Pseudoconvex non-Stein domains in primary Hopf surfaces
aa r X i v : . [ m a t h . C V ] N ov PSEUDOCONVEX NON-STEIN DOMAINS IN PRIMARYHOPF SURFACES
CHRISTIAN MIEBACHA
BSTRACT . We describe pseudoconvex non-Stein domains in pri-mary Hopf surfaces using techniques developed by Hirschowitz.
1. I
NTRODUCTION
Let H be a primary Hopf surface. In [LY12] Levenberg and Ya-maguchi characterize locally pseudoconvex domains D ⊂ H ha-ving smooth real-analytic boundary that are not Stein. In this notewe generalize their result to arbitrary pseudoconvex domains usingideas developed by Hirschowitz in [Hir74] and [Hir75]. For the rea-ders’ convenience these ideas are reviewed in a slightly generalizedform in Section 2. In Section 3 we review the structure of primaryHopf surfaces in order to describe a certain (singular) holomorphicfoliation F of H . This allows us to formulate the following MainTheorem, which is proven in Sections 4 and 5. Main Theorem.
Let D ⊂ H be a pseudoconvex domain. If D is not Stein,then D contains with every point p ∈ D the topological closure F p of theleaf F ∈ F passing through p. I would like to thank Karl Oeljeklaus for helpful discussions on thesubject of this paper and Stefan Nemirovski for a suggestion on howto prove Lemma 5.3. I am also grateful to Peter Heinzner and theSFB/TR 12 for an invitation to the Ruhr-Universit¨at Bochum wherea part of this paper has been written.2. A
REVIEW OF H IRSCHOWITZ ’ METHODS
In this section we present the methods developed by Hirschowitzin [Hir75] in a slightly more general setup.Let X be a complex manifold with holomorphic tangent bundle TX → X , and let π : P TX → X be the projectivized holomorphictangent bundle. A continuous function on X is called strictly pluri-subharmonic on X if it is everywhere locally the sum of a continuousplurisubharmonic and a smooth strictly plurisubharmonic function. Mathematics Subject Classification.
Definition 2.1.
Let ϕ ∈ C ( X ) be plurisubharmonic. Then we define S ( ϕ ) to be the set of [ v ] ∈ P TX such that ϕ is in a neighborhoodof π [ v ] the sum of a plurisubharmonic function and a smooth func-tion that is strictly plurisubharmonic on any germ of a holomorphiccurve defining [ v ] . Lemma 2.2.
Let ϕ ∈ C ( X ) be plurisubharmonic.(1) The set S ( ϕ ) is open in P TX.(2) If S ( ϕ ) = P TX, then ϕ is strictly plurisubharmonic on X.(3) If ∑ k ϕ k converges uniformly on compact subsets of X where ϕ k ∈C ( X ) , then we have S (cid:0) ∑ k ϕ k (cid:1) ⊃ S k S ( ϕ k ) .Proof. This is [Hir75, Proposition 1.3]. (cid:3)
For any plurisubharmonic function ϕ ∈ C ( X ) we define C ( ϕ ) tobe P TX \ (cid:8) [ v ] ∈ P TX ; ϕ is smooth around π [ v ] and ∂ϕ ( v ) = (cid:9) and then set(2.1) C ( X ) : = \ ϕ ∈C ( X ) plurisubharmonic C ( ϕ ) .Every set C ( ϕ ) (and thus C ( X ) ) is closed in P TX . The next lemma isa slight generalization of [Hir75, Proposition 1.5]. Lemma 2.3.
Let X be a complex manifold and let Ω : = X \ π (cid:0) C ( X ) (cid:1) .Then there exists a plurisubharmonic function ψ ∈ C ( X ) which is strictlyplurisubharmonic on Ω .Proof. Since π : P TX → X is proper, the set Ω is open in X . If Ω is empty, there is nothing to prove. Therefore let us suppose that Ω is a non-empty open subset of X . Consequently, P TX \ C ( X ) isnon-empty.For every [ v ] ∈ P TX \ C ( X ) we find a plurisubharmonic function ϕ [ v ] ∈ C ∞ ( X ) with ∂ϕ [ v ] v =
0. We claim that the function ψ [ v ] : = exp ◦ ϕ [ v ] is strictly plurisubharmonic in the direction of [ v ] . To seethis, we calculate ∂∂ψ [ v ] = e ϕ [ v ] (cid:0) ∂ϕ [ v ] ∧ ∂ϕ [ v ] + ∂∂ϕ [ v ] (cid:1) .In other words, we obtain [ v ] ∈ S ( ψ [ v ] ) . Since X has countable topo-logy, we get an open covering P TX \ C ( X ) ⊆ ∞ [ k = S ( ψ k ) .It is possible to find λ k > ∑ ∞ k = λ k ψ k converges uniformlyon compact subsets of X . To prove this, choose a countable exhaus-tion X = S j K j by compact sets with K j ⊂ ˚ K j + . For every j there are SEUDOCONVEX NON-STEIN DOMAINS IN PRIMARY HOPF SURFACES 3 λ k , j > ∞ ∑ k = λ k , j k ψ k k K j converges. Since k ψ k k K j ≤ k ψ k k K j + for every j , we may suppose that λ k , j ′ ≤ λ k , j for all j ≤ j ′ . Defining λ k : = λ k , k and noting that everycompact subset K ⊂ X is contained in K j for some j , we conclude ∞ ∑ k = λ k k ψ k k K ≤ ∞ ∑ k = λ k k ψ k k K j ≤ j ∑ k = λ k k ψ k k K j + ∞ ∑ k = j + λ k , j k ψ k k K j < ∞ which proves the claim. It follows that the limit function ψ : = ∑ k ψ k is continuous and satisfies S ( ψ ) ⊃ S k S ( ψ k ) ⊃ P TX \ C ( X ) , hence itis strictly plurisubharmonic on Ω . (cid:3) In the following we say that a complex manifold X is pseudocon-vex if there is a continuous plurisubharmonic exhaustion function ρ : X → R > . Lemma 2.4.
Let X be a pseudoconvex complex manifold and let γ : U → X be the integral curve of a holomorphic vector field on X where U is adomain in C . If γ ′ ( U ) meets C ( X ) , then γ ′ ( U ) is contained in C ( X ) . IfX admits a smooth plurisubharmonic exhaustion function, then γ ′ ( U ) ⊂ C ( X ) implies that γ ( U ) is relatively compact in X. In particular, in thiscase we have U = C .Proof. Let ξ be the holomorphic vector field on X with integral curve γ and suppose that γ ′ ( ) = ξ ( x ) ∈ C ( X ) . It is enough to show that0 is an inner point of the set of t ∈ U with γ ′ ( t ) ∈ C ( X ) , for thenthe closed set ( γ ′ ) − (cid:0) C ( X ) (cid:1) is also open, hence equal to U . In otherwords, we must prove that for every plurisubharmonic function ϕ ∈C ( X ) smooth in a neighborhood of x t : = γ ( t ) we have ξ ( ϕ )( x t ) = | t | is sufficiently small.To do this, choose α ∈ R > such that x = γ ( t ) ∈ X α : = (cid:8) x ∈ X ; ρ ( x ) < α (cid:9) where ρ is a continuous plurisubharmonic exhaustionfunction of X . Let Φ ξ be the holomorphic local flow of ξ . For | t | sufficiently small we have Φ ξ t ( X α + ) ⊃ X α ∋ x t : = Φ ξ t ( x ) .Since Φ ξ t : X α → X α + is holomorphic, ϕ t : = ϕ ◦ Φ ξ t is continuousplurisubharmonic on X α and smooth in a neighborhood of x foreach plurisubharmonic function ϕ ∈ C ( X α + ) that is smooth in aneighborhood of x t . Following the proof of [Hir75, Proposition 1.6]we construct a continuous plurisubharmonic function ψ t on X which CHRISTIAN MIEBACH coincides with ϕ t in a neighborhood of x . Choose β ∈ R such that ϕ t ( x ) < β < α and note that K : = ρ − ( β ) ⊂ X α is compact. Thenchoose a convex increasing function χ on R fulfilling χ (cid:0) ρ ( x ) (cid:1) < ϕ t ( x ) and χ ( β ) > k ϕ t k K .Finally, define ψ t : X → R by ψ t ( x ) : = ( max (cid:0) ϕ t ( x ) , χ ◦ ρ ( x ) (cid:1) : ρ ( x ) ≤ βχ ◦ ρ ( x ) : ρ ( x ) ≥ β .One checks directly that ψ t is continuous plurisubharmonic and co-incides with ϕ t in some neighborhood of x . Consequently, we maycalculate ξ ϕ ( x t ) = dds (cid:12)(cid:12)(cid:12)(cid:12) t ϕ (cid:0) Φ ξ s ( x ) (cid:1) = ξ ϕ t ( x ) = ξψ t ( x ) = ξ x = γ ′ ( ) ∈ C ( X ) . Therefore we see that γ ′ ( t ) ∈ C ( X ) forevery t ∈ U sufficiently close to 0, which proves the first part of thelemma.If ρ is smooth, then choosing ϕ = ρ in the argument given above,we see that γ ( U ) lies in a fiber of ρ , hence is relatively compact. (cid:3)
3. S
TATEMENT OF THE M AIN T HEOREM
Let us fix a , a ∈ C such that 0 < | a | ≤ | a | <
1. The automor-phism ϕ : C \ { } → C \ { } , ( z , z ) ( a z , a z ) , generates afree proper Z -action on C \ { } . By definition, the compact com-plex surface H a : = ( C \ { } ) / Z for a = ( a , a ) is a primary Hopfsurface . We will write [ z , z ] : = π ( z , z ) where π : C \ { } → H a isthe quotient map.The torus T = C ∗ × C ∗ acts holomorphically on H a with threeorbits. More precisely, we have H a = E ∪ H ∗ a ∪ E where H ∗ a : =( C ∗ × C ∗ ) / Z is the open T -orbit, and where E : = ( C ∗ × { } ) / Z = T · [
1, 0 ] and E : = ( { } × C ∗ ) / Z = T · [
0, 1 ] are elliptic curves.Note that H ∗ a is a connected Abelian complex Lie group which thuscan be represented as C / Γ where Γ is a discrete subgroup of rank3 of C . The map p : C → C / Γ ∼ = H ∗ a is the universal coveringof H ∗ a . Let V be the real span of Γ and set W : = V ∩ iV . There aretwo possibilities. Either p ( W ) is dense in V / Γ ∼ = ( S ) , or p ( W ) isclosed, hence compact, hence an elliptic curve E . In the first case, wehave O ( H ∗ a ) = C , i.e., H ∗ a is a Cousin group , while in the second case H ∗ a ∼ = C ∗ × E .For the following result we refer the reader to [BHPV04, Chap-ter V.18]. SEUDOCONVEX NON-STEIN DOMAINS IN PRIMARY HOPF SURFACES 5
Proposition 3.1.
The open orbit H ∗ a is not Cousin if and only if a k = a k for some relatively prime k , k ∈ Z .Remark. If there exist relatively prime integers k , k with a k = a k ,then we have the elliptic fibration H a → P , [ z , z ] [ z k : z k ] . Thegeneric fiber is the elliptic curve E = C ∗ / ( z ∼ cz ) where c : = a k = a k . Note that for a generic choice of a = ( a , a ) the open subset H ∗ a is a Cousin group.Suppose that H ∗ a is Cousin and let ξ ∈ t be the generator of therelatively compact one parameter subgroup p ( W ) . Let ξ H a be theholomorphic vector field induced by the T -action on H a . One checksdirectly that ξ H a has no zeros in H a , hence defines a holomorphicfoliation of H a . Note that the open subset H ∗ a is saturated with re-spect to F and that the leaves of F | H ∗ a are relatively compact in H ∗ a .The closure of a leaf F ⊂ H ∗ a in H ∗ a is a Levi-flat compact smoothhypersurface. In fact, these Levi-flat hypersurfaces are the fibers ofthe pluriharmonic function [ z , z ] log | z | log | a | − log | z | log | a | defined on H ∗ a ,see [LY12]. If H a is elliptic, then it is foliated by elliptic curves. Again, H ∗ a is saturated with respect to this foliation and the leaves are com-pact in H ∗ a . This shows that in both cases we obtain a (singular) holo-morphic foliation F of H a such that the leaves of F | H ∗ a are relativelycompact in H ∗ a .We now state the main result of this note. Theorem 3.2.
Let H a be a primary Hopf surface and let D ⊂ H a be apseudoconvex domain. If D is not Stein, then D contains with every pointp ∈ D the topological closure F p of the leaf F ∈ F passing through p.Remark. For locally pseudoconvex domains having smooth real-ana-lytic boundary this result has been obtained by Levenberg and Ya-maguchi using the theory of c -Robin functions, see [LY12].4. E XISTENCE OF PLURISUBHARMONIC EXHAUSTIONS
In this section we will show that every smoothly bounded locallypseudoconvex domain D ⊂ H a admits a continuous plurisubhar-monic exhaustion function. For this we will modify Hirschowitz’proof of [Hir74, Th´eor`eme 2.1]. Proposition 4.1.
Let D ⊂ H a be locally pseudoconvex and suppose thatneither E nor E is a component of ∂ D. Then D admits a continuousplurisubharmonic exhaustion function.Remark.
The hypothesis of Proposition 4.1 is fulfilled if D is locallypseudoconvex and smoothly bounded. Hence, Theorem 3.2 indeedgeneralizes the main result of [LY12]. CHRISTIAN MIEBACH
Proof.
We define Ω : = (cid:8) ( x , ξ ) ∈ D × t ; exp ( ξ ) · x ∈ D (cid:9) . By defi-nition, Ω is an open subset of D × t containing D × { } . Since D islocally pseudoconvex in H a , it follows that Ω is locally pseudocon-vex in D × t .We define the boundary distance d : D → R > by d ( x ) : = sup (cid:8) r > { x } × B r ( ) ⊂ Ω (cid:9) .It is elementary to check that d is lower semicontinuous. Since Ω islocally pseudoconvex in D × T , every point x ∈ D has an open Steinneighborhood U such that Ω ∩ ( U × t ) is pseudoconvex. Due to aresult of Lelong, see [Lel68, Theorem 2.4.2], the function − log d isplurisubharmonic on U ∩ D and therefore everywhere on D .Note that − log d ≡ − ∞ if and only if D contains H ∗ a . Since neither E nor E is a component of ∂ D , this implies D = H a so that we mayexclude this case in the following.For every x ∈ H a the orbit map t → T · x , ξ exp ( ξ ) · x , is openinto its image T · x . Thus we see that − log d ( x ) goes to infinity as x approaches a point in ∂ D . Since ∂ D is compact, this implies that − log d is an exhaustion.To end this proof, one verifies directly that d is upper semicontin-uous in any point x ∈ D such that d ( x ) = ∞ . Therefore, for everyconstant C > ( C , − log d ) is a continuous plurisubhar-monic exhaustion of D . (cid:3) Remark.
The proof of Proposition 4.1 shows that the polar set givenby {− log d = − ∞ } is non-empty if and only of D contains E or E . Remark.
In [DF82], Diederich and Fornæss give an example of a re-latively compact pseudoconvex domain in a P -bundle over a Hopfsurface that has smooth real-analytic boundary but does not admitan exhaustion by pseudoconvex subdomains.5. P ROOF OF T HEOREM
Lemma 5.1.
The domain H ∗ a admits a smooth plurisubharmonic exhaus-tion function. Consequently, if D ⊂ H a is pseudoconvex, then D ∗ : = D ∩ H ∗ a is likewise pseudoconvex. If H ∗ a is Cousin, this lemma follows from [Hu10, Proposition 2.4].If H ∗ a is not Cousin, then H ∗ a ∼ = C ∗ × E clearly has a smooth plurisub-harmonic exhaustion.Let D ⊂ H a be a pseudoconvex domain which is not Stein. If thepseudoconvex domain D ∗ ⊂ H ∗ a is not Stein, then D ∗ is saturatedwith respect to the foliation F | H ∗ a , see [GMO13, Theorem 3.1]. Since SEUDOCONVEX NON-STEIN DOMAINS IN PRIMARY HOPF SURFACES 7 the leaves of F are relatively compact orbits of a one parameter sub-group of T , continuity of the action map T × H a → H a implies that D is saturated with respect to F in this case. Therefore, let us assumethat D = D ∗ is Stein. We will complete the proof of Theorem 3.2 byshowing that then D is Stein as well.We note first that D cannot contain E or E if D ∗ is Stein. In-deed, due to the continuity of the leaves of F remarked above, if E was contained in D , then some of the relatively compact leaves of F would lie in D ∗ , contradicting the assumption that D ∗ is Stein.Let us consider the subset C ( D ) ⊂ P TD defined in (2.1) where π : P TD → D is the projectivized tangent bundle. The proof of thefollowing lemma relies essentially on the explicit knowledge of thestructure of primary Hopf surfaces. Lemma 5.2.
Let D ⊂ H a be a pseudoconvex domain. If D ∗ = D ∩ H ∗ a isStein, then we have π (cid:0) C ( D ) (cid:1) ⊂ D ∗ \ D.Proof.
Suppose that π (cid:0) C ( D ) (cid:1) meets D ∗ . Since H ∗ a is an Abelian com-plex Lie group, it has a biinvariant Haar measure. Therefore, we canapply the usual convolution technique in order to approximate thecontinuous plurisubharmonic exhaustion of D uniformly on com-pact subsets by smooth ones. This allows us to apply Lemma 2.4 toprove existence of a complex one parameter subgroup A of T and apoint x ∈ D ∗ such that A · x is relatively compact in D . Note that A · x is not relatively compact in D ∗ since the latter is assumed to beStein.We claim that that the closure of such a curve A · x in D (and hence D itself) would have to contain E or E , which then, as noted above,will contradict our assumption that D ∗ is Stein. In order to provethis claim, consider A = (cid:8) ( e tz , e tz ) ; t ∈ C (cid:9) where z = ( z , z ) ∈ C \ { } . If z = z =
0, we see directly that the closure of A · x contains E or E . Hence, suppose that z , z =
0. If z z / ∈ R ,already the closure of A in C \ { } contains { z = } ∪ { z = } ,thus the closure of A · x in D contains E ∪ E as well in this case.Therefore, we are left to deal with the case A = (cid:8) ( e t , e λ t ) ; t ∈ C (cid:9) where λ = z z ∈ R . Consider the smooth map π A : C ∗ × C ∗ → R > defined by π A ( w , w ) : = | w | λ | w | . The closure A of A in T = C ∗ × C ∗ iscontained in the kernel of π A . Since A · x is closed and non-compactin D ∗ , we conclude that π A ( a , a ) = | a | λ | a | is closed and non-compactin R > ; in particular we have | a | λ = | a | . Now choose t m ∈ C suchthat a m e t m = c for all m ∈ Z . It follows that (cid:12)(cid:12)(cid:12) a m e λ t m (cid:12)(cid:12)(cid:12) = (cid:18) | a || a | λ (cid:19) m | c | λ . CHRISTIAN MIEBACH
Since | a || a | λ =
1, we see that (cid:0) a m e λ t m (cid:1) converges to 0 for m → ∞ or m → − ∞ . This proves again that the closure of A · x in D contains E or E . (cid:3) Combining the Lemmas 5.2 and 2.3, we see that there exists aplurisubharmonic function ϕ on D which is strictly plurisubharmo-nic on D ∗ . Remark.
Due to [Ri68] (see the formulation given in [Dem12, ChapterI.5.E]), we may assume without loss of generality that ϕ is smooth on D ∗ .As we have noted above, D cannot contain E or E , so that D ∩ E and D ∩ E are closed Stein submanifolds of D . Therefore, thefollowing lemma implies that D is Stein, which then completes theproof of Theorem 3.2. Lemma 5.3.
Let X be a connected complex manifold endowed with a pluri-subharmonic exhaustion ϕ . Suppose that there exists a closed Stein sub-manifold A of X with at most finitely many connected components suchthat ϕ is strictly plurisubharmonic on X \ A. Then X is Stein.Proof.
For α ∈ R write X α : = { ϕ < α } . In the first step we will showthat X n is Stein for all n ≥
1. Since A is Stein, every A n : = A ∩ X n isa closed Stein submanifold of X n .Due to [Siu77], we find an open Stein neighborhood U n + of A n + in X n + . Consequently, there exists a strictly plurisubharmonic ex-haustion function ψ n + on U n + . Let us choose a relatively compactopen neighborhood V n of A n in U n + as well as a cutoff function χ n + which is identically 1 on V n and which vanishes near ∂ U n + . Then χ n + ψ n + : X → R is strictly plurisubharmonic in a neighborhood of A n and its Levi form is uniformly bounded from below on X n . More-over, let ρ n be a plurisubharmonic exhaustion of X n which is strictlyplurisubharmonic and smooth on X n \ A n . Then, for k sufficientlylarge the function χ n + ψ n + | X n + ke ρ n is a smooth strictly plurisub-harmonic exhaustion function of X n , proving that X n is Stein.Since we know now that X n is Stein, we can apply [Nar62, Corol-lary 1] which implies that X n − is Runge in X n for every n ≥ X = S n ≥ X n is a Runge exhaustion of X by relativelycompact Stein open subsets, hence X is Stein, see [GuRo65, Theo-rem VII.A.10]. (cid:3) R EFERENCES [BHPV04] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van deVen,
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