Non straightenable complex lines in C 2

(1)

Ark. Mat., 34 (1996), 97-101

Non straightenable complex lines in C 2

Franc Forstneric(1), Josip Globevnik(2) and Jean-Pierre Rosay(1)

Abhyankar and Mob, in [1, (1.6), p. 151], and M. Suzuki, in [11, w proved that if P is a polynomial embedding of C into C 2, then there exists ~b, a polynomial automorphism of C 2, such that ( ~ b o P ) ( C ) = C x {0}. The corresponding and much easier result has been proved for polynomial embeddings of C into C '~, for n > 4 by Z. Jelonek [7] (more generally, Jelonek treats the case of embeddings of C k into C ~, for n > 2 k + 2 ) . The case of polynomial embeddings of C into C 3 seems open. See also

[8].

The main goal of this paper is to show that the above results do not gener- alize to holomorphic embeddings of C. Another goal is an interpolation theorem (Proposition 2 below).

P r o p o s i t i o n 1. Let n > l . There exists a proper holomorphic embedding H: C - ~ C n such that for no automorphism ~ of C ~, ( ~ o H ) ( C ) = C x {0} c C n.

Notice however that it has been proved in [4, (4.1)] that for every R > 0 and s > 0 there exists ~b, an automorphism of C ~, such that I ( r 1 6 2 for every E C, Ir R. So, compact subsets of the complex line H ( C ) can be "approximately straightened".

The above proposition has been known for some time, for n > 3 . In [4, (7.8)]

this is pointed out as being (non stated but) clear in [10]. For n = 2 , we keep the same approach as in [10]. But we can now take advantage of the ground breaking work by Anders@n and Lempert [2], as further developed in [4].

. P r o o f o f P r o p o s i t i o n 1

Proposition 1 is an immediate consequence of the following two propositions:

(1) Partially supported by NSF grant.

(2) Partially supported by a grant from the Ministry of Science of the Republic of Slovenia

(2)

98 Franc Forstneric, Josip Globevnik and Jean-Pierre Rosay

P r o p o s i t i o n 2. Let n > l . Let (OLj)jE N be a discrete sequence in C n, (i.e.

I ~ j ] - - ~ - ~ as j--*co). There exists H: C--+C n, a proper holomorphic embedding of C into C n, such that a j E H ( C ) for every j E N .

The case n > 2 is t r e a t e d in [10] (where in addition it is shown that, for n > 2 , the preimages of the points (~j can be arbitrarily prescribed), so the result is new only for n = 2 . We wish to point out that the technique in the proof of Proposition 2, without any substantial modification, allows one to embed C k into C n, so that the image of C k contains an arbitrary given discrete sequence, if 1 < k < n .

P r o p o s i t i o n 3. (Theorem 4.5 in [9].) Let n > l . There exists a discrete se- quence ( a j ) j e N in C n such that for no automorphism r of C n, g ) ( a j ) E C • (for all j 's).

In [9], such sequences are called "non tame". The existence of non tame sequences is not immediate. Indeed, any sequence is the union of two tame sequences.

An interesting, more detailed, study of non tame sequences is to be found in [6].

All that is left is therefore to prove Proposition 2.

2. P r o o f of P r o p o s i t i o n 2

As already said, our work depends on an extension of the work of Andershn Lempert, as given in [4]. Also, our proof is inspired from [5]. From [4], we shall need only the following:

L e m m a 1. Let K be a polynomially convex compact set in C n ( n > l ) . Let p and q E C n \ K . For every E>O, there exists r an automorphism of C n, such that r and I r for every z E K . In addition we can fix arbitrarily chosen points p l , . . . ,Ps in g (i.e. r

Proof of n e m m a 1. Let ~/: [0, 1 ] - - * c n \ g be an arc, ~/(0)=p, ~ ( 1 ) = q . Apply Theorem 2.1 in [4] to the following situation: In Theorem 2.1 replace K by K U { p } and consider ~ a sufficiently small neighborhood of K U ( p } . Take ~t to be the identity on K , and to be ~ ( z ) = z § near p. Fixing finitely many given points is a trivial addition.

Remark. If K is convex, the lemma is very simple to prove with really elemen- t a r y tools. Without any intent to look for more generality we now simply state the following.

L e m m a 2. Let K be a polynomially convex compact set in C n. Let H be a proper holomorphie embedding of C into C '~. Let R > 0 and L R = { z = H ( ~ ) : ~ E C , I~I<R}. Then the polynomial hull of K U L R is contained in K U H ( C ) .

(3)

Non straightenable complex lines in C 2 99

Proof of Lemma

2. Let p E C n. Assume that

p~KUH(C).

Let f be a polynomial such that f ( p ) = l , but Ifl < 1 on K . Let g be an entire function which vanishes identically on H ( C ) , but such that

g(p)r

(The existence of such a g follows from Cartan's Theorem A, but in the application we can explicitly exhibit such a g, a polynomial. See the remark at the end of the paper.) Now, for N large enough, we have

IfNy(p)I>SUPKuH(C)IfNgl.

So, p is not in the polynomial hull of

KULR.

We now begin the proof of Proposition 2 itself.

Proof of Proposition

2. We start with the embedding H0: C--*C '~, H0(~)=

((, 0) and 0o=0. In the

jth

step of the construction we shall find

Oj

>0, Q EC, and then construct a proper holomorphic embedding Hi: C--+C n such that:

(i) H i ( ~ l ) = a l , / e { 1 , . . . ,j}, (ii) I H j ( ~ ) ] > l a j l - 1 if I(l>Qj,

(iii) I H j ( ( ) - H j _ l ( ( ) l _ < e j < 2 - j for I(I-<Qj with

ej

to be chosen small enough, depending on previous choices,

(iv) ej>_0j_l+l.

Once this is done, we set H = l i m

Hj

(uniform convergence on compact sets)9 The inequality

]Hj-Hj_ll<2-J

in condition (iii) shows that the sequence of maps

Hj

does converge, and that the limit g satisfies: I g ( ~ ) - H j (()1< 1 for I(I ~ ~j+l. Prom (ii) we get that if

~j~-I~l~j+l,

then I H ( ~ ) l > l a J l - 2 . So g is proper9 And (i) implies t h a t H ( ( t ) = a l , for any I.

Finally we have to explain the choice of e j, so as to make sure t h a t H is an embedding 9 Let R > 0 , and let G be any holomorphic embedding of C (or of the disk {I(I<R}) into C n, and 0 < r < R . Then, there exists 7 > 0 (depending on G, r and R) such t h a t if

G'

is any holomorphic map from the disk {1(I < R } into C ~ satisfying

IG-G'I

~ , on this disk, then the restriction of G' to the smaller disk {1~1 < r } is an embedding 9 In the ( j - l ) st step of the construction, the radius

Oj-1

and the map

Hi-1

have been chosen9 In the

jth

step the radius

Oj

will be chosen first, as will be explained below9 We then apply the above to

G=Hj-1, R=~j,

r = Q i _ l , to get

~ = ~ i If for every j E N , 9 ~ z = i ez ~ ? i , then the limit map H will be an embedding 9 + ~ Indeed, the restriction of H to any disk {1~1< 0i-1} will be an embedding, since the inequality I H -

H i_ 11 <- ~i

will hold on the disk {1(] < 0i }. A possible choice of

e i

^is

therefore: e i = 2 - i minl<y (1, ~?l).

Here is a way to find Qj, Q and construct Hi:

We already have H i _ l ( ~ ) = a l for / e { 1 , . . . , j - l } . If

aj=Hj_l(~)

^{for some}

( c C , we just take ~j =~,

Hi=Hi_l

and 0i large enough so that (ii) and (iv) hold9 Otherwise (the general case), we choose Qi >- ~Y- 1 + 1 so large that I H i _ 1 (~)1> l ayl for every CEC, I(l>0i. Let F be defined by:

F = {z e C~: Izl < l a y ] - l / 2 } U H j - l { l ~ l

<_ Qj}.

(4)

100 Franc Forstneric, Josip Globevnik and Jean-Pierre Rosay

T h e polynomial hull of F does not contain a j since a j ~ H j _ I ( C ) , and according to L e m m a 2. Take ~j so t h a t H j _ I ( ~ j ) does not belong to this hull (it is enough to take I~jl large enough). By L e m m a 1 we can find r an a u t o m o r p h i s m of C n, fixing c~1 ,... , a j - 1 , as close as we wish to the identity on F and such t h a t r In particular we take r close enough to the identity on F so t h a t the image of the ball

{IzI<Iajl- 1}

contains the ball {Iz[_<Iajl-1}. So if I~lkt)j then I H j _ l ( ~ ) l > ] a j l , hence I ~ j ( H / _ I ( 4 ) ) I > I c u I - 1 .

We set H j = r o H j _ l . Properties (i)-(iv) are immediate to check.

This ends the proof. We just add the following r e m a r k with respect to the proof of L e m m a 2. One has Hj = r ... r176 and H 0 ( ~ ) = ( ~ , 0). So if Zk denotes the k th coordinate function in C n, the functions Zkor -1 . . . r for k E { 2 , . . . , n}, have precisely H j (C) as their c o m m o n zero set.

Note. Y~rther examples, related to Proposition 1, have been given by G. Buz- zard and J. E. Fornaess ([3]). In particular, they give the example of a complex line e m b e d d e d in C 2, whose complement is hyperbolic.

R e f e r e n c e s

1. ABHYANKAR, S. S. and MOH, T. T., Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166.

2. ANDERSI~.N, E. and LEMPERT, L., On the group of holomorphic automorphisms of C n, Invent. Math. 110 (1992), 371-388.

3. BUZZARD, G. and FORNAESS, J. E., An embedding of C in C 2 with hyperbolic complement, Preprint.

4. FORSTNERIC, F. and ROSAY, J-P., Approximation of biholomorphic mappings by automorphisms of C n, Invent. Math. 112 (1993), 323 349. Erratum in Invent.

Math. 118 (1994), 573-574

5. GLOBEVNIK, J. and STENSONES~ S., Holomorphic embeddings of planar domains into C 2, Math. Ann. 303 (1995), 579 597.

6. GRUMAN, L., L' image d'une application holomorphe, Ann. Fac. Sci. Toulouse Math.

12 (1991), 75-101.

7. JELONEK, Z., The extension of regular and rational embeddings, Math. Ann. 277 (1987), 113-120.

8. KRAFT, H., Challenging Problems on A]fine n-space, Bourbaki Seminar 47, 802, June 95, 1994-95.

9. ROSAY, J-P. and RUDIN, W., Holomorphic maps from C n to C n, Trans. Amer. Math.

Soc. 310 (1988), 47-86.

10. ROSAY, J-P. and RUDIN, W., Holomorphic embeddings of C in C ~, in Several Com- plex Variables: Proceedings of the Mittag-LeJ~er Institute, 1987-88 (Fornaess, J. E., ed.), pp. 563-569. Math. Notes 38, Princeton Univ. Press, Princeton, N. J., 1993.

(5)

Non straightenable complex lines in C 2 I01

11. SUZUKI, M., Propri@t@s topologiques des polyn0mes de deux variables complexes, et automorphismes alg@briques de l' espace C 2, J. Math. Soc. Japan 26 (1974), 241-257.

Received April 26, 1995 Franc Forstneric

Department of Mathematics University of Wisconsin Madison, WI 53706 U.S.A.

Josip Globevnik

Institute of Mathematics Physics and Mechanics Jadranska 19

61111 Ljubljana Slovenia

Jean-Pierre Rosay

Department of Mathematics University of Wisconsin Madison, WI 53706 U.S.A.