Approximation of holomorphic maps with a lower bound on the rank

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$ is an integer satisfying $(n-r+1) (p-r+1)\geq 2$ then every holomorphic map from a neighborhood of $K$ to $Y$ with rank $\ge r$ at every point of $K$ can be approximated uniformly on $K$ by entire maps $\C^n\to Y$ with rank $\ge r$ at every point of $\C^n$.


Introduction
In this paper we consider the following problem of approximating holomorphic maps with a lower bound on their rank. Let K be a closed polydisc (or a closed ball) in a complex Euclidean space C n , and let f be a holomorphic map from an open neighborhood of K to a complex manifold Y such that rank z f := rank (df z ) ≥ r for all z ∈ K, where r is an integer satisfying 1 ≤ r ≤ min{n, dim Y }. Is it possible to approximate f uniformly on K by entire maps f : C n → Y satisfying rank z f ≥ r at every point z ∈ C n ?
The answer clearly depends on the complex analytic properties of Y . If Y is Kobayashi hyperbolic [17], this fails already when r = 1 and K is a disc in C. More generally, if Y is Eisenman k-hyperbolic for some 1 ≤ k ≤ dim Y [5] then Y admits no holomorphic maps C n → Y of rank ≥ k, and hence the answer is negative for r ≥ k. More precise quantitative obstructions to the existence of large polydiscs in complex manifolds were obtained by Kodaira [18].
In the positive direction, Forster proved that holomorphic maps C n → Y = C p satisfy the jet transversality theorem [6], which gives a positive answer if r is sufficiently small compared to n and p (see Theorem 1.4 below). If r = n < p, the above rank condition is satisfied by immersions C n → C p , and in this case an affirmative answer follows from the h-principle due to Eliashberg and Gromov [14]. If n > r = p, the rank condition is satisfied by submersions, and the approximation result follows from the h-principle proved by Forstnerič [9]. The Runge approximation problem for holomorphic immersions C n → C n in the equidimensional case is still open.
In this paper we consider maps to certain algebraic manifolds. We shall say that a p-dimensional complex manifold is of Class A 0 if it is quasi projective algebraic and is covered by finitely many Zariski open sets biregularly isomorphic to C p . Examples include all complex projective spaces and Grassmanians. Class A will consist of all algebraic manifolds of the form Y = Y \A where Y is a manifold of class A 0 and A is a closed algebraic subvariety of Y of complex codimension at least two. (See Def. 2.1 in §2.) The following is our main result. Theorem 1.1. Let K ⊂ C n be a closed polydisc, a closed ball, or a product of a (lower dimensional) closed polydisc and a ball. Let Y be a p-dimensional manifold of Class A. Assume that 1 ≤ r ≤ min{n, p}, and that r < n if n = p. Every holomorphic map f from a neighborhood of K to Y and satisfying rank z f ≥ r at every point z ∈ K can be approximated uniformly on K by entire maps C n → Y with rank ≥ r at every point of C n .
We wish to emphasize that Theorem 1.1 does not follow from the jet transversality theorem, except for values of r which are small compared to n and p; compare with Theorem 1.4 below. The following special case may be of particular interest; the analogous result for submersions (when n > dim Y ) was proved in [9]. Corollary 1.2. Let K ⊂ C n and Y be as in Theorem 1.1. If n < dim Y then every holomorphic immersion from a neighborhood of K to Y can be approximated uniformly on K by entire immersions C n → Y .
Manifolds of Class A were considered by Gromov under the name Ellregular manifolds [13, §3.5], and by Forstnerič [11, §2]; in those papers the reader can find many further examples. Both classes are stable with respect to blowing up points. Every such manifold Y enjoy the following properties which will play an important role in our proof: -holomorphic maps from (neighborhoods of) compact convex sets in C n to Y can be approximated by regular algebraic maps (morphisms) C n → Y [11, Corollary 1.2]; -algebraic maps from affine algebraic manifolds to Y enjoy a version of the jet transversality theorem (see [11,Sect. 5 (1) In [19, §5] Rosay and Rudin constructed discrete sets D ⊂ C n such that the only holomorphic map F : C n → C n with non degenerate Jacobian (JF ≡ 0) and satisfying F (C n \D) ⊂ C n \D is the identity map F (z) = z (z ∈ C n ); thus any holomorphic map F : C n → C n \D has rank F < n at each point.
(2) In [7, §6] a proper holomorphic embedding F : C m ֒→ C m+n is constructed for any pair of integers m, n ∈ N such that the image of any holomorphic map G : C n → C m+n satisfying rank z G = n at some point z ∈ C n intersects the submanifold A = F (C m ) ⊂ C m+n infinitely many times. It follows that any entire map C n → C m+n \A has rank < n at each point.
(3) According to Corollary 2 in [4] the complement P 2 \A of a very generic algebraic curve A of degree d ≥ 21 in the projective plane P 2 is hyperbolic, i.e., there exist no nonconstant holomorphic maps C → P 2 \A, and hence Theorem 1.1 fails for r = 1.
We now give another result whose main ingredient is the jet transversality theorem for holomorphic maps. In many analytic applications it is important to know the dimension of 'degeneration sets' of a generically chosen holomorphic map f : X → Y between a given pair of complex manifolds. Denote by H(X, Y ) the space of holomorphic maps X → Y equiped with the compact-open topology. By J k (X, Y ) we denote the manifold of all kjets of holomorphic maps X → Y . Given f ∈ H(X, Y ) and an integer r ∈ N we set Σ f,r = {x ∈ X : rank x f < r}.
We shall say that rank f ≥ r on a set K ⊂ X if rank x f ≥ r for all x ∈ K; if we do not specify K, it will be understood that K = X.
Following [10] we say that a complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K ⊂ C m (m ∈ N) to Y can be approximated uniformly on K by entire maps C m → Y . By the main result of [10] CAP is equivalent to the classical Oka property. Examples of complex manifolds with CAP include complex Lie groups and complex homogeneous spaces. Theorem 1.4. Let X be a Stein manifold and let Y be a complex manifold satisfying CAP. Let dim X = n, dim Y = p, and let r be an integer satisfying r ≤ min(n, p). Set d = (n − r + 1)(p − r + 1).   [16] that every holomorphic mapping f : X → Y from a Stein manifold X to any complex manifold Y can be approximated on any compact set K ⊂ X by holomorphic maps from a neighborhood of K to Y whose k-jet extension is transversal to a given analytic subset of the jet manifold J k (X, Y ). See also [11,Theorem 4.8]. This gives the following analogue of Theorem 1.4: Let X be a Stein manifold, and let Y be a complex manifold. Let n, p, r, d be as in Theorem 1.4. Given a compact set K ⊂ X and a holomorphic map f : X → Y , there is a holomorphic map f from an open neighborhood of K in X to Y which approximates f on K as close as desired and satisfies (2) if d > n then rank f ≥ r on K.

Preliminaries
Recall [12] that a projective algebraic set (or variety) is a closed subset of a complex projective space P n of the form where the p j 's are homogeneous holomorphic polynomials on C n+1 . Such A is a closed complex analytic subvariety of P n , and every closed complex analytic subvariety of P n is of this form by Chow's theorem [3, p. 74]. Occasionally we shall omit the adjective 'projective'. The topology on P n in which the closed sets are exactly the projective algebraic sets is called the Zariski topology on P n . A quasi-projective algebraic set is a difference Y \Y ′ of two closed algebraic subvarieties Y, Y ′ ⊂ P n . A (quasi) projective algebraic manifold is a (quasi-) projective algebraic set without singularities.
Let U ⊂ P n be a quasi-projective algebraic set. A function f : where P and Q are homogeneous polynomials on C n+1 of the same degree and Q(Z) = 0 for all Z ∈ U . A continuous map F : U → P N is a regular map if its components with respect to any affine chart C N ⊂ P N are regular functions on U ∩ F −1 (C N ). If U ′ ⊂ P N is another quasi-projective algebraic set then a bijective map F : U → U ′ is a biregular isomorphism if both F and F −1 are regular maps.
A is an algebraic set in Y with complex codimension at least two.
Manifolds of class A were used by Gromov under the name Ell-regular manifolds [13, §3.5]; our terminology conforms to the one by Forstnerič [11].
Example 2.2. Complex affine and projective space, as well as complex Grassmanians, are manifolds of Class A 0 . Further examples are the rational surfaces, i.e., complex surfaces birationally equivalent to P 2 [2, p. 244]. Apart from P 2 these include the Hirzebruch surfaces Σ n , n ∈ Z + .
We recall a few relevant notions regarding transversality of mappings.
we say that f is transverse to B, and denote it by f ⋔ B.
For latter application we state a couple of known transverality lemmas. The first one provides a lot of transversal maps to choose from a transversal family of maps; a proof consists of a reduction to Sard's theorem and can be found in [1] or [22].
The following lemma, together with Sard's theorem, implies a jet version of the Transversality Lemma for holomorphic maps. See [11,Lemma 4.5] or [16]; here we supply some additional details to the proof. Recall that J k (X, Y ) denotes the complex manifold of all k-jets of holomorphic maps X → Y between a pair of complex manifolds. Lemma 2.5. Let X be a Stein manifold of dimension r, embedded as a closed complex submanifold of C n , let Y be a complex manifold of dimension p, and let F : X × C N → Y be a holomorphic map such that for every x ∈ X the map F (x, · ) : C N → Y is a submersion at 0 ∈ C N . Let W denote the vector space of all holomorphic polynomial maps P : Proof. We need to prove that H is a submersion at points ( It is enough to show that H ′ is a submersion at (Φ(x), 0). Therefore we can assume that X = C r ⊂ C n and also Y = C p . Given x 0 ∈ C r we denote by W x 0 the set of all polynomials P ∈ W such that P (x 0 ) = 0. For a fixed x ∈ C r ⊂ C n the map W x → C M (r,N,k) , P → ∂ k x P (x), is a submersion. Here J k (C r , C N ) = C r × C N × C M (r,N,k) for some M (r, N, k) ∈ N and ∂ k x P (x) denotes all partial derivatives of P of order less or equal than k without the 0-th derivative. For every multiindex I = (i 1 , . . . , i r ) we can write Here R contains derivatives of P of order lower than |I| and derivatives of F . For a fixed x = x 0 we get where R(x 0 ) depends linearly on the components of j . We also see that H(x 0 , P ) is a block-wise lower triangular linear map in the base is a submersion, and hence H is a submersion at (x 0 , 0). Definition 2.6. Let X and Y be complex manifolds. Holomorphic maps X → Y satisfy Condition Ell 1 if for every map f ∈ H(X, Y ) there is a holomorphic map H : X × C N → Y for some N ∈ N, satisfying (1) H(x, 0) = f (x) for all x ∈ X, and (2) the map H(x, · ) : C N → Y is a submersion at 0 ∈ C N for every x ∈ X.
The Ell 1 condition is useful when combined with the (Jet) Transversality Lemma in approximating a given holomorphic map by a holomorphic map transversal to a given submanifold. Condition Ell 1 holds for holomorphic maps from any Stein manifold to any complex manifold Y which enjoys the CAP property [11,Proposition 4.6 (b)]. In short the idea is to construct a finite collection of sprays on Y using bundles described in Lemma 2.10 and combining them into map H from the definition of Ell 1 . In particular, Ell 1 holds for maps of Stein manifolds to manifolds of Class A since these enjoy the CAP property.
The following result from [11] follows from Lemma 2.5 and Sard's theorem.
Lemma 2.7. ([11, Theorem 4.2]) Let X be a Stein manifold and let Y be a complex manifold such that holomorphic maps X → Y satisfy Condition Ell 1 . Choose a distance function d on Y . Let Z be a closed complex submanifold (or a closed complex subvariety) in J k (X, Y ). Given a compact set K ⊂ X, a holomorphic map f : X → Y and an ǫ > 0, there is a holomorphic map f 1 : X → Y such that We will need the following lemma which was also used in the proof of Proposition 2 in [6]. Proof. Let A ∈ M r (n, m). Change bases in C n and C m such that A takes fixed, this is just a translation, and therefore Φ is a submersion.
To conclude the proof it now suffices to show M r (n, m) ∩ U = Φ −1 (0).
which has rank r if and only if E ′ − D ′ B ′−1 C ′ = 0, and this is equivalent to A ′ ∈ Φ −1 (0). Definition 2.9 (Spray on a manifold). A spray on a complex manifold X is a holomorphic map s : E → X from total space of a holomorphic vector bundle p : E → X satisfying s(0 x ) = x for all x ∈ X. The spray is algebraic if p : E → X is an algebraic vector bundle and s : E → X is algebraic map.
The following lemma is due to Gromov [13] (Lemmas 3.5B and 3.5C); see also [8,Lemma 1.3]. Here we supply additional details of the proof.
Proof. We can't just extend s to Y × C n because of the singularities on Λ. However, we will show that any point y ∈ Λ admits a Zariski neighborhood C n ≃ V ⊂ Y such that s extends to E| V for m ∈ N large enough. Since Y = ∪ r j=1 U j for Zariski open sets U j biregularly isomorphic to C n , with U 1 = U , we will get the desired extension by choosing the largest m.
Let ϕ j : U j → C n , 1 ≤ j ≤ r be biregular isomorphisms; the collection {(U j , ϕ j ) : 1 ≤ j ≤ r} is then an algebraic atlas on Y . Choose y 0 ∈ Λ\U ; without loss or generality we may assume that y 0 ∈ U 2 and ϕ 2 (y 0 ) = 0. Recall that the spray s is given in the local chart U 1 × C n on Y × C n by s ′ 1 (z, t) = z + t. In the local chart U 2 × C n the same spray is of the form 2 )} and has singularities in the complement. In particular, s ′ 2 is holomorphic at all points (z, 0) with z ∈ ϕ 2 (U 1,2 ). Since U 2 \U 1 ⊂ P n \U 1 ⊂ Λ, we have Ω := C n \ϕ 2 (Λ ∩ U 2 ) ⊂ ϕ 2 (U 1,2 ) and hence s ′ 2 is holomorphic on a neighborhood of Ω × {0}.
For a fixed z ∈ Ω we can write s ′ 2 (z, t) = z + ∞ |α|=1 f α (z)t α , where α is a multiindex and f α are matrices with rational functions as elements. Note that the transition maps of the bundle E → Y are Φ ij : U i,j ×C n → U i,j ×C n where Φ ij (y, t) = (y, (b j (y)/b i (y)) m t). Here b j is a regular defining function for Λ ∩ U j and b i is a regular defining function for Λ ∪ U i . The bundle E| U 1 is trivial and can be identified with U 1 × C n . Denote bys ′ 2 the map s in the local chart U 2 × C n on E. Theñ . For z ∈ U 2 ∩ ϕ 2 (U 1,2 ) this can be written as using the above series expansion for s ′ 2 . By the Cauchy formula for the coefficients of a power series for the rational map s ′ 2 , holomorphic on a neighborhood of (z, 0), the maximum of the degrees of the poles of f α (z) is bounded by some integer which is independent of z and α. Hence there is m ∈ N such that b 2 (ϕ −1 2 (z)) m f α (z) is holomorphic on C n and equals zero when z ∈ ϕ 2 (Λ ∩ U 2 ), |α| ∈ N. This shows that for such m the maps is holomorphic on E at points 0 ∈ E y , y ∈ Λ. For other points t ∈ C n we can still write s ′ 2 as power series, since the intersection of the singular set of s ′ 2 with {z} × C n ≡ C n is nowhere dense in C n . Furthermore, because of the factors b 2 (ϕ −1 2 (z)) m we can extends ′ 2 to a continuous, locally bounded map on a neighborhood of hypersurface ϕ 2 (Λ ∩ U 2 ) × C n withs ′ 2 (z, t) = z for z ∈ ϕ 2 (Λ ∩ U 2 ), t ∈ C n . By Riemann extension theorems ′ 2 extends to a holomorphic map on a neighborhood of ϕ 2 (Λ ∩ U 2 ) × C n .

Proofs of main theorems
The following is Lemma 3.4 in [9, p. 156] with r = n − 2, s = 2 and D × L instead of L.
Lemma 3.1. Let K ⊂ C n be a product of a closed polydisc and a ball, and let Σ ⊂ C n \K be an algebraic set with dim Σ ≤ n − 2. Let D = π(K), where π : C n−2 × C 2 → C n−2 is a standard projection and L ⊂ C 2 is a compact polydisc such that K ⊂ D × L. Given ǫ > 0 there exists an automorphism Ψ of C n of the form Ψ(z ′ , z ′′ ) = (z ′ , ψ(z ′ , z ′′ )) (z ′ ∈ C n−2 , z ′′ ∈ C 2 ) such that The following lemma is the main ingredient in the proof of Theorem 1.1.
Lemma 3.2. Let Y be a manifold of class A with dim Y = p. Choose a distance function d on Y . Let K ⊂ C n = C n−2 × C 2 , L ⊂ C 2 , D ⊂ C n−2 be as in Lemma 3.1. Let r ∈ N satisfy (n − r + 1)(p − r + 1) ≥ 2. Given a holomorphic map f : K → Y satisfying rankf ≥ r on K and an ǫ > 0, there exists an algebraic mapf : D × L → Y such that (i) d(f (z), f (z)) < ǫ for all z ∈ K, and (ii) rankf ≥ r on D × L.
Proof. By Corollary 3.2 in [11] we can approximate the map f : K → Y with an algebraic map C n → Y . So we can assume that f is algebraic, defined on whole C n and with rank f ≥ r on K. Let Σ f,r = {z ∈ C n : rank z f < r}. Then Σ f,r ∩ K = ∅ provided the above approximation was good enough.
If dim Σ f,r ≤ n − 2, Lemma 3.1 furnishes an automorphism Ψ of C n which approximates the identity on K and satisfies Ψ(D ×L) ⊂ C n \Σ f,r . The map we are looking for isf = f • Ψ. Now suppose that dim Σ f,r = n − 1. We will reduce this to the previous case dim Σ f = n − 2. This reduction is similiar to the one used in the proof of Proposition 5.4 in [11]. By the definition of a Class A manifold we have Y = Y \A where Y is a manifold of Class A 0 and A an algebraic subset of codimension at least two in Y . We will approximate f on D × L by an algebraic map f 0 : C n → Y such that dim Σ f 0 ,r ≤ n − 2. By approximating well enough we will also get f 0 (D × L) ⊂ Y \A. In each step of the approximation a given map f will be replaced by a nearby algebraic map f 1 such that the corresponding set Σ f 1 ,r ⊂ C n has less n−1 dimensional irreducible components than Σ f . Choose a point z 0 ∈ Σ f belonging to exactly one (n − 1)-dimensional irreducible component Σ ′ of Σ f,r . Set y 0 = f (z 0 ). By the definition of a class A manifold there is a Zariski open neigborhood U of y 0 in Y which is biregularly isomorphic to C p , where p = dim Y . Hence there is a biregular isomorphism ϕ : U = Y \Λ → C n = P n \H where H is the plane at infinity in P n . Since ϕ has poles at Λ it can be viewed as a holomorphic map Y → P n . Let ϕ(z 0 ) ∈ V ⊂ P n where V ≡ C n . There is a polynomial q defined on V which vanishes on ϕ(Λ) ∩ V but q(ϕ(y 0 )) = 0. The closure of the zero locus of q in P n is an algebraic set. Denote by Λ its ϕ-preimage. Then Λ is an algebraic set in Y of pure dimension p − 1 such that Λ ∪ U = Y and y 0 ∈ Λ.
Let L be a holomorphic line bundle over Y defined by the divisor of Λ. Using Lemma 2.10 we get a spray s : Since f is algebraic, f * E is an algebraic vector bundle over C n . By Serre's theorem A [20] the bundle f * E is generated by finitely many algebraic sections, and hence there is a surjective algebraic vector bundle map g : C n × C q → f * E for some q ∈ N. We can write g(z, t) = q j=1 g j (z)t j where g j : C n → f * E are sections. Set Φ = s • ι • g, Λ = f −1 ( Λ), V = C n \Λ. From the above statements it is easy to conclude the following properties of the algebraic bundle map H : C n × C q → Y : -H(z, 0) = f (z) for z ∈ C n , -H(z, t) = z for z ∈ Λ and t ∈ C q , and -H(z, ·) is a submersion on C q for every z ∈ V .
Let W denote the space of all quadratic polynomial maps C n → C q . Set f P (z) = H(z, P (z)) for P ∈ W . By Lemma 2.8 the set Z j = {(z, y, α) ∈ J 1 (C n , Y ) : rank α = j} is a submanifold of J 1 (C n , Y ) of codimension (n − j)(p − j) and Z = ∪ r−1 j=0 Z j is a closed subvariety of codimension (n − r + 1)(p−r +1) ≥ 2. By Lemma 2.5 for every P in some open dense subset of W we get j 1 f P ⋔ Z . If we choose P close to 0 from subset of polynomials with up to first degree terms equal zero, we can conclude the following about the map f P : C n → Y : (1) f P (z) = f (z), df P (z) = df (z) for z ∈ Λ, (2) f P approximates f on D × L, and (3) the algebraic set Σ f P ,r = (j 1 f P ) −1 (Z) has dimension less than n − 1 at every point of V .
By (3) the only remaining irreducible n − 1 dimensional components of f P are those lying in Λ, where they are equal those of Σ f,r ∩ Λ. Hence the number of n − 1 dimensional components intersecting with Λ has not increased. At least one component Σ ′ from Σ f,r is missing since z 0 / ∈ Λ. Set Σ 1 = Σ f P ,r , f 1 = f P . The map f 1 : C n \Σ 1 → Y has rank ≥ r and Σ 1 has less (n − 1)-dimensional components than Σ f,r . By repeating this procedure we obtain in finitely many steps the desired algebraic map f 0 with dim Σ f 0 ≤ n − 2.
Lemma 3.3. Let Y be a manifold of class A with dim Y = p. Let K ⊂ C n be a product of a closed ball and a closed polydisc, and let Q ⊂ C n be a closed polydisc containing K. Let r be such that (n − r + 1)(p − r + 1) ≥ 2. Every holomorphic map f : K → Y with rank f ≥ r on K can be approximated unikformly on K by algebraic mapsf : Q → Y satisfying rankf ≥ r on Q.
Proof. If n is even, we write C n = C 2 × · · · × C 2 (n/2 factors) and let π j : C n → C n−2 be the projection whose kernel is the j-th factor. Let Q = L 1 × · · · × L m where L j ⊂ C 2 are polydiscs. Using Lemma 3.2 we approximate f = f 0 on K = K 0 by f 1 : L 1 × π 1 (K) → Y . Set K 1 = L 1 ×π 1 (K) and approximate f 1 on K 1 by f 2 : L 2 ×π 2 (K 1 ) → Y using Lemma 3.2 (for purposes of shorter notation the coordinates have been permuted). By continuing in this fashion we get the desired approximation in m = n/2 steps. In the case of odd n we use an extra disc.
Proof of Theorem 1.1. By the definition of a class A manifold we have Y = Y \A, where Y is a manifold of Class A 0 and A is a closed algebraic subset of Y of codimension at least two. Choose a distance function d on Y induced by a complete Riemannian metric.
Suppose that K is a closed polydisc in C n . Choose an exhaustion of C n with closed polydiscs Q j , j ∈ Z + , where Q 0 = K ⊂ Q 1 .
By Corollary 3.2 in [11] we can approximate f uniformly on K by an algebraic map f 0 : C n → Y .
By Lemma 2.7 we can approximate f 0 uniformly on Q 0 by a holomorphic map transversal to A on Q 0 , and hence we can assume that f 0 ⋔ A on Q 0 . Choose a positive number δ 0 > 0 such that every holomorphic map g : C n → Y with d(g(z), f 0 (z)) < δ 0 for all z ∈ Q 1 satisfies rankg ≥ r on Q 0 and g ⋔ A on Q 0 . Using Lemma 3.3 and Corollary 3.2 in [11] we get a holomorphic map f 1 : C n → Y satisying rankf 1 ≥ r on Q 1 and d(f 1 (z), f 0 (z)) < min(ǫ/2, δ 0 /2) for all z ∈ Q 0 .
Proceeding inductively we get a sequence of holomorphic maps f j : C n → Y and a decreasing sequence of positive numbers δ j > 0 satisfying the following: (i) d(f j+1 (z), f j (z)) < min(ǫ/2 j+1 , δ j /2) for all z ∈ Q j and j ≥ 0, (ii) rank f j ≥ r on Q j and f j ⋔ A on Q j , and (iii) every holomorphic map g : C n → Y with d(g(z), f j (z)) < δ j for all z ∈ Q j+1 satisfies rank g ≥ r on Q j and g ⋔ A on Q j .
The sequence of holomorphic maps f j converges uniformly on the compacts in C n to a holomorphic map F : C n → Y satisfying d(F (z), f (z)) < ǫ for all z ∈ Q 0 = K (a consequence of (i)) and d(F (z), f j (z)) < δ j for all z ∈ Q j and j ≥ 0 (because of (ii) and the definition of the numbers δ j ).
This implies F ⋔ A on C n and rankF ≥ r on C n . To show that F (C n ) ⊂ Y = Y \A suppose F (z) ∈ A for some z ∈ C n . The transversality condition F ⋔ A implies rank z F + dim z A ≥ p = dim Y . This and F ⋔ z A implies g(U ) ∩ A = ∅ for all holomorphic maps g : U → Y close enough to F on some neighborhood U of z in C n . Since f j : C n → Y for all j ≥ 0, we have a contradiction. This completes the proof of Theorem 1.1.
Choose an exhaustion of X by compact sets K l , l ∈ N, and let Ω l = {f ∈ H(X, Y ) : j 1 f ⋔ Z on K l }.
Then Ω = ∩ l∈N Ω l . Since H(X, Y ) is a Baire space, it is enough to show that Ω l is open and dense in H(X, Y ). Openess follows directly from the definition of transversality and the fact that Z is closed (if we perturb f on neighborhood of K a little, transversality condition will still be satisfied on K by Cauchy inequality for derivatives). To prove density choose g ∈ H(X, Y ) and a compact subset L ⊂ X. Using Lemma 2.7 we get a holomorphic map f which approximates g on L ∪ K l and satisfies the transversality condition in the definition of Ω l .
Case (2): The assumed inequality implies dim X +dim Z < dim J 1 (X, Y ). Therefore j 1 f ⋔ Z implies j 1 f (X) ∩ Z = ∅ which is equivalent to rank f ≥ r on X. The set of such f is open and dense by the proof of case (1).