × n matrices of rank exactly r. (Alternately, one can approximate analytic maps into a Grassmannian.) Theorem 1.1 is a generalization of (the complex form of) this approximation result of Nash. In fact, Theorem 1.1 can be used to obtain the following form of the Tancredi-Tognoli theorem with a more explicit description of the equivalent Nash algebraic vector bundle:

Proposition 3.2. — Let Ω be a Runge domain in an n-dimensional affine algebraic variety S and let E → Ω be a holomorphic vector bundle. Then for every relatively compact domain Ω_{0} ⊂⊂ Ω there exist:

(i) an n-dimensional projective algebraic variety Z,

(ii) a Nash algebraic injection i : Ω_{0} ↪ Z,

(iii) an algebraic vector bundle Ẽ → Z such that i^{⋆}Ẽ is isomorphic to E_{|Ω_{0}},

(iv) an ample line bundle L → Z with trivial restriction i^{⋆}L.

Moreover, if S is smooth, then Z can be taken to be smooth.

Proof. — By Corollary 2.6, there is a holomorphic map Φ : Ω → G into a Grassmannian, such that E ≃ Φ^{⋆}Q where Q is the universal quotient vector bundle. We construct a holomorphic map of the form

f = (Φ, Φ′) : Ω → X = G × IP^{N},

where Φ′ : S → IP^{N} is chosen in such a way that f is an embedding and X has an ample line bundle L with trivial pull-back f^{⋆}L (using the same argument as in the reduction of Theorem 1.1 to Lemma 3.1). Then E ≃ f^{⋆}Q_{X}, where Q_{X} is the pull-back of Q to X.

Choose a Runge domain Ω_{1} with Ω_{0} ⊂⊂ Ω_{1} ⊂⊂ Ω. By Theorem 1.1, f_{|Ω_{1}} is a uniform limit of Nash algebraic embeddings f_{ν} : Ω_{1} → X. Hence the images f_{ν}(Ω_{1}) are contained in algebraic subvarieties A_{ν} ⊂ X of dimension n = dim S. Let f′ = f_{μ|Ω_{0}}, where μ is chosen to be sufficiently large so that f′ is homotopic to f_{|Ω_{0}}. Then the holomorphic vector bundle E′ := f^{′⋆}Q_{X} is topologically isomorphic to E_{|Ω_{0}}. By a theorem of Grauert [Gr58], E′ ≃ E_{|Ω_{0}} as holomorphic vector bundles. If S is singular, we take Z = A_{μ}, i = f′, and Ẽ = E′.

If S is smooth, we must modify A_{μ} to obtain an appropriate smooth variety Z. To accomplish this, let σ : A′′_{μ} → A_{μ} be the normalization of A_{μ} and let f′′ : Ω_{0} → A′′_{μ} be the map such that f′ = σ∘f′′; since f′ is an embedding into X, we see that f′′(Ω_{0}) is contained in the set of regular points of A′′_{μ}. (The reason for the introduction of the normalization is that f′(Ω_{0}) can contain multiple points of A_{μ}.) By Hironaka’s resolution of singularities [Hi64], there exists a desingularization π : Z → A′′_{μ} with center contained in the singular locus of A′′_{μ}. Let i : Ω_{0} ↪ Z denote the embedding given by f′′ = π ∘ i. We note that the exceptional divisor of Z does not meet i(Ω_{0}). The algebraic vector bundle Ẽ := (σ ∘ π)^{⋆}Q_{X} then satisfies

i^{⋆}Ẽ = (σ ∘ π ∘ i)^{⋆}Q_{X} = f^{′⋆}Q_{X} = E′ ≃ E_{|Ω_{0}}.

Finally, we note that the line bundle L′′ := σ^{⋆}L_{|A_{μ}} is ample on A′′_{μ} and thus there is an embedding A′′_{μ} ⊂ IP^{m} such that O(1)_{|A′′_{μ}} = L^{b} for some b ∈ IN (where O(1) is the hyperplane section bundle in IP^{m}). We can suppose that π is an embedded resolution of singularities with respect to this embedding; i.e., there is a modification π : X̃ → IP^{m} such that X̃ is smooth and Z ⊂ X̃ is the strict transform of A′′_{μ}. Then there is an exceptional divisor D ≥ 0 of the modification π : X̃ → IP^{m} such that L̃ := π^{⋆}O(a) ⊗ O(−D) is ample on X̃ for a ≫ 0. (We can take D = π^{⋆}[π(H)] − [H], for an ample hypersurface H ⊂ X̃ that does not contain any component of the exceptional divisor of X̃.) We can assume that μ has been chosen sufficiently large so that f^{′⋆}L ≃ (f^{⋆}L)_{|Ω_{0}} is trivial. By construction i^{⋆}O_{Z}(D) is trivial, and hence so is i^{⋆}(L̃_{|Z}).

Remark 3.3. — Here, it is not necessary to use Grauert’s theorem in its full strength. The special case we need can be dealt with by means of the results obtained in Section 2. In fact, for μ large, there is a one-parameter family of holomorphic maps F : Ω_{1} × U → X, where U is a neighborhood of the interval [0, 1] in C, such that F (z, 0) = f(z) and F (z, 1) = f_{μ}(z) on Ω_{1} ; this can be seen by taking a Stein neighborhood V of f(Ω) in X and a biholomorphism φ = ψ^{−1} from a neighborhood of the diagonal in V × V onto a neighborhood of the zero section in the normal bundle (using Lemma 2.1 (i)); then F (z, w) = ψ(w φ(f(z), f_{μ}(z))) is the required family. Now, if Y is a Stein manifold and U is connected, Lemma 2.4 and an easy connectedness argument imply that all slices over Y of a holomorphic vector bundle B → Y × U are isomorphic, at least after we take restriction to a relatively compact domain in Y. Hence

E′ = f_{μ}^{⋆}Q_{X|Ω_{0}} = (F^{⋆}Q_{X})_{|Ω_{0}×{1}} ≃ (F^{⋆}Q_{X})_{|Ω_{0}×{0}} = f^{⋆}Q_{X|Ω_{0}} ≃ E_{|Ω_{0}}.

Remark 3.4. — In general, the embedding i : Ω_{0} ↪ Z in Proposition 3.2 will not extend to a (univalent) rational map S - - → Z, but will instead extend as a multi-valued branched map. For example, let S = X ^{\} H, where H is a smooth hyperplane section of a projective algebraic manifold X with some nonzero Hodge numbers off the diagonal. By Cornalba-Griffiths [CG75, § 21] (see also Appendix 2 in [CG75]) there is a holomorphic vector bundle E′ → S such that the Chern character ch(E′) ∈ H^{even}(S, Q) is not the restriction of an element of H^{even}(X, Q) given by (rational) algebraic cycles. Choose Ω_{0} ⊂⊂ Ω ⊂ S such that the inclusion Ω_{0} ↪ S induces an isomorphism H^{even}(S, Q) ≈ H^{even}(Ω_{0}, Q), and let E = E′_{|Ω}.

Now let i : Ω_{0} ↪ Z and Ẽ → Z be as in the conclusion of Proposition 3.2. We assert that i cannot extend to a rational map on S. Suppose on the contrary that i has a (not necessarily regular) rational extension i′ : X - - → Z. By considering the graph Γ_{i′} ⊂ X × Z and the projections pr_{1} : Γ_{i′} → X, pr_{2} : Γ_{i′} → Z, we obtain a coherent algebraic sheaf F = pr_{1⋆}O(pr_{2}^{⋆}Ẽ) on X with restriction F_{|Ω_{0}} = i^{⋆}Ẽ ≃ E_{|Ω_{0}}, and therefore ch(F)_{|Ω_{0}} = ch(E′)_{|Ω_{0}}. Now ch(F) is given by algebraic cycles in X, but by the above, ch(F)_{|S} = ch(E′), which contradicts our choice of E′. In particular, E_{|Ω_{0}} is not isomorphic to an algebraic vector bundle on S.

Problem 3.5. — The method of proof used here leads to the following natural question. Suppose that f : Ω → X is given as in Theorem 1.1, and assume in addition that S is nonsingular and dim X ≥ 2 dim S + 1. Is it true that f can be approximated by Nash algebraic embeddings f_{ν} : Ω_{ν} → X such that f_{ν}(Ω_{ν}) is contained in a nonsingular algebraic subvariety A_{ν} of X with dim A_{ν} = dim f_{ν}(Ω_{ν}) = dim S? If dim X = 2 dim S, does the same conclusion hold with f_{ν} being an immersion and A_{ν} an immersed nonsingular variety with simple double points ?

Our expectation is that the answer should be positive in both cases, because there is a priori enough space in X to get the smoothness of f_{ν}(Ω_{ν}) by a generic choice of f_{ν}, by a Whitney-type argument. The difficulty is to control the singularities introduced by the Nash algebraic retractions. If we knew how to do this, the technical point of using a desingularization of A_{μ} in the proof of Proposition 3.2 could be avoided.

4. Nash algebraic approximations omitting ample divisors

In this section we use L^{2} approximation techniques to give the following conditions for which the Nash algebraic approximations in Theorem 1.1 can be taken to omit ample divisors:

Theorem 4.1. — Let Ω be a Runge domain in an affine algebraic manifold S, and let f : Ω → X be a holomorphic embedding into a projective algebraic manifold X (with dim S < dim X). Suppose that there exists an ample line bundle L on X with trivial restriction f^{⋆}L to Ω. Then for every relatively compact domain Ω_{0} ⊂⊂ Ω, one can find sections h_{ν} ∈ H^{0}(X, L^{⊗μ(ν)}) and Nash algebraic embeddings f_{ν} : Ω_{0} → X ^{\} h_{ν}^{−1}(0) converging uniformly to f_{|Ω_{0}}. Moreover, if we are given a positive integer m and a finite set of points (t_{j}) in Ω_{0}, then the f_{ν} can be taken to have the same m-jets as f at each of the points t_{j}.

Corollary 4.2. — Let Ω_{0} ⊂⊂ Ω ⊂ S be as in Theorem 4.1. Suppose that H^{2}(Ω; ZZ) = 0. Then for every holomorphic embedding f : Ω → X into a projective algebraic manifold X (with dim S < dim X), one can find affine Zariski open sets Y_{ν} ⊂ X and Nash algebraic embeddings f_{ν} : Ω_{0} → Y_{ν} converging uniformly to f_{|Ω_{0}}. Moreover, if we are given a positive integer m and a finite set of points (t_{j}) in Ω_{0}, then the f_{ν} can be taken to have the same m-jets as f at each of the points t_{j}.

The main difficulty in proving Theorem 4.1 is that the submanifold f(Ω) might not be contained in an affine open set. To overcome this difficulty, we first shrink f(Ω) a little bit and apply a small perturbation to make f(Ω) smooth up to the boundary with essential singularities at each boundary point. Then f(Ω) becomes in some sense very far from being algebraic, but with an additional assumption on f(Ω) it is possible to show that f(Ω) is actually contained in an affine Zariski open subset (Corollary 4.9). The technical tools needed are Hörmander’s L^{2} estimates for \overline{∂} and the following notion of complete pluripolar set.

Definition 4.3. — Let Y be a complex manifold.

(i) A function u : Y → [−∞, +∞[ is said to be quasi-plurisubharmonic (quasipsh for short) if u is locally equal to the sum of a smooth function and of a plurisubharmonic (psh) function, or equivalently, if \sqrt{−1}∂\overline{∂}u is bounded below by a continuous real (1, 1)-form.

(ii) A closed set P is said to be complete pluripolar in Y if there exists a quasi-psh function u on Y such that P = u^{−1}(−∞).

Remark 4.4. — Note that the sum and the maximum of two quasi-psh functions is quasi-psh. (However, the decreasing limit of a sequence of quasi-psh functions may not be quasi-psh, since any continuous function is the decreasing limit of smooth functions and there are many continuous non-quasi-psh functions, e.g., − log^{+} |z|.) The usual definition of pluripolar sets deals with psh functions rather than quasi-psh functions; our choice is motivated by the fact that we want to work on compact manifolds, and of course, there are no nonconstant global psh functions in that case. It is elementary to verify that if u is a quasi-psh function on a Stein manifold S, then u+φ is psh for some smooth, rapidly growing function φ on S, and hence our definition of complete pluripolar sets coincides with the usual definition in the case of Stein manifolds. Finally, we remark that the word “complete” refers to the fact that P must be the full polar set of u, and not only part of it.

Lemma 4.5. — Let P be a closed subset in a complex manifold Y.

(i) If P is complete pluripolar in Y, there is a quasi-psh function u on Y such that P = u^{−1}(−∞) and u is smooth on Y ^{\} P.

(ii) P is complete pluripolar in Y if and only if there is an open covering (Ω_{j}) of Y such that P ∩ Ω_{j} is complete pluripolar in Ω_{j}.

Proof. — We first prove result (i) locally, say on a ball Ω = B(0, r_{0}) ⊂ C^{n}, essentially by repeating the arguments given in Sibony [Sib85]. Let P be a closed set in Ω, and let v be a psh function on Ω such that P = v^{−1}(−∞). By shrinking Ω and subtracting a constant from v, we may assume v ≤ 0. Select a convex increasing function χ : [0, 1] → IR such that χ(t) = 0 on [0, 1/2] and χ(1) = 1. We set

w_{k} = χ( exp(v/k)).

Then 0 ≤ w_{k} ≤ 1, w_{k} is plurisubharmonic on Ω and w_{k} = 0 in a neighborhood of P. Fix a family (ρ_{ε}) of smoothing kernels and set Ω′ = B(0, r_{1}) ⊂⊂ Ω. For each k, there exists ε_{k} > 0 small such that w_{k} ⋆ ρ_{ε_{k}} is well defined on Ω′ and vanishes on a neighborhood of P ∩ Ω′. Then

h_{k} = max {w_{1} ⋆ ρ_{ε_{1}}, …, w_{k} ⋆ ρ_{ε_{k}}}

is an increasing sequence of continuous psh functions such that 0 ≤ h_{k} ≤ 1 on Ω′ and u_{k} = 0 on a neighborhood of P ∩ Ω′. The inequalities

h_{k} ≥ w_{k} ⋆ ρ_{ε_{k}} ≥ w_{k} = χ( exp(v/k))

imply that lim h_{k} = 1 on Ω′ ^{\} P. By Dini’s lemma, (h_{k}) converges uniformly to 1 on every compact subset of Ω′ ^{\} P. Thus, for a suitable subsequence (k_{ν}), the series

h(z) := |z|^{2} + ∑_{ν=0}^{+∞}(h_{k_{ν}} (z) − 1)  (4.1)

converges uniformly on every compact subset of Ω′ ^{\} P and defines a strictly plurisubharmonic function on Ω′ which is continuous on Ω′ ^{\} P and such that h = −∞ on P ∩ Ω′. Richberg’s regularization theorem [Ri68] implies that there is a psh function u on Ω′ such that u is smooth on Ω′ ^{\} P and h ≤ u ≤ h + 1. Then u^{−1}(−∞) = P ∩ Ω′ and Property (i) is proved on Ω′.

(ii) The “only if” part is clear, so we just consider the “if” part. By means of (i) and by taking a refinement of the covering, we may assume that we have open sets Ω̃_{j} ⊃⊃ Ω_{j} ⊃⊃ Ω′_{j} where (Ω′_{j}) is a locally finite open covering of Y and for each j there is a psh function u_{j} on Ω̃_{j} such that P ∩ Ω̃_{j} = u_{j}^{−1}(−∞) with u_{j} smooth on Ω̃_{j} ^{\} P. We define inductively a strictly decreasing sequence (t_{ν}) of real numbers by setting t_{0} = 0 and

t_{ν+1} ≤ inf \{ ⋃_{j,k≤ν} u_{j}( \overline{Ω}_{j} ∩ \overline{Ω}_{k} ^{\} u_{k}^{−1}([−∞, t_{ν}[))\} .  (4.2)

The infimum is finite since u_{j} and u_{k} have the same poles. Now, if we take in addition t_{ν+1} − t_{ν} < t_{ν} − t_{ν−1}, there exists a smooth convex function χ : IR → IR such that χ(t_{ν}) = −ν. Choose functions φ_{j} ∈ C^{∞}(Ω_{j}) such that φ_{j} ≡ 0 on Ω′_{j}, φ_{j} ≤ 0 on Ω_{j}, and φ_{j}(z) → −∞ as z → ∂Ω_{j}. We set

v_{j} = χ ∘ u_{j} + φ_{j} .

Then v_{j} is quasi-psh on Ω_{j}. Let

v(z) = max_{Ω_{j}∋z} v_{j}(z), z ∈ V .  (4.3)

Clearly v^{−1}(−∞) = P. To show that v is quasi-psh it suffices to verify that if z_{0} ∈ ∂Ω_{k}, then for z ∈ Ω_{k} ^{\} P sufficiently close to z_{0} we have v_{k}(z) < v(z). This is obvious if z_{0} ∉ P since v_{k} → −∞ as z → z_{0}. So consider z_{0} ∈ ∂Ω_{k} ∩ P ∩ Ω′_{j}. Suppose that z ∈ Ω_{k} ∩ Ω′_{j} ^{\} P is sufficiently close to z_{0} so that

max{u_{j}(z), u_{k}(z)} < min{t_{j}, t_{k}} and φ_{k}(z) ≤ −3 .

Then we have u_{j}(z) ∈ [t_{ν+1}, t_{ν}[ and u_{k}(z) ∈ [t_{μ+1}, t_{μ}[ with indices μ, ν ≥ max{j, k}. Then by (4.2), u_{j}(x) ≥ t_{μ+2} and u_{k}(x) ≥ t_{ν+2}; hence |μ − ν| ≤ 1 and |χ ∘ u_{j}(z) − χ ∘ u_{k}(z)| ≤ 2. Therefore

v(z) ≥ v_{j}(z) = χ ∘ u_{j}(z) ≥ χ ∘ u_{k}(z) − 2 ≥ v_{k}(z) + 1 ,

which completes the proof of (ii).

To obtain the global case of property (i), we replace the max function in (4.3) by the regularized max functions

max_{ε} = max ⋆ ρ_{ε}^{m} : IR^{m} → IR ,

where ρ_{ε}^{m} is of the form ρ_{ε}^{m}(x_{1}, …, x_{m}) = ε^{−m}ρ(x_{1}/ε) ⋯ ρ(x_{m}/ε). This form of smoothing kernel ensures that

x_{m} ≤ max{x_{1}, …, x_{m−1}} − 2ε ⇒ max_{ε}(x_{1}, …, x_{m}) = max_{ε}(x_{1}, …, x_{m−1}) . (Of course, max_{ε} is convex and is invariant under permutations of variables.) The function v_{ε} obtained by using max_{ε} in (4.3) with any ε ≤ 1/2 is quasi-psh on Y, smooth on Y ^{\} P, and has polar set P.

Now we show that for any Stein submanifold M of a complex manifold Y, there are complete pluripolar graphs of sections K → N_{M} over arbitrary compact subsets K ⊂ M.

Proposition 4.6. — Let M be a (not necessarily closed) Stein submanifold of a complex manifold Y, with dim M < dim Y, and let K be a holomorphically convex compact subset of M. Let ψ : U → V be as in Lemma 2.1(i). Then there is a continuous section g : K → N_{M} which is holomorphic in the interior K^{∘} of K, such that g(K) ⊂ U and the sets K_{ε} = ψ(εg(K)) are complete pluripolar in Y for 0 < ε ≤ 1.

Here εg(K) denotes the ε-homothety of the compact section g(K) ⊂ N_{M}; thus K_{ε} converges to K as ε tends to 0.

Proof. — By 4.5(ii), the final assertion is equivalent to proving that εg(K) is complete pluripolar in N_{M}, and we may of course assume that ε = 1. The holomorphic convexity of K implies the existence of a sequence (f_{j})_{j≥1} of holomorphic functions on M such that K = ∩{|f_{j}| ≤ 1}. By Lemma 2.5, there is a finite sequence (g_{k})_{1≤k≤N} of holomorphic sections of N_{M} without common zeroes. We consider the sequence (j_{ν}, k_{ν})_{ν≥1} of pairs of positive integers obtained as the concatenation Λ_{1}, Λ_{2}, …, Λ_{ℓ}, … of the finite sequences

Λ_{ℓ} = (j_{q}, k_{q})_{ν(ℓ)≤q<ν(ℓ+1)} = ((1, 1), …, (1, N), …, (ℓ, 1), …, (ℓ, N)) (where ν(ℓ) = Nℓ(ℓ − 1)/2 + 1). We define g to be the generalized “gap sequence”

g(z) = ∑_{ν=1}^{∞} e^{−p_{ν}} f_{j_{ν}} (z)^{p_{ν}^{2}} g_{k_{ν}} (z) ∀z ∈ K ,  (4.4)

where (p_{ν}) is a strictly increasing sequence of positive integers to be defined later. The series (4.4) converges uniformly on K (with respect to a Hermitian metric on N_{M}) and thus g is continuous on K and holomorphic on K^{∘}. To construct our psh function with polar set g(K) we first select a smooth Hermitian metric ∥ ∥ on N_{M} such that ζ ↦ log ∥ζ∥ is psh on N_{M} (e.g., take ∥ζ∥^{2} = ∑ |g_{i}^{⋆}(z) · ζ|^{2} for ζ ∈ N_{M,z}, where (g_{i}^{⋆}) is a collection of holomorphic sections of N_{M}^{⋆} without common zeroes). We consider the global holomorphic sections

s_{q} = ∑_{ν=1}^{q} e^{−p_{ν}} f_{j_{ν}}^{p_{ν}^{2}} g_{k_{ν}}  (4.5)

of N_{M}, which converge to g uniformly on K exponentially fast. We consider the psh functions u_{ℓ} on N_{M} given by

u_{ℓ}(ζ) = max_{ν(ℓ)≤q<ν(ℓ+1)} \frac{1}{p_{q}^{4}} log ∥ζ − s_{q}(z)∥ , ∀ζ ∈ N_{M,z}, ∀z ∈ M .  (4.6)

We shall show that the infinite sum

u = ∑_{ℓ=1}^{∞} max(u_{ℓ}, −1)  (4.7)

is psh on N_{M} with u^{−1}(−∞) = g(K). For z ∈ K, ζ = g(z) ∈ g(K), we obtain from (4.4) the estimate

∥ζ − s_{q}(z)∥ ≤ ce^{−p_{q+1}}

where c is independent of the choice of z ∈ K. Thus if we choose the p_{q} with p_{q+1} ≥ 2p_{q}^{4}, we have u_{ℓ} < −1 on g(K) for ℓ large, and thus u ≡ −∞ on g(K). Next we show that u is psh. For this it suffices to show that u ≢ −∞ and

∑_{ℓ=1}^{∞} max(sup_{A} u_{ℓ}, 0) < +∞  (4.8)

for all compact subsets A of N_{M} (since (4.8) implies that u_{|A^{∘}} can be written as a decreasing sequence of psh functions). Let M_{q} be an exhausting sequence of compact subsets of M. When ζ ∈ N_{M,z} with z ∈ M_{q}, we have

log ∥ζ − s_{q}(z)∥ ≤ log(1 + ∥ζ∥) + (p_{q}^{2} + 1) log(1 + C_{q})

where C_{q} is the maximum of ∥g_{1}∥, …, ∥g_{N}∥, |f_{1}|, …, |f_{q}| on M_{q}. In addition to our previous requirement, we assume that p_{q} ≥ log(1 + C_{q}) and hence

u_{ℓ}(ζ) ≤ \frac{2 + log(1 + ∥ζ∥)}{p_{ν(ℓ)}}, ∀ζ ∈ N_{M,z}, ∀z ∈ M_{ν(ℓ)} .

Since p_{ν(ℓ)} ≥ p_{ℓ} ≥ 2^{ℓ−1} (in fact, the growth is at least doubly exponential), the bound (4.8) follows.

We finally show that u(ζ) is finite outside g(K). Fix a point ζ ∈ N_{M,z}−g(K). If z ∈ K, then the logarithms appearing in (4.6) converge to log ∥ζ − g(z)∥ > −∞, and hence u(ζ) > −∞. Now suppose z ∉ K. Then one of the values f_{j}(z) has modulus greater than 1. Thus for ℓ sufficiently large, there is a q = q(ℓ) in the range ν(ℓ) ≤ q < ν(ℓ + 1) such that |f_{j_{q}}(z)| > 1 and ∥g_{k_{q}}(z)∥ = max ∥g_{i}(z)∥ > 0. Assume further that |f_{j_{q}}(z)| = max_{1≤i≤ℓ} |f_{i}(z)|. For this choice of q we have

∥ζ − s_{q}(z)∥ ≥ e^{−p_{q}}|f_{j_{q}}(z)|^{p_{q}^{2}}|g_{k_{q}}(z)| − \frac{|f_{j_{q}}(z)|^{p_{q−1}^{2}+1}|g_{k_{q}}(z)|}{|f_{j_{q}}(z)| − 1} − ∥ζ∥ .

Since p_{q} ≥ 2p_{q−1}^{4}, we easily conclude that ∥ζ − s_{q(ℓ)}(z)∥ > 1 for ℓ sufficiently large and hence u_{ℓ}(z) is eventually positive. Therefore u(z) > −∞; hence u^{−1}(−∞) = g(K), and so g(K) is complete pluripolar in N_{M}.

Remark. — If one chooses p_{ν} to be larger than the norm of the first ν derivatives of f_{1}, …, f_{ν} on K in the above proof, the section g will be smooth on K.

Proposition 4.7. — Let K ⊂ M ⊂ Y be as in Proposition 4.6, and let I be a finite subset of the interior K^{∘} of K. Then for every positive integer m we can select the section g : K → N_{M} in Proposition 4.6 with the additional property that the m-jet of g vanishes on I (and thus K_{ε} is tangent to K of order m at all points of I).

Proof. — The proof of Proposition 4.6 used only the fact that the sections g_{1}, …, g_{N} do not vanish simultaneously on M ^{\} K. Hence, we can select the g_{j} to vanish at the prescribed order m at all points of I.

Next, we prove the following simple approximation theorem based on Hörmander’s L^{2} estimates (see Andreotti-Vesentini [AV65] and [Hö66]).

Proposition 4.8. — Let X be a projective algebraic manifold and let L be an ample line bundle on X. Let P be a complete pluripolar set in X such that L is trivial on a neighborhood Ω of P; i.e., there is a nonvanishing section s ∈ H^{0}(Ω, L). Then there is a smaller neighborhood V ⊂ Ω such that every holomorphic function h on Ω can be approximated uniformly on V by a sequence of global sections h_{ν} ∈ H^{0}(X, L^{⊗ν}); precisely, h_{ν}s^{−ν} → h uniformly on V.

Proof. — By Lemma 4.5(i), we can choose a quasi-psh function u on X with u^{−1}(−∞) = P and u smooth on X^{\}P. Fix a Kähler metric ω on X and a Hermitian metric on L with positive definite curvature form Θ(L). Let (U_{α})_{0≤α≤N} be an open cover of X with trivializing sections s_{α} ∈ H^{0}(U_{α}, L). We write ∥s_{α}∥^{2} = e^{−φ_{α}} so that for v = v^{α}s_{α}(z) ∈ L_{z} we have ∥v∥^{2} = |v^{α}|^{2}e^{−φ_{α}(z)}. We note the curvature formula, Θ(L)_{|U_{α}} = \sqrt{−1}∂\overline{∂}φ_{α}. In the following, we shall take U_{0} = Ω and s_{0} = s. We shall also drop the index α and denote the square of the norm of v simply by ∥v∥^{2} = |vs^{−1}|^{2}e^{−φ(z)}.

By multiplying u with a small positive constant, we may assume that Θ(L) + \sqrt{−1}∂\overline{∂}u ≥ εω for some ε > 0. Then, for ν ∈ IN large, Hörmander’s L^{2} existence theorem (as given in [De92, Theorem 3.1]) implies that for every \overline{∂}-closed (0, 1)-form g with values in L^{⊗ν} such that

∫_{X} ∥g∥^{2}e^{−νu}dV_{ω} < +∞

(with dV_{ω} = Kähler volume element), the equation \overline{∂}f = g admits a solution f satisfying the estimate

∫_{X} |fs^{−ν}|^{2}e^{−ν(φ+u)}dV_{ω} = ∫_{X} ∥f∥^{2}e^{−νu}dV_{ω} ≤ \frac{1}{λ_{ν}} ∫_{X} ∥g∥^{2}e^{−νu}dV_{ω},

where λ_{ν} is the minimum of the eigenvalues of

ν(Θ(L) + \sqrt{−1}∂\overline{∂}u) + Ricci(ω)

with respect to ω throughout X. Clearly λ_{ν} ≥ εν + ρ_{0} where ρ_{0} ∈ IR is the minimum of the eigenvalues of Ricci(ω). We apply the existence theorem to the (0, 1)-form g = \overline{∂}(hθs^{ν}) = h\overline{∂}θs^{ν}, where θ is a smooth cut-off function such that

θ = 1 on {z ∈ Ω : φ(z) + u(z) ≤ c},
Supp θ ⊂ {z ∈ Ω : φ(z) + u(z) < c + 1} ⊂⊂ Ω

for some sufficiently negative c < 0. Then we get a solution of the equation \overline{∂}f_{ν} = h\overline{∂}θs^{ν} on X with

∫_{X} |f_{ν}s^{−ν}|^{2}e^{−ν(φ+u)}dV_{ω} ≤ \frac{1}{εν + ρ_{0}} ∫_{X} |h|^{2}|\overline{∂}θ|^{2}e^{−ν(φ+u)}dV_{ω}.

Since Supp(\overline{∂}θ) ⊂ \{c ≤ φ(z) + u(z) ≤ c + 1\}, we infer

e^{−ν(c−1)} ∫_{\{φ+u<c−1\}} |f_{ν}s^{−ν}|^{2}dV_{ω} ≤ \frac{e^{−νc}}{εν + ρ_{0}} ∫_{Ω} |h|^{2}|\overline{∂}θ|^{2}dV_{ω}.

Thus f_{ν}s^{−ν} → 0 in L^{2}({φ+u < c−1}). Since the f_{ν}s^{−ν} are holomorphic functions on {φ + u < c − 1}, it follows that f_{ν}s^{−ν} → 0 uniformly on the neighborhood V = {φ + u < c − 2} ⊂⊂ Ω of P. Hence h_{ν} = hθs^{ν} − f_{ν} is a holomorphic section of L^{⊗ν} such that h_{ν}s^{−ν} converges uniformly to h on V.

Corollary 4.9. — Let X be a projective algebraic manifold and let M be a Stein submanifold with dim M < dim X, such that there exists an ample line bundle L on X with trivial restriction L_{|M}. Let K be a holomorphically convex compact subset of M, and let (K_{ε})_{0<ε≤1} be the complete pluripolar sets constructed in Proposition 4.6. Then for each sufficiently small ε, there is a positive integer ν(ε) and a section h_{ν(ε)} ∈ H^{0}(X, L^{⊗ν(ε)}) such that K_{ε} does not intersect the divisor of h_{ν(ε)}; i.e., K_{ε} is contained in the affine Zariski open set X ^{\} h_{ν(ε)}^{−1}(0).

Proof. — By Lemma 2.4, L is trivial on a neighborhood Ω of M. For ε > 0 small, we have K_{ε} ⊂ Ω and we can apply Proposition 4.8 to get a uniform approximation h_{ν(ε)} ∈ H^{0}(X, L^{⊗ν(ε)}) of the function h = 1 such that |h_{ν(ε)}s^{−ν(ε)} − h| < 1/2 near K_{ε}. Then K_{ε} ∩ h_{ν(ε)}^{−1}(0) = ∅.

We now use Corollary 4.9 to prove Theorem 4.1:

Proof of Theorem 4.1. — Choose a domain Ω_{1} with Ω_{0} ⊂⊂ Ω_{1} ⊂⊂ Ω such that Ω_{1} is a Runge domain in Ω (and hence in S). By Corollary 4.9 applied to the compact set

K = holomorphic hull of f(\overline{Ω}_{1}) in the Stein variety f(Ω) ⊂ X,

we get an embedding j_{ε} : K → X of the form j_{ε}(z) = ψ(εg(z)), converging to Id_{K} as ε tends to 0, such that j_{ε}(K) is contained in an affine Zariski open subset Y_{ν,ε} = X ^{\} h_{ν,ε}^{−1}(0), h_{ν,ε} ∈ H^{0}(X, L^{⊗μ(ν,ε)}). Moreover, by Proposition 4.7, we can take j_{ε} tangent to Id_{K} at order k at all points t_{j}. We first approximate f by F_{ν} = j_{ε_{ν}} ∘ f on Ω_{1}, choosing ε_{ν} so small that sup_{Ω_{1}} δ(F_{ν}(z), f(z)) < 1/ν with respect to some fixed distance δ on X. We denote Y_{ν} = Y_{ν,ε_{ν}} in the sequel. By construction, F_{ν}(Ω_{1}) ⊂ j_{ε_{ν}} (K) ⊂ Y_{ν}.

Let Y_{ν} ⊂ C^{N} be an algebraic embedding of Y_{ν} in some affine space. Then F_{ν} can be viewed as a map

F_{ν} = (F_{1,ν}, …, F_{N,ν}) : Ω_{1} → C^{N}, F_{ν}(Ω_{1}) ⊂ Y_{ν}.

By Lemma 2.1, there is a retraction ρ_{ν} : V_{ν} → Y_{ν} defined on a neighborhood V_{ν} of Y_{ν}, such that the graph Γ_{ρ_{ν}} is contained in an algebraic variety A_{ν} with dim A_{ν} = N. Now, since Ω_{1} is a Runge domain in S, there are polynomial functions P_{k,ν} in the structure ring C[S], such that F_{k,ν} − P_{k,ν} is uniformly small on \overline{Ω}_{0} for each k = 1, …, N. Of course, we can make a further finite interpolation to ensure that P_{k,ν} − F_{k} vanishes at the prescribed order m at all points t_{j}. We set

P_{ν} = (P_{1,ν}, …, P_{N,ν}) : S → C^{N}, f_{ν} = ρ_{ν} ∘ P_{ν} : Ω_{0} → Y_{ν} ,

and take our approximations P_{k,ν} sufficiently close to F_{k,ν}, in such a way that f_{ν} is well defined on Ω_{0} (i.e. P_{ν}(Ω_{0}) ⊂ V_{ν}), is an embedding of Ω_{0} into Y_{ν}, and satisfies δ(f_{ν}, F_{ν}) < 1/ν on Ω_{0}. Then δ(f_{ν}, f) < 2/ν on Ω_{0}. Now, the graph of f_{ν} is just the submanifold

Γ_{f_{ν}} = (Ω_{0} × Y_{ν}) ∩ (P_{ν} × Id_{Y_{ν}})^{−1}(Γ_{ρ_{ν}}).

Each connected component of Γ_{f_{ν}} is contained in an irreducible component G_{ν,j} of dimension n = dim S of the algebraic variety (P_{ν} × Id_{Y_{ν}})^{−1}(A_{ν}), hence Γ_{f_{ν}} is contained in the n-dimensional algebraic variety G_{ν} = ⋃ G_{ν,j}.

Remark 4.10. — The existence of an ample line bundle on X with trivial restriction to M is automatic if H^{1}(M, O_{M}^{⋆}) = H^{2}(M, ZZ) = 0, since in that case every holomorphic line bundle on M is trivial. In Corollary 4.9, the conclusion that K admits small perturbations K_{ε} contained in affine Zariski open sets does not hold any longer if the hypothesis on the existence of an ample line bundle L with L_{|M} trivial is omitted. In fact, let M be a Stein manifold and let K be a holomorphically convex compact set such that the image of the restriction map H^{2}(M, ZZ) → H^{2}(K, ZZ) contains a nontorsion element. Then there exists a holomorphic line bundle E on M such that all positive powers E_{|K}^{⊗m} are topologically nontrivial. Let g_{0}, …, g_{N} ∈ H^{0}(M, E) be a collection of sections which generate the fibres of E and separate the points of M. By the proof of Corollary 2.6, we find a holomorphic embedding φ : M → IP^{N} into some projective space, such that E ≃ φ^{⋆}O(1). Since φ^{⋆}O(m)_{|K} ≃ E_{|K}^{⊗m}, we infer that O(m)_{|φ(K)} is topologically nontrivial for all m > 0. It follows that φ(K) cannot be contained in the complement IP^{N} ^{\} H of a hypersurface of degree m, because O(m) is trivial on these affine open sets. The same conclusion holds for arbitrary homotopic deformations φ(K)_{ε} of φ(K).

To give a specific example, let M = C^{⋆} × C^{⋆} and K = ∂Δ × ∂Δ. We consider the torus T = C/\{1, \sqrt{−1}\}ZZ, which has a holomorphic embedding g : T → IP^{2}. Let φ : M → IP^{8} be the holomorphic embedding given by φ(z, w) = ψ(g(t), (1 : z : w)) where ψ : IP^{2} × IP^{2} → IP^{8} is the Segre embedding and t = \frac{\sqrt{−1}}{2π} log z + \frac{1}{2π} log w. Since φ(K) is a C^{∞} deformation of the elliptic curve A = ψ(g(T) × {(1 : 0 : 0)}) ⊂ IP^{8}, then for any deformation φ(K)_{ε} of φ(K) and for any algebraic hypersurface H ⊂ IP^{8} we have φ(K)_{ε} · H = A · H > 0 (where · denotes the intersection product in the homology of IP^{8}), and therefore φ(K)_{ε} ∩ H ≠ ∅.

5. Exhaustion of Stein manifolds by Runge domains of affine algebraic manifolds

The methods developed in this paper are used in this section to give a new proof of Stout’s Theorem 1.6 by methods substantially different from those in Stout [St84] and in Tancredi-Tognoli [TT89]. These methods are then used to obtain Theorem 1.8 from Proposition 3.2. To prove Theorem 1.6, we first show the existence of proper embeddings of arbitrarily large Stein domains into affine algebraic manifolds, such that the normal bundle of the embedding is trivial. Secondly, when such an embedding is given, the embedded manifold is a global complete intersection, so it can be approximated through an approximation of its defining equations by polynomials.

We begin with an elementary lemma about Runge neighborhoods.

Lemma 5.1. — Let S be a closed submanifold of a Stein manifold Y. Then there is a fundamental system of neighborhoods Ω of S which are Runge open sets in Y.

Proof. — Let f_{1} = … = f_{N} = 0 be a finite system of defining equations of S in Y, and let u be a strictly plurisubharmonic smooth exhaustion function of Y. For every convex increasing function χ ∈ C^{∞}(IR), we set

Ω_{χ} = { ∑_{1≤j≤N} |f_{j}(z)|^{2}e^{χ(u(z))} < 1}.

Since Ω_{χ} is a sublevel set of a global psh function on Y, we infer that Ω_{χ} is a Runge domain in Y by [Hö66, Th. 5.2.8]. Clearly {Ω_{χ}} is a fundamental system of neighborhoods of S.

Lemma 5.2. — Let S be a Stein manifold. For any relatively compact Stein domain S_{0} ⊂⊂ S, there is a proper embedding j : S_{0} → A into an affine algebraic manifold, such that the image j(S_{0}) has trivial normal bundle N_{j(S_{0})} in A.

Proof. — By the Bishop-Narasimhan embedding theorem, we can suppose that S is a closed submanifold of an affine space C^{p}. Since any exact sequence of vector bundles on a Stein manifold splits, T_{S} ⊕ N_{S/C^{p}} is isomorphic to the trivial bundle T_{C^{p}|S} = S × C^{p}. Hence, to make the normal bundle of S become trivial, we need only to embed a Stein neighborhood Ω_{0} of S_{0} (in C^{p}) into an affine algebraic manifold A so that the restriction to S_{0} of the normal bundle of Ω_{0} in A is isomorphic to T_{S_{0}}.

For this, we choose by Lemma 2.1 a holomorphic retraction ρ : Ω → S from a neighborhood Ω of S in C^{p} onto S. We can suppose Ω to be a Runge open subset of C^{p}, by shrinking Ω if necessary (apply Lemma 5.1). Let Ω_{0} ⊂⊂ Ω be a Runge domain in C^{p} containing \overline{S}_{0}. By Proposition 3.2 applied to the holomorphic vector bundle ρ^{⋆}T_{S} on Ω, there is an open embedding φ : Ω_{0} → Z into a projective algebraic manifold Z, an algebraic vector bundle E → Z such that φ^{⋆}E ≃ (ρ^{⋆}T_{S})_{|Ω_{0}}, and an ample line bundle L → Z such that φ^{⋆}L is trivial. We consider the composition of embeddings

S ∩ Ω_{0} ⟶ Ω_{0} {φ \atop ⟶} Z ⟶ E ⟶ Ê,

where Z ↪ E is the zero section and Ê = P (E ⊕ C) is the compactification of E by the hyperplane at infinity in each fibre. As N_{Z/_{E}} ≃ E, we find

(N_{Z/_{E}})_{|S∩Ω_{0}} ≃ (φ^{⋆}E)_{|S∩Ω_{0}} ≃ (ρ^{⋆}T_{S})_{|S∩Ω_{0}} = (T_{S})_{|S∩Ω_{0}} ,
N_{S∩Ω_{0}/_{E}} = (N_{S/C^{p}})_{|S∩Ω_{0}} ⊕ (N_{Z/_{E}})_{|S∩Ω_{0}}
≃ (N_{S/C^{p}} ⊕ T_{S})_{|S∩Ω_{0}} ≃ (S ∩ Ω_{0}) × C^{p} .

Now, Ê is equipped with a canonical line bundle O_{E}(1), which is relatively ample with respect to the fibres of the projection π : Ê → Z. It follows that there is an integer m ≫ 0 such that L̂ := π^{⋆}L^{⊗m} ⊗ O_{E}(1) is ample on Ê. Now, the zero section of E embeds as P (0 ⊕ C) ⊂ P (E ⊕ C) = Ê, hence O_{E}(1)_{|Z} ≃ Z × C. Since φ^{⋆}L is trivial, we infer that L̂_{|S∩Ω_{0}} is also trivial. Apply Corollary 4.9 to X = Ê with the ample line bundle L̂ → Ê, and to some holomorphically convex compact set K in the submanifold S ∩ Ω_{0} ⊂ Ê, such that K ⊃ \overline{S}_{0}. We infer the existence of a holomorphic embedding i : S_{0} → Y into an affine Zariski open subset Y of Ê. By construction, this embedding is obtained from the graph of a section K → N_{S∩Ω_{0}/_{E}}, hence

N_{S_{0}/Y} ≃ (N_{S∩Ω_{0}/_{E}})_{|S_{0}} ≃ S_{0} × C^{p}.

It remains to show that we can find a proper embedding j satisfying this property. We simply set

j = (i, i′) : S_{0} ⟶ A = Y × C^{q},

where i′ : S_{0} → C^{q} is a proper embedding of S_{0} into an affine space. Clearly, j is proper. Moreover

N_{j(S_{0})/Y ×C^{q}} ≃ (T_{Y} ⊕ C^{q})/Im(di ⊕ di′)

with di being injective, hence we get an exact sequence

0 ⟶ C^{q} ⟶ N_{j(S_{0})/Y ×C^{q}} ⟶ N_{i(S_{0})/Y} ⟶ 0 .

As N_{i(S_{0})/Y} is trivial, we conclude that N_{j(S_{0})/Y ×C^{q}} is also trivial.

Lemma 5.3. — Let S_{0} be a closed Stein submanifold of an affine algebraic manifold A, such that N_{S_{0}/A} is trivial. For any Runge domain S_{1} ⊂⊂ S_{0} and any neighborhood V of S_{0} in A, there exist a Runge domain Ω in A with S_{0} ⊂ Ω ⊂ V, a holomorphic retraction ρ : Ω → S_{0}, and a closed algebraic submanifold Y ⊂ A with the following properties:

(i) Y ∩ ρ^{−1}(S_{1}) is a Runge open set in Y;

(ii) ρ maps Y ∩ ρ^{−1}(S_{1}) biholomorphically onto S_{1}.

Proof. — By Lemmas 2.1 and 5.1, we can find a Runge domain Ω in A with S_{0} ⊂ Ω ⊂ V and a holomorphic retraction ρ : Ω → S_{0}. Let I be the ideal sheaf of S_{0} in A. Then I/I^{2} is a coherent sheaf supported on S_{0}, and its restriction to S_{0} is isomorphic to the conormal bundle N_{S_{0}/A}^{⋆}. Since this bundle is trivial by assumption, we can find m = codim S_{0} global generators g_{1}, …, g_{m} of I/I^{2}, as well as liftings g̃_{1}, …, g̃_{m} ∈ H^{0}(A, I) (A is Stein, thus H^{1}(A, I^{2}) = 0). The system of equations g̃_{1} = … = g̃_{m} = 0 is a regular system of equations for S_{0} in a sufficiently small neighborhood. After shrinking Ω, we can suppose that

S_{0} = \{z ∈ \overline{Ω} : g̃_{1}(z) = … = g̃_{m}(z) = 0\}, dg̃_{1} ∧ … ∧ dg̃_{m} ≠ 0 on \overline{Ω}.

Let P_{j,ν} be a sequence of polynomial functions on A converging to g̃_{j} uniformly on every compact subset of A. By Sard’s theorem, we can suppose that the algebraic varieties Y_{ν} = ⋂_{1≤j≤m} P_{j,ν}^{−1}(0) are nonsingular (otherwise, this can be obtained by adding small generic constants ε_{j,ν} to P_{j,ν}). It is then clear that Y_{ν} ∩ Ω converges to S_{0} uniformly on all compact sets. In particular, the restriction

ρ_{ν} : Y_{ν} ∩ ρ^{−1}(S_{1}) ⟶ S_{1}

of ρ is a biholomorphism of Y_{ν} ∩ ρ^{−1}(S_{1}) onto S_{1} for ν large. This shows that S_{1} can be embedded as an open subset of Y_{ν} via ρ_{ν}^{−1}.

It remains to show that Y_{ν} ∩ρ^{−1}(S_{1}) is a Runge domain in Y_{ν} for ν large. Let h be a holomorphic function on Y_{ν}∩ρ^{−1}(S_{1}). Then h∘ρ_{ν}^{−1} is a holomorphic function on S_{1}. Since S_{1} is supposed to be a Runge domain in S_{0}, we have h ∘ ρ_{ν}^{−1} = lim h′_{k} where h′_{k} are holomorphic functions on S_{0}. Then h′_{k} ∘ ρ is holomorphic on Ω, and its restriction to Y_{ν} ∩ ρ^{−1}(S_{1}) converges to h uniformly on compact subsets. However, Ω is also a Runge domain in A, so we can find a holomorphic (or even algebraic) function h_{k} on A such that |h_{k} −h′_{k} ∘ρ| < 1/k on \overline{Y_{ν} ∩ ρ^{−1}(S_{1})}, which is compact in Ω for ν large. Then the restriction of h_{k} to Y_{ν} converges to h uniformly on all compact subsets of Y_{ν} ∩ ρ^{−1}(S_{1}), as desired.

Proof of Theorem 1.6. — Take a Stein domain S_{0} ⊂⊂ S such that Ω ⊂⊂ S_{0}. Then S_{1} = Ω is also a Runge domain in S_{0}, and Theorem 1.6 follows from the combination of Lemmas 5.2 and 5.3.

We can now use Theorem 1.6 and Lemmas 5.2 and 5.3 to infer Theorem 1.8 from Proposition 3.2.

Proof of Theorem 1.8. — By Theorem 1.6, we can take a Runge domain Ω̃ ⊂⊂ S such that Ω̃ ⊃⊃ Ω and embed Ω̃ as a Runge domain in an affine algebraic manifold M. By applying Proposition 3.2 with M playing the role of S, Ω̃ the role of Ω, and Ω the role of Ω_{0}, we obtain a projective algebraic manifold Z containing Ω and an algebraic vector bundle Ẽ → Z such that Ẽ_{|Ω} ≃ E_{|Ω}. Moreover, there is an ample line bundle L on Z such that L_{|Ω} is trivial.

Only one thing remains to be proved, namely that Ω can be taken to be a Runge subset in an affine open subset Z ^{\} D. (As mentioned earlier, this would be immediate if Problem 3.5 had a positive answer.) To reach this situation, we modify the construction as follows. Embed Ω as Ω × {0} ⊂ Z × IP^{1}. This embedding has trivial normal bundle and moreover L×O_{IP}^{1}(1)_{|Ω×{0}} is trivial. By Corollary 4.9, for every holomorphically convex compact set K ⊂ Ω, there is a holomorphic map g : K^{∘} → C ⊂ IP^{1} which is smooth up to the boundary, such that the graph of g is contained in an affine open set Z_{1} ^{\}D_{1}, where D_{1} is an ample divisor of Z_{1} := Z × IP^{1} in the linear system m|L×O_{IP}^{1}(1)|, m ≫ 0. Let Ω′ ⊂ K^{∘} be a Runge open subset of Ω and let j_{N} : Ω′ → C^{N} be a proper embedding. We consider the embedding

j′ = (Id_{Ω′}, g, j_{N}) : Ω′ ⟶ Z_{2} := Z × IP^{1} × IP^{N}.

Its image j′(Ω′) is a closed submanifold of the affine open set (Z_{1} ^{\} D_{1}) × C^{N} = Z_{2} ^{\} D_{2}, where D_{2} = (D_{1} × IP^{N}) + (Z_{1} × IP_{∞}^{N−1}) is ample in Z_{2}. Moreover, the normal bundle of j is trivial (by the same argument as at the end of the proof of Lemma 5.2). Let Ω′′ ⊂⊂ Ω′ be an arbitrary Runge domain in Ω′. By Lemma 5.3, we can find a nonsingular algebraic subvariety Z′′ ⊂ Z_{2} such that Ω′′ is biholomorphic to a Runge open set in Z′′ ^{\} (Z′′ ∩ D_{2}), via a retraction ρ from a small neighborhood of j′(Ω′) onto j′(Ω′). Let E_{2} = pr_{1}^{⋆}Ẽ → Z_{2} and let E′′ be the restriction of E_{2} to Z′′. Since ρ^{⋆}E_{2} ≃ E_{2} on a neighborhood of j′(Ω′) by Lemma 2.4, we infer that E′′_{|Ω′′} ≃ E_{|Ω′′}. Therefore the conclusions of Theorem 1.8

hold with Ω′′, Z′′, D′′ = Z′′ ∩ D_{2} and E′′ in place of Ω, Z, D and Ẽ. Since Ω′′ can be taken to be an arbitrary Runge open set in S, Theorem 1.8 is proved.

References

[AV65] Andreotti, A., Vesentini, E. — Carleman estimates for the Laplace-Beltrami equation in complex manifolds, Publ. Math. I.H.E.S., 25 (1965), 81-130.

[Bi61] Bishop, E. — Mappings of partially analytic spaces, Amer. J. Math., 83 (1961), 209-242.

[CG75] Cornalba, M., Griffiths, P. — Analytic cycles and vector bundles on noncompact algebraic varieties, Invent. Math., 28 (1975), 1-106.

[De90] Demailly, J.-P. — Cohomology of q-convex spaces in top degrees, Math. Zeitschrift, 203 (1990), 283-295.

[De92] Demailly, J.-P. — Singular Hermitian metrics on positive line bundles, Complex algebraic varieties, Lecture Notes in Math., Vol. 1507, Springer-Verlag, Berlin, 1992, 87-104.

[De94] Demailly, J.-P. — Algebraic criteria for the hyperbolicity of projective manifolds, manuscript in preparation.

[Dr82] van den Dries, L. — A specialization theorem for analytic functions on compact sets, Proc. Koninklijke Nederlandse Akademie van Wetenschappen, Ser. A, 85 (1982), 391-396.

[Ei70] Eisenman, D. — Intrinsic measures on complex manifolds and holomorphic mappings, Mem. Amer. Math. Soc., No. 96, (1970).

[EG92] Eliashberg, Y., Gromov, M. — Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n/2 + 1, Ann. of Math., 136 (1992), 123-135.

[Gr58] Grauert, H. — Analytische Faserungen über holomorphvollständigen Räumen, Math. Ann., 135 (1958), 263-273.

[GA74] Griffiths, P., Adams, J. — Topics in algebraic and analytic geometry, Princeton Univ. Press, Princeton, NJ, 1974.

[GR65] Gunning, R.C., Rossi, H. — Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, NJ, 1965.

[Hi64] Hironaka, H. — Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., 79 (1964), 109-326.

[Hö66] Hörmander, L. — An introduction to complex analysis in several variables, 1st edition, Elsevier Science Pub., New-York, 1966; 3rd revised edition, North-Holland Math. Library, Vol. 7, Amsterdam, 1990.

[Ko67] Kobayashi, S. — Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan, 19 (1967), 460-480.

[Ko70] Kobayashi, S. — Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970.

[La86] Lang, S. — Hyperbolic and diophantine analysis, Bull. Amer. Math. Soc., 14 (1986), 159-205.

[La87] Lang, S. — Introduction to complex hyperbolic spaces and diophantine analysis, Springer-Verlag, Berlin, 1987.

[Na60] Narasimhan, R. — Embedding of holomorphically complete spaces, Amer. J. Math., 82 (1960), 913-934.

[Nh52] Nash, J. — Real algebraic manifolds, Ann. of Math., 56 (1952), 405-421.

[NO90] Noguchi, J., Ochiai, T. — Geometric function theory, American Mathematical Society, Providence, RI, 1990.

[Oka37] Oka, K. — Sur les fonctions de plusieurs variables, II. Domaines d’holomorphie, J. Sci. Hiroshima Univ., 7 (1937), 115-130.

[Ri68] Richberg, R. — Stetige streng pseudokonvexe Funktionen, Math. Ann., 175 (1968), 257-286.

[Ro71] Royden, H. — Remarks on the Kobayashi metric, Proceed. Maryland Conference on Several Complex Variables, Lecture Notes in Math., Vol. 185, Springer-Verlag, Berlin, 1971.

[Sib85] Sibony, N. — Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J., 52 (1985), 157-197.

[Siu76] Siu, Y.-T. — Every Stein subvariety has a Stein neighborhood, Invent. Math., 38 (1976), 89-100.

[St84] Stout, E.L. — Algebraic domains in Stein manifolds, Proceedings of the Conference on Banach algebras and several complex variables (New Haven, 1983), Contemporary Math., Vol. 32, Amer. Math. Soc., Providence, RI, 1984.

[TT89] Tancredi, A., Tognoli, A. — Relative approximation theorems of Stein manifolds by Nash manifolds, Boll. Un. Mat. Ital., Serie-VII, 3-A (1989), 343-350.

[TT90] Tancredi, A., Tognoli, A. — On the extension of Nash functions, Math. Ann., 288 (1990), 595-604.

[TT93] Tancredi, A., Tognoli, A. — Some remarks on the classification of complex Nash vector bundles, Bull. Sci. Math, 117 (1993), 173-183.

[We35] Weil, A. — L’intégrale de Cauchy et les fonctions de plusieurs variables, Math. Ann., 111 (1935), 178-182.

[Wh65] Whitney, H. — Local properties of analytic varieties, Differential and combinatorial topology (a symposium in honor of Marston Morse), Princeton University Press, Princeton, NJ, 1965, 205-244.

[Yau78] Yau, S.-T. — A general Schwarz lemma for Kähler manifolds, Amer. J. Math., 100 (1978), 197-203.