Math. Ann. 227, 61--66 (1977) @ by Springer-Verlag 1977

Holomorphie Fibrations of Bounded Domains

Alan Huckleber ry* Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA

One of the most striking differences between one and several complex variables is that in the latter case the p rob lem of uniformizat ion is immense. The first evidence of this was Poincare 's observat ion that the ball, B : '-- {(z, w)l Izl 2 + Iwl 2 < 1}, and the bicyclinder, A : ' - {(z, w)l Iz] < l, Iwl < 1 }, are not b iholomorphica l ly equivalent. One way of proving this is as follows : Assume that ~0 :A ~ B is such a map, ~o =((p~, q~2), where A=D~ x D 2 is the natural product structure. Consider the derivatives

(?(Pi - - x ( O , w ) a s f u n c t i o n s o f w e D a f o r i = l , 2 . Suppose tha t , f o r i = l , ) tim (0,w) ( ' 2 - " u , - ~ w o U 2

= 0 for all woe~D z. Then (?z (0, w)= 0, i = 1, 2, and the jacobian of ~0 vanishes on

the disk '{0} × D 2. Since p was assumed to be biholomorphic , this can' t happen. I f

#Pl (0, w , ) ~ a + 0 as w , - , w 0, then we may apply there exists w 0 e ? D ; such that (3Z

Montel 's Theorem to the sequence of functions ~b, :D~ ~ B , qS,(z) : = q(z, w,), finding

a convergent subsequence 4 , - ~ 4 " D ~ - - , ? B with ~ - ( 0 ) = a . In part icular q5 is not

identically constant. But there exists a function j • O(IF2I such that . [(4(0))= 1 and }J(P)I < 1 for all p • B with p + qS(0). Thus j '4 violates the m a x i m u m principle on D~. Hence a contradic t ion has been reached.

Using Montel 's theorem to "push out" the disks and Rado ' s theorem for the desired contradict ion, R e m m e r t and Stein [3] prove the following local version of the above (also see N a r a s i m h a n [2] for this and other related topics):

Theorem (Remmert-Stein) . Let D be a domain in q" =~'"~ x IF "~. Suppose that there exists aE c?D contained in a product neighborhood U = U~ x U 2 C tI~", Ui C ~F"', such that

U c~ D = D 1 × D2, Di open in (F '~', with I) 2 (~ U 2 4 = U 2. Let f2 be any bounded domain with strictly pseudoconcex bmmdary. Theft there is flo proper hotomorphic map j rom D into f2.

* Partially supported by NSF grant MCS75-07086 A01

62 A. Huckleberry

Roughly speaking, the basic idea of Remmert and Stein is the fo l lowing: / j a bounded domain is fibered then its boundary should be fibered. The following is typical of what we can prove in this direction:

Let E be a bounded domain oj holomorphy in 117". Suppose that E is the total space oJ a holomorphic fiber bundle whose base and fiber are positive dimensional. Then the "smooth" part oJ the boundary oJ E contains an open dense set which is joliated by complex maniJolds.

A more general version of this as well as several other related remarks are given in Section 3. For example, it is shown that {[ E is a relatively compact open subset oJ a Stein man{[old and is the total space oJ a holomorphic fiber bundle, then gE is not smooth (i.e. at least C2). We note that the techniques for the proofs are totally elementary, making the most out of Montel's theorem and the maximum principle.

It is reasonable to ask if there are "interesting" fiber bundles whose total spaces are bounded domains. O fcourse an example is given (Section 2), but we do note the following :

Let E be a relatively compact domain in a Stein maniJold. IJ E is the total space oJ a holomorphic fiber bundle, then the transition Junctions are constant. Although this result is well-known, for the sake of completeness we sketch a quick proof at the end of this note.

1 am particularly indebted to A. Howard for his numerous helpful remarks during the genesis of this work.

1. Preparations

We use F~E--*B to denote a holomorphic fiber bundle whose base B and fiber F are positive dimensional. In this situation we often say that the total space E is holomorphicalIy fibered. We will always let g:E-~B be the projection map with F ~ : = ~ - 1(b). It should be emphasized that we only assume that such bundles are hotomorphically locally trivial. Thus, if A is a polydisk in B over which the bundle is trivial, then there exists a (surjective) biholomorphic map q0 :F x A--,~-I(A) such ~(q~(p, -))=id a for all p~F. We use ~ 0 ~ : F ~ - l(z) to denote the map defined by p ~q~(p, z). If the bundle is trivial over A 1 and A 2 (via the biholomorphic maps q~ 1 and q)2) then, assuming A C A ~ c~A 2, we have the map ~p ~ 2 : = @2 1 ~;(p 1 :F x A --, F x A. Of course ~p~2 =~b x id a, where the holomorphic map ~b : F x A--,F has the property that 4)(-,z)E AutoF for all zeA. Such q5 :F x A ~ F are called transition mappings.

Since our results have to do with boundaries, we must also fix the notation for these considerations. For this let E be a relatively compact domain in some complex manifold X. We say that c~E is smooth at pot ~?E if there exists an open neighborhood U of p0 in X and a real valued twice continuously differentiable function ff : U--,R with dE 4=0 on U such that Ec~ U = {pc UIQ(p)<0} (i.e. {~ =0} agrees with 0E in some neighborhood of p0 ). Ifpo is a smooth boundary point then the (real) tangent plane of E at Po contains a unique complex tangent plane which is given in local

(~2~) (Po))restricted to coordinates by {y] ~(po)(Zi -z i (po))=O}. The Hessian \~z /~ i

this complex tangent plane is called the Levi Jorm oJ Q at Po, L~(po). The signature of

Holomorphic Fibrations 63

Lo(Po ) is a biholomorphic invariant of the surface 8EcrU, If P0 is a smooth boundary point and Le(Po)>O for some defining function ~, then P0 is said to be a pseudoconvex boundary point. If the stronger condition Le(Po)>0 is fulfilled then Po is called a strictly pseudoconvex (s.~.c.) boundary point, Suppose that Po is a smooth boundary point with U and o as above, and furthermore assume that L~(p) >_0_ for all pc bTc~c,E.'~ I f no pc Ur~SE is s.t#.c, then U n S E contains an open dense subset which is Jotiated by complex mani folds the dimension of which is at least the smallest nullity of Le(p), pe U. To make matters easier, we define this "foliation" to mean that through every point in the set there is a positive dimensional complex manifold which is also contained in the set. Of course in this situation there will be open sets where the rank of the Levi form is constant. These foliations are smoothly parameterized I-5].

2. An Example

Since the purpose of this paper is to discuss holomorphic fibrations of bounded domains, we should give at least one non-trivial example. For this let E" : ~{(Z, ]4')Et~ 2t 1 ~ IZI <~ 1, tt4'12 .~ IZ [}. Then E is a bounded domain with piecewise smooth pseudoconvex boundary. We take B= '{~cC)½ <I~t < 1 } to be the base, with lr : E ~ B being defined by ~(z, w) = z. The fibers of this map are all equivalent to the unit disk D in the complex plane. To see that this fibration is holomorphically

locally trivial, take A 1 : =B\IR + and A 2 " = B\IR-. Denoting by [/~1 and ~ - 2 any particular square-root Junctions on A~ and A 2 respectively, we have the maps

q)~ : D x A i ~ z - l(Ai) defined by (pi(r/, ~) : = (~, ~ ' r/). It follows immediately that D--*E--*B is a holomorphic fiber bundle. It is easy to check that this is not globally holomorphically trivial, because any such trivialization D x B ~ E would require the use of a globally defined square-root on the annulus B. We remark that, with a bit more work, one can construct non-trivial holomorphic fibrations of bounded domains where the base is itself not a bounded domain.

3. Extension to the Boundary

Our first observation is that, if a bounded domain E were the total space of a holomorphic fiber bundle, then there would be very strange behavior at the s.t/,,c. boundary points.

Proposition. Suppose that E is a relatively compact domain in a complex maniJold X. lJ E is the total space oJ a holomorphicfiber bundle and Po c 8E is a s.tt~.c, boundary point, then, jor all b~B, the closure oJ F~ in E, Fb, contains Po.

Pro@ Let (p:F x A ~ z - l ( A ) C E be some local trivialization of the bundle, where A is an (n-k)-dimensional polydisk and F has dimension k. Let U be an open neighborhood of P0 in X such that every pc Uc~OE is s.tp.c. Suppose there exists a sequence (po(X,) = p~ converging to Po. Define q0, :A - , E by q0,(z) : = qo~(x,). Then {q~,} is a uniformly bounded family of maps of A into X. Consequently, by Montel 's theorem, there exists a convergent subsequence qo, --+qS, where ~ : A ~ O E is a hotomorphic map with (b(0) = Po- Since Uc~SE is a s.~t~.c, hypersurface, it contains no

64 A. Huckleberry

positive dimensional analytic sets. Thus ~b(z)=po. Therefore, if PoCFho [say F~o = q~o(F) in the above trivialization], then, for all b "near" b o, poe F~. More precise- ly, taking F~ = q)~(F) for ze A, lim~o:(x,~) = no. Since B is connected, one can join b o to any beB by a curve which is covered by finitely many polydisks over which the bundle is trivial. Continuing the above argument from polydisk to polydisk, one sees that, if poe~'~o, then poeF~ for every beB.

The above argument shows that, in order to complete the proof, we only need Po e F~o for someboeB. Furthermore, if there exist points pe Uc~(~E arbitrarily near Po for which p6 F~tet for b(p)e B, then pe Fb,, for a fixed b o (independent of p) and consequently poeF~o. Thus, by taking U smaller if necessary, we only need to consider the case where F b ~ ( U ~ ? E ) = ~ for all be B. But the strong pseudocon- vexity of U~?~E rules this out. To see this, let {p,} _(_ Uc~E converge to Po. Let F, be the fiber of the bundle which contains p, and t a k e ~ to be the connected component o f F s ~ U containing p,. For the following argument, I]" [Is is the sup-norm on the set S. Since U ca (?E is s.~t,,.c., we can find (taking U smaller if necessary)je O(U) such that I l f l l£~v=f(po)=l and ]]fll,~w~__<½. By assumption ,~,c~(Uc~(?E)=qS. Thus Ilfl[~ = ]lfll,~,~a:<~-. But this is contrary to the fact that p,e ~-~, and P,-'Po, where f(Po)= 1.

[]

We are now in a position to prove the main result of this note.

Theorem. Let E be a relatively compact domain in a complex man(fold X. Suppose there exists a s.tt;.c, boundary point po~?;E. Then E is not the total space o[ a holomorphic fiber bundle.

Pro@ Suppose that E is the total space of a holomorphic fiber bundle, F-~E--,B. Let q) :F x A --,~z- I(A) be some trivialization, where A is a polydisk with coordinates z=(z I ..... z,_k). Take U to be a coordinate neighborhood of Po in X such that U~c~E is a s.y,.c, hypersurface. When necessary, we consider U to be an open set in 07". Let ~ = (Imq) o :F--,E)~ U. Thus the restriction of q)o maps ~00- I(j~) biholom- orphically onto ,:N. We call this map ~k' = (~1 ..... t;,) and, recalling that we have ~o:

• ~ ' i] O(q)(Tt(~),but, composing with ~, -1 we for all ze A, we let glj. = ~ ] , = o" Thus ,qije

consider g~je O(,N)_.Our goal is to show that, for some (~ ~N, gq(~,)= 0 for all i and j. For this, let K : = , Y T = ~ u ( U ~ E ) u ( c ~ U ~ ) . By shrinking U if necessary, we may assume that glj extends continuously to points of K which are in (')U~E. If p,=~0o(X,) converges to p e U c~dE, then, since all convergent sequences q) , : =q)(x,~,. ):A ~ E must converge to the map which is identically p, g~j(p,)---,O. Thus for all i and ,j, gij continues to a function which is continuous on K. Again, by taking U smaller if necessary, there exists leO((7) which is nowhere zero in C ~ such that !lfl[6~f=J'(po)=l and lifl[,-v~E<½. Define V:='{~eU[Ifff,)[>3}. Then V is an open neighborhood of Po with Vc~E relatively compact in Uc~/~. Supposing that g~j4:0, we define the following sequence of functions on K:

h , : - go" ,(~-~/" II.qi/l~ \ 3 / "

Holomorphic Fibrations 65

Recall that g o - 0 on Kc~.E. Thus

I l h . t l ~ = II . l i . ~ v t , < - <

However,aij = l i g j ~-h. Thusilgi~[IK~,v<~[gijtKtlh.!IK.Thlsbeingtrueforaltn,

9i~-0 on ~c~V for all i and j. In particular this contradicts the fact that, since ~0 :F x A-+~- ~(A) is biholomorphic, its jacobian has maximal rank. []

Corollary 1. Let E be a relatively compact domain in a Stein maniJold X. Assume that OE is smooth. Then E is not the total space of a holomorphic fiber bundle.

Pro@ Since X is a Stein manifold, it has a strictly plurisubharmonic exhaustion

function 0 :X--+IR +. Let 0(P0)= max 0(P). Then P0 e 0E and, by comparing the Levi peE

forms of 0 and the defining function for c~E, one sees that Po is a s.g,.c, boundary point. []

Corollary 2. Let E be a relatively compact domain oJ holomorphy in a complex man!lold X and suppose that E is the total space oJ a holomorphic fiber bundle. Then the smooth part of the boundary oJ E contains an open dense set which isJoliated by positive dimensional complex man{lolds.

Pro@ By [5], it is enough to show that the Levi form has a positive dimensional null space at each smooth boundary point. However, since it is positive semi-definite and positive definiteness is ruled out by the theorem above, this is immediate. []

Corollary 3. Let F ~ E ~ B be a holomorphic fiber bundle, f2 a relatively compact domain oJ holomorphy in a complex man!;Jold X, and f: E ~ (2 a proper, light, subjective holomorphic map. Then the smooth part o[ the boundary oJ (2 contains a dense open set which is Joliated by positive dimensional complex mani[otds.

Pro@ The only argument above which does not immediately carry over to this situation is the one which involves defining the derivatives g~j on the biholomorphic image of the fiber. In this case we only have a proper image of the fiber. To

circumvent this, we consider instead the map 7' = q ; x ( ~ ) , where ,# is that map \ ~ - - - ] /

defined in the proof of the theorem above. It is important to note that this is a bounded map and thus the closure of the image of 7', K, is compact. One now uses

this K and applies the same techniques as above to show that ~zj -=0 for all i and./. []

We remark that the hypothesis of"light" in Corollary 3 can be replaced by any weaker condition which implies that the bundle structure on E is not destroyed by,l [e.g. rank J > max(dimF, dim B)]. The reader should also note that even a proper image in a Stein manifold cannot have smooth boundary. (I.e. the obvious generalization of Corollary 1 is valid.)

66 A. Huckleberry

3. The Transition Mappings

Royden [4] has proved that the transition mappings o f a holomorphic fiber bundle with hyperbolic fiber are constant (i.e. ~0:f 'x A ~ F is independent o f the A- variable). I f the bundle is modeled on a holomorphic principal bundle, one finds this in W. Kaup 's original paper on hyperbolic manifolds [1]. We give a p roo f o f this fact in our setting.

Theorem. Let E be a bounded domain in a Stein manifold and suppose that E is the total space of a holomorphic fiber bundle. Then the transition mappings are constant.

Pro@ By the embedding theorem for Stein manifolds, we may assume that F is contained in a bounded domain in 117". Let 4) :F x A --,F be some transition mapping. For every ze A, define c~(z)e AutoF by c((z): = 4~(', z) and let a"(z) be its n-th iterate. We may assume that c~(0)=ide AutoF. Let ArC F be a coordinate polydisk with coordinates ~=(~l .... ,~k). Take z = ( z I . . . . . Z,_k) to be the coordinates on A. It is enough to show that the ~-jacobian e(z),(0) is the identity for all zeA . So let e(~),(0) = I + A(z), where A is a holomorphic matrix valued function on A such that A(0) = 0. Thus, since c~(0) = id, e"(z),(0) = I + nA(z) + o(A(z)). But F is contained in a bounded

" 0 domain, Thus all derivatives ~-z{ ~ ( z ) , ( ) must be uniformly bounded independent of

n (Cauchy Inequalities). This cannot happen unless A(z ) -O ,

Corollary. Let E be as in the above theorem, Then the bundle structure comesjrom a representation oJ ~zl(B ) in AutoF. A simply connected E is thereJore biholomorphically equivalent to a product.

References

t. Kaup,W.: Hyperbolische Komplex R~ume. Ann. Inst. Fourier 18, 303 -330 (1968) 2. Narasimhan, R. : Several complex variables. Chicago: University of Chicago Press 1971 3. Remmert, R., Stein,K. : Eigentliche holomorphe Abbildungen. Math. Zeit. 73, 159-- 189 (1960) 4. Royden, H.E. : Holomorphic fiber bundles with hyperbolic fiber. Proc. of AMS. 43, 31L--312 (1974) 5. Sommer, F. : Komplex-analytische BlS, nerung reeller Hyperfl~ichen in II TM. Math. Ann. 137, 392 411

(1959)

Received July 18, 1976