Proper holomorphic embeddings with small limit sets

Let $X$ be a Stein manifold of dimension $n\ge 1$. Given a continuous positive increasing function $h$ on $\mathbb R_+=[0,\infty)$ with $\lim_{t\to\infty} h(t)=\infty$, we construct a proper holomorphic embedding $f=(z,w):X\hookrightarrow \mathbb C^{n+1}\times \mathbb C^n$ satisfying $|w(x)|<h(|z(x)|)$ for all $x\in X$. In particular, $f$ may be chosen such that its limit set at infinity is a linearly embedded copy of $\mathbb{CP}^n$ in $\mathbb{CP}^{2n}$.


The main result
A theorem of Remmert [23], Narasimhan [21], and Bishop [4] states that every Stein manifold X of dimension n ≥ 1 admits a proper holomorphic map to C n+1 , a proper holomorphic immersion to C 2n , and a proper holomorphic embedding in C 2n+1 .(See also [18, Chap.VII.C].)We are interested in the question how much space proper holomorphic embeddings or immersions X → C N need, and how small can their limit sets at infinity be.
By Remmert [22], the image A = f (X) ⊂ C N of a proper holomorphic map f : X → C N is a closed complex subvariety of pure dimension n = dim X.Such an A is algebraic if and only if it is contained, after a unitary change of coordinates on C N , in a domain of the form for some C > 0 (see Chirka [5,Theorem 2,p. 77]).Equivalently, if H = CP N \ C N ∼ = CP N −1 denotes the hyperplane at infinity and A ∞ = A ∩ H, where A is the topological closure of A in CP N , then A is algebraic if and only if there is a linear subspace L ∼ = CP N −n−1 of H ∼ = CP N −1 such that L ∩ A ∞ = ∅.If this holds then A and A ∞ are algebraic subvarieties of pure dimension n and n − 1, respectively.If X is not algebraic then the image of any proper holomorphic immersion f : X → C N is not algebraic either, so its limit set f (X) ∞ ⊂ CP N −1 has a nonempty intersection with every linear subspace We construct proper holomorphic embeddings with images in small Hartogs domains.
Furthermore, given a compact O(X)-convex set K in X, an open neighbourhood U ⊂ X of K, and a holomorphic map f 0 = (z 0 , w 0 ) : U → C n+1 × C p satisfying (1.1) for all x ∈ K, we can approximate f 0 uniformly on K by a proper holomorphic embedding

1).
The function h in Theorem 1.1 can be chosen to grow arbitrarily slowly, and hence the image f (X) may be arbitrarily close to the subspace C n+1 × {0} n in the Fubini-Study metric on CP 2n+1 .Choosing h such that lim t→∞ h(t)/t = 0 gives the following corollary.
Corollary 1.2.Every Stein manifold X of dimension n ≥ 1 admits a proper holomorphic embedding f : X ֒→ C 2n+1 whose limit set f (X) ∞ = f (X) ∩ H is a linearly embedded copy of CP n in H = CP 2n+1 \ C 2n+1 ∼ = CP 2n .In particular, every open Riemann surface X admits a proper holomorphic embedding in C 3 whose limit set is a projective line CP 1 ⊂ CP 2 .The analogous result holds for proper holomorphic immersions X → C 2n .
By the preceding discussion, the limit set f (X) ∞ intersects every projective subspace L ⊂ CP N −1 of dimension N − n − 1, unless f (X) is algebraic.Therefore, the nonalgebraic embeddings given by Corollary 1.2 have the smallest possible limit sets.
Given a nonalgebraic complex subvariety X of C N , its closure X ⊂ CP N and the limit set X ∞ ⊂ CP N −1 need not be analytic subvarieties, and for any pair of integers 1 ≤ n < N there are n-dimensional closed complex submanifolds X ⊂ C N with X ∞ = CP N −1 .(This always holds if N = n + 1 and X is nonalgebraic.)Indeed, if X is a closed complex subvariety of C N (N > 1) then for any closed discrete set B = {b j } j∈N ⊂ C N there exist a domain Ω ⊂ C N containing X and a biholomorphic map Φ : Ω → C N such that B ⊂ Φ(X) (see [13,Theorem 6.1] or [14, Theorem 4.17.1 (i)]).Note that X ′ = Φ(X) is a closed complex subvariety of C N .Choosing B such that its closure in CP N contains the hyperplane at infinity implies X ′ ∞ = CP N −1 .A characterization of the closed subsets of CP N −1 which are limit sets of closed complex subvarieties of C N of a given dimension does not seem to be known.

The corollary is especially interesting in dimension n = 1.
A long standing open question (the Forster conjecture [11], also called the Bell-Narasimhan conjecture [3,2]) asks whether every open Riemann surface, X, admits a proper holomorphic embedding in C 2 .Recent surveys of this subject can be found in [14, Secs.9.10-9.11]and the preprint [1]  It was recently shown by Drinovec Drnovšek and Forstnerič [8,Theorem 1.3] that, under a mild condition on an unbounded closed convex set E ⊂ C N , proper holomorphic embeddings f : X ֒→ C N from any Stein manifold X with 2 dim X < N such that f (X) ⊂ Ω = C N \ E are dense in the space O(X, Ω) of all holomorphic maps X → Ω.A similar result holds for immersions if 2 dim X ≤ N .Their proof relies on the fact, proved by Forstnerič and Wold [17], that such Ω is an Oka domain.(See [14, Definition 5.4.1 and Theorem 5.4.4] for the definition and the main result concerning Oka manifolds.)Note that the domains in Theorem 1.1 are much smaller than those in [8, Theorem 1.3] when the codimension is at least 2. On the other hand, Theorem 1.1 does not pertain to proper maps in codimension 1 (the case p = 0).We do not know whether a Hartogs domain of the form which appears in Theorem 1.1, is an Oka domain, except if p = 1, the function h > 0 on R + grows at least linearly at infinity, and log h(|z|) is plurisubharmonic on C n+1 (see Forstnerič and Kusakabe [15, Proposition 3.1]).Our proof does not require the Oka property of Ω in (1.2).
The proofs are very delicate and depend on Oka theory.We do not know whether one can expect a similar control of the range of the embedding in these dimensions.

Proof of Theorem 1.1
Our proof of Theorem 1.1 relies on the following technical result, which is a special case of [9, Theorem 1.1] by Drinovec Drnovšek and Forstnerič.(See also [16,Theorem 6], which is based on the same result.)Similar results were obtained earlier by Dor [6,7].
Theorem 2.1.Assume that X is a Stein manifold of dimension n ≥ 1, D is a relatively compact, smoothly bounded, strongly pseudoconvex domain in X, K is a compact set contained in D, t 0 is a real number, σ : C n+1 → R is a strongly plurisubharmonic exhaustion function which has no critical points in the set {σ ≥ t 0 }, and g 0 : D → C n+1 is a continuous map that is holomorphic in D and satisfies g 0 D \ K ⊂ {σ > t 0 }.Given numbers t 1 > t 0 and ǫ > 0, there is a holomorphic map g : D → C n+1 satisfying the following conditions: The analogous result holds much more generally, and we only stated the case that will be used here.For condition (b), see [9, Lemma 5.3], which is the main inductive step in [9, proof of Theorem 1.1].We remark that a map from a compact set in a complex manifold is said to be holomorphic if it is holomorphic in an open neighbourhood of the said set.
Let Ω ⊂ C N be a domain of the form (1.2) with coordinates (z, w) ∈ C n+1 × C p where p ≥ n, N = n + 1 + p, and the function h : [0, ∞) → (0, ∞) is as in the theorem.We shall use Theorem 2.1 with the exhaustion function σ(z) = |z| on C n+1 ; the nonsmooth point at the origin will not matter.We denote by B the open unit ball in C n+1 .
Since the set K ⊂ X is compact and O(X)-convex, there exist a smooth strongly plurisubharmonic Morse exhaustion function ρ : X → R + and a sequence 0 < c 0 < c 1 < • • • with lim i→∞ c i = +∞ such that every c i is a regular value of ρ and, setting where U ⊂ X is a neighbourhood of K as in the theorem (see [19,Theorem 5.1.6,p. 117]).Note that the set D i is O(X)-convex for every i = −1, 0, 1, . ... We may assume that the given holomorphic map f 0 = (z 0 , w 0 ) : U → Ω satisfies condition (1.1) for all x ∈ D 0 and z 0 (x) = 0 for x ∈ bD 0 .(We shall use the subscript in z i and w i as an index in the induction process; a notation for the components of these maps will not be needed.)Pick a number t 0 ∈ R with 0 < t 0 < inf Choose a sublevel set By the Oka-Weil theorem, we may approximate the map w 0 : U → C p uniformly on D 0 by a holomorphic map w 1 : X → C n such that (z 0 , w 1 )(D 0 ) ⊂ Ω.
We shall now construct a holomorphic map z 1 : D 1 → C n+1 such that the holomorphic map f 1 = (z 1 , w 1 ) : D 1 ֒→ Ω enjoys suitable properties to be explained in the sequel.This will be the first step of an induction procedure.
Pick a number t 1 ≥ t 0 + 1 so big that (2.1) (Such a number exists since lim t→∞ h(t) = +∞.)Fix ǫ > 0 whose precise value will be determined later.Let z0 : D 0 → C n+1 be a holomorphic map given by Theorem 2.1 (with z0 = g in the notation of that theorem, applied to the map g 0 = z 0 , the compact set D −1 ⊂ D 0 , and the numbers ǫ and t 0 < t 1 ).Condition (b) in Theorem 2.1 gives Since the function h in (1.2) is continuous, it follows that if ǫ > 0 is small enough then the map (z 0 , w 1 ) : D 0 → C N has range in Ω, and we have that We now use the fact that C n+1 \ t 1 B is an Oka domain (see Kusakabe [20,Corollary 1.3]).Hence, the main result of Oka theory gives a holomorphic map z 1 : (See [12,Theorem 1.3] for a precise statement of a more general result.In the special case at hand, this was proved by a more involved argument in the paper [16] by Forstnerič and Ritter, predating Kusakabe's work [20].)If the number ǫ > 0 is chosen small enough, it follows from (2.1)-(2.3)and the definition of Ω (1.2) that (2.4) Since the dimension of the target space C N is at least 2 dim X + 1, we may assume after a small perturbation that the map f 1 = (z 1 , w 1 ) : D 1 → Ω is an embedding satisfying the above conditions (see [14,Corollary 8.9.3]).Assuming as we may that all approximations are close enough, we also have that |f 1 − f 0 | < ǫ 0 on D −1 for a given ǫ 0 > 0.
Continuing inductively, we obtain an increasing sequence and a sequence of holomorphic embeddings f i = (z i , w i ) : D i ֒→ C 2n+1 satisfying the following conditions for i = 1, 2, . ...
Note that the conditions (i) and (ii) also holds for i = 0 by the assumptions on f 0 , and the conditions (i)-(v) hold for i = 1 by the construction of the map f 1 .
The inductive step is similar to the one from i = 0 to i = 1, which was explained above.Assume inductively that conditions (i)-(v) hold for some i ∈ {1, 2, . ..}. Pick a number ǫ i satisfying conditions (vi) and (vii).Also, fix a number ǫ > 0 whose precise value will be determined during this induction step.By the Oka-Weil theorem, there is a holomorphic map If ǫ > 0 is chosen small enough then Theorem 2.1, applied to the map g 0 = z i : (This is an analogue of the condition (2.3).)Finally, we perturb the holomorphic map slightly to make it an embedding.If all approximations are close enough then f i+1 satisfies conditions (i)-(iv), and (v) holds by the choice of t i+1 .This completes the induction step.
Conditions (iv) and (vi) imply that the sequence f i converges to the limit map f = (z, w) = lim i→∞ f i : X → C N satisfying |f − f i | < 2ǫ i on D i−1 for every i = 0, 1, . ... (In particular, we have that |f − f 0 | < 2ǫ 0 on K.) Conditions (i) and (vii) then imply that f is a holomorphic embedding with f (X) ⊂ Ω.Finally, conditions (ii)-(iv) and (vi) imply that the map z : X → C n+1 is proper, and hence f is proper as map to C N .
by Alarcón and López, where the authors constructed a proper harmonic embedding of any open Riemann surface in C × R 2 ∼ = C 2 with a holomorphic first coordinate function.Note that if X → C 2 is a proper holomorphic map with nonalgebraic image then f (X) ∞ = CP 1 .(There are algebraic open Riemann surfaces which do not embed as smooth proper affine curves in C 2 .)Corollary 1.2 gives proper holomorphic embeddings f : X ֒→ C 3 whose images are arbitrarily close to the subspace C 2 × {0} in the Fubini-Study metric on CP 3 , and f (X) ∞ = CP 1 .