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Finite determinacy of dicritical singularities in $(\mathbb{C}^2,0)$

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Jones, Peter

Zinsmeister, Michel

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In memory of Alain Dufresnoy, a friend
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ANNALES
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L’INSTITUT FOURIER
Jean-Pierre ROSAY
Extension of holomorphic bundles to the disc (and Serre’s Problem on Stein
bundles)
Tome 57, no 2 (2007), p. 517-523.
<http://aif.cedram.org/item?id=AIF_2007__57_2_517_0>
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Ann. Inst. Fourier, Grenoble
57, 2 (2007) 517-523
EXTENSION OF HOLOMORPHIC BUNDLES TO THE
DISC (AND SERRE’S PROBLEM ON STEIN BUNDLES)
by Jean-Pierre ROSAY (*)
Abstract. — Holomorphic bundles, with fiber Cn, defined on open sets in C
by locally constant transition automorphisms, are shown to extend to holomorphic
bundles on the Riemann sphere. In particular, it allows us to give an example
of a non-Stein holomorphic bundle on the unit disc, with polynomial transition
automorphisms.
Résumé. — On montre que des fibrés holomorphes, à fibre Cn, définis sur des
ouverts de C par des automorphismes de transition localement constants se pro-
longent en fibrés holomorphes définis sur la sphère de Riemann. Ceci permet en
particulier d’obtenir un exemple de fibré non de Stein sur le disque, avec automor-
phismes de transition polynomiaux.
1. Introduction
We shall study (locally trivial) holomorphic bundles over open sets in
C (in which, the variable will be denoted by ζ), with fiber Cn. We shall
assume that, in appropriate local trivializations, the bundle is given by
gluing (transition) fiber automorphisms that are locally independent of the
base point and that belong to a group G of automorphisms of Cn.
Of course, such a bundle is trivial over any simply connected region. If
U is an open set in C covered by simply connected domains (Uj), over U
any such fiber bundle is obtained by gluing the trivial bundles Uj ×Cn, by
using fiber automorphisms belonging to G and locally independent of the
base point.
Our goal is to extend such bundles to bundles defined over larger open
sets in C, and in fact over the whole Riemann sphere. For the extended
Keywords: Holomorphic bundles, Stein manifolds, groups of automorphisms of Cn.
Math. classification: 32L05, 32Q28.
(*) Partly supported by NSF grant.
518 Jean-Pierre ROSAY
bundles, we must allow the fiber automorphisms to (even locally) depend
on the base point.
Definition 1.1. — We shall say that a group G of automorphisms
of Cn is generated by one parameter groups if there are one (complex)
parameter groups (St)t∈C (i.e., Ss+t = Ss ◦ St) of automorphisms of Cn
whose elements generate G.
Examples to have in mind are:
(A) The group of all polynomial automorphisms of C2.
(B) For any n, the group of polynomial automorphisms generated by the
invertible affine maps and the polynomial shears, i.e., maps of the type
(z1, · · · , zn) 7→ (z1, · · · , zn−1, zn + Q(z1, · · · , zn−1)),
where Q is an arbitrary polynomial in (n − 1) variables. A corresponding
one parameter group is
(z1, · · · , zn) 7→ St(z1, · · · , zn) = (z1, · · · , zn−1, zn + tQ(z1, · · · , zn−1)).
If n = 2, this group (generated by affine maps and shears) is the group of
all polynomial automorphisms of C2 , by Jung’s Theorem [11].
(C) The group of automorphisms of Cn generated by the invertible affine
maps, the shears (as in (B) but with Q entire) and the ‘over-shears’ (in the
terminology of Andersén-Lempert), i.e., maps of the type
(z1, · · · , zn) 7→ (z1, · · · zn−1, zneQ(z1,··· ,zn−1)).
In that case, set
St(z1, · · · , zn) = (z1, · · · zn−1, znetQ(z1,··· ,zn−1)).
One parameter groups of automorphisms of Cn are given by the flow
of complete (i.e., integrable for all time) holomorphic vector fields on Cn.
There are several notions of completeness for holomorphic vector fields: in
positive time, in real time or in complex time. In Cn, these three notions
coincide [1]. On the topic of Example (C) and of complete holomorphic
vector fields, see [3] and [9].
Proposition 1.2. — Let D be a bounded domain in C bounded by a
Jordan curve, and for j = 1, · · ·N , let Dj be a region bounded by a Jordan
curve. Assume that Dj ⊂ D, Dj ∩Dk = ∅, if j 6= k. Let Λ = D \ ∪Nj=1Dj .
Let S denote the Riemann sphere S = C ∪ {∞}.
Let Π : X → Λ be a (locally trivial) holomorphic fiber bundle with fiber
Cn and (in appropriate local trivializations) with gluing automorphisms
of the fibers locally independent of the base point and in a group G of
ANNALES DE L’INSTITUT FOURIER
HOLOMORPHIC BUNDLES 519
automorphisms of Cn generated by one parameter groups. Then there exists
a holomorphic fiber bundle X̃ → S with fiber Cn and (in appropriate local
trivializations) with gluing automorphisms of the fibers in G (allowed to
depend on the base point), whose restriction over Λ is X.
Corollary 1.3. — (a) The fiber bundle in Skoda’s original exam-
ple of non-Stein bundle [13] (and see [12]) extends to a holomorphic
bundle over the Riemann sphere (and so to a holomorphic bundle
over the disc).
(b) The possibly non-Stein fiber bundles constructed by Demailly [6]
on annuli, with fiber C2 and the gluing automorphism
(z1, z2) 7→ (z2,−z1 + zk2 ) ,
extend to holomorphic bundles on the Riemann sphere with poly-
nomial automorphisms of the fibers.
The Corollary is immediate, and the consequences for the problem of
Stein bundles are as follows.
(α) There are two versions of Skoda’s example. In the first, more trans-
parent, one (Theorem in [13]), the base is made of the unit disc with 8 discs
removed. In the second one (final remark in [13] and Theorem 1 in [12])
the base is the disc with 2 discs removed. In the first example, around each
deleted disc the bundle is defined by a gluing automorphism which, up to a
possible permutation of the coordinates, is an over-shear as in Example C.
In the second example, around one of the deleted disc the gluing is given
by the automorphism (z1, z2) 7→ (z1, z2ez1), and for the other deleted disc
by (z1, z2) 7→ (iz2, z1).
So, starting from Skoda’s example, one gets a new construction of a
non-trivial, non-Stein, holomorphic bundle on the unit disc (and on C).
This construction seems to me to be much less technical than the original
example of Demailly [6] and [7]. Furthermore the construction relies only
on the simplest case in the proof of Section II, with a trivial filling of each
hole, since around each hole the gluing is made by automorphisms that are
elements of a one parameter group of automorphisms.
(β) The maps used by Demailly are of Henon type. The map (z1, z2) 7→
(z2,−z1 + zk2 ) is the composition of the linear map (z1, z2) 7→ (z2,−z1),
and of the shear (z1, z2) 7→ (z1, z2 + zk1 ). Each of these two maps is an
element of a one parameter group of polynomial automorphisms. But, if
k > 2, the map (z1, z2) 7→ (z2,−z1 + zk2 ) itself is not even an element of a
one parameter group of (possibly non-polynomial) automorphisms of C2,
since the proof of Case 1 in Section II would then show that the Demailly
TOME 57 (2007), FASCICULE 2
520 Jean-Pierre ROSAY
bundles would be trivial. Following Section II, the extension of the bundle
from the annulus to the disc is then obtained by first extending the bundle
to the disc with 2 (smaller) holes, and then extending trivially to each hole.
Starting from the polynomial examples of Demailly (non-Stein for a given
annulus if k is large enough, and non-Stein for a given k > 2 if the annulus is
thick enough), one gets examples of holomorphic bundles on the unit disc
(and on C), with fiber C2, and with gluing polynomial automorphisms,
that are non trivial and non-Stein. This answers a question in [6]. The
question was asked again recently by H. Skoda, who considered it to be
the last question in the Serre Problem on Stein bundles, left open after the
counterexamples of Skoda [13] [12], Demailly [6] [7], and Coeuré-Loeb [5].
If one takes k = 2, all the polynomial automorphisms to be used in
the proof of the Proposisiton can be chosen of degree 6 2. So, there are
examples of non-Stein bundles on the disc, with polynomial gluing auto-
morphisms of the fiber of degree 6 2, while all bundles on the disc with
affine gluing automorphisms are trivial.
2. Proof of the Proposition
It is of course enough to extend the given bundle X over each “hole” Dj ,
and over the component D0 of S \D, containing ∞. Fix j ∈ {0, 1, · · · , N},
and let V be a neighborhood of bDj in Λ that is topologically an annulus.
We shall decompose V into the union V + ∪ V − of two simply connected
regions with intersection V + ∩ V − = ω ∪ ω′, where ω and ω′ are disjoint
connected open sets. We wish to extend the bundle X|Π−1(V ) to a bundle
over Dj ∪ V , defined by automorphisms of the fibers in the group G. Un-
needed for the easy Case 1 below, un-necessary but helpful for the general
case, we shall be more specific. For r > 0, let ∆r be the open disc in R2
of radius r centered at the origin. We can take V and V ± so that for some
homeomorphism χ of ∆2 into S: Dj = χ(∆1), and V = χ(∆2\∆1), V + and
V − being respectively the image of the regions Im ζ > − 12 and Im ζ <
1
2 .
Since we start with locally constant gluing fiber automorphisms, and
since V + and V − are simply connected, the fiber bundle X is trivial over
V + and over V −. Over V , it can be defined by gluing V +×Cn and V −×Cn
over ω ∪ ω′, with the identification of (ζ, z) ∈ V + × Cn with (ζ, T0z) ∈
V − × Cn if ζ ∈ ω, and with (ζ, T1z) ∈ V − × Cn if ζ ∈ ω′, where T0 and
T1 are automorphisms of Cn that belong to G. Of course we could take
T0 = 1 (but we don’t, in order to simplify the exposition in the second case
below). We now split the proof in two cases.
ANNALES DE L’INSTITUT FOURIER
HOLOMORPHIC BUNDLES 521
2.1. Case 1
This is the easy case when there is in G a one (complex) parameter
group of automorphisms (St)t∈C such that T1 ◦T−10 = S1. In that case, the
bundle extends trivially over Dj , and the trivialization is by means of fiber
automorphisms in G. (If one wished to get a version of the Proposition for
bundles defined by gluing automorphisms depending on the base point, one
would need here that both T0 and T1 be defined on V +∪V −, as holomorphic
function of the base point, and the possibity of a holomorphic choice of the
one parameter group).
For identifying the trivial bundle (Dj ∪ V ) × Cn with the restriction of
X over V , we need to define fiber automorphisms:
Φ+ : V + × Cn → V + × Cn
and
Φ− : V − × Cn → V − × Cn ,
such that
(∗) (Φ−)−1 ◦ T̃0 ◦ Φ+ = 1 on ω × Cn ,
(∗∗) (Φ−)−1 ◦ T̃1 ◦ Φ+ = 1 on ω′ × Cn ,
where T̃j(ζ, z) = (ζ, Tj(z)). Let L+ and L− be holomorphic functions de-
fined respectively on V + and V −, such that L+ +L− = 0 on ω, L+ +L− =
−1 on ω′. This is a classical Cousin problem that here is elementary to
solve (use logarithms!). Set
Φ+(ζ, z) =
(
ζ, T−10 S
L+(ζ)(z)
)
Φ−(ζ, z) =
(
ζ, S−L
−(ζ)(z)
)
.
(Here comes the local dependence on ζ.)
For ζ ∈ ω:
[S−L
−(ζ)]−1 ◦ T0 ◦ (T−10 ◦ SL
+(ζ)) = SL
−(ζ)+L+(ζ) = S0 = 1 ,
so (∗) is satisfied.
For ζ ∈ ω′:
(2.1) [S−L
−(ζ)]−1 ◦ T1 ◦ (T−10 ◦ SL
+(ζ)) = SL
−(ζ) ◦ S1 ◦ SL
+(ζ)
= SL
−(ζ)+1+L+(ζ) = S0 = 1,
so (∗∗) is satisfied.
That ends the proof of Case 1. Notice that in case a trivial extension is
possible over each Dj , it does not give a global trivial extension, even just
on C or D. Skoda’s example illustrates that.
TOME 57 (2007), FASCICULE 2
522 Jean-Pierre ROSAY
2.2. General Case
The general case can be reduced to the previous case by introducing a
higher connectivity (replacing Dj by finitely many smaller holes) before
filling each hole individually trivially. Recall that the ‘hole’ Dj is parame-
terized by ∆1, under the map χ, and that Dj ∪ V is parameterized by ∆2.
In general one has
T1 ◦ T−10 = Ek ◦ · · · ◦ E1 ,
where each Ep (p = 1, · · · , k) is an element of a one parameter group of
automorphisms in G.
Fix points
(2.2) b0 = −2 , a1 = −1 < b1 < a2 < b2 < · · · < ap < bp < ap+1
< · · · < bk = 1 , ak+1 = 2.
Let W0 be the complement in ∆2 of the circles of diameter [ap, bp], p =
1, · · · , k, and let W = χ(W0). So, W is a multiply connected region in S
that contains V and that is contained in V ∪Dj . Set W+ = χ(W0∩{Im ζ >
0}), and W− = χ(W0 ∩ {Im ζ 6 0}). We first extend the restriction of
the given fiber bundle X over V , to a fiber bundle X ′ over W . For that
purpose, consider the fiber bundle X ′ obtained by gluing W+ × Cn with
W− × Cn over each component χ([bp, ap+1]) of the intersection, by using
the automorphism Ep ◦ · · · ◦E1 ◦ T0 (T0 if p = 0, T1 if p = k). Now, for the
extended bundle X ′, over each hole of W (the images under χ of the discs
of diameter [ap, bp]), we are in Case 1, since
[Ep ◦ · · ·E1 ◦ T0] ◦ [Ep−1 ◦ · · ·E1 ◦ T0]−1 = Ep .
For p = 1, read [E1 ◦ T0] ◦ T−10 = E1. Therefore, according to Case 1, X ′
extends to a bundle over Λ∪Dj , with gluing automorphisms in G (trivially
over each hole of W , but not trivially over Dj as illustrated by Demailly’s
example).
Remark 2.1. — It has been pointed out that the map (z1, z2) 7→ (z2,−z1
+zk2 ), for k > 2, is not an element of a one parameter group of automor-
phisms of C2, i.e., it is not the time-1 map of a complete vector field on C2.
The reason given was rather indirect. For a more direct approach to this
question, see [2] and [4]. F. Forstnerič pointed out to me the interesting
fact that, although they are far from being trivial since they can even be
non-Stein, all holomorphic fibers bundles with fiber Cn over a Stein base
have holomorphic sections [10], [8], and they satisfy the “Oka Principle”.
ANNALES DE L’INSTITUT FOURIER
HOLOMORPHIC BUNDLES 523
BIBLIOGRAPHY
[1] P. Ahern, M. Flores & J.-P. Rosay, “On R+ and C complete holomorphic vector
fields”, Proc. A.M.S. 128 (2000), p. 3107-3113.
[2] P. Ahern & F. Forstnerič, “One parameter automorphism groups on C2”, Com-
plex Varables Theory Appl. 27 (1995), p. 245-268.
[3] E. Andersén & L. Lempert, “On the group of holomorphic automorphisms of Cn”,
Inven. Math. 110 (1992), p. 371-388.
[4] G. Buzzard & J. E. Fornaess, “Complete holomorphic vector fields and time-1
maps”, Indiana Math. J. 44 (1995), p. 1175-1182.
[5] G. Coeuré & J.-J. Loeb., “A counterexample to the Serre problem with a bounded
domain in C2,”, Ann. Math. 122 (1985), p. 329-334.
[6] J.-P. Demailly, “Différents exemples de fibrés holomorphes non de Stein”, in Sémi-
naire P. Lelong, H. Skoda 1976/77 (Springer, ed.), vol. 694, L.N. in Math., 1978,
p. 15-41.
[7] ——— , “Un exemple de fibré holomorphe non de Stein à fibre C2 ayant pour base
le disque ou le plan”, Inven. Math. 48 (1978), p. 293-302.
[8] F. Forstnerič & J. Prezelj, “Oka’s principle for holomorphic bundles with
sprays”, Math. Ann. 317 (2000), p. 117-154.
[9] F. Forstnerič & J.-P. Rosay, “Approximation of biholomorphic mappings by
automorphisms of Cn”, Inven. Math. 112 (1993), p. 323-349, Erratum in Inven.
Math. 118 (1994), 573-574.
[10] M. Gromov, “Oka’s principle for holomorphic sections of elliptic bundles”, Journal
A.M.S. 2 (1989), p. 851-897.
[11] H. Jung, “Über ganze birationale Transformationen der Ebene”, Reine Angew.
Math. 184 (1942), p. 161-174.
[12] H. Skoda, “Fibrés holomorphes à base et à fibre de Stein”, Inven. Math. 43 (1977),
p. 97-107.
[13] ——— , “Fibrés holomorphes à base et fibre de Stein”, C. R. Acad. Sci. Paris Série
AB 284 (1977), no. 19, p. 1199-1202.
Manuscrit reçu le 26 janvier 2006,
accepté le 30 janvier 2006.
Jean-Pierre ROSAY
University of Wisconsin
Department of Mathematics
Madison WI 53706 (USA)
jrosay@math.wisc.edu
TOME 57 (2007), FASCICULE 2


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