On minimal kernels and Levi currents on weakly complete complex manifolds
aa r X i v : . [ m a t h . C V ] F e b ON MINIMAL KERNELS AND LEVI CURRENTSON WEAKLY COMPLETE COMPLEX MANIFOLDS
FABRIZIO BIANCHI AND SAMUELE MONGODI Abstract.
A complex manifold X is weakly complete if it admits a continuousplurisubharmonic exhaustion function φ . The minimal kernels Σ kX , k ∈ [0 , ∞ ] (theloci where are all C k plurisubharmonic exhaustion functions fail to be strictly plurisub-harmonic), introduced by Slodkowski-Tomassini, and the Levi currents, introducedby Sibony, are both concepts aimed at measuring how far X is from being Stein. Wecompare these notions, prove that all Levi currents are supported by all the Σ kX ’s,and give sufficient conditions for points in Σ kX to be in the support of some Levicurrent.When X is a surface and φ can be chosen analytic, building on previous workby the second author, Slodkowski, and Tomassini, we prove the existence of a Levicurrent precisely supported on Σ ∞ X , and give a classification of Levi currents on X .In particular, unless X is a modification of a Stein space, every point in X is in thesupport of some Levi current. Introduction
Given an abstract (and possibly very complicated) manifold, a natural question iswhether it is possible to see it as a subset of a simpler space. In the real category, afundamental theorem by Nash states that this is always possible, and in a very strongsense: every Riemannian manifold can be isometrically embedded in some R N . Whenmoving to the complex category, we can then ask the following natural question: is itpossible to embed any complex manifold in some C N , by means of a holomorphic map?We call Stein a manifold for which the above holds true. This time, the rigidity ofholomorphic functions readily provides negative examples: for instance, the maximumprinciple implies that any holomorphic map on a compact complex manifold must beconstant, and thus the manifold cannot be Stein. A central question is then to under-stand when a given complex manifold is Stein. More specifically, given a dimension n , one would like to understand the obstructions for an n -dimensional manifold to beStein.A major advance in this direction was provided by Grauert [3]: a complex manifold isStein if and only if it admits a C strictly plurisubharmonic (psh for short) exhaustionfunction. The C assumption was relaxed to C by Narasimhan [12, 13]. In viewof these results, it is natural to tackle the question by studying the positive conePsh e ( X ) of all continuous psh exhaustion functions on X (or more generally the conePsh ke ( X ) := Psh e ( X ) ∩ C k for some k ∈ [0 , ∞ ], and in particular to find obstructionsfor them to be strictly psh. As a rough idea, such obstructions must correspond to thepresence of some sets in X along which all continuous psh functions must necessarily be Date : February 11, 2021.2010
Mathematics Subject Classification. pluriharmonic. As a prototypical example, the blow-up of a point and its correspondingexceptional divisor give precisely this kind of obstruction.A precise study of this kind of phenomena was started by Slodkowski and Tomassiniin [24] in the setting of weakly complete complex manifolds , i.e., manifolds admitting acontinuous psh exhaustion function. A crucial definition is the following: for k ∈ [0 , ∞ ]the minimal kernel of a manifold X (with respect to Psh ke ) is(1) Σ kX := { x ∈ X : i∂∂u is degenerate at x ∀ u ∈ Psh ke ( X ) } , i.e., the subset of X where no element of Psh ke can be strictly psh. A key result of [24]is that, whenever Psh ke ( X ) is not empty, there actually exists a function φ ∈ Psh ke ( X )(called minimal ) which fails to be strictly psh precisely on the minimal kernel Σ kX .Moreover, the minimal kernels are local maximum sets (see Definition 3.1). Somefiner properties are also established (some requiring at least the C regularity, see forinstance [24, Theorem 3.9]). Observe that the Σ kX ’s are increasing in k , but it is notknown whether equalities should occur in general, see for instance [23, Section 5.10].In [9, 10], the second author, Slodkowski, and Tomassini showed that, if X hascomplex dimension 2 and Psh ∞ e contains at least one real analytic function, the minimalkernel is either a union of countably many compact (and negative) curves or equal tothe whole manifold, by giving a full classification of the possible structures that sucha manifold can present. An important point here is that, although in general theminimal kernel does not have a priori an analytic structure, however its intersectionwith any level of a psh exhaustion function does (at least in dimension 2).In [20], Sibony introduced the notion of Levi current (see Definition 2.1), which isrelated to the (non-)existence of strictly psh functions on a complex manifold and thusto the problem of determining whether a given manifold is Stein, see also [15, 19, 21].Extremal Levi-currents are supported on sets where all continuous psh functions areconstant. In the case of infinitesimally homogeneous manifolds, a foliation is con-structed and linked to the obstructions to Steinness.Our goal here is to compare these two approaches, and in particular to use thenotion of Levi current on X to study the analytic structure of the minimal kernels Σ kX .In order to do this, let us denote by Psh k , for 0 ≤ k ≤ ∞ , the cone of C k psh functionon X and define the distribution E k in T X as(2) E k := { ( x, v ) ∈ T X : ( dϕ ) x ( v ) = 0 ∀ ϕ ∈ Psh k ( X ) } . A distribution is a subset of T X whose intersection with T x X is a (real) vector subspaceof the latter for every x ∈ X . In general, E k will not be a subbundle of T X , as dim E kx is not constant; however it is a closed subset of T X , hence the function x dim E kx isupper semicontinuous.The following is our main result. Theorem 1.1.
Let X be a weakly complete complex manifold and T a Levi current on X . Denote by Σ kX the minimal kernels of X and by F the union of the supports of allLevi currents on X . Then F ⊆ Σ kX for all k ≥ and (1) if T has compact support K T , then K T is a local maximum set; (2) if K is a local maximum set, then there exists a Levi current supported on K .In particular, K ∩ F = ∅ . N MINIMAL KERNELS AND LEVI CURRENTS 3
Moreover, if X is a surface, φ ∈ C k a psh exhaustion function, and Y a regularconnected component of a level set { φ = c } , (3) if ≤ k ≤ ∞ and U ⊆ Σ kX is an open set in Y and there exists x ∈ U suchthat dim E kx = 2 , then X is a union of compact complex curves. In particular, F = Σ kX = X ; (4) if ≤ k ≤ ∞ and Y ⊆ Σ kX , then there exists c ′ < c such that the connectedcomponent of φ − ([ c ′ , c ]) containing Y is contained in F . In particular, Y ⊆ F . Moreover, given any psh function u ∈ Psh ( X ), any Levi current T can be naturallydisintegrated as T = R T c dµ , where µ is a positive measure on R and T c is a Levicurrent supported on { u = c } , see Corollary 2.3. This in particular gives examplesof Levi currents for which Item (1) applies. Notice that, whenever two level sets of u, v ∈ Psh ( X ) do not coincide, this allows to further refine the description of theextremal Levi current. This motivates the definition (2) of the distributions E k .It follows from [9, 24] that, when k ≥
2, for any level set Y of an exhaustion function φ ∈ Psh e , the set Σ kX ∩ Y is a local maximum set (or empty), see Lemma 3.4. HenceItem (2) applies for instance to such sets. Finally, the manifold X = C × P providesan example where the Items (3) and (4) apply. Remark 1.2.
It would be interesting to know if the equality holds in Item (2) (andin particular for the intersections between levels sets of a psh exhaustion function andthe minimal kernel). Namely if, for any point in a local maximum set K , there existsa Levi current T such that x ∈ spt T .The paper is organized as follows. In Section 2 we recall the definition of Levicurrents and the properties that we will need in the sequel. In Section 3 we proveItems (1) and (2) of Theorem 1.1. The first item is established for Σ kX ∩ { φ = c } (where c is an attained value for a continuous minimal function φ and k ≥
2) in[24, Theorem 3.6], and is actually a consequence of [19, Theorem 3.1], where it isproved through an integration by parts, see also [20, Proposition 4.2] for an analogousstatement for F . We give here a different proof by means of a characterization ofthe local maximum property due to Slodkowski [22] which allows to bypass the useof Brebermann functions and Jensen measures as in [24]. In Section 4 we study therelation between the minimal kernels Σ kX and distributions in the tangent bundle T X given by directions satisfying some degeneracy condition. This leads to the proof ofItem (3). The proof of Theorem 1.1 is completed in Section 5, where we establish Item(4). In Section 6 we consider the case where X is a surface and the exhaustion functionin Theorem 1.1 can be chosen analytic. By exploting the main result in [9], we deducea classification of Levi currents in this case. Acknowledgements.
The authors would like to thank Nessim Sibony for the refer-ences and for very useful comments that also helped to improve Item (2) in the maintheorem, and Zbigniew Slodkowski for very helpful remarks on a preliminary version ofthis paper. They also would like to thank Viˆet-Anh Nguyˆen and Giuseppe Tomassinifor useful observations and discussions.This work was supported by the Research in Pairs 2019 program of the CIRM (Cen-tro Internazionale di Ricerca Matematica), Trento and the FBK (Fondazione BrunoKessler). The authors would like to warmly thank CIRM-FBK for their support,
F. BIANCHI AND S. MONGODI hospitality, and excellent work conditions. This project also received funding fromthe I-SITE ULNE (ANR-16-IDEX-0004 ULNE), the LabEx CEMPI (ANR-11-LABX-0007-01) and from the CNRS program PEPS JCJC 2019.2.
Levi currents on complex manifolds
In this section we recall the definition of Levi current and give the properties thatwe need in the sequel. These results are essentially contained in [20, Section 4] and[21, Section 3], we sketch here the proofs for completeness. We let X be any com-plex manifold and we denote by Psh ( X ) the space of continuous plurisubharmonicfunctions on X . Definition 2.1 (Sibony [20]) . A current T on X is called Levi current if(1) T is non zero;(2) T is of bidimension (1 , T is positive;(4) i∂∂T = 0;(5) T ∧ i∂∂u = 0 for all u ∈ Psh ( X ).A Levi current T is extremal if T = T = T whenever T = ( T + T ) / T , T Levi currents.
Lemma 2.2.
Take u ∈ Psh ( X ) and let T be a Levi current. The currents T ∧ ∂u , T ∧ ∂u , and T ∧ ∂u ∧ ∂u are all well defined and vanish identically on X .Proof. The currents in the statement are well defined when u is smooth, and thearguments from [2, Section 2] and [20, Section 4] prove the good definition for u ∈ Psh ( X ).If u ∈ Psh ( X ), then also exp( u ) ∈ Psh ( X ). So, by Definition 2.1 of Levi current, T ∧ i∂∂ exp( u ) vanishes identically. Hence, we have0 = exp( u ) T ∧ i∂∂u + i exp( u ) T ∧ ∂u ∧ ∂u . Given that T ∧ i∂∂u = 0, we conclude that T ∧ ∂u ∧ ∂u = 0. This gives the lastidentity. We prove now the first one, the proof for the second one is similar. Since T is positive, for any (0 , α by Cauchy-Schwarz’s inequality we get |h T, ∂u ∧ α i| ≤ c h T, ∂u ∧ ¯ ∂u i · h T, α ∧ ¯ α i where c is a constant independent of T, u , and α . Since the first factor in the RHS iszero by the first part of the proof, the assertion follows. (cid:3) By a standard disintegration procedure, we obtain the following consequence.
Corollary 2.3.
Suppose T is a Levi current and u ∈ Psh ( X ) ; then there exists ameasure µ on R and a collection of currents T c , c ∈ R such that • T c is supported on Y c = { x ∈ X : u ( x ) = c } for all c ∈ R ; • T c is non zero for µ -almost every c ∈ R ; • whenever T c = 0 , T c is a Levi current; N MINIMAL KERNELS AND LEVI CURRENTS 5 • for every -form α on X we have h T, α i = Z R h T c , α i dµ ( c ) . Moreover, if u ∈ Psh ( X ) and c is a regular value for u , then T c = j ∗ S c , where j isthe inclusion of Y c in X and S c a current on the real manifold Y c . Notice that T c needs not be extremal, as it is easily seen considering X = C × P . Remark 2.4.
Suppose now that X is weakly complete and let φ be a psh exhaustionfunction. By Corollary 2.3, every Levi current is obtained by averaging Levi currentswhich are supported on the level sets of φ ; as the latter is an exhaustion of X , itslevel sets are compact, so every Levi current on X is an integral average of compactlysupported Levi currents, i.e., positive currents of bidimension (1 ,
1) which are ∂∂ -closedand compactly supported. Corollary 2.5. If T is a Levi current and u ∈ Psh ( X ) , then the vector field associatedto T is tangent to the kernel of ∂u ∧ ∂u , whenever the latter is non-zero (and the formeris defined). Moreover, if there exists v ∈ Psh ( X ) which is strictly plurisubharmonicat a point x ∈ X , then x spt T for any Levi current T .Proof. The first statement is equivalent to T ∧ ∂u ∧ ∂u = 0, hence follows from Lemma2.2. Suppose now that we have v ∈ Psh ( X ) which is strictly psh at x and a Levicurrent T . First, by Richberg [16, Satz 4.3] we can assume that v is C ∞ and strictlypsh near x . Then, if ρ ∈ C ∞ ( X ) is supported in a neighbourhood of x where v is strictlypsh and k ρ k C is small enough, then also v + ρ is psh. In particular, we can choose ρ i , ≤ i ≤ n such that the ker( ∂ ( v + ρ i ) ∧ ∂ ( v + ρ i )) x are independent (over R ). Thisproperty holds true in a neighbourhood of x . Hence, as the vector field associated to T (on a full measure subset of the support T for the mass measure) should belong to allthese subspaces, the only possibility is that x spt T . This concludes the proof. (cid:3) Lemma 2.6.
Let T be a Levi current such that K = spt T is compact. If u is definedand plurisubharmonic in an open neighbourhood V of K and strictly plurisubharmonicat x ∈ V , then x spt T .Proof. Let V ′ ⋐ V be an open neighbourhood of K containing x . Let χ ∈ C ∞ ( V )be such that χ | V ′ ≡
1, then χu is defined on X and psh on V ′ . As spt T ⊆ V ′ , alsospt( T ∧ i∂∂u ) ⊆ V ′ , so T ∧ i∂∂u is positive; moreover, as T is a Levi current, we have i∂∂T = 0, hence 0 = h i∂∂T, u i = h T, i∂∂u i = h T ∧ i∂∂u, i . Therefore, T ∧ i∂∂u = 0 as a (positive) measure.Since ( i∂∂u ) x >
0, this happens in a neighbourhood of x , so T ∧ i∂∂u is strictlypositive in a neighbourhood of x unless T is zero there. This gives x spt T andconcludes the proof. (cid:3) Lemma 2.7.
Suppose that a current T satisfies requests − of Definition 2.1 and T has compact support. Then T is a Levi current.Proof. Given that T is compactly supported, so are uT , T ∧ ∂u , T ∧ ∂u , and T ∧ i∂∂u for all u ∈ Psh ( X ). Moreover, as T is positive and u is psh, T ∧ i∂∂u is a positivemeasure on X ; therefore, it is zero if and only if h T ∧ i∂∂u, i = 0. F. BIANCHI AND S. MONGODI
Notice that, by Stokes’ theorem, we have h i∂∂ ( uT ) , i , hence0 = −h i∂∂u ∧ T, i + h ui∂∂T, i + h i∂ ( ∂u ∧ T ) , i − h i∂ ( ∂u ∧ T ) , i . We have i∂∂T = 0 by hypothesis, while h i∂ ( ∂u ∧ T ) , i and h i∂ ( ∂u ∧ T ) , i vanish byanother application of Stokes’ theorem. Therefore T ∧ i∂∂u = 0, that is, T is a Levicurrent. (cid:3) Local maximum sets
We establish here Items (1) and (2) of Theorem 1.1. We recall the following defini-tion, see also [24, Section 2] and [17].
Definition 3.1.
Let X be a complex manifold and K ⊂ X be compact. We say that K is a local maximum set if every x ∈ K has a neighbourhood U with the followingproperty: for every compact set K ′ ⊂ U and every function ψ which is strictly psh ina neighbourhood of K ′ , we have max K ∩ K ′ ψ = max K ∩ bK ′ ψ. Proposition 3.2.
Suppose that X is weakly complete. If T is a Levi current withcompact support, then spt T is a local maximum set.Proof. Suppose that K := spt T is not a local maximum set. By [22, Proposition 2.3]there exist x ∈ K , a neighbourhood B of x , with local coordinates z with origin in x ,such that B ≡ { z ∈ C n : k z k < } , and ψ : B → R strictly psh with ψ ( x ) = 0 and ψ ( y ) ≤ − ǫ k z ( y ) k for all y ∈ K ∩ B . Up to replacing ψ by an element of a continuousapproximating sequence, we can directly assume that ψ is continuous. By taking apossibly smaller ball a x , we can also assume that − ǫ k z ( y ) k − ǫ/ ≤ ψ ( y ) on K . Set A := { y ∈ B : k z ( y ) k > / } and V = { y ∈ B : | ψ ( y ) + ǫ k z ( y ) k | < ǫ/ } . By the continuity of ψ and the bounds above, V is an open subset of B containing K .Consider u ∈ Psh ( X ) such that u ( x ) = − ǫ/ B | u | < ǫ/ X ). Since K ∩ B ⊂ V , there also exists χ ∈ C ∞ ( X ) besuch that χ | K ≡ χ ∩ B ⊂ V . Define the function v : X → R as v = ( χ max { u, ψ } on B,χu on X \ B. We claim that v = χu on A . Indeed, for every p ∈ spt χ ∩ A , we have that p ∈ V , andso ψ ( p ) < − ǫ/ ǫ/ − ǫ/
2. Hence, ψ ( p ) < u ( p ) and v ( p ) = ( χu )( p ).By construction, v coincides with ψ in a neighbourhood of x . It follows that v ispsh in a neighbourhood of K and strictly psh in x . Therefore, we have x spt T byLemma 2.6. This gives a contradiction with the choice of x ∈ K and completes theproof. (cid:3) Proposition 3.3.
Let K ⊂ X be a local maximum set. There exists a Levi current T such that spt T ⊆ K .Proof. By [21, Theorem 3.1] (see also [20, Section 4]) and Lemma 2.7, if there areno Levi currents supported on K , there exists a smooth strictly psh function u onsome open neighbourhood U of K . By slightly perturbing u , for every x ∈ K wecan construct 2 n continuous strictly psh functions on some neighbourhood K ⊂ U ′ ⊆ N MINIMAL KERNELS AND LEVI CURRENTS 7 U such that du , . . . , du n are linearly independent at x . This implies that, in aneighbourhood of x , we have { u = u ( x ) } ∩ . . . { u n = u n ( x ) } = { x } . By [23, Corollary 1.11] and [22, Theorem 4.2], for every family of continuous pshfunctions on U ′ there exists a local maximum set K ′ ⊆ K with the property that allfunctions of the family are constant on K ′ . Choosing a point x ∈ K ′ , the previousparagraph gives that x is isolated in K ′ . This is a contradiction, and the proof iscomplete. (cid:3) We conclude the section with the following result, that we will need to prove Item(3) of Theorem 1.1.
Lemma 3.4.
Let X be a weakly complete complex surface and Y a regular level for a C exhaustion function φ . Then, for all k ≥ , Σ kX ∩ Y is a local maximum set and, forall local maximum sets K ⊆ Σ kX , K is foliated by holomorphic discs, i.e., it is locallya union of disjoint holomorphic discs.Proof. The intersection Σ kX ∩ Y is a local maximum set by [9, Theorem 3.2]. Asobserved in [9, Proposition 3.5], the proof of the lemma is then essentially given in[24, Lemma 4.1], see in particular the Assertion in the proof of that lemma. (cid:3) We point out that [24, Lemma 4.1] relies on a result by Shcherbina [18], which holdstrue only in dimension 2; this is the reason for restricting ourselves to the case ofsurfaces. it would be interesting to prove (or disprove) a similar statement in higherdimension. 4.
Kernels and tangent directions
In this section we let X be a weakly complete complex manifold of dimension n andassume that Psh k ( X ) contains at least one exhaustion function φ : X → R for some2 ≤ k ≤ ∞ . Recall that the minimal kernel Σ kX of X is defined as in (1) and thedistribution E k of T X as in (2). We consider further the distribution S k of T X givenby(3) S k := { ( x, ξ ) ∈ T X : ξ ∈ T x X, ( i∂∂u ) x ( ξ, ξ ) = 0 ∀ u ∈ Psh k ( X ) } . Similar objects have already appeared in relation to the study of the Levi problem, seefor instance [5] in the case of homogeneous manifolds and [4, 24]. We also set E kℓ := { x ∈ X : dim E kx ≥ ℓ } and S k := { x ∈ X : dim S kx ≥ } . By definition, E kℓ ⊆ E kℓ − and E kℓ is closed in E kℓ − for all ℓ ≥
1. Observe moreoverthat S k is a complex distribution. Remark 4.1.
Let T be a Levi current. Then, for almost every point of the supportof T (with respect to the mass measure), the vector field associated to T at x belongsto the fibre S kx of S k at x . Proposition 4.2.
We have S k ⊆ E k , and S k = E k = E k = Σ kX . F. BIANCHI AND S. MONGODI
Proof.
It follows from the definition of E k that E k = E k . Moreover, if ( x, v )
6∈ E k ,there exists ψ ∈ Psh k ( X ) such that ( dψ ) x ( v ) = 0; then i∂∂ exp( ψ ) = exp( ψ ) i∂∂ψ + i exp( ψ ) ∂ψ ∧ ∂ψ ≥ , which implies that ( i∂∂ exp( ψ )) x ( v, v ) ≥ exp( ψ ( x )) | ∂ψ x ( v ) | > x, v )
6∈ S k .It follows that S k ⊆ E k .We now prove that E k = Σ kX . If x Σ kX , then there exists ψ ∈ Psh ke ( X ) which isstrictly psh around x ; therefore, given any ρ : X → R smooth with compact supportnear x , there exists ǫ > ψ + ǫρ is still psh. So, we can construct psh functionsof class C k whose differentials span the tangent space at x , which implies that thesedifferentials do not have any nontrivial common kernel in T x X . So E kx = { } , hence E k ⊆ Σ kX .On the other hand, if E kx = { } , given v , . . . , v n ∈ T x X linearly independent, we canchoose psh functions ψ ij , i, j = 1 , . . . , n of class C k and such that ( dψ ij ) x ( v i + v j ) = 0.Therefore, the function ψ := P ni,j =1 ψ ij has positive defined Levi form at x . Addingto the exhaustion function φ suitable multiples of ψ , we see that x Σ kX . This gives E k ⊇ Σ kX , hence E k = Σ kX .In order to conclude, we need to prove that S k ⊇ Σ kX . Take x ∈ Σ kX and supposeby contradiction that, for every v ∈ T x X there is ϕ v : X → R which is C k , psh, andsuch that ( dd c ϕ v ) x ( v, v ) >
0. Then, as above, we can construct a C k function ψ whichis strictly psh at x . This gives the desired contradiction and completes the proof. (cid:3) The following result gives Item (3) of our main Theorem 1.1.
Proposition 4.3.
Let X be a weakly complete complex surface, φ a C k , ≤ k ≤ ∞ ,exhaustion psh function and Y a regular connected component of a level set { φ = c } of φ . Suppose that U ⊆ Y is an open set in Y and U ⊆ Σ kX . If there exists x ∈ U such that dim E kx = 2 , then X is a union of compact complex curves. In particular, Σ kX = F = X = E k , and E k is contained in a (possibly empty) analytic subset of thesingular levels for φ . We will need the following theorem by Nishino, see [14, Proposition 9 and Th´eor`emeII] and [11, Section 2.2.1].
Theorem 4.4 (Nishino) . Let X be a weakly complete or compact surface that containsan uncountable family F of disjoint connected compact complex curves. Then thereexist a Riemann surface R and a meromorphic map h : X → R with compact fibers.Proof of Proposition 4.3. By Theorem 4.4, to prove the first assertion it is enough toshow that X contains uncountably many disjoint compact complex curves.Since x ∈ U is such that dim E kx = 2, there exists ψ ∈ Psh ke ( X ) such that ( dφ ) x and ( dψ ) x are linearly independent; hence, the map ψ | Y : Y → R is not constant.Since k ≥
4, by Sard’s theorem we can find regular values b for ψ arbitrarily closeto b := ψ ( x ), therefore the sets C b = { y ∈ Y : ψ ( y ) = b } intersect the open set U ⊆ Σ kX .For any y ∈ C b ∩ Σ kX , by Proposition 4.2 we have T y C b = E ky = S ky . Therefore, C b ∩ Σ kX is a complex curve, being a real, smooth 2-dimensional manifold with complex tangentspace. On the other hand, the set C b \ Σ kX is open in C b . Let z ∈ C b be a boundarypoint (with respect to C b ); as z ∈ Σ kX , by Lemma 3.4 there is a holomorphic disc N MINIMAL KERNELS AND LEVI CURRENTS 9 f : D → X such that f ( D ) ⊂ Y and f (0) = z . If ζ ∈ D is close enough to 0, then,setting w = f ( ζ ), we have w ∈ Σ kX , and ( dφ ) w and ( dψ ) w independent. This gives w ∈ E k \ E k , which in turn implies that E kw = T w C ψ ( w ) . Therefore f ( D ) coincideslocally with a leaf C b ′ . Hence C b is contained in Σ kX , so it is a compact complex curve.As b was taken arbitrarily among the regular values close enough to b , we finduncountably many disjoint (since they correspond to distinct values) compact complexcurves in X , as desired.In order to conclude, we need to prove the final assertion on E k . We proved abovethat there exists a meromorphic map h : X → R with compact fibres, where R isRiemann surface. It is enough to prove that E k ⊆ { h ′ = 0 } .Let x ∈ X be such that h ′ ( x ) = 0. Consider a strictly psh exhaustion function ψ for R (which we can assume to be C ∞ near x by [16]) and the family of functions F := { ψ + ǫρ } , where ρ is a smooth function compactly supported near h ( x ). Forevery such ρ , ψ + ǫρ is still strictly psh for ǫ sufficiently small. Thus, we can obtain aset of generators for the tangent space given by differentials at h ( x ) of psh functionsin F . Pre-composing the corresponding functions with h , we obtain that the space ofdifferentials at x of psh functions on X has dimension at least 2. Hence, x / ∈ E k , andthe proof is complete. (cid:3) Remark 4.5.
Suppose that X is a surface and Y a regular level for an exhaustionfunction φ ∈ Psh e ( X ). Let K ⊆ Y ∩ Σ X be a local maximum set. By Lemma 3.4, Y and K are foliated by holomorphic discs. For every such disk, its tangent bundleis exactly the restriction of S . By [1, Theorem 1.4], there exists a ∂∂ -closed positivecurrent of bidimension (1 , S , supported in K . By Lemma 2.7, suchcurrent is a Levi current. This gives a different proof of Item (2) when dim X = 2.5. End of the proof of Theorem 1.1
It follows from Corollary 2.5 (or Lemma 2.6) that spt T ⊆ Σ kX for every Levi current T and all k ≥
0. Thus, we have F ⊆ Σ kX for all k ≥
0. Moreover, Items (1), (2), and(3) follow from Propositions 3.2, 3.3, and 4.3, respectively.Let now Y be a regular connected component of a level set for an exhaustion pshfunction φ ∈ Psh ke ( X ) for some k ≥
2. The remaining item follows from the nextproposition.
Proposition 5.1. If k ≥ and Y ⊆ Σ kX , there exists c ′ < c such that the connectedcomponent of φ − ([ c ′ , c ]) containing Y is contained in F .Proof. We assume for simplicity that the level { φ = c } is regular and connected, theargument is similar otherwise. Since k ≥
2, by [24, Theorem 3.9] there is c ′ < c suchthat, setting K = { x ∈ X : c ′ ≤ φ ( x ) ≤ c } , the form ( dd c φ ) vanishes on the interior of K , hence on K . So, we have K ⊆ Σ kX .Consider the current T given by T := i∂φ ∧ ∂φ. It is clear that T is a current of bidimension (1 , φ . Moreover, i∂∂T is induced by the form i∂∂ ( i∂φ ∧ ∂φ ) = − ( ∂∂φ ) . So, i∂∂T vanishes where φ is not strictly psh, hence on Σ kX . Let B be the interior of K , then the restriction of T to B is a current of bidimension (1 , ∂∂ -closed(in B ); moreover, given u ∈ Psh ( X ), we have that T ∧ ∂∂u = 0 on Σ kX , so T is a Levicurrent.By construction and Lemma 2.2 we have T ∧ ∂φ = 0, so we can disintegrate T alongthe levels of φ , see Corollary 2.3: there exist currents T s with s ∈ ( c ′ , c ), such that, for α a 2-form with spt α ⊂ B , h T, α i = Z cc ′ h T s , α i dµ ( s ) for some measure µ on ( c ′ , c ) . Since φ ∈ C , the measure µ is absolutely continuous with respect to the Lebesguemeasure on ( c ′ , c ).As T is ∂∂ -closed in B , so is µ -almost every T s in B ; therefore, for a dense open setof s ∈ ( c ′ , c ), T s is a positive, ∂∂ -closed current of bidimension (1 ,
1) andspt T s = { x ∈ X : φ ( x ) = s } . The set in the RHS is compact since φ is an exhaustion function. By Lemma 2.7, T s is a Levi current.In conclusion, the level set { x ∈ X : φ ( x ) = s } is contained in F for almost all s ∈ ( c ′ , c ), so φ − ([ c ′ , c ]) ⊆ F , as F is closed. In particular, Y ⊆ F . (cid:3) The proof of Theorem 1.1 is complete.6.
Real analytic exhaustion function
A classification of those weakly complete complex surfaces X admitting an analyticexhaustion function is given in [9]. As a direct consequence, we can get an analogouscomplete classification of the possible Levi currents in this setting.First notice that each exceptional divisor V in X corresponds to an extremal Levicurrent given by the current of integration [ V ]. Without loss of generality, to simplifyour next statement, we can thus assume that X has no such divisors on the regularlevels of α . The statement for a general X is then a direct consequence. Theorem 6.1.
Let X be a weakly complete complex surface admitting an analyticexhaustion function α . Assume that X has no exceptional divisors on the regularlevels of α . Then one of the following possibilities hold: (1) X is Stein (and so, admits no Levi currents); (2) F = Σ ∞ X = X = ∪ V i , where all the V i are (disjoint) connected compact curves,and all extremal Levi currents are of the form λ [ V ′ i ] for some positive λ , with V ′ i an irreducible component of some V i ; (3) F = Σ ∞ X = X , every regular level Y c of α is foliated by curves U i , and thesupport of any extremal Levi currents on Y c is equal to (a connected componentof ) Y c . Observe also that, although a priori we would only have Σ kX ⊆ Σ ∞ X for all k ≥ kX = Σ ∞ X for all k ≥ Proof.
It follows from [9, Theorem 1.1] that one of the following possibilities holds:(1) X is a Stein space; N MINIMAL KERNELS AND LEVI CURRENTS 11 (2) X is proper over a (possibly singular) complex curve;(3) the connected components of the regular levels of α are foliated with densecomplex curves.In the first and second cases, the assertion follows from the characterization of Levicurrents given in Section 2. In the third case, a Levi current can be constructed, forinstance, by means of [1, Theorem 1.4]. By proposition 3.2, the support of any Levicurrent is a local maximum set. By [7, Lemma 3.3], a local maximum set contained ina Levi-flat hypersurface must be a union of leaves of the Levi foliation.Hence, in the third case, any Levi current on a regular level set of the exhaustionfunction is supported on the whole level set, as all the leaves of the Levi foliation aredense. This in particular applies to extremal Levi currents. The proof is complete. (cid:3) References [1] B. Berndtsson and N. Sibony,
The ∂ -equation on a positive current , Inventiones Mathematicae (2002), no. 2, 371–428, DOI 10.1007/s002220100178.[2] T.-C. Dinh and N. Sibony, Pull-back of currents by holomorphic maps , Manuscripta Mathematica (2007), no. 3, 357–371, DOI 10.1007/s00229-007-0103-5.[3] H. Grauert,
On Levi’s problem and the imbedding of real-analytic manifolds , Annals of Mathe-matics (1958), no. 2, 460–472, DOI 10.2307/1970257.[4] T. Harz, N. Shcherbina, and G. Tomassini, On defining functions for unbounded pseudoconvexdomains , Mathematische Zeitschrift (2017), 987–1002, DOI 10.1007/s00209-016-1792-9.[5] A. Hirschowitz,
Le probleme de L´evi pour les espaces homogenes , Bulletin de la Soci´et´eMath´ematique de France (1975), 191–201, DOI 10.24033/bsmf.1801.[6] S. Mongodi,
Weakly complete domains in Grauert type surfaces , Annali di Matematica Pura edApplicata (1923 -) (2019), no. 4, 1185–1189, DOI 10.1007/s10231-018-0814-0.[7] S. Mongodi and Z. Slodkowski,
Domains with a continuous exhaustion in weakly complete surfaces ,Mathematische Zeitschrift (2020), 1011–1019, DOI 10.1007/s00209-020-02466-z.[8] S. Mongodi, Z. Slodkowski, and G. Tomassini,
On weakly complete surfaces , Comptes RendusMathematique (2015), no. 11, 969 – 972, DOI 10.1016/j.crma.2015.08.009.[9] ,
Weakly complete complex surfaces , Indiana University Mathematics Journal (2018),no. 2, 899 – 935, DOI 10.1512/iumj.2018.67.6306.[10] , Some properties of Grauert type surfaces , International Journal of Mathematics (2017),no. 8, 1750063 (16 pages), DOI 10.1142/S0129167X1750063X.[11] S. Mongodi and G. Tomassini, Minimal kernels and compact analytic objects in complex surfaces ,Advancements in Complex Analysis, 2020, pp. 329–362, DOI 10-1007/978-3-030-40120-7 9.[12] R. Narasimhan,
The Levi problem for complex spaces , Mathematische Annalen (1961), no. 4,355–365, DOI 10.1007/BF01451029.[13] ,
The Levi problem for complex spaces II , Mathematische Annalen (1962), no. 3, 195–216, DOI 10.1007/BF01470950.[14] T. Nishino,
L’existence d’une fonction analytique sur une vari´et´e analytique complexe `a deuxdimensions , Publications of the Research Institute for Mathematical Sciences (1982), no. 1,387–419, DOI 10.2977/prims/1195184029 (French).[15] T. Ohsawa and N. Sibony, Bounded psh functions and pseudoconvexity in K¨ahler manifold , NagoyaMathematical Journal (1998), 1–8, DOI 10.1017/S0027763000006516.[16] R. Richberg,
Stetige streng pseudokonvexe Funktionen , Mathematische Annalen (1967), no. 4,257–286, DOI 10.1007/BF02063212.[17] H. Rossi,
The local maximum modulus principle , Annals of Mathematics (1960), no. 1, 1–11,DOI 10.2307/1970145.[18] N. Shcherbina, On the polynomial hull of a graph , Indiana University Mathematics Journal (1993), no. 2, 477–503, DOI 10.1512/iumj.1993.42.42022.[19] N. Sibony, Pfaff systems, currents and hulls , Mathematische Zeitschrift (2017), no. 3-4, 1107–1123, DOI 10.1007/s00209-016-1740-8. [20] ,
Levi problem in complex manifolds , Mathematische Annalen (2018), 1047–1067, DOI10.1007/s00208-017-1539-x.[21] ,
Pseudoconvex domains with smooth boundary in projective spaces , MathematischeZeitschrift, posted on 2020, DOI 10.1007/s00209-020-02613-6.[22] Z. Slodkowski,
Local maximum property and q-plurisubharmonic functions in uniform algebras ,Journal of mathematical analysis and applications (1986), no. 1, 105–130, DOI 10.1016/0022-247X(86)90027-2.[23] ,
Pseudoconcave decompositions in complex manifolds , Contemporary Mathematics, Ad-vances in Complex geometry, 2019, pp. 239–259, DOI 10.1090/conm/735/14829.[24] Z. Slodkowski and G. Tomassini,
Minimal kernels of weakly complete spaces , Journal of FunctionalAnalysis (2004), no. 1, 125–147, DOI 10.1016/S0022-1236(03)00182-4. CNRS, Univ. Lille, UMR 8524 - Laboratoire Paul Painlev´e, F-59000 Lille, France
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