The union problem on complex manifolds

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THE UNION PROBLEM ON COMPLEX MANIFOLDS

PATRICK W. DARKO

Received 14 May 2001

LetΩbe a relatively compact subdomain of a complex manifold, exhaustable by Stein open sets. We give a necessary and sufficient condition forΩto be Stein, in terms ofL2_-estimates for the ¯∂-operator, equivalent to the condition of Markoe (1977) and Silva (1978).

2000 Mathematics Subject Classification: 32E10, 32C35, 35N15.

1. Introduction. As indicated in [7], from the beginning of the theory of Stein spaces, the following question has held great interest: is a complex space, which is exhaustable by a sequenceX1X2 ··· of Stein subspaces, itself Stein?

In [1], the following is proved: every domain inCmwhich is exhaustable by a se-quence of Stein domainsB1B2 ··· is itself Stein, and this is shown to hold more generally for unramified Riemann domainᏮoverCm_{in [6]. In [11], the following is} proved: letXbe a reduced complex space andX1X2 ··· be an exhaustion ofX by Stein domains, if every pair(Xj,Xj+1)is Runge thenX=UXj is Stein. Recently, Markoe [9] and Silva [10] proved the following: letX be reduced andX1X2 ··· be an exhaustion ofXby Stein domains. ThenXis Stein if and only ifH1_(X,ᏻ)=0 (ᏻbeing the structure sheaf ofX).

More recently the following has been proved in [12]: letΩ1⊂Ω2⊂ ···be a sequence of open Stein subsets of a Stein spaceX,Ω=∞j=1Ωj, and dimH1(Ω,ᏻ) <∞. ThenΩ is Stein.

Fornæss [4] produced an example to show that ifX1X2 ···is a sequence of Stein manifolds, the limit manifoldX=Xj, in which eachXjis an open submanifold, need not be Stein. But it is known that if the limit manifold is itself an open submanifold of a Stein manifold then the limit manifold is necessarily Stein.

This led Fornæss and Narasimhan to pose the following problem [5]: let X be a Stein space andΩ1Ω2 ··· an increasing sequence of Stein open sets inX. IsΩj Stein? As indicated above this is the case whenXis a Stein manifold, but this question remains open in the general case.

In this paper, we consider the case whereXis a general complex manifold andΩ1

Ω2 ··· an increasing sequence of open Stein manifolds inXsuch thatΩ=Ωjis relatively compact inX. We give a condition forΩto be Stein, equivalent to Markoe’s and Silva’s condition and involvingL2_{-estimates for the ¯∂operator.}

2. Preliminaries. LetXbe ann-dimensional complex manifold with aC∞_Hermitian

metric. The spaceL2

is a Hilbert space under the scalar product,

(f ,g)=

Xf∧∗g,¯ (2.1)

where∗is the Hodge∗-operator associated with the metric and orientation ofX. LetΩ1Ω2 ···be an increasing sequence of Stein open sets inXsuch that their unionΩ=∞j=1Ωjis relatively compact inX.

The following theorem is our main result.

Theorem2.1. The unionΩis Stein if and only if given anf∈L2

(p,q)(Ω), which is ¯

∂-closed in the sense of distributions, there is au∈L2_(p,q−1)(Ω)such that∂u¯ =fin the sense of distributions and

uL2

(p,q−1)(Ω)≤KfL2(p,q)(Ω), q >0, (2.2)

whereKdepends onΩ.

Let U be a bounded open set in Cn_{, and ᏻ the structure sheaf of C}n_{. A section} f=(f1,...,fp)∈Γ(U,ᏻp), wherep >0 is an integer, isL2-bounded if

fL2_(U)= f1L2_(U)+···+bfpL2_(U)<∞. (2.3)

We then denote all sections ofᏻpoverUthat areL2_{-bounded byΓ}

2(U,ᏻp).

For the definition ofL2_{-bounded sections of coherent analytic sheaves, we require} the coherent analytic sheafᏲto be defined on a simply connected polycylinder neigh-borhoodVof the closure ofU. Then by [8, Theorem 5, Section F, Chapter VI], there is anᏻ-homographic in another simply connected polycylinder neighborhoodV of the closure ofU,

ᏻp λ_{→Ᏺ →0, (2.4)}

wherep >0 is some integer; andf∈Γ(U,Ᏺ)isL2_{-bounded iff∈Γ}

2(U,Ᏺ):=λ(Γ2(U, ᏻp_{)). It can be shown thatΓ}

2(U,Ᏺ)is independent ofλandp, so thatΓ2(U,Ᏺ)is well defined.

Now letΩbe a relatively compact subdomain of ann-dimensional complex manifold X. An open subsetY ofΩis said to be admissible for the coherent analytic sheafᏲ defined in the neighborhood of the closure ofΩinX, ifYis Stein. There is a coordinate neighborhoodVinXof the closure, ¯Y ofY such thatV is biholomorphic to a simply connected polycylinderV inCn_{, and ¯Y is contained in the neighborhood of ¯Ωwhere} Ᏺis defined asf∈Γ(Y ,Ᏺ)which isL2_{-bounded if}

f∈Γ2(Y ,Ᏺ):=

g∈Γ(Y ,Ᏺ):η_∗(g)∈Γ2

η(Y ),η_∗(Ᏺ), (2.5)

whereηis the restriction of the biholomorphic mapV→V1_{toY, andη}

∗(Ᏺ)is the

zero direct image ofᏲonY.

where theUj’s are as above, we say thatᐂis a finite admissible cover ofΩforᏲand we define theL2_{(alternate)q-cochains ofᐂwith values inᏲas those cochains,}

c=cα∈Cq(ᐂ,Ᏺ)=

α∈Iq+1

ΓUα,Ᏺ,

Uα=Ui0∩···∩Uiq, α=

i0,...,iq,

(2.6)

which are alternate and satisfycα∈Γ2(Uα,Ᏺ)for allα∈Iq+1. We denote byC2q(ᐂ,Ᏺ) the space ofL2_{-bounded cochains.}

The coboundary operator,

δ:Cq_{(ᐂ,Ᏺ) →C}q+1_{(ᐂ,Ᏺ), (2.7)}

mapsC₂q(ᐂ,Ᏺ)intoCq+1

2 (ᐂ,Ᏺ). IfZ q

2(ᐂ,Ᏺ)= {c∈C q

2(ᐂ,Ᏺ):δc=0}andB q

2(ᐂ,Ᏺ)= δCq−1

2 (ᐂ,Ᏺ), then as usualB q

2(ᐂ,Ᏺ)⊆Z q

2(ᐂ,Ᏺ)and we defineH q

2(ᐂ,Ᏺ):=Z q 2(ᐂ,Ᏺ)/ B2q(ᐂ,Ᏺ)and call it theL2-bounded cohomology ofᐂwith values inᏲ. We then have the following theorem.

Theorem2.2. For anyq >0, the natural map

H2q(ᐂ,Ᏺ)→Hq(Ω,Ᏺ) (2.8)

is an isomorphism.

We useTheorem 2.2as a pivot to proveTheorem 2.1, but the proof ofTheorem 2.2 is not given here, since it is similar to that of [2, Theorem].

3. A triangle of isomorphisms. LetΩbe as inTheorem 2.1. By the end of the sec-tionTheorem 2.1 will be proved. If U≠∅is an open set in ¯Ω, thenᏮp_Ω(U)is the Hilbert space of holomorphicp-formshonΩ∩Usuch that

hL2

(p,0)(Ω∩U)<∞. (3.1)

IfVis open in ¯Ωwith∅≠V⊂U, the restriction mapγVU:ᏮpΩ(U)→ᏮpΩ(V )is defined.

ThenᏮp0= {Ꮾ p

Ω(U),γVU}is the canonical presheaf ofL2-holomorphicp-forms on ¯Ω. The associated sheafᏮp2 is the sheaf of germs ofL2-holomorphicp-forms on ¯Ω. We then have the following lemma.

Lemma3.1. LetᏰp be the sheaf of germs of holomorphicp-forms on X, andᐂ a finite admissible cover ofΩ forᏰp_{. Then the following diagram is an isomorphism} triangle of cohomology groups:

Hq2

ᐂ,Ᏸp ∼ _Hq_Ω,Ᏸp

Hq_Ω¯_,Ꮾp₂

(3.2)

Proof. FromTheorem 2.2and the fact that any finite cover of ¯Ωhas a refinement ᐁ= {Vj}j∈Jsuch thatᐁΩ= {Vj∩Ω}j∈J is a finite admissible cover ofΩforᏰp, the lemma follows.

Now, using Hörmander’sL2_{-estimates locally we get the following lemma.}

Lemma3.2. The cohomology groupHq_(Ω¯,Ꮾp

2)is isomorphic to the quotient space _g

:g∈L2

(p,q)(Ω)and∂g¯ =0

_/∂h¯ :h∈L2

p,q−1(Ω)and∂h¯ ∈L2(p,q)(Ω)

_,

(3.3)

whereΩis as inTheorem 2.1.

Also the following lemma is proved in [3].

Lemma 3.3. If Ω X is Stein, where X is a complex manifold, then given f ∈ L2

(p,q)(Ω)with∂f¯ =0, there isu∈L2(p,q−1)(Ω)such that

¯

∂u=f , uL2

(p,q−1)(Ω)≤KfL2(p,q)(Ω), (3.4)

whereKdepends onΩ.

To finish with the proof ofTheorem 2.1 we remark that Ᏸ0_{=ᏻis the structure} sheaf ofX(X,Ωas inTheorem 2.1), thereforeTheorem 2.1follows from Lemmas3.1, 3.2, and3.3, and from Markoe’s and Silva’s condition.

References

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[3] ,L2_{estimates for the∂operator on Stein manifolds, Math. Proc. Cambridge Philos.} Soc.129(2000), no. 1, 73–76.

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[5] J. E. Fornæss and R. Narasimhan,The Levi problem on complex spaces with singularities, Math. Ann.248(1980), no. 1, 47–72.

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[7] ,Theory of Stein Spaces, Springer-Verlag, Berlin, 1979.

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[9] A. Markoe,Runge families and inductive limits of Stein spaces, Ann. Inst. Fourier (Greno-ble)27(1977), no. 3, 117–127.

[10] A. Silva,Rungescher Satz and a condition for Steinness for the limit of an increasing sequence of Stein spaces, Ann. Inst. Fourier (Grenoble)28(1978), no. 2, 187–200 (German).

[11] K. Stein, Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math. 7

[12] L. M. Tovar,Open Stein subsets and domains of holomorphy in complex spaces, Topics in Several Complex Variables (Mexico, 1983), Pitman, Massachusetts, 1985, pp. 183– 189.

Patrick W. Darko: Department of Mathematics and Computer Science, Lincoln Uni-versity, Lincoln UniUni-versity, PA19352, USA