CONTINUITY PRINCIPLE AND EXTENSION PROPERTIES OF MEROMORPHIC MAPPINGS WITH VALUES IN NON KÄHLER

MANIFOLDS

S. IVASHKOVICH

Abstract. In this paper we are proving an analogue of E. Levi Continuity Principle for meromorphic mappings with values in general complex spaces. We also describe the singularities of meromorphic mappings into complex spaces carrying pluriclosed Hermitian metric forms and geometrical obstructions to removing them.

Contents

0. Introduction 2 0.1. Continuity principle 2 0.2. Hartogs-type extension theorem and spherical shells 5 0.3. Kähler case: Griffiths approach revisited 8 0.4. Applications, generalisations, open questions 9 1. Continuity principle. 11 1.1. Cycle space associated to a meromorphic map 11 1.2. Analyticity of C_{f,C} 14 1.3. Proof of the Continuity principle 15 1.4. Spaces from G_{k} have bounded cycle geometry 17 1.5. Construction of Example 1 18 2. Hartogs-type extension and spherical shells 19 2.1. Generalities on pluripotential theory 19 2.2. Metrics on complex spaces and proof of Theorem 2 in dimension two 22 2.3. Meromorphic families of analytic sets 26 2.4. Proof of Theorem 2 in higher dimensions 28 2.5. Complex Plateau problem for three-dimensional contours 33 3. Generalisations, applications, open questions 36 3.1. Multivalued mappings and singular domains of definition 37 3.2. Construction of Example 2: appearance of shells in symmetric products 37 3.3. Extension theorems for meromorphic correspondences 38 3.4. Complex Plateau problem for higher-dimensional contours 38 3.5. Coverings of compact complex manifolds 40 4. Open questions 41 References 42

Date: April 1997. 1991 Mathematics Subject Classification. 32D15. Key words and phrases. meromorphic map, Continuity principle, Hartogs extension theorem,

spherical shell, complex Plateau problem. Research during the author’s stay at MSRI was supported in part by NSF grant DMS-9022140.

0. Introduction

0.1. Continuity principle. Let us start with recalling the ”Continuity principle” for meromorphic functions due to E. Levi, see [Lv]. Consider a meromorphic function f, which is defined on a ring domain Δ^{n} × A(r, 1). Here A(r, 1) = {z_{n+1} ∈ ℂ : r < |z_{n+1}| < 1} is an annulus in ℂ and Δ^{n} is a unit polydisk. Denote by S the set of points s ∈ Δ^{n}, such that the restriction f_{s} := f |_{{s}×A(r,1)} is well defined and extends as a meromorphic function of one variable onto the whole disk Δ. Theorem (E. Levi). If S is not contained in a countable union of locally closed proper analytic subsets of Δ^{n} then f meromorphically extends onto the whole polydisk Δ^{n+1}.

Recall that by a locally closed analytic subset of Δ^{n} one means an analytic set in some open subset of Δ^{n}.

In this paper we are interested wether this theorem can be generalised to the case of meromorphic mappings with values in more or less arbitrar complex spaces.

Recall that a meromorphic mapping f : D → X between normal complex spaces D and X is given by a holomorphic map f : D \ I(f) → X of the complement to an analytic set I(f) of codimension at least two, such that the closure Γ̄_{f} of the graph of f is an analytic set in the product D × X. One requires also that Γ̄_{f} should be proper over D, i.e. restriction π |Γ̄_{f}: Γ̄_{f} → D of natural projection π : D × X → D onto Γ̄_{f} is proper. As I(f) one takes usually the minimal analytic subset of D such that f is holomorphic on D \ I(f) and calles I(f) the set of points of indeterminancy of f.

In the case X = ℂℙ^{1} this gives exactly a meromorphic function on D, see [Re]. One cannot expect for the direct generalisation of the E. Levi theorem to the

mappings into general complex spaces as the following couple of examples show.

1. Let X = ℂ^{2} \ {0}/(z ∼ 2z) be a Hopf surface. And let π : ℂ^{2} \ {0} → X be a natural projection. Embedd the unit bidisk i : Δ^{2} → ℂ^{2} in a standart way, and define f = π ∘ i : Δ^{2} \ {0} → X. Then for all s ∈ S = Δ \ {0} f_{s} is holomorphic on Δ, but f_{0} doesn’t extends to zero.

2. In [Hs-1] A. Hirschowitz constructed an example of compact complex surface X (of class V II_{0}) and holomorphic mapping π : 𝔹_{∗}^{2} → X from the puctured ball into X, such that for any complex curve C ∋ 0 the restriction π |_{C\{0}} extends holomorphically onto C, but π cannot be meromorphically extended to origin. If we put f = π ∘ i : Δ_{∗}^{2} → X as above, we obtain a mapping f : Δ × Δ_{∗} → X such that for all s ∈ Δ f_{s} is holomorphic on Δ, but f is not a meromorphic mapping of the couple (z_{1}, z_{2}).

3. In the example of M. Kato, see [Ka-1] or consrtruction of our Example 1, S can be made equal to Δ ∩ {z : |z − 1| < 1}, and for all s ∈ Δ \ S f_{s} doesn’t extends holomorphically onto Δ. I.e. the maximal set S of points for which f_{s} extends onto Δ can be “essentially” smaller then Δ!

Hovewer, let us follow the main steps of the proof of E. Levi theorem. Step 1 . Localisation. For an integer p denote by S_{p} the set os thouse s ∈ S that the extension f_{s} is well defined and has at most p poles counting with multiplicities. Then one immediately sees that there is a point z_{0} ∈ Δ^{n} and a natural p such that for any neighborhood U ∋ z_{0} the set S_{p} ∩ U is not contained in a proper analytic subset of U. Step 2. “Lewisches Einsatz”. Take such z_{0} as above and take ε_{1} > ε_{2} > 0 small enough to find a neighborhood of U ∋ z_{0} such that f is holomorphic in Ū × A(1 − ε_{1}, 1 − ε_{2}). Then f meromorphically extends onto U × Δ. Step 3. Globalisation. If U ≠ Δ^{n} one can (obviously) allwayse find z_{1} ∈ ∂U ∩Δ^{n} and p such that the set U_{p} of z ∈ U such that f_{z} has at most p poles is not analytic in any neighborhood of z_{1}. This shows that the maximal U such that f is meromorphic on U × Δ is equal to Δ^{n}.

Our first observation is that the statement of Step 2 (i.e. main ingredient of E. Levi Theorem) remains true for the mappings into an “almost” arbitrary complex space instead of ℂℙ^{1}. One only need to express the “boundedness of the number of poles” in other terms. Taking into account that the winding number of f_{s} |_{∂Δ} is uniformly bounded, say by N, one easily sees that to say that f_{s} takes the value ∞ not more then p times is equivalent to say that the area of the image of f_{s}, counted with multiplicities, is not more then N · (p + 1). Here ℂℙ^{1} is equipped with the usual spherical metric of total area one. Definition 0.1. We say that a complex space X is disk-convex in dimension k if for any compact K ⊂⊂ X there is another compact K̂ such that for every meromorphic mapping f : Δ̄^{k} → X with f(∂Δ^{k}) ⊂ K one has f(Δ̄^{k}) ⊂ K̂.

All compact spaces are of course disk-convex. More generally all k- convex spaces are disk-convex in dimension k.

Put A^{k}(r, 1) = {z ∈ ℂ^{k} : r < ∥z∥ < 1} and A_{s}^{k}(r, 1) := {s} × A^{k}(r, 1) for s ∈ Δ^{n}. Let f : Δ^{n} × A^{k}(r, 1) → X be a holomorphic mapping into a normal complex space X. Denote by S the set of points s ∈ Δ^{n} such that the restriction f_{s} := f |_{A_{s}^{k}(r,1)} extends meromorphically onto the polydisk Δ_{s}^{k} := {s} × Δ^{k}. Theorem 1 (Continuity principle). Let f : Δ^{n} × A^{k}(r, 1) → X be a holomorphic mapping into a normal, disk-convex in dimension k complex space X. Suppose that there is a constant C_{0} < ∞ and a compact K ⊂ X such that for s in some subset S ⊂ Δ^{n}, which is thick at origin:

(a) the restriction f_{s} := f |_{A_{s}^{k}(r,1)} is well defined and extends meromorphically onto the polydisk Δ_{s}^{k} := {s} × Δ^{k}, and Vol(Γ_{f_{s}}) ≤ C_{0} for all s ∈ S;

(b) f(Δ^{n} × A^{k}(r, 1)) ⊂ K and f_{s}(Δ^{k}) ⊂ K for all s ∈ S. Then:

1. If n = 1 then there is a neighborhood U ∋ 0 in Δ such that f extends meromorphically onto U × Δ^{k}.

2. If n ≥ 2 and X has bounded cycle geometry in dimension k, then again there is a neighborhood U ∋ 0 in Δ^{n} and a meromorphic extension of f onto U × Δ^{k}.

Here Γ_{f_{s}} denotes the graph of f_{s} and the volumes are taken with respect to some Hermitian metric h on X and a standart Euclidean metric on ℂ^{k}. The condition of finitness clearly doesn’t depends on the particular choice of the metrics. As usually saying that a set S ⊂ Δ^{n} is thick at the point z_{0} we mean that for any neighborhood U ∋ z_{0} S ∩ U is not contained in a proper analytic subset of U.

Denote by B_{k}(X) the Barlet space of compact analytic cycles of dimension k in X. This is an analytic space, which has not more then countable number of components. Definition 0.2. We say that a complex space X has bounded cycle geometry in dimension k if all connected components of the Barlet space B_{k}(X) are compact.

In other words X has bounde cycle geometry in dimension k if all irreducible components of B_{k}(X) are compact and all connected components of B_{k}(X) are just the finite unions of irreducible ones. Note again that the property to have bounded cycle geometry doesn’t depend on the choice of Hermitian metric.

There are two points to discuus here. The first one is the boundedness of cycle geometry condition in the case n ≥ 2 on the contrary to the case n = 1. The second - local character of the extension obtained, on the contrary to the classical case X = ℂℙ^{1}, i.e. what are the obstructions to the globalisation (Step 3)?

Cocerning the first point we shall construct a following example, showing that if n ≥ 2 the condition of boundedness of cycle geometry cannot be removed. Denote by z_{1}, z_{2}, z_{3}, z_{0} the coordinates in ℂ^{4} = ℂ_{z}^{3} × ℂ_{z_{0}}^{1}, z = (z_{1}, z_{2}, z_{3}). Example 1. There is a compact complex 4-dimensional manifold X and a holomorphic mapping f : Δ^{3} × A^{1}(r, 1) → X such that:

(1) for any s ∈ S = {(z_{1}, z_{2}, z_{3}) ∈ Δ^{3} : |z_{1}|^{2} > |z_{2}|^{2} + |z_{3}|^{2}} the restriction f_{s} = f |_{A_{s}^{1}(r,1)} holomorphically extends onto Δ_{s};

(2) for any t > 1 there is a constant C_{t} < ∞ such that for all s ∈ S_{t} = {(z_{1}, z_{2}, z_{3}) ∈ Δ^{3} : |z_{1}|^{2} > t · (|z_{2}|^{2} + |z_{3}|^{2})} one has area(Γ_{f_{s}}) ≤ C_{t};

(3) but for all z ∈ Δ^{3} \ S̄ = {(z_{1}, z_{2}, z_{3}) ∈ Δ^{3} : |z_{1}|^{2} < |z_{2}|^{2} + |z_{3}|^{2}} every point of the innner circle of the annulai A_{z}^{1}(r, 1) := {z_{0} ∈ Δ_{z} : 1 > |z_{0}|^{2} > |z_{2}|^{2} + |z_{3}|^{2} − |z_{1}|^{2}} consists of essentially singular points of f_{z} : A_{z}^{1}(r, 1) → X, here r^{2} = |z_{2}|^{2} + |z_{3}|^{2} − |z_{1}|^{2}.

In particular this f doesn’t extend meromorphically to any open set of the type U × Δ, where U is a neighborhood of the origin in Δ^{3}. The cycle geometry of this X is not bounded in dimension one. If one wishes to have an example of such type with n = 2, one can take a restriction of f onto {z_{3} = 0}.

Our idea for the proof of the Continuity principle is rufly the following. First, adapting the Barlet construction to our (noncompact) case, one shows that there is a finite dimensional normal complex space C_{f}, parametrising (analytically) the cycles Z in Δ^{n+k} × X which:

1) project onto {z} × Δ^{k} for some z in the neighborhood of zero in Δ^{n};

2) Z ∩ (Δ^{n} × A^{k}(r, 1) × X) ⊂ Γ_{f}.

From the condition on the thickness of S in the Theorem 1 one gets that dim C_{f} = n. Take now the universal space Z → C_{f} and the evaluation map F : Z → Δ^{n+k} × X. For a neighborhood U ∋ 0 denote Z_{U} = (π ∘ F)^{−1}(U). Here π : Δ^{n+k} × X → Δ^{n} is a natural projection. If there exists a neighborhood U ∋ 0 in Δ^{n} such that F |_{Z_{U}}: Z_{U} → U × Δ^{k} × X is proper then F (Z_{U}) will be an analytic subset in U × Δ^{k} × X extending the graph of f |_{U×A^{k}(r,1)}.

For n = 1 F is allwayse proper, because a nonconstant holomorphic map from the unit disk to the complex space is alwayse proper. For n ≥ 2 the condition of boundedness of cycle geometry naturally comes out. There is one more case, important in the applications, when F is proper. Corollary 1. Let f : Δ^{n} × A^{k}(r, 1) → X be a holomorphic map into a normal, disk-convex in dimension k complex space X. Suppose that:

(1) for every s ∈ Δ^{n} outside of thin set, the restriction f_{s} extends meromorphically onto Δ_{s}^{k};

(2) there is a compact K ⊂⊂ X such that f_{s}(Δ_{s}^{k}) ⊂ K for all s and f(Δ^{n} × A^{k}(r, 1)) ⊂ K;

(3) the volumes of the graphs Γ_{f_{s}} are uniformly bounded in Δ^{n}, i.e. there exists C_{0} < ∞ s.t. vol(Γ_{f_{s}}) ≤ C_{0} for all s. Then f meromorphically extend onto Δ^{n+k}.

Note that here one doesn’t needs the boundedness of cycle geometry of X. It is worth, probably to point out one case when the boundedness of cycle geometry is satisfied automatically - when k = dim X − 1. Really the cycle space of divisors is allwayse compact (provided X is compact).

0.2. Hartogs-type extension theorem and spherical shells. The second point, which should be discussed in concern with the Continuity principle, is the local character of the extension obtained. Namely a mapping into a general space extends only to U × Δ^{k}. While in the case X = ℂℙ^{1} U = Δ^{n}. And in fact in general the maximal U such that the map can be extended onto U × Δ^{k} can be smaller then Δ^{n}, as we had see above.

The reason is that the volume of Γ_{f_{z}} can tend to +∞ when z approach ∂U ∩ Δ^{n}. Take for exapmle the Hopf surface X = (ℂ^{2} \ {0})/(z ∼ 2z) and a canonical projection f : ℂ^{2} \ {0} → X. Concider its restriction onto Δ^{2} \ {0}. (1, 1)-form w = i/2\frac{dz_{1}∧dz̄_{1}+dz_{2}∧dz̄_{2}}{∥z∥^{2}} defines a canonical metric on X. One easily sees that area(Γ_{f_{s}}) ∼ log |s| when s → 0.

Such type of behavoir means a violation of the Hartogs extension phenomenon for the mapppings into this X. Denote by

H_{n}^{k}(r) := {(z′, z′′) ∈ Δ^{n+k} : 1 − r < ∥z′′∥ < 1or∥z′∥ < r} =

= Δ^{n} × A^{k}(1 − r, 1) ∪ Δ^{n}(r) × Δ^{k}  (0.1)

the k-concave Hartogs figure in ℂ^{n+k}.

Definition 0.3. We say that the meromorphic mappings into the space X have the Hartogs-type extension property in bidimension (n, k) if any meromorphic map f : H_{n}^{k}(r) → X extends meromorphically onto Δ^{n+k}.

So let us look now for the reasons of falue of Hartogs-type extension of meromorphic mappings with values in (normal) complex spaces. When the image space is Kähler there are no obstructions for the Hartogs-type extension, see [Iv-3]. Here we shall try to propose a general conjecture and state a result in the direction of proving it. Our point of depart will be the following observation:

every compact complex manifold of dimension k + 1 carries a Hermitian metric form w with dd^{c}w^{k} = 0. Really, the condition to carry dd^{c}-closed strictly positive (k, k)-form for a compact

complex manifold is alternative to that of carrying a bidimension (k + 1, k + 1)- current T with dd^{c}T ≥ 0 but ≢ 0. This in the case of dim X = k + 1 is a nonconstant plurisubharmonic function, which on compact X doesn’t exist.

Let us introduce the class G_{k} of normal complex spaces, carrying a nondegenerate positive dd^{c}-closed strictly positive (k, k)-forms. Note that the sequence {G_{k}} is rather exaustive: G_{k} contains all compact complex manifolds of dimension k + 1.

Note also that compact spaces from G_{k} have bounded cycle geometry in dimension k, see 1.4. We conjecture that meromorphic mappings into the spaces of class G_{k} are ”almost Hartogs-extendable” in bidimension (n, k) for all n ≥ 1:

Cojecture. Every meromorphic map f : H_{n}^{k}(r) → X, where X ∈ G_{k} and is diskconvex in dimension k, extends to a meromorphic map from Δ^{n+k} \ A to X, where A is an analytic subvariety of Δ^{n+k} (may be empty) of pure codimension k + 1. Moreover, if A ≠ ∅, then for every sphere 𝕊^{2k+1} embedded into Δ^{n+k} \ A in such a way that [𝕊^{2k+1}] ≠ 0 in H_{2k+1}(Δ^{n+k} \ A, ℤ), f(𝕊^{2k+1}) also is not homologous to zero in X.

In this paper we shall prove this conjecture in the case k = 1.

Definition 0.4. Let us call a Hermitian form w on X plurinegative if dd^{c}w ≤ 0.

The class of normal complex spaces admiting plurinegative Hermitian metrik form we shall denote by P_{−}.

Theorem 2. Let f : H_{n}^{1}(r) → X be a meromorphic map into a disk-convex complex space X which admits a plurinegative Hermitian metrik form. Then:

(1) f extends to a meromorphic map f̂: Δ^{n+1} \ A → X, where A is closed (n − 1)- polar subset of Δ^{n+1}.

(2) If moreover, w is pluriclosed then A is a analytic subvariety of Δ^{n+1} of pure codimension two (may be empty). If A ≠ ∅ then for every sphere 𝕊^{3} embedded into Δ^{n+1} \ A in such a way that [𝕊^{3}] ≠ 0 in H_{3}(Δ^{n+1} \ A, ℤ), its image f(𝕊^{3}) also is not homologous to zero in X. Remarks. 1. One can estimate the number of irreducible components of the singularity set A in this theorem meating a compact subset P ⊂⊂ Δ^{n+1}. Namely, let a compact K ⊂ X, which contains cl[f(P \ S)], is chosen to be a finite subcomplex of CW - complex X. Choose a point z′ ∈ Δ^{n−1} such that A intersects Δ_{z}^{2} := {z} × Δ^{2} by discrete set A_{z′}. Let A_{z′} ∩ ∂P = ∅. Then

| A_{z′} ∩ P |≤ | ∫_{∂(P ∩Δ_{z′}^{2})} d^{c}w| · [inf{| ∫_{γ} d^{c}w| : γ ∈ H_{3}(K, ℤ), ∫_{γ} d^{c}w ≠ 0}]^{−1}. (0.2)

In other words the number of branches of singular set (and moreover, their existence) is bounded by the differential geometry of X. Remark that the subset {| ∫_{γ} d^{c}w| : γ ∈ H_{3}(K, ℤ), ∫_{γ} d^{c}w ≠ 0} ⊂ ℝ is separated from zero, see (2.2.14).

2. Let us call a spherical shell of dimension k + 1 in complex space X an image Σ of the standard sphere 𝕊^{2k+1} ⊂ ℂ^{k+1} under the meromorphic map of some neighborhood of 𝕊^{2k+1} into X, such that Σ is not homologous to zero in X. This notion is close to the notion of the global spherical shell, introduced by Kato, see [Ka-3]. Thus we obtain the following Corollary 2. Let X be a disk-convex complex space which possess a pluriclosed Hermitian metric form. Then the following is equivalent:

(a) X possesses a meromorphic extension property in bidimension (n, 1) for all n ≥ 1,

and thus in all bidimensions (n, k).

(b) X contains no two-dimensional spherical shells.

3. A wide class of complex manifolds without two-dimensional spherical shells is for example a class of such manifolds X for which the Hurewicz homomorphism π_{3}(X) → H_{3}(X, Z) vanishes.

4. Corollary 2 for the case when X is compact complex surface was proved in [Iv-2] using the classification of surfaces.

5. A number of applications will show that characterisation of the obstructions for the extendibility of meromorphic mappings in terms of spherical shells is useful.

Theorem 2 was proved in [Iv-2] under an additional (very restrictive) assumption: manifold X doesn’t containe rational curves. In this case meromorphic maps into X are just holomorhpic . Let us explaine the difference. One can prove, see [Iv-4], that if holomorphic mappings into a complex space X are Hartogs extendable in bidimension (n, k), then they are Hartogs extendable in bidimension (n+1, k). Thus, for example, to prove the Hartogs-type extension for holomorphic mappings under the assumption of absence of rational curves, it is sufficient to prove the extendibility of holomorphic mappings with values in X from H_{1}^{1}(r) onto the unit bidisk.

Hovewer such reduction fail for the meromorphic maps, i.e. in the presnce of rational curves. In [Iv-4] we had constructed the following example, shoving that increasing of dimension in general for meromorphic mappings doesn’t works.

Example 2. There exists a compact complex three-fold X such that:

(a) For every domain D in ℂ^{2} every meromorphic mapping f : D ⟶ X extends to a meromorphic mapping f̂: D̂ ⟶ X. Here D̂ stands for the envelope of holomorphy of D.

(b) But there exists a meromorphic mapping F : B^{3} \ {0} ⟶ X from punctured threeball into X which does not extend to the origin.

This example shows that for the meromorphic mappings the Hartogs-type extendibility in bidimension (1, 1) doesn’t imply the extendibility in bidimension (2, 1) and, what is more suprusing, it doesn’t imply the extendibility in bidimension (1, 2)! Of course this X doesn’t carries a pluruiclosed metric form.

Continuity principle will be applied in the proof of Theorem 2 in the following context. First we shall prove in 2.2 this theorem for n=1. Then in the case n ≥ 2 first extend the mapping onto the set Δ^{n} \ S, where S is closed (n-1)-polar subset of Δ^{n+1}. Taking an appropriate projections we shall find ourselves in the assumtion of the Corollary 1 with k = 2. (In fact we cold also use the C.P.with k = 1 together with an observation that spaces carrying a plurinegative metrik-forms have bounded cycle geometry in dimension one.) This will give the result.

In fact extension of the map from the complement of the thin set makes the major diffuculty here. In Kähler case it was done by Y.-T. Siu using his theorem on analyticity of upper level sets of Lelong numbers for the closed positive currents. In our case currents (i.e. preimages of the metric form under the mapping f ) are not longer closed, but only pluriclosed, even only plurinegative. For such currents the Theorem of Siu is not longer valid. And this was a motivation for us to develope the approach based on the Continuity principle.

0.3. Kähler case: Griffiths approach revisited. As the title of this article says and as we had just explained above, our object here are meromorphic mappings into non Kähler complex manifolds. Hovewer let us make one remark, which is probably interesting especially in the Kähler case. First result on extension of meromorphic mappings is due to P. Griffiths, who proved in [Gr] that every holomorphic mapping from a punctured ball 𝔹_{∗}^{n} ⊂ ℂ^{n} into a compact Kähler manifold extends meromorphically to origin. He conjectured then that if A is a codimension two analytic subvariety in complex manifold D and G a sufficiently small neighborhood of A then any meromorphic map f : D \ Ḡ → X into a compact Kähler manifold X extends meromorphically onto the whole D. He also proposed an approach to the proof of this statement in the case when the ampping f is defined on D \ A. Namely, he proposed to estimate the integrals

∫_{K\A}(f^{∗}w)^{q}, q = 1, ..., n

where K is a relatively compact subdomain in D, and f^{∗}w is a pull back (as a current) of the Kähler form w from X. This whould meen the estimate of the volume of the graph of f in the neighborhood of A × X. So the application of the Bishop extension theorem for analytic sets would give the statement. Indeed, his proof in the case A = {point} was exactly this. In fact, as one can see in the proof of the Theorem 2 (and it was known in the Kähler case), for q = 1, 2 one can estimate thouse integrals in the same way as Griffiths did. But then B. Shiffman and Taylor constructed an example of a bedegree (1,1) current T in Δ^{3} \ {line} such that ∫_{Δ^{3}\{line}} T^{3} = ∞, see [Si-3]. As we already had mentioned the codimension two singularities where succefully removed by Y.-T. Siu in [Si-3] using a different approach.

Hovewer our Continuity Principle says exactly that it is enough to estimate just ∫_{K\A} f^{∗}w! Thus we obtain another proof of the theorem of Siu, which works also in non Kähler case.

There was an attempt in [Sb] to realise the Griffiths approach by means of Fubini theorem. Unfortunatly it containes an unrecouverable gap.

0.4. Applications, generalisations, open questions: meromorphic correspondences, complex Plateau problem, coverings of compact complex manifolds. It is natural to consider also the extension of meromorphic mappings from singular spaces. This is equivalent to considering a multivalued meromorphic correspondences from smooth domains, and this to single valued maps into symmetric powers of the image space, see 3.1 for details. However one pays price for such a reductions. In this direction we construct in 3.2 the following

Example 3. There is a compact complex surface X such that:

(a) every meromorphic map f : H^{2}(r) → X extends meromorphically onto Δ^{2}, but (b) there exists a two-valued meromorphic correspondence Z between punctured two-ball 𝔹_{∗}^{2} ∈ ℂ^{2} and X which cannot be extended to origin.

The reason is in fact, that sym^{2}(X) containes a two-dimensional spherical shell, while X not. In 3.3 we restate our results for multivalued meromorphic correspondences between (singular) complex spaces.

Results and techniques of this paper can be applied to several questions in complex analysis and geometry. We shall give two such applications.

Recall that a complex Plateau problem for a compact real submanifold M of a complex manifold X consists in finding an analytic chain A ⊂ X\M with “boundary” M, see 2.5 for details.

Corollary 3. Let M be a strongly pseudoconvex, maximally complex compact CRmanifold in a disk-convex complex manifold X from class G_{1}. Suppose that M bounds an abstract smooth Stein domain.

(a) If dim M ≥ 5 then the complex Plateau problem for M ⊂ X has a solution.

(b) If dim M = 3 then the complex Plateau problem for M ⊂ X has a solution iff M is

homologous to zero in X. Remarks. 1. Let H^{2} := ℂ^{2} \ {0}/z ∼ 2z be a Hopf surface. Take M to be an image of a standard unit sphere from ℂ^{2} under the natural projection π : ℂ^{2} \ {0} → H^{2}. M is not homologous to zero in H^{2} (i.e. it is a spherical shell in H^{2}!), so a complex Plato problem has no solution for this M.

In the case (a), i.e. when dim M ≥ 5 the spherical shells in X are not an obstructions for finding a film with boundary M because we have “enough concavity”.

2. Consider a Hopf three-fold H^{3} := ℂ^{3} \ {0}/(z ∼ 2z). In this case take a sphere 𝕊^{3} in a hyperplane {z_{1} = 0}. Its image M under the natural projection will be homologous to zero but will not bound any analytic set in H^{3}. The reason here is that H^{3} doesn’t belongs to G_{1} but only to P_{−}.

3. If one doesn’t requires strict pseudoconvexity of a “contour” M then counterexamples are known already in ℂℙ^{3}, see [Db].

4. The condition on M to bound an abstract Stein smooth Stein domain is really restrictive in dimension 3, while for bigger dimension one has the Rossi theorem guaranteeing the existence (but in general not smooth) abstract Stein domain with boundary M, see [Rs].

The proof consists in extension of a CR-embedding of M into X onto the Stein domain bounded by this M. We do it along the levels of appropriate plurisubharmonic Morse exhaustion function, see 2.5.

Consider now a domain D in complex manifold Ω, which covers (without ramifications) a compact complex manifold X. Corollary 4. Suppose that X ∈ G_{1}. Then:

(a) if dim X ≥ 3 then D is equal to a Levi-pseudoconvex domain minus a (possibly empty) variety of pure codimension two;

(b) if X is Kähler then D is Levi-pseudoconvex.

Part (b) was conjectured in [C-H] and was stated without proof in [Iv-3] as a simple corollary of the Hartogs-type extension theorem for mappings into Kähler manifolds. The proof is obtained by applying the extension results to the covering map D → X, see 3.5.

It is interesting to consider the case when Ω is compact and D = Ω \ E, where E is removable, i.e. holomorphic functions from V \ E extend onto V for the Stein open subsets V ⊂ Ω. For example if the Haussdorff (dim_{ℝ} Ω − 2)-measure of E is zero. Then, provided X carries a pluriclosed (negative) metric, one gets a further information about E, ex. analyticity or pluripolarity. Such examples really occur, see [Ka-2], [La].

In the end of this paper in §4 we give some open questions.

1. Continuity principle.

1.1. Cycle space associated to a meromorphic map. We shall need some notions and results from the theory of cycle spaces developed by D.Barlet, see [Ba]. Partially recalling thouse facts we adapt them to our situation. For the english spelling of the Barlet terminology we refer to [F].

Recall that an analytic cycle of dimension k in complex space Y is a formal sum Z = ∑_{j} n_{j}Z_{j}, where {Z_{j}} is a locally finite sequence of analytic subsets (allwayse of pure dimension k) and n_{j} are positive integers called multiplicities of Z_{j}. |Z| := ⋃_{j} Z_{j}- support of Z. All complex spaces in this paper are reduced, normal and countable at infinity.

With a given meromorphic mapping f : Δ^{n} ×A^{k}(r, 1) → X, satifying conditions of the Theorem 1 we shall associate the following space of cycles. Fix some 0 < c < 1. Consider a set Cf,C^{′} of all analytic cycles Z in Y := Δ^{n+k} × X of pure dimension k, such that:

(a) Z ∩[Δ^{n} ×A^{k}(r, 1)] = Γ_{f_{z′}} ∩{z}×A^{k}(r, 1)×X for some z′ ∈ Δ^{n}(c). This means, in particular, that for this z′ f_{z′} extends meromorphically from A_{z′}^{k}(r, 1) onto Δ_{z′}^{k}.

(b) vol(Z) < C, where C is a some constant, C > C_{0}, C_{0} beeing from Theorem 1. Define C̄_{f,C} to be a closure of Cf,C^{′} in the usual topology of currents, see below.

We shall show that C_{f,C} := {Z ∈ C̄_{f,C} : vol(Z) < C} is an analytic space of finite dimension in the neighborhood of each of its points.

Let Z be an analytic cycle of dimension k in (reduced, normal) complex space Y. In our applications Y will be Δ^{n+k} × X. By a coordinate chart adapted to Z we shall understand an open neighborhood V in Y such that V ∩ |Z| ≠ ∅ together with an isomorphism j of V onto a closed subvariety Ṽ in the neighborhood of Δ̄^{k} × Δ̄^{q}, such that j^{−1}(Δ̄^{k} × ∂Δ^{q}) ∩ |Z| = ∅. We shall denote such a chart by (V, j). By an image j(Z) of cycle Z under isomorphism j (or by any other isomorphism) we shall understand the image of underlying analytic set together with multiplicities.

Sometimes we shall, following Barlet, denote: Δ^{k} = U, Δ^{q} = B and give to the quadriple (V, j, U, B) the name scale adapted to Z.

If π : ℂ^{k} × ℂ^{q} → ℂ^{k} is a natural projection, then π |_{j(Z)}: j(Z) → Δ^{k} is a branching covering of degree say d. Number q depends on the embedding dimension of Y (or X in our case). We shall skeep sometimes j in our notations. The branched covering π |_{Z}:= π : Z∩(Δ^{k}×Δ^{q}) → Δ^{k} defines in a natural way a mapping ϕ : Δ^{k} → Sym^{d}(Δ^{q})

- symmetric power of Δ^{q} of degree d - as ϕ(z) = (π |_{Z})^{−1}(z). This allouds represent a cycle Z ∩ Δ^{k+q} with |Z| ∩ (Δ̄^{k} × ∂Δ^{q}) = ∅, as a graph of d-valued holomorphic map.

Let S be a normal complex space.

Definition 1.1.1. A holomorphic map Φ : S × Δ̄^{k} → Sym^{d}(Δ^{q}) we shall call an analytic family of k-dimensional subvarities in Δ^{k} × Δ^{q} parametrised by S.

We shall need the following Changing of the projection Theorem, due to Barlet. Let (V, j, U, B) be a scale on Δ̄^{k} × Δ^{q} adapted to the cycle Z. Let Z be included in an analytic family Φ in Δ^{k} × Δ^{q} as above, i.e. there is a holomorphic map Φ : S × Δ̄^{k} → Sym^{d}(Δ^{q}) such that Φ(s_{0}, ·) represents Z for some s_{0} ∈ S. Then for some neighborhood S_{0} ∋ s_{0} V is adapted to all Z_{s}, s ∈ S_{0}. In particular this defines a map Ψ : S_{0} × U → Sym^{d_{1}}(B).

Theorem (D. Barlet). Mapping Ψ is holomorphic.

For the proof see [Ba], Chapitres I and II. Here finite dimensionality and normality of S are essential.

Withough loss of generality we suppose that our mapping f is defined on Δ^{n} × A^{k}(r, b) with b > 1. Now each Z ∈ C_{f,C} can be covered by a finite number of adapted neighborhoods (V_{α}, j_{α}). Their union ⋃_{α} V_{α} we shall denote by W_{Z}. Taking this covering (V_{α}, j_{α}) to be small enough, we can suppose that:

(c) if V_{α_{1}}∩V_{α_{2}} ≠ ∅ then on every irreducible component of the intersection Z∩V_{α_{1}}∩V_{α_{2}} a point x_{1} is fixed such that

(c_{1}) either there is a polycilinder neighborhood Δ_{1}^{k} ⊂ Δ^{k} of π_{1}(j_{1}(x_{1})) such that the chart V_{12} = j_{α_{1}}^{−1}(Δ_{1}^{k} ×Δ^{q}) is adapted to Z and is contained in V_{2}, here V_{12} is given the same embedding j_{α_{1}};

(c_{2}) or this is fulfilled for V_{2} instead of V_{1}.

(d) if V_{α} ∋ y with p(y) ∈ Δ̄^{n}(c) × A^{k}(\frac{r+1}{2}, 1) then p(V̄_{α}) ⊂ Δ̄^{n}(\frac{c+1}{2}) × A^{k}(r, 1).

Here we denote by p : Δ^{n+k} × X → Δ^{n+k} and by π : Δ^{n} × Δ^{k} × X → Δ^{n} the natural projections. Case (c_{1}) can be fulfiled when embedding dimension of V_{α_{1}} is smaller or equal of that of V_{α_{2}}, and (c_{2}) in the opposite case, see [B].

Note that we have allwayse that p(Z) = Δ_{z_{0}}^{k} for some z_{0} ∈ Δ^{k}(c) if Z ∈ C_{f,C}, because Z is a limit of Z_{n} ∈ Cf,C^{′}.

The subsets W_{Z} together with the topology of convergence on compact subsets on Hol(Δ^{k}, Sym^{d}(Δ^{q})) define a (metrizable) topology on our cycle space C_{f,C}.

Each cycle Z ∈ C_{f,C} can be viewed as a current of integration and thus C_{f,C} can be also equiped with the corresponding locally flat topology. Cycles Z_{n} are converging in this topology to Z if for any smooth (k, k) -form w with compact support in Y

∫_{Z_{n}} w → ∫_{Z} w.

The both topologies are equivalent, see [F]. For us will be important that C̄_{f,C} is compact. This generalisation of Bishop’s theorem is due to Harvey and Shiffman, see [H-S].

Family of cycles Z_{s} parametrised by normal complex space S we call analytic if for any s_{0} ∈ S, and any coordinate chart (V, j) adapted to Z_{s_{0}} the family Z_{s} ∩ V is analytic in V for s in the neighborhood of s_{0} in the sence of Definition 1.1.1.

We shall need a criterium for the analyticity of changing the projections map in the case when the parameter space S is of infinite dimension.

Denote by S_{m}(ℂ^{q}) (resp. by Λ^{i}(ℂ^{q})) the m-th symmetric (resp. exterior) pover of ℂ^{q}. Put F_{i} := Hom(Λ^{i}(ℂ^{k}), Λ^{i}(ℂ^{q})).

Let (S, s_{0}) be a germ of Banach analytic space and let

Φ : (S, s_{0}) → H(Δ̄^{k}, Sym^{d}(Δ^{q}))

be a germ of analytic map.

Fix some s ∈ S and denote by Z_{s} the analytic set in Δ^{k} × Δ^{q} defined by Φ_{s} := Φ(s, ·) : Δ^{k} → Sym^{d}(Δ^{q}). Let R(Z_{s}) the locus of ramification of π |_{Z_{s}}: Z_{s} → Δ^{k}. For z ∈ Δ^{k} \ R(Z_{s}) define

T_{m}^{i}(Φ_{s}) : Δ^{k} → F_{i} ⊗ S_{m}(ℂ^{p})  (1.1.1)

as

T_{m}^{i}(Φ_{s})(z) = Σ_{j=1}^{d}Λ^{i}(DΦ_{s,j}(z)) ⊗ Φ_{s,j}(z)^{m}.  (1.1.2)

Here Φ_{j}, j = 1, ..., d are the local branches of Φ. T_{m}^{i}(Φ_{s}) extend holomorphically onto the whole polydisk Δ^{k}, see [B] Proposition 1, Chap.2. This defines a maps

T_{m}^{i}(Φ) : S × Δ^{k} → F_{i} ⊗ S_{m}(ℂ^{p}).  (1.1.3)

Following Barlet we call the family isotropic if thouse maps are analytic for all i, m. The statement of Canging the Projection Theorem remains true for the case of Banach analytic sets S, provided the family Φ is isotropic, see Theorem 4 in [B]. In fact one needs to check analyticity only for 1 ≤ m ≤ d − 1. Note also that F_{i} = 0 if i > max{k, q}.

Now let E = (V, j, U, B) be a scale on the complex space Y. Let us denote by Hol_{Y} (Ū, sym^{d}(B)) the Banach analytic set of all d-sheeted analytic subsets on Ū × B, which are contained in j(Y). We need to make the tautological family Hol_{Y} (Ū, sym^{d}(B)) × U → sym^{d}(B) isotropic. Put F := Σ_{i}F_{i} and S := Σ_{m}S_{m}(ℂ^{q}). Fix some polydisk U′ ⊂⊂ U and call the data E = (V, j, U, U′, B) the double scale on Y. Consider a continuous map

T : Hol(Ū, sym^{d}(B)) → Hol(Ū′, F ⊗ S)

given by

h ∈ Hol(Ū, sym^{d}(B)) → Σ_{i,m}T_{m}^{i}(h)  (1.1.4)

Denote by Ĥol(Ū, sym^{d}(B)) the graph Γ_{T} of (1.1.4) in Hol(Ū, sym^{d}(B))×Hol(Ū′, F ⊗ S).

Let Ĥol_{Y} (Ū, sym^{d}(B)) be the restriction of Γ_{T} onto Hol_{Y} (Ū, sym^{d}(B)). Then the set Ĥol_{Y} (Ū, sym^{d}(B)) is a Banach analytic subset of Hol(Ū, sym^{d}(B)) × Hol(U′, F ⊗ S), and the tautological family

Ĥol(Ū, sym^{d}(B)) × U → sym^{d}(B)

is (obviously) isotropic! See [B]. Definition 1.1.2. The family Z of analytic cycles in an open set W ⊂ Y, parametrised by a Banach analytic set S, is called analytic in the neighborhood of s_{0} ∈ S if for any scale V adapted to Z_{s_{0}} there is a neighborhood U ∋ s_{0} s.t. {Z_{s} : s ∈ U} is isotropic in V.

1.2. Analyticity of C_{f,C}. Let f : Δ^{n} × A^{k}(r, 1) → X be our map. Denote by C_{0} the subset of C̄_{f,C} consisting of cycles which are limits of {Γ_{f_{sn}}} for s_{n} → 0, s_{n} ∈ S. This is a compact subset (by Bishop’s theorem) of the topological space C_{f,2C}. For every cycle Z ∈ C_{0} define its neighborhood W_{Z} as above. Let W_{Z_{1}}, ..., W_{Z_{N}} be a finite covering of C_{0}. Remark that there is an ε_{0} > 0 such that for any s ∈ S ∩ Δ^{n}(ε_{0}) we have Γ_{f_{s}} ⊂ ⋃_{j=1}^{N} W_{Z_{j}}.

We whant to show now that C_{f,2C} is an analytic space of finite dimension in the neighborhood of C_{0}. It is enough to prove this for the W_{Z_{1}} for example.

Let W_{Z_{1}} = ⋃_{α=1}^{N_{1}} V_{α}. We divide V_{α} -s into two types. Type 1. For such V_{α} as in (d) put

H_{α} := ⋃_{z}{[Γ_{f_{z}} ∩ A^{k}(r, 1) × X] ∩ V_{α}} ⊂ Hol_{Y} (Δ^{k}, Sym^{d_{α}}(Δ^{p})).  (1.2.1)

Union is taken over all z ∈ Δ^{n} for which V_{α} is adapted to Γ_{f_{z}}. Type 2. For all others V_{α} we put H_{α} := Ĥol_{Y} (Δ̄^{k}, Sym^{d_{α}}(Δ^{p})).

All H_{α} are open sets in complex Banach spaces and, for V_{α} of first type they are of dimension n and smooth.

For every irreducible component of V_{α} ∩ V_{β} ∩ Z_{1} we fix some point x_{αβl} on this component (indice l indicates the component), and some chart V_{α}∩V_{β} ⊃ (V_{αβl}, ϕ_{αβl}) ∋ x_{αβl} adapted to this component as in (c). Put H_{αβl} := Ĥol(Δ^{k}, Sym^{d_{αβl}}(Δ^{p})).

Consider a finite products Π_{(α)}H_{α} and Π_{(αβl)}H_{αβl}. In the second product we take only triples with α < β. They are Banach analytic spaces and by Changing the projection Theorem of Barlet, for each pair α < β we have two holomorphic mappings Φ_{αβ} : H_{α} → Π_{(l)}H_{(αβl)} and Ψ_{αβ} : H_{β} → Π_{(l)}H_{αβl}. This defines two holomorphic maps Φ, Ψ : Π_{(α)}H_{α} → Π_{α<β,l}H_{αβl}. Kernel A_{1} of this pair, i.e. thouse h = {h_{α}} that Φ(h) = Ψ(h), consists exactly from analytic cycles in a neighborhood W_{Z_{1}} of Z_{1}. This kernel is a Banach analytic set and moreover, the family A_{1} is an analytic family in W_{Z_{1}}. Lemma 1.2.1. A_{1} is of finite dimension.

Proof of the Lemma. Take a smaller covering {Vα^{′}, j_{α}} of Z_{1}. Namely Vα^{′} = V_{α} for the V_{α} of the first type and Vα^{′} = j_{α}^{−1}(Δ_{1−ε}^{k} × Δ^{p}) for the second. In the same manner we obtain a Banach analytic set A1^{′}. We have a holomorphic mapping K : A_{1} → A1^{′} defined by the restrictions. Diffe rential dK ≡ K of this map is a compact operator. We also have an in verse map F because of the isotropy of the family A_{1}, see above. Thus id − dK ∘ dF is Fredholm. Because A1^{′} ⊂ {h ∈ Π_{(i)}Hi^{′} : (id − K ∘ F)(h) = 0} we obtain that A1^{′} is an analytic subset in the complex manifold of finite dimension.

Lemma is proved

The same holds for any point in Δ^{n} instead of zero. Thus C_{f,C} is finite dimensional analytic space.

1.3. Proof of the Continuity principle. Now we are prepared to prove our Continuity principle. Consider a universal family Z := {Z_{a} : a ∈ C_{f,2C_{0}}}, this time constant C_{0} is taken from Theorem 1. This is complex space of finite dimension . We have an evaluation map

F : Z → Δ^{n+k} × X  (1.3.1)

defined by Z_{a} ∈ Z → Z_{a} ⊂ Δ^{n+k} × X.

Case n = 1. Consider the union Ĉ_{0} of thouse components of C_{f,2C} which intersect C_{0}. At least one of thouse components, say K, containes two points s_{1} and s_{2} s.t. Z_{s_{1}} projects onto Δ_{0}^{k} and Z_{s_{2}} projects onto Δ_{s}^{k} with s ≠ 0. Consider the restriction Z |_{K} of the universal space onto K. This is an irreducible complex space of finite dimension. Take points z_{1} ∈ Z_{s_{1}} and z_{2} ∈ Z_{s_{2}} and join them by an analytic disk ϕ : Δ → Z |_{K}, ϕ(0) = z_{1}, ϕ(1/2) = z_{2}. Then the composition ψ = π ∘ F ∘ ϕ : Δ → Δ is not degenerate because ψ(0) = 0 ≠ s = ψ(1/2). Thus ψ is proper and obviously so is the map F : Z |_{ϕ(Δ)}→ F (Z |_{ϕ(Δ)}) ⊂ Δ^{n+k} × X. Thus F : Z |_{ϕ(Δ)}) is an analytic set in U × Δ^{k} × X for small enough U extending Γ_{f} by the reason of dimension. Case n ≥ 2. Consider an increasing family of complex spaces {C_{f,C} : C ≥ C_{0}}, where C_{0} is from the Theorem 1. Note that for C_{1} < C_{2} C_{f,C_{1}} is an open subset of C_{f,C_{2}}. This allouds correctly define an irreducible components of the analytic space C_{f} := ⋃_{C≥C_{0}} C_{f,C}.

As above by C_{0} denote the compact subset of C_{f}, which consists from the limits of Γ_{f_{s}}, s → 0, s ∈ S, vol(Γ_{f_{s}}) ≤ C_{0}. Clearly only finite number of components of C_{f} intersect C_{0}. Denote them by K_{1}, ..., K_{N}.

Note that if all this components would have zero dimension it would contradict the thickness of S at origin. Take any component, say K_{1} of positive dimension. If evaluation map F restricted onto Z |_{K_{1}} is proper, then again F (Z |_{K_{1}}) is an analytic set in the neighborhood U ×Δ^{k} ×X. If for all K_{j} F (Z |_{K_{j}}) is proper then, because we only have only finitely many this components , we shall have a neighborhood U ∋ 0 in Δ^{n} and analytic set F (Z |_{⋃ K_{j}}) in U × Δ^{k} × X. Would this set be of dimension less then n + k, it would contradict the thickness of S at zero. Thus in this case we have again an extension F (Z |_{⋃ K_{j}} of Γ_{f} onto U × Δ^{k} × X.

It remaine to prove that for any K_{j} the restriction of F onto Z |_{K_{j}} is proper. Suppose now that there is a component, say K_{1}, such that F restricted to Z |_{K_{1}} is not proper. Consider a map ϕ : K_{1} → Δ^{n} given by ϕ(s) = π(F (Z_{s})). If ϕ^{−1}(0) is compact, then there are a neighborhoods W_{1} ⊃ ϕ^{−1}(0) in K_{1} and V_{1} ∋ 0 in Δ^{n} such that ϕ |_{W}: W_{1} → V_{1} ⊂ Δ^{n} is proper. Thus F : Z |_{W_{1}}→ V_{1} × Δ^{k} × X is proper. If this is the case for all K_{1}, ..., K_{N} then again F : Z |_{⋃ W_{j}}→ ⋂ V_{j} × Δ^{k} × X is proper and we are done.

So, suppose that ϕ^{−1}(0) is not compact in K_{1}. Each Z_{s}, s ∈ ϕ^{−1}(0) has a form {0} × B_{s} ∪ Γ_{f_{0}}, where B_{s} is a compact cycle in X. Thus an appropriate connected component of ϕ^{−1}(0) parametrizes a noncompact connected and closed subvariety in B_{k}(X). This contradicts the bounded cycle geometry condition on X.

Theorem 1 is proved.

Proof of the Corollary 1. In the proof of Corollary 1 we shall need some resuts on the meromorphic families of analytic subsets. They will be used also in the proof of Theorem 2. That’s why we had collected them in 2.3. For the proofs of thouse results we refer to [Iv-4].

Denote by ν_{j} the minimal volume of compact j-dimensional analytic subset in K ⊂⊂ X - compact from Corollary 1. Put

ν = min{vol(A_{k−j} · ν_{j} : j = 1, ..., k},  (1.3.2)

where A_{k−j} runs over all (k − j)-dimensional analytic subsets of Δ^{k}, which intersect Δ_{r}^{k}. Denote by W the maximal open subset of Δ^{n} such that f meromorphically extends onto Δ^{n} × A^{k}(r, 1) ∪ W × Δ^{k}. Put S = Δ^{n} \ W. Let

S_{l} = \{z ∈ S : vol(Γ_{f_{z}} ≤ l · \frac{ν}{2}\}.  (1.3.3)

Lemma 2.3.2 tells us that S_{l+1} \ S_{l} are pluripolar and by the Josefson theorem so is S.

Consider a function v(z) = vol(Γ_{f_{z}}). We know that v ≥ 0, v ≤ C_{0} and v is lower semicontinuous . The latter follows from Lemma 2.3.1.(4). Let v^{∗} be its upper regularisation, i.e. v^{∗} = lim sup_{z→z_{0}} v(z). Then v^{∗} is upper semi-continuous and bounded by C_{0}. Consider an analytic space

C_{f,2C_{0},c} := {Z ∈ C_{f,2C_{0}} : ∥π(Z)∥ < c},  (1.3.4)

where 0 < c < 1 is fixed. Consider the following relatively compact in C_{f,2C_{0}} and closed in C_{f,2C_{0},c} set:

C_{f,v^{∗}+ \frac{ν}{4} ,c} = \{Z ∈ C_{f,2C_{0},c} : vol(Z) ≤ v^{∗}[π(Z)] + \frac{ν}{4}\},  (1.3.5)

where, as above π : Δ^{n} × Δ^{k} × X → Δ^{n} is a natural projection. Function vol(Z) is contunious on C_{f,2C_{0}}. Thus for any Z_{0} ∈ C̄_{f,v^{∗}+ \frac{ν}{4} ,c} the set W_{Z_{0}, \frac{ν}{4}} := \{Z : | vol(Z) − vol(Z_{0})| < \frac{ν}{4}\} is an open neighborhood of Z_{0} in C_{f,2C_{0}}. Closures and neighborhoods are taken in C_{f,2C_{0},c}. Put

W_{f,v^{∗}, \frac{ν}{2}} := ⋃_{Z_{0}∈C̄_{f,v^{∗}+ \frac{ν}{4} ,c}} W_{Z_{0}, \frac{ν}{4}}  (1.3.6)

and

W_{f,v^{∗}, \frac{3ν}{4}} := ⋃_{Z_{0}∈C̄_{f,v^{∗}+ \frac{ν}{4} ,c}} W_{Z_{0}, \frac{ν}{2}} .  (1.3.7)

We are going to show now that W̄_{f,v^{∗}, \frac{ν}{2}} = W_{f,v^{∗}, \frac{3ν}{4}} .

Find a point z_{0} ∈ W which has a neighborhood, say V, such that for all z ∈ V |vol(Γ_{f_{z}}) − vol(Γ_{f_{z_{0}}})| < \frac{ν}{2}. Such z_{0} exists because W is not pluripolar. Lemma 2.3.2 shows that the analytic spaces W_{f,v^{∗}, \frac{ν}{2}} and W_{f,v^{∗}, \frac{3ν}{4}} coincide with ⋃_{z∈V} Γ_{f_{z}} in the neighborhood of z_{0}. The same argument shows that the following is true. Take the irreudcible components of analytic spaces W_{f,v^{∗}, \frac{ν}{2}} and W_{f,v^{∗}, \frac{3ν}{4}} which contain ⋃_{z∈V} Γ_{f_{z}}. We denote them in the same manner W_{f,v^{∗}, \frac{ν}{2}} and W_{f,v^{∗}, \frac{3ν}{4}} respectively. The holomorphis mapping π ∘ F : W_{f,v^{∗}, \frac{ν}{2}} → Δ^{n} is surjective. The same for π ∘ F : W_{f,v^{∗}, \frac{3ν}{4}} → Δ^{n}. There is a pluripolar set Ŝ ⊃ S such that π ∘ F : W̄_{f,v^{∗}, \frac{ν}{2}} \ (π ∘ F)^{−1}(Ŝ) → Δ^{n}\Ŝ is biholomorphic. And the same for π∘F : W_{f,v^{∗}, \frac{3ν}{4}} \(π∘F)^{−1}(Ŝ) → Δ^{n} \ Ŝ. But this means that W_{f,v^{∗}, \frac{3ν}{4}} \ W_{f,v^{∗}, \frac{ν}{2}} ⊂ (π ∘ F)^{−1}(Ŝ). The latter is a pluripolar subset of an irreducible analytic space, and thus has dense complement. So the statement is proved.

Note now that if 0 < c_{1} < c then W̄_{f,v^{∗}, \frac{ν}{2}} ∩ (π ∘ F)^{−1}(Δ̄_{c_{1}}^{n}) is compact in W_{f,v^{∗}, \frac{3ν}{4}} . This gives the properness of π ∘ F restricted to W_{f,v^{∗}, \frac{3ν}{4}} , and thus the properness of F there.

q.e.d.

1.4. Spaces from G_{k} have bounded cycle geometry. Let us proove here that spaces from G_{k} have bounded cycle geometry in dimension k. In fact somewhat stronger statement is true. Denote by P_{−}^{k} the class of complex spaces which carry a strictly positive (k, k)- forms Ω^{k,k} which are dd^{c}-negative, i.e. dd^{c}Ω^{k,k} ≤ 0. Proposition 1.4.1. Let X ∈ P_{−}^{k} be disk-convex in dimension k and let K be an irreducible component of B_{k}(X). Then

1) K is compact.

2) If Ω^{k,k} os some dd^{c}-negative (k, k)-form on X then ∫_{Z_{s}} Ω^{k,k} ≡ const for s ∈ K.

3) X has bounded cycle geometry.

Proof. Let F : Z |_{K}→ X be the evaluation map, and let Ω^{k,k} be a sitrictly positive dd^{c}-negative (k, k)-form on X. Then ∫_{Z_{s}} Ω^{k,k} measures the volume of Z_{s}. Let us prove that the function v(s) = ∫_{Z_{s}} Ω^{k,k} is plurisuperharmonic on K. Take an analytic disk ϕ : Δ → K. Then by Stokes theorem and reasons of bedegree for any nonnegative test function ψ ∈ D(Δ) :

< ψ, Δϕ^{∗}(v) >= ∫_{Δ} Δψ · ∫_{Z_{ϕ(s)}} Ω^{k,k} = ∫_{Z|_{ϕ(Δ)}} dd^{c}(π^{∗}ψ) ∧ Ω^{k,k} =

= ∫_{ϕ(Δ)} π^{∗}ψ ∧ dd^{c}Ω^{k,k} ≤ 0.

Here π : Z |_{K}→ K is a natural projection. So Δϕ^{∗}(v) ≤ 0 for any analytic disk in K in the sence of distributions. Thus v is plurisuperharmonic.

Note that by Harvey-Shiffman generalisation of Bishops theorem v(s) → ∞ while s → ∂K. So by the minimum principle v ≡ const and K is compact again by Bishop theorem.

2) The same computation shows that ∫_{Z_{s}} Ω^{k,k} is plurisuperharmonic for any dd^{c}- negative (k, k)-form. While K is proved to be compact, we get the statement.

3) Let R be any connected component of B_{k}(X). Write R = ⋃_{j} K_{j}. From (1) we have that v is constant on R. So if {K_{j}} is not finite then R has an accumulation point s = lim s_{j}, where all s_{j} belong to different components K_{j} of R. This contradicts to the fact that B_{k}(X) is a complex space.

q.e.d.

1.5. Construction of Example 1. We start with recalling one example due to M. Kato, see [Ka-1].

In ℂℙ^{3} with homogeneous coordinates [z_{0} : z_{1} : z_{2} : z_{3}] consider a domain D = {z ∈ ℂℙ^{3} : |z_{0}|^{2} + |z_{1}|^{2} > |z_{2}|^{2} + |z_{3}|^{2}}. Group Sp(1, 1) naturally acts on ℂℙ^{3} and preserves D, i.e. g(D) = D for all g ∈ Sp(1, 1). Action of Sp(1, 1) is transitive on D and Kato proved, using the result of Vinberg, that there exist a discrete subgroup Γ ⊂ Sp(1, 1), acting properly and discontinuously on D, and such that D/Γ = X^{3} is a compact complex manifold, see [Ka-1] for details.

Projective plane ℂℙ^{2} = {z_{3} = 0} intersects D by the complement to the unit ball [z_{0} : z_{1} : z_{2}] ∈ ℂℙ^{2} : |z_{2}|^{2} < |z_{0}|^{2} + |z_{1}|^{2}}. If π : D → X^{3} is a natural projection, then its restriction π |_{ℂℙ^{2}∩D}: ℂℙ^{2} \ 𝔹̄^{4} → X^{3} defines a holomorphic map from the complement to the closed unit ball to X^{3}, which has singularity at each point of ∂𝔹^{4}!

Blow up ℂℙ^{4} at origin of its affine part. Denote by ℂℙ_{0}^{4} the resulting manifold. There is a natural holomorphic projection. p : ℂℙ_{0}^{4} → ℂℙ^{3}-concidered as an exeptional divisor. Γ beeing a group of 4 × 4 matrices acts naturally on affine part ℂ^{4} of ℂℙ^{4}. This action obviously extends onto ℂℙ^{4} and lifts onto ℂℙ_{0}^{4}. Moreover the actions of Γ on ℂℙ^{3} and ℂℙ_{0}^{4} are equivariant with respect to the projection p. Put D̂ := p^{−1}(D) and X̂ := D̂/Γ. If take now f := {z ∈ ℂ^{4} : |z_{0}|^{2}+|z_{1}|^{2} > |z_{2}|^{2}+|z_{3}|^{2}} → X̂ to be a restriction of quotient map, we get our example.

2. Hartogs-type extension and spherical shells

2.1. Generalities on pluripotential theory. We start with some well known facts from pluripotential theory . Let D be an open subset of ℂ^{n} and S subset of D. Consider the following class of functions:

U(S, D) = {u ∈ P_{+}(D) : u|_{S} ≥ 1}  (2.1.1)

where by P_{+}(D) we denote the class of nonnegative plurisuperharmonic functions in D. Definition 2.1.1. The lower regularization w_{∗} of the function

w(ζ, S, D) = inf{u(ζ) : u ∈ U(S, D)}  (2.1.2)

is called a P- measure of S in D, i.e.

w_{∗}(z, S, D) = lim_{ζ→z} inf w(ζ, S, D)  (2.1.3)

Note that w_{∗} is plurisuperharmonic in D . Definition 2.1.2. A point s_{0} ∈ S is called a locally regular point of S if w^{∗}(s_{0}, S ∩ Δ^{n}(s_{0}, ε), Δ^{n}(s_{0}, ε)) = 1 for all ε > 0.

We shell also say that the set S is locally regular at s_{0}.

Recall that subset S ⊂ D is called pluripolar if there exists a plurisubharmonic function u in D, u ≢ −∞, such that u |_{S}≡ −∞. S is complete pluripolar if S = {z : u(z) = −∞}. We shall repeatedly use the following statement:

if subset S ⊂ D (D is now pseudoconvex) is not locally regular at all its points then S

is pluripolar, see [B-T],[Sd]. We shall repeteadly use the following immediate corollary from the famous Josefson theorem, see [Kl]: Let Ω be a pseudoconvex set in ℂ^{n}, and let S_{n} be a sequence of subsets of Ω such that:

1) S_{1} is closed and pluripolar in Ω;

2) S_{n+1} ⊂ Ω \ S_{n} is closed in Ω \ S_{n} and pluripolar; Then S := ⋃_{n=1}^{∞} S_{n} is pluripolar in Ω.

Denote by D^{k,k}(Ω) the space of C^{∞}-forms of bidegree (k, k) with compact support on complex manifold Ω. ϕ ∈ D^{k,k}(Ω) is real if ϕ̄ = ϕ. The dual space D_{k,k}(Ω) is the space of currents of bidimension (k, k) (bidegree (n − k, n − k), n = dim Ω). T ∈ D_{k,k}(Ω) is real if < T, ϕ̄ >= \overline{< T, ϕ >} for all ϕ ∈ D^{k,k}(Ω).

Definition 2.1.3. Current T ∈ D_{k,k}(Ω) is called positive if for all ϕ_{1}, ..., ϕ_{k} ∈ D^{1,0}(Ω)

< T, \frac{i}{2}ϕ_{1} ∧ ϕ̄_{1} ∧ ... ∧ \frac{i}{2}ϕ_{k} ∧ ϕ̄_{k} >≥ 0.

T is negative if −T is positive. Definition 2.1.4. We shall say that the current T ∈ D_{k,k}(Ω) is pluripositive (- negative) if T is positive and dd^{c}T is positive (-negative).

T is pluridefinite if it is either pluripositive or plurinegative. Definition 2.1.5. The current T (not necessarily positive) is pluriclosed if dd^{c}T = 0.

Let ϕ = Σ_{J}ϕ_{J}(x)dx_{J} ∈ D^{k}(Ω), where Ω open subset of ℝ^{n}. Euclidean norm of ϕ at x is

∥ϕ(x)∥ = (Σ_{|J|=k}|ϕ_{J}(x)|^{2})^{1/2}.

Further if T ∈ D_{k}(Ω) and U is open in Ω then the mass of T in U is a number

∥T ∥(U) = sup{| < T, ϕ > | : ϕ ∈ D^{k}(U), ∥ϕ(x)∥ ≤ 1, x ∈ U}.  (2.1.4)

Let K ∈ Ω be closed and T ∈ D_{k}(Ω \ K). We say that T has locally finite mass in the neighborhood of K if for any open relatively compact U ∈ Ω ∥T ∥(U \ K) < ∞. In this case a trivial extension T̃ of T onto Ω is defined in the following way. Take a sequence u_{n} ∈ C^{∞}(Ω), 0 ≤ u_{n} ≤ 1, u_{n} ≡ 0 in the neighborhood of K, and u_{n} ↗ χ_{Ω\K} uniformly on compacts in Ω \ K. Here χ_{A} is a characteristic function of the set A. Then for ϕ ∈ D^{k}(Ω) put

< T̃, ϕ >= lim_{n→∞} < u_{n}T, ϕ >  (2.1.5)

This correctly defines a current on Ω, see [Lg].

If K is complete pluripolar compact in strictly pseudoconvex domain Ω ⊂ ℂ^{n} and T closed, positive current on Ω\K, then T has locally finite mass in the neighborhood of K, see [Iv-2], Lemma 2.1 . Lemma 2.1.1. (a) Let K be a complete, pluripolar compact in strictly pseudoconvex domain Ω ∈ ℂ^{n} and T be a pluridefinite current of bidegree (1,1) on Ω \ K of locally finite mass in the neighborhood of K, and such that dT has coefficients measures in Ω \ K. Then dd^{c}T̃ has coefficients measures on Ω.

(b) If K is of Haussdorff dimension zero and n = 2 then χ_{K} · dd^{c}T̃ is negative. Proof. Part (a) of this lemma is proved in [Iv-2], Proposition 2.3 for the currents of bidimension (1,1) (the condition on dT was forgotten there). If T is of bedegree (1,1) then take T ∧ (dd^{c}∥z∥^{2})^{n−2} to get the same conclusion.

(b) Let {u_{n}} be a sequence of smooth p.s.h.functions in Ω, equal to zero in the neighborhood of K, 0 ≤ u_{n} ≤ 1 and such that u_{n} ↗ χ_{Ω\K} uniformly on compacts in Ω \ K, see Lemma 1.2 from [Sb]. Put v_{n} = u_{n} − 1.

Let w_{e} denote the Euclidean volume-form in ℂ^{2}. Put { \atop dd^{c}T} = μ_{0} · w_{e}^{n}. Then μ_{0} is a measure on Ω. According to part (a) the distribution μ defined from dd^{c}T̃ = μ · w_{e}^{n} is a measure. Write

μ = χ_{K} · μ + χ_{Ω\K} · μ  (2.1.6)

Where obviously χ_{Ω\K} = μ_{0}. The measure χ_{K} · μ we denote by μ_{1}. We shall prove that the measure μ_{1} is nonpositive.

Take a ball in ℂ^{n} centered at s_{0} ∈ K such that ∂B ∩ K = ∅. One has

μ_{1}(B ∩ K) = − lim_{k→∞} ∫_{B} v_{k} · μw_{e}^{n} = − lim_{k→∞} < v_{k}, dd^{c}T̃ >=

= − lim_{k→∞} < dd^{c}v_{k}, T̃ >≤ 0,  (2.1.7)

because T̃ is positive and dd^{c}v_{k} ≥ 0. So for any such ball we have

μ_{1}(B ∩ K) ≤ 0  (2.1.8)

All that left, is to use the following Vitali-type theorem for a general measures, see [Fd], p.154 . Let D be an open set in ℂ^{n} and σ a finite positive Borel measure on D. Let further B be a family of closed balls of positive radii such, that for any point x ∈ D the family B contains the balls of arbitrarily small radii centered at x. Then one can find a countable subfamily {B_{i}} of balls in B such that

σ(D \ ⋃_{(i)} B_{i}) = 0.  (2.1.9)

Represent our measure μ_{1} as a difference μ_{1} = μ_{1}^{+} − μ_{1}^{−} of two nonnegative measures. Take some relatively compact open subset D ⊂ Ω. As B take the family of all balls that ∂B ∩ K = ∅. Because of zero dimensionality of K it is a Vitali-type covering. Let {B_{i}} are such that μ_{1}^{+}(D \ ⋃_{(i)} B_{i}) = 0. Then μ_{1}^{+}(D) = μ_{1}^{+}(D \ ⋃_{(i)} B_{i}) + ∑_{(i)} μ_{1}^{+}(B_{i}) = ∑_{(i)} μ_{1}^{+}(B_{i}). Consequently

μ_{1}(D) = μ_{1}^{+}(D) − μ_{1}^{−}(D) ≤ μ_{1}^{+}(⋃_{(i)} B_{i}) − μ_{1}^{−}(⋃_{(i)} B_{i}) =

= ∑_{i} μ_{1}^{+}(B_{i}) − ∑_{i} μ_{1}^{−}(B_{i}) = ∑_{i} μ_{1}(B_{i}) ≤ 0  (2.1.10)

by (2.1.8). Thus μ_{1}(D) ≤ 0 for any relatively compact open D in Ω. So the measure μ_{1} is negative.

q.e.d.

Remark. As we shall see in the proof of the Theorem 2 (Lemma 2.6.1) the conclusion of the part (b) of Lemma 2.1.1 remains true for arbitrary n and complete, pluripolar closed set of Haussdorff dimension 2n − 4.

Recall, see [Sb], that a subset K ⊂ Ω is called (complete) p-polar if for any a ∈ Ω there exists a neighborhood V of a and such coordinates z_{1}, ..., z_{n} in V that the sets

K_{z_{I}^{0}} = K ∩ {z_{i_{1}} = z_{i_{1}^{0}}, ..., z_{i_{p}} = z_{i_{p}^{0}}}  (2.1.11)

are (complete) pluripolar in subspaces V_{z_{i}^{0}} := {z ∈ V : z_{i_{1}} = z_{i_{1}^{0}}, ..., z_{i_{p}} = z_{i_{p}^{0}}} for almost all z_{I}^{0} = (z_{i_{1}}^{0}, ..., z_{i_{p}}^{0}) ∈ π^{I}(V), where I runs over a finite set of multiindices with |I| = p, such that {(π^{I})^{∗}w_{e}}_{I} generates the space of bedegree (p, p)-forms. Here π^{I}(z_{1}, ..., z_{n}) = (z_{i_{1}}, ..., z_{i_{p}}) stands for the projection onto the space of variables (z_{i_{1}}, ..., z_{i_{p}}).

2.2. Metrics on complex spaces and proof of Theorem 2 in dimension two. A Hermitian metric form on the complex space X we define in the following way. Let an open covering U_{α} of X is given together with proper holomorphic injections ϕ_{α} : U_{α} → V_{α} into a domains V_{α} ⊂ ℂ^{n(α)}. Let Uα^{′} be the images of U_{α}. Let {w_{α}} are positive (1,1)-forms on V_{α}. {w_{α}} defines a Hermitian metric form on X if (ϕ_{α} ∘ ϕ_{β}^{−1})^{∗}w_{β} = w_{α} for all α, β. Note that ϕ_{α} ∘ ϕ_{β}^{−1} is defined in some neighborhood of ϕ_{β}(U_{α} ∩ U_{β}) in ℂ^{n(β)}. We say that the metric w is Kähler if dw_{α} = 0 on V_{α} for all α. Definition 2.2.1. w is pluriclosed (negative) if dd^{c}w_{α} = 0 (dd^{c}w_{α} ≤ 0) in V_{α} for all α.

Let a meromorphic mapping f : H^{2}(r) → X from two-dimensional Hartogs figure into a disk-convex complex space is given. Let w be a plurinegative metric form on X. By W denote the maximal open subset of the unit disk Δ such that f meromorphically extends onto H_{W} (r) := W × Δ ∪ Δ × A_{1−r,1}. Here by A_{1−r,1} we denote the annulus {z ∈ ℂ : 1 − r < |z| < 1}. Let I(f) be the fundamental set of f and by f̂ denote the mapping f̂(z) = (z, f(z)) into the graph. For z ∈ W define

μ_{t}(z) = area f̂(Δ_{z}(t)) = ∫_{Δ_{z}(t)}(dd^{c}|λ|^{2} + f |_{Δ_{z}(t)}^{∗} w).  (2.2.1)

Here Δ_{z}(t) = {(z, λ) : |λ| < t}. We start with the following simple observation. Denote by ν_{1} = ν_{1}(K) the infimum of areas of compact complex curves which are contained in compact K ⊂ X. ν_{1} > 0, see Lemma 2.3.1 below. Lemma 2.2.1. Let f : Δ × A_{1−r,1} → X be a holomorphic mapping into a diskconvex complex space X. Suppose that for a sequence of points {s_{n}} ⊂ Δ, s_{n} → 0, the following holds:

(a) f_{s} := f |_{{s}×A_{1−r,1}} extends holomorphically onto Δ_{s} := {s} × Δ;

(b) For a compact K in X, which contains f[(Δ(1/2)×A_{1−2/3·r,1−1/3·r})∪⋃_{(n)}{s_{n}}× Δ_{1−1/3·r}], one has

| area f̂(Δ_{s_{n}}(1 − 1/3 · r)) − area f̂(Δ_{0}(1 − 1/3 · r)| ≤ \frac{1}{2} · ν_{1}(K).  (2.2.2)

Then f holomorphically extend onto V × Δ for some open V ∋ 0. Proof. First of all let us show that H−lim_{n→∞} f̂(Δ̄_{s_{n}}(1−1/3·r)) = f̂(Δ̄_{0}(1−1/3·r)), i.e. the sequence of graphs {f̂(Δ̄_{s_{n}}(1 − 1/3 · r))} converges in Hausdorff metric to the graph of the limit. If not, there would be a subsequence (still denoted as {f̂(Δ̄_{s_{n}}(1 − 1/3 · r))}), such that H − lim_{n→∞} f̂(Δ̄_{s_{n}}(1 − 1/3 · r)) = f̂(Δ̄_{0}(1 − 1/3 · r)) ∪ ⋃_{j=1}^{N} C_{j}, where ⋃_{j=1}^{N} C_{j} is a union of compact curves, see Lemma 2.3.1 below. Thus by (2.1.2) area f̂(Δ̄_{s_{n}}(1−1/3·r)) ≥ area f̂(Δ̄_{0}(1−1/3·r))+N ·ν_{1}(K). This contradicts (2.2.2).

Take a Stein neighborhood V of f̂(Δ̄_{0}(1 − 1/3 · r)), see [Si-1]. Then for δ > 0 small enough we have that f(Δ_{δ} × A_{1−1/3r−δ,1−1/3r+δ}) ⊂ V and f(Δ_{s_{n}}(1 − 1/3r)) ⊂ V if s_{n} ∈ Δ_{δ}. From Hartogs theorem for holomorphic functions we get that f extends to a holomorphic map from Δ_{δ} × Δ_{1−1/3r−δ} to V.

q.e.d.

Lemma 2.2.2. If the metric form w is plurinegative and W is maximal then ∂W ∩Δ is complete polar in Δ. Proof. Take a point z_{0} ∈ ∂W ∩ Δ. Choose relatively compact in Δ neighborhood U of z_{0} and 1 − r < t < 1 such that I(f) ∩ Ū × ∂Δ(t) = ∅. Denote by ϕ = iϕ^{αβ}dz_{α} ∧ dz̄_{β} the current f^{∗} + dd^{c}∥z∥^{2}. The area function from (1.2.1) can be written now as

μ_{t}(z_{1}) = i · ∫_{|z_{2}|≤t} ϕ_{22}(z_{1}, z_{2})dz_{2} ∧ dz̄_{2}.  (2.2.3)

The condition that dd^{c}ϕ is negative means that

\frac{∂^{2}ϕ_{11}}{∂z_{2}∂z̄_{2}} + \frac{∂^{2}ϕ_{22}}{∂z_{1}∂z̄_{1}} − \frac{∂^{2}ϕ_{12}}{∂z_{2}∂z̄_{1}} − \frac{∂^{2}ϕ_{21}}{∂z_{1}∂z̄_{2}} ≤ 0  (2.2.4)

on H_{W} (r). Now we can estimate the Laplacian of μ_{t}:

Δμ(z_{1}) = i ∫_{|z_{2}|≤t} \frac{∂^{2}ϕ_{22}}{∂z_{1}∂z̄_{1}} dz_{2} ∧ dz̄_{2} ≤ i ∫_{|z_{2}|≤t}(−\frac{∂^{2}ϕ_{11}}{∂z_{2}∂z̄_{2}} + \frac{∂^{2}ϕ_{12}}{∂z_{2}∂z̄_{1}} + \frac{∂^{2}ϕ_{21}}{∂z_{1}∂z̄_{2}} )dz_{2} ∧ dz̄_{2} =

= i ∫_{|z_{2}|=t} \frac{∂ϕ_{11}}{∂z_{2}} dz_{2} + i ∫_{|z_{2}|=t} \frac{∂ϕ_{12}}{∂z̄_{1}} dz̄_{2} − i ∫_{|z_{2}|=t} \frac{∂ϕ_{21}}{∂z_{1}} dz_{2} = ψ(z_{1}).  (2.2.5)

Inequality (2.2.5) holds for z_{1} ∈ U ∩ W. But the right hand side ψ is smooth in the whole U. Let Ψ be the smooth solution of ΔΨ = ψ in U. Put μ̂(z) = μ_{t}(z) − Ψ(z). Then μ̂ is superharmonic and bounded from below in U ∩ W, after shrinking of U.

Denote further by E the set of points z_{1} ∈ ∂W ∩ U such that μ_{t}(z) → +∞ as z ∈ W, z → z_{1}. Note that μ̂(z) also tends to +∞ in this case. For any point z_{∞} ∈ [∂W ∩ U] \ E we can find a sequence {z_{n}} ⊂ W, z_{n} → z_{∞} such that μ_{t}(z_{n}) ≤ C. So by Lemma 2.2.1 f |_{Δ_{z∞}\Δ_{z∞}(1−r)} extends onto Δ_{z_{∞}}. Thus we can define

μ_{t}(z) = area f̂(Δ_{z}(t)) = ∫_{Δ_{z}(t)}(dd^{c}|λ|^{2} + f |_{Δ_{z∞}(t)}^{∗} w).  (2.2.6)

Let ν_{1} be from Lemma 2.2.1. Set E_{j} = \{z ∈ ∂W ∩ U : μ_{t}(z) ≤ \frac{j}{2}ν_{1}\} for j = 1, 2, .... From Lemma 2.3.1 we see that E_{j} are closed subsets of ∂W ∩ U, E_{j} ⊂ E_{j+1} and we have that ∂W ∩ U = E ∪ ⋃_{j=1}^{∞} E_{j}.

Further from Lemma 2.2.1 we see that E_{j+1} \ E_{j} is a discrete subset of U \ E_{j}, say U \ E_{j} = {a_{ij}}. Now put

u_{1}(z) = − ∑_{i,j} c_{ij} log |z − a_{ji}|.  (2.2.7)

Here the positive constants c_{ij} are chosen in such a manner that ∑_{i,j} c_{ij} < +∞. Then u_{1}(z) is superharmonic in U, u_{1}(z) → +∞ as z → ⋃_{j=1}^{∞} E_{j} and u_{1}(z) ≠ +∞ for all z ∈ U ∩ W. Now put u_{2}(z) = μ̂(z) + u_{1}(z). Note that u_{2} is superharmonic in W ∩ U and u_{2}(z) → +∞ as z → ∂W ∩ U. Define

u_{n}(z) = min{n, u_{2}(z)}  (2.2.8)

for n ≥ 3. Note that u_{n} are superharmonic in U, because u_{n} ≡ n in the neighborhood of ∂W ∩ U. Put now u(z) = lim_{n→∞} u_{n}(z). Then u is superharmonic in U as a nondecreasing limit of superharmonic functions. Using the fact that μ̂ is finite on W, we obtain that u(z) = u_{2}(z) ≠ +∞ for any z ∈ U ∩ W and u |_{U\W} ≡ +∞, i.e. ∂W ∩ Δ is complete polar in Δ. So the Lemma is proved.

q.e.d.

In what follows we shall use the fact that the closed set of zero harmonic measure on the plain has zero Hausdorff dimension, see [Gl].

(a) Let us finish the proof of the part (1) of Theorem 2 in dimension two. Put S_{1} = Δ \ W where W is a maximal domain in Δ such that our map f extends meromorphically onto H_{W} (r). We had proved that S_{1} is of harmonic measure zero. In particular S_{1} is zero dimensional. For any δ > 0 we can find 0 < δ_{1} < δ such that ∂Δ_{1−δ_{1}} ∩ S_{1} = ∅. Now we can change coordinates z_{1}, z_{2} and consider the Hartogs figure H = {(z_{1}, z_{2}) ∈ Δ^{2} : 1 − r < |z_{2}| < 1, |z_{1}| < 1 or |z_{2}| < 1, 1 − δ_{1} − ε < |z_{1}| < 1 − δ_{1} + ε}, where ε is small enough. Applying once more Lemma 2.2.1 we extend f onto Δ × (Δ \ S_{2}) where S_{2} is of harmonic measure zero and obtain the statement of part (1) of Theorem 2 in the case of dimension two.

(b) Suppose now that our metric form w on X is pluriclosed. Adding to S := S_{1} × S_{2} the discrete in Δ^{2} \ S set of points of indeterminacy of f we can suppose that f is holomorphic on Δ^{2} \ S. Denote by T the positive (1,1) current (in fact smooth form) f^{∗}w on Δ^{2} \ S. By Lemma 3.3 from [Iv-2] we have that T has locally summable coefficients on the whole Δ^{2} and dd^{c}T̃ is a negative measure supported on S. Lemma 2.2.3. Suppose that the metric form w is pluriclosed and take a ball B ⊂⊂ Δ^{2} such that ∂B ∩ S = ∅.

(i) If f(∂B) is homologous to zero in X then dd^{c}T̃ = 0 on B.

(ii) If dd^{c}T̃ = 0 then f meromorphically extends onto B.

In [Iv-2] Lemma 4.4 ] this statement is proved for the case when S ∩ B = {0}. One can easily check that the same proof holds for the case when S ∩ B is closed zero dimensional.

So statement (2) of Theorem 2 is proved in this case.

Let us prove the Remark 1, stated after the Theorem 2 in Introduction. Take a relatively compact open subset P ⊂ Δ^{2} with smooth boundary and choose a finite subcomplex K of CW-complex X to contain the cl[f(P̄ \ S)]. Let θ_{1}, ..., θ_{N} be the generators of H^{3}(K, ℤ) and ψ_{1}, ..., ψ_{L} be the generators of H_{3}(K, ℤ). Take a real numbers r_{1}, ..., r_{N} such that

d^{c}w = r_{1}θ_{1} + ... + r_{N}θ_{N}  (2.2.9)

Take a ball B ⊂⊂ Δ^{2} with ∂B ∩ S = ∅. Then there are integers z_{1}, ..., z_{L} such that

f(∂B) = z_{1}ψ_{1} + ... + z_{L}ψ_{L}  (2.2.10)

in H_{3}(K, ℤ). For the measure μ defined from dd^{c}T̃ = μ · (\frac{i}{2})^{2}dz ∧ dz̄ we have that

μ(B ∩ S) = ∫_{B} dd^{c}T̃ = ∫_{∂B} d^{c}T.  (2.2.11)

Using that, we can write

μ(B ∩ S) = ∫_{f(∂B)} d^{c}w = ∑_{k=1}^{N} ∑_{i=1}^{L} z_{i}r_{k} ∫_{ψ_{i}} θ_{k}.  (2.2.12)

Put c^{ik} = ∫_{ψ_{i}} θ_{k} ∈ ℤ. Now if we put z̃^{k} = ∑_{i=1}^{L} z_{i}c^{ik} ∈ ℤ then

μ(B ∩ S) = ∑_{k=1}^{N} z̃^{k}r_{k}  (2.2.13)

So for any such a ball μ(B ∩ S) is a linear combination with integer coefficients of a given reals r_{1}, ..., r_{N}. By induction one easily prove that there is an ε = ε(r_{1}, .., r_{N}) > 0 such that each linear combination of r_{1}, ..., r_{N} with integer coefficients is either zero or greater with modulus then ε. We denote this ε as ε(w, K).

Now we obtain that

|S ∩ P | ≤ ε(w, K)^{−1} · | ∫_{∂P} d^{c}w|.  (2.2.14)

q.e.d.

2.3. Meromorphic families of analytic sets. When one studies meromorphic mappings from domains of dimension more then two, one needs some analog of Lemma 2.2.1 for “meromorphic polydisks”. That will be given in this paragraph, see Lemmas 2.3.1 and 2.3.2. For the proof of this two Lemmas we refer to [Iv-4].

Fix a complex space X, equipped with some Hermitian metric h. By w_{h}, or simply by w denote the (1, 1)-form canonically associated with h. Let Δ^{q} be a polydisk in ℂ^{q} with standard Euclidean metric e. The associated form will be denoted by w_{e} = dd^{c}∥z∥^{2} = i/2 ∑_{j=1}^{q} dz_{j} ∧ dz̄_{j}. By p_{1} : Δ^{q} × X ⟶ Δ^{q} and p_{2} : Δ^{q} × X ⟶ X we denote the projections onto the first and second factors. On the product Δ × X we consider the metric form w = p_{1}^{∗}w_{e} + p_{2}^{∗}w_{h}.

Definition 2.3.1. By a meromorphic q-disk in the complex space X we shall understand a meromorphic mapping ϕ : Δ^{q} ⟶ X , which is defined in some neighbourhood of the closure Δ̄^{q}.

It will be convenient for us to consider instead of mappings ϕ : Δ^{q} ⟶ X their graphs Γ_{ϕ}. By ϕ̂ = (z, ϕ(z)) we shall denote the mapping into the graph Γ_{ϕ} ⊂ Δ^{q}×X. The volume of the graph Γ_{ϕ} of the mapping ϕ is given by

vol(Γ_{ϕ}) = ∫_{Γ_{ϕ}} w^{q} = ∫_{Δ^{q}}(ϕ^{∗}w_{h} + dd^{c}∥z∥^{2})^{q}  (2.3.1)

Here by ϕ^{∗}w_{h} we denote the preimage of w_{h} under ϕ, i.e. ϕ^{∗}w_{h} = (p_{1})_{∗}p_{2}^{∗}w_{h}.

Recall that the Hausdorff distance between two subsets A and B of the metric space (Y, ρ) is a number ρ(A, B) = inf{ε : A^{ε} ⊃ B, B^{ε} ⊃ A}. Here by A^{ε} we denote the ε-neighborhood of the set A, i.e. A^{ε} = {y ∈ Y : ρ(y, A) < ε}.

Further, let {ϕ_{r}} be the sequence of meromorphic mappings of the complex space D into the complex space X.

Definition 2.3.2. We shall say that {ϕ_{r}} converge on the compacts in D to the meromorphic mapping ϕ : D ⟶ X, if for every relatively compact open D_{1} ⊂⊂ D the graphs Γ_{ϕ_{r}} ∩ (D_{1} × X) converge in the Hausdorff metric on D_{1} × X to the graph Γ_{ϕ} ∩ (D_{1} × X).

First we shall prove the following

Lemma 2.3.1. Let {ϕ_{r}} be a sequence of meromorphic q-disks in complex space X. Suppose that there exists a compact K ⊂ X and a constant C < ∞ such that:

a) ϕ_{r}(Δ^{q}) ⊂ K for all r;

b) vol(Γ_{ϕ_{r}}) ≤ C for all r.

Then there exists a subsequence {ϕ_{r_{j}}} and a proper analytic set A ⊂ Δ^{q} such that:

1) the sequence {Γ_{ϕ_{r_{j}}} } converges in the Hausdorff metric to the analytic subset Γ of Δ^{q} × X of pure dimension q;

2) Γ = Γ_{ϕ} ∪ Γ̂,where Γ_{ϕ} is the graph of some meromorphic mapping ϕ : Δ^{q} ⟶ X,and Γ̂ is a pure q-dimensional analytic subset of Δ^{q} × X ,mapped by the projection p_{1} onto A;

3) ϕ_{r_{j}} ⟶ ϕ on compacts in Δ^{q} \ A;

4) one has

lim_{j⟶∞} vol(Γ_{ϕ_{r_{j}}}) ≥ vol(Γ_{ϕ}) + vol(Γ̂).  (2.3.2)

5) For every 1 ≤ p ≤ dimX − 1 there exists a positive constant ν_{p} = ν_{p}(K, h) such that the volume of every pure p-dimensional compact analytic subset of X which is contained in K is not less then ν_{p}.

6) Put Γ̂ = ⋃_{p=0}^{q−1} Γ_{p}, where Γ_{p} is a union of all irreducible components of Γ̂ such that dim[p_{1}(Γ_{p})] = p. Then

vol_{2q}(Γ̂) ≥ ∑_{p=0}^{q−1} vol_{2p}(A_{p}) · ν_{q−p}  (2.3.3)

where A_{p} = p_{1}(Γ_{p}).

Let S be a set. By Δ^{q}(b) we always denote a polydisk of radii b in ℂ^{n} centered at origin. Polydisk is equipped with the usual Euklidean metric from ℂ^{n}. Definition 2.3.3. By a family of q-dimensional polydisks in complex space X we shell understand an subset F ⊂ S × Δ^{q}(b) × X such that, for every s ∈ S the set F_{s} = F ∩ {s} × Δ^{q}(b) × X is a graph of a meromorphic mapping of Δ^{q}(b) into X.

Suppose further that the set S is equipped with topology and let our space X be equipped with some Hermitian metric h. Definition 2.3.4. We shall say that the family F is continuous at point s_{0} ∈ S if H − lim_{s→s_{0}} F_{s} = F_{s_{0}}.

Here by H−lim_{s→s_{0}} F_{s} we denote the limit of closed subsets of F_{s} in the Haussdorff metric on Δ^{q}(b) × X. F is continuous if it is continuous at each point of S. If Ω is open in Δ^{q}(b) then the restriction F_{Ω} is naturally defined as F ∩ (S × Ω × X).

When S is a complex space itself, we give the following

Definition 2.3.5. Call the family F meromorphic if the closure F̂ of the set F is an analytic subset of S × Δ^{q}(b) × X.

Consider a meromorphic mapping f : Δ^{p} × Δ^{q}(a) ⟶ X into a complex space X. Let S be some closed subset of Δ^{p} and s_{0} ∈ S some accumulation point of S. Suppose that for each s ∈ S the restriction f_{s} = f|_{{s}} × Δ^{q}(a) meromorphically extends onto a q-disk Δ^{q}(b). Let, as in Lemma 2.3.1 ν_{j} denotes the minima of volumes of j-dimensional compact analytic subsets contained in some compact K ⊂ X. Fix some a < c < b and put

ν = min{vol(A_{q−j}) · ν_{j} : j = 1, ..., q},  (2.3.6)

where A_{q−j} are running over all (q − j)-dimensional analytic subsets of Δ^{q}(b), intersecting Δ̄^{q}(c). Here a < c < b. Clearly ν > 0. In the following Lemma the volumes of graphs over polydisks Δ(b) are taken. Lemma 2.3.2. Suppose that there exists a neighbourhood U ∋ s_{0} in Δ^{p} such that, for all s_{1}, s_{2} ∈ S ∩ U

| vol(Γ_{f_{s_{1}}}) − vol(Γ_{f_{s_{2}}})| < ν/2  (2.3.7)

Then the family {Γ_{f_{s}} : s ∈ U} is continuous at s_{0}. If, moreover s_{0} is a locally regular point of S then there exists a neighbourhood V_{c} ∋ s_{0} in Δ^{p} such, that f meromorphically extends onto V_{c} × Δ^{q}(c).

2.4. Proof of Theorem 2 in higher dimensions. Proof. (1) Let f : H_{n}^{1}(r) → X be our map. For an open subset W ⊂ Δ^{n} denote by

H_{W} (r) = (W × Δ) ∪ (Δ^{n} × A_{1−r,1})

the Hartogs figure over W. Step 1. f extends to a holomorphic map of ⋃_{z′∈Δ_{r}^{n−1}\R_{1}}(Δ_{z′}^{2} \ S_{z′}) into X, where R_{1} is contained in locally finite union of locally closed proper subvarieties of Δ_{r}^{n−1} and S_{z′} is zerodimensional and pluripolar in Δ_{z′}^{2}. Proof of Step 1. For z′ = (z_{1}, ..., z_{n−1}) ∈ Δ_{r}^{n−1} by H_{z′}^{2}(r) denote the two-dimensional Hartogs domain {z′} × H^{2}(r) in bidisk Δ_{z′}^{2} = {z′} × Δ^{2} ∈ ℂ^{n+1}. Shrinking H_{n}^{1}(r) if necessary, we can suppose that I(f) consists of finitely many irreducible components. Denote by R_{1} the set of z′ ∈ Δ_{r}^{n−1} such that dim[H_{z′}^{2} ∩ I(f)] > 0. R_{1} is clearly contained in finite union of locally closed proper analytic subsets of Δ_{r}^{n−1}. For z′ ∈ Δ_{r}^{n−1} \ R_{1} by 2.1 f |_{H_{z′}^{2}(r)} extends to a holomorphic map f_{z′} : Δ_{z′}^{2} \ S_{z′} → X, where S_{z′} is zerodimensional and complete pluripolar in Δ_{z′}^{2}. Also S_{z′} ⊃ Δ_{z′}^{2} ∩ I(f).

Take a point z′ ∈ Δ_{r}^{n−1} \R_{1} and a point z_{n} ∈ Δ\π_{n}(S_{z′}). Here π_{n} : {z′}×Δ×Δ → {z′} × Δ is the projection onto the variable z_{n}. Take a domain U ⊂⊂ {z′} × Δ × {0} which is biholomorphic to the unit disk, doesn’t contains points from π_{n}(S_{z′}) and contains the points u := (z′, 0, 0) and v := (z′, z_{n}, 0). We take also U to intersect A(r, 1). If {z′}×{0} is in π_{n}(S_{z′} the take as u some point close to (z′, 0, 0) in {z′}×Δ. Find a Stein neighbourhood V of the graph Γ_{f|_{{z′}×Ū×Δ}}. Let w ∈ ∂U ∩A(r, 1) be some point. We have that f({z′, w} × Δ) ⊂ V and f({z′} × ∂U × Δ) ⊂ V. So the usual continuity principle for holomorphic functions gives us the holomorphic extension of f to the neighborhood of {z′} × Ū × Δ in Δ^{n+1}. Changing little bit a bend of z_{n+1} and repeating the arguments as above we obtain a holomorphic extension of f onto the neighborhood of {z′} × (Δ \ S_{z′}) for each z′ ∈ Δ_{r}^{n−1} \ R_{1}.

Step 2. f extends holomorphically onto (Δ_{r}^{n−1} × Δ^{2}) \ R, where R is a closed subset of Δ_{r}^{n−1} × Δ^{2} of Hausdorff codimension 4. Proof of Step 2. Consider a subset R_{2} ⊂ R_{1} consigning of such z′ ∈ Δ_{r}^{n−1} that dim[H_{z′}^{2} ∩ I(f)] = 2, i.e. H_{z′}^{2} ⊂ I(f). This is a finite union of locally closed subvarieties of Δ_{r}^{n−1} of complex codimension at least two. Thus ⋃_{z′∈R_{2}} Δ_{z′}^{2} has Hausdorff codimension at least four.

For z′ ∈ R_{1} \ R_{2} = {z′ ∈ Δ_{r}^{n−1} : dim[H_{z′}^{2}(r) ∩ I(f)] = 1} using Theorem 1 we can extend f_{z′} holomorphically onto Δ_{z′}^{2} minus zerodimensional polar set. Repeating the arguments from Step 1 we can extend f holomorphically to the neighborhood of Δ_{z′}^{2} \ C_{z′} in Δ_{r}^{n−1} × Δ^{2}. Here C_{z′} is a complex curve containing one dimensional components of H_{z′}^{2}(r) ∩ I(f).

⋃_{z′∈R_{1}\R_{2}} C_{z′} has Hausdorff codimension at least four. Thus the proof of Step 2 will be completed when we put R = ⋃_{z′∈R_{1}\R_{2}} C_{z′} ∪ ⋃_{z′∈R_{2}} Δ_{z′}^{2}. Step 3. We shall state this step in the form of the Lemma. Lemma 2.6.1. There is a closed (n-1)-polar subset R_{0} ⊂ R and a holomorphic extension of f onto (Δ_{r}^{n−1} × Δ^{2}) \ R_{0} such that the current T := f^{∗}w has locally summable coefficients in the neighborhood of R_{0}. Moreover dd^{c}T̃ is negative.

Take a point z_{0} ∈ R and using the fact that R is of Hausdorff codimension four in ℂ^{n+1}, find a neighbourhood V ∋ z_{0} with a coordinate system (z_{1}, ..., z_{n+1}) such that V = Δ^{n−1} × Δ^{2} is those coordinates and for all z′ ∈ Δ^{n−1} one has R ∩ ∂Δ_{z′}^{2} = 0. By 2.1 the restrictions f_{z′} extend holomorphically onto Δ_{z′}^{2} \ R_{0}(z′), where R_{0}(z′) are closed complete pluripolar in Δ_{z′}^{2} of Hausdorff dimension zero. By the arguments similar to that ones from Step1 f extends holomorphically to the neighborhood of V \ R_{0}, R_{0} := ⋃_{z′∈Δ^{n−1}} R_{0}(z′).

Consider now a current T = f^{∗}w defined on (Δ^{n−1} × Δ^{2}) \ R. Note that T is smooth, positive and dd^{c}T ≤ 0 there. By the part (b_{2}) of Lemma 1.1.3 every restriction T_{z′} := T |_{Δ_{z′}^{2}}∈ L_{loc}^{1}(Δ_{z′}^{2}), z′ ∈ Δ^{n−1}. We shall use the following Oka-type inequality for plurinegative currents, proved in [F-Sb]:

there is a constant C_{ρ}, such that for any plurinegative current T in Δ^{2} one has

∥T ∥(Δ^{2}) + ∥dd^{c}T ∥(Δ^{2}) ≤ C_{ρ}∥T ∥(Δ^{2} \ Δ̄_{ρ}^{2}).  (2.4.1)

Here 0 < ρ < 1.

Use (2.4.1) for the trivial extensions T̃_{z′} of T_{z′}, which are plurinegative by (b) of Lemma 2.1.1, to obtain that masses ∥T̃_{z′}∥(Δ^{2}) are uniformly bounded on z′ on compacts in Δ^{n−1}. On L^{1} the mass norm coincides with the L^{1}-norm. So taking the second factor in Δ^{n−1} × Δ^{2} with different bends and using Fubini theorem we obtain that T ∈ L_{loc}^{1}(Δ_{r}^{n−1} × Δ^{2}).

All that left to prove is that dd^{c}T̃ is negative. It is enough to prove that for any collection L of (n−1) linear functions \{l_{1}, ..., l_{n−1}\} the measure dd^{c}T̃∧ \frac{i}{2}∂l_{1}∧ ¯∂l_{1}∧...∧ \frac{i}{2}∂l_{n−1} ∧ ¯∂l_{n−1} is nonpositive, see [Hm]. Complete those functions to the coordinate system {z_{1} = l_{1}, ..., z_{n−1} = l_{n−1}, z_{n}, z_{n+1}} and note that for almost all z′ ∈ Δ^{n−1} the set Δ_{z′}^{2} ∩ R_{0} is of Hausdorff dimension zero. Thus T̃ |_{z′} is plurinegative for all such z′. Take a nonnegative function ϕ ∈ D(Δ^{n+1}). We have that

< dd^{c}(T̃ ∧ (dd^{c}∥L∥^{2})^{n−1}), ϕ >= ∫_{Δ^{n+1}} T̃ ∧ (dd^{c}∥z′∥^{2})^{n−1} ∧ dd^{c}ϕ =

= ∫_{Δ^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{Δ^{2}}(T̃)_{z′} ∧ dd^{c}ϕ == ∫_{Δ^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{Δ^{2}} T̃_{z′} ∧ dd^{c}ϕ ≤

≤ ∫_{Δ^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{Δ^{2}} dd^{c}(T̃)_{z′} ∧ ϕ ≤ 0.

We had used here Fubini theorem for L^{1}-functions, the fact that (T̃)_{z′} = T̃_{z′} for currents from L_{loc}^{1} which are smooth outside of suitably situated set R_{0}, and finally the plurinegativity of T̃_{z′}.

Thus T̃ is plurinegative. Putting S = R_{0} we get the statement of (a) for Δ_{r}^{n−1} ×Δ^{2} instead of Δ^{n+1}. But this obviously implies it for Δ^{n+1}.

(2) Now suppose that the metric form w is pluriclosed. Write V = B^{n−1} × B^{2} for some neighborhood of point a ∈ S such that π |_{S}: S → B^{2} is proper and S_{z′} = S ∩B_{z′}^{2} is zerodimensional pluripolar compact in B_{z′}^{2} for all z′ ∈ B^{n−1}.

For every z′ ∈ B^{n−1} let S_{z′}^{0} be a finite, by Theorem 1, set of points s ∈ B_{z′}^{2} such that dd^{c}T̃_{z′} has nonzero mass at s. |S_{z′}^{0}| is uniformly bounded on z′ on B̄^{n−1}. The points s ∈ S_{z′}^{0} could be also characterized by the condition that for any 3-sphere S_{r}(s) ⊂ B_{z′}^{2} centered at s ∫_{S_{r}(s)} d^{c}T_{z′} = C_{s} < 0. Number C_{s} doesn’t depend on r sufficiently small. From this one immediately gets the closeness of the set S^{0} := ⋃_{z′∈B^{n−1}} S_{z′}^{0}. Step 4. f extends meromorphically onto V \ S^{0}. Proof of Step 4. Let b ∈ B_{z′}^{2} \ S_{z′}^{0}. Find a neighborhood W ≅ B^{n−1} × B^{2} of b such that W ∩ S_{0} = ∅ and π_{2} |_{S}: S ∩ W → B^{2} is proper. Here S is a minimal closed subset of V such that f : V \ S → X is holomorphic, see part (1).

First we prove the following: Lemma 2.6.2. Suppose that the metric form w on X is pluriclosed and for all z′ ∈ B^{n−1} f(∂B_{z′}^{2}) ∼ 0 in X. Then:

(i) dd^{c}T̃ = 0 in the sense of distributions.

(ii) There is a (1, 0)-current γ in W, smooth in W \ R_{0}, such that T̃ = i(∂γ̄ − ¯∂γ).

Proof. (i) Let T̃_{ε} be a smoothing of T̃ by convolution. Then T̃_{ε} are plurinegative and T̃_{ε} → T in D_{n,n}(W). We have that

∫_{W} dd^{c}T̃_{ε} ∧ (dd^{c}∥z′∥^{2})^{n−1} = ∫_{∂W} d^{c}T̃_{ε} ∧ (dd^{c}∥z′∥^{2})^{n−1} =

= ∫_{∂B^{n−1}×B^{2}} d^{c}T_{ε} ∧ (dd^{c}∥z′∥^{2})^{n−1} + ∫_{B^{n−1}×∂B^{2}} d^{c}T_{ε} ∧ (dd^{c}∥z′∥^{2})^{n−1}.  (2.6.2)

The first integral vanishes by the degree reason. So

∥dd^{c}T̃_{ε} ∧ (dd^{c}∥z′∥^{2})^{n−1}∥(W) = − ∫_{B^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{∂B_{z′}^{2}} d^{c}T̃_{ε}.  (2.6.3)

Remark now that ∫_{∂B_{z′}^{2}} d^{c}T̃_{ε} → ∫_{∂B_{z′}^{2}} d^{c}T_{ε} = ∫_{f(∂B_{z′}^{2})} d^{c}w = 0 because f(∂B_{z′}^{2}) ∼ 0 in X. So the right hand side of (2.6.3) tends to zero as ε ↘ 0. We get that

∥dd^{c}T̃ ∧ (dd^{c}∥z′∥^{2})^{n−1}∥(W) = lim_{ε↘0} ∥dd^{c}T̃_{ε} ∧ (dd^{c}∥z′∥^{2})^{n−1}∥(W) = 0.  (2.6.4)

Taking sufficiently many such coordinate systems we see that ∥dd^{c}T̃∥(W) = 0. (ii) ∂T̃ is a ¯∂-closed and ∂-closed (2, 1)-current. So, if ϕ ∈ D_{n−2,2}(W) is ∂-closed and such that ¯∂ϕ = T̃ then ϕ is smooth on W \ S by elliptic regularity of ¯∂. We have now dT̃ = ∂T̃ + ¯∂T̃ = ¯∂ϕ + ∂ϕ̄. Thus d(T̃ − ϕϕ̄) = 0. So T̃ − ϕ − ϕ̄ is d-closed current of degree two on W. Consider an elliptic system in W:

dγ = T̃ − ϕ − ϕ̄
d^{∗}γ = 0  (2.6.5)

Then γ has a solution in W. Indeed, let γ_{1} be any solution of the first equation. Find a distribution γ on W with ∗d ∗ dγ = Δγ = ∗d ∗ γ_{1} and put γ_{2} = γ_{1} − dγ. γ_{2} is smooth on W \ S because Δγ_{2} = d^{∗}dγ_{2} + dd^{∗}γ_{2} = d^{∗}(T̃ − ϕ + ϕ̄) and γ_{2} satisfies (3.4.6). Write γ_{2} = γ^{1,0} + γ̄^{1,0}. Then we have ∂γ^{1,0} = −ϕ and ¯∂γ̄^{0,1} = −ϕ̄, so

T̃ = dγ_{2} + ϕ + ϕ̄ = ∂γ̄^{1,0} + ¯∂γ^{0,1}  (2.6.6)

with γ^{1,0} having needed regularity.

q.e.d.

Lemma 2.6.3. If T̃ is pluriclosed then the volumes Γ_{f_{z}} ∩ 𝔹_{z}^{2} × X are uniformly bounded for z ∈ 𝔹_{r}^{n−1}. Proof. Now let γ^{1,0} be as in (2.6.6). Smoothing by convolutions we still have T̃_{ε} = ∂γ̄_{ε}^{1,0} + ¯∂γ_{ε}^{1,0}. Take 0 < r_{1} < r_{2} in such a way that ∂(B_{r}^{n−1} × B_{r_{2}}^{2}) ∩ S ⊂ ∂B_{r}^{n−1} × B_{r_{2}}^{2} for all r < r_{1} , then:

∫_{B_{r}^{n−1}×B_{r}^{2}\S} T^{2} ∧ (dd^{c}∥z′∥^{2})^{n−1} ≤ ∫_{B_{r}^{n−1}×B_{r_{2}}^{2}\S} T^{2} ∧ (dd^{c}∥z′∥^{2})^{n−1} =

= lim_{ε↘0} ∫_{B_{r}^{n−1}×B_{r_{2}}^{2}\S} T̃_{ε}^{2} ∧ (dd^{c}∥z′∥^{2})^{n−1} ≤ lim_{ε↘0} ∫_{B_{r}^{n−1}×B_{r_{2}}^{2}} T̃_{ε}^{2} ∧ (dd^{c}∥z′∥^{2})^{n−1} =

= lim_{ε↘0} ∫_{B_{r}^{n−1}×B_{r_{2}}^{2}}(∂γ̄_{ε}^{1,0} + ¯∂γ_{ε}^{1,0})^{2} ∧ (dd^{c}∥z′∥^{2})^{n−1} ≤

≤ lim_{ε↘0} ∫_{B_{r}^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{B_{r_{2}}^{2}} d(γ̄_{ε}^{1,0} + γ_{ε}^{1,0}) ∧ d(γ̄_{ε}^{1,0} + γ_{ε}^{1,0}) =

= lim_{ε↘0} ∫_{B_{r}^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{B_{r_{2}}^{2}}(γ̄_{ε}^{1,0} + γ_{ε}^{1,0}) ∧ d(γ̄_{ε}^{1,0} + γ_{ε}^{1,0}) =

= ∫_{B_{r}^{n−1}}(dd^{c}∥z′∥^{2})^{n−1} ∫_{B_{r_{2}}^{2}}(γ̄^{1,0} + γ^{1,0}) ∧ d(γ̄^{1,0} + γ^{1,0}) ≤ c · r^{2(n−1)}.  (2.6.7)

In the second inequality we had used the positivity of T. In the third the fact that ¯∂γ̄_{ε}^{1,0}∧∂γ_{ε}^{1,0} is positive and ¯∂γ̄_{ε}^{1,0}∧ ¯∂γ̄_{ε}^{1,0} = 0. Finally γ_{ε}^{1,0} → γ^{1,0} on B̄_{r}^{n−1}×∂B_{r_{2}}^{2} while γ^{1,0} is smooth there. This gives a needed bound for the ∫_{B_{r}^{n−1}×B_{r}^{2}\S} T^{2}∧(dd^{c}∥z′∥^{2})^{n−1}.

q.e.d.

Theorem 1 gives us now the proof of Step 4. Step 5. The set S^{0} is analytic of pure codimension two. Proof of Step 5. We had proved that f meromorphically extends onto V \ S^{0}, where V ≅ B^{n−1} × B^{2} and S^{0} is a graph of N- valued continuous mapping of B^{n−1} to B^{2}. N- valued means that |S_{z′}^{0}| ≤ N for every z′ ∈ B̄^{n−1} and there exist z0^{′} such |S_{z′}^{0}| = N. Continuous - simply that S^{0} is closed. Remark that T̃ is L^{1}-current on V with dd^{c}T̃ ≤ 0 and supported on S^{0}. Lemma 2.6.4. Let S^{0} be the graph of N- valued continuous mapping of Δ̄^{k} to Δ^{l} and let R be a closed positive current in Δ^{k+l} of bidimension (k, k) supported on S^{0}. Then S^{0} is a pure k-dimensional analytic variety in Δ^{k+l}. Proof. Write R = R_{K,J̄}(\frac{i}{2})^{k} \frac{∂}{∂z^{K}} ∧ \frac{∂}{∂z̄^{J}}, where K and J are multiindices of length k. Consider a measures R_{K,J̄}. Disintegrate this measures with respect to the natural projection π : Δ^{k} × Δ^{l} → Δ^{k}, see [Bk], p.58. Denote by μ_{K,J̄} = π_{∗}(R_{K,J̄}) their direct images. Then disintegration means that one has probability measures ν_{K,J̄,z′} on Δ_{z′}^{l} := {z′} × Δ^{l} with the property that for every continuous function h in Δ^{k+l}

< R_{K,J̄}, h >= ∫_{Δ^{k}}(∫_{Δ_{z′}^{l}} h̄ |_{Δ_{z′}^{l}} dν_{K,J̄,z′})dμ_{K,J̄},  (2.6.8)

see [Bk] or [D-M].

Let Ω be the maximal open subset of Δ^{k} such that the multivalued map s, which is given by its graph S^{0} takes exactly N different values (and N is maximal). First we shall prove that S^{0} ∩ (Ω × Δ^{l}) is analytic.

Let further Ω_{1} be some simply connected open subset of Ω. Then s |_{Ω_{1}} decomposes to N well-defined single valued maps s^{1}, ..., s^{N}. So it is enough to consider the case when s is single valued. Put s(z′) = (s_{1}(z′), ..., s_{l}(z′)). Note that in this case ν_{K,J̄,z′} = δ_{{z′′−s(z′)}}. We can write for the coefficients of our current R and for ϕ ∈ C^{∞}(Δ^{k+l}) such that π(supp ϕ) ⊂⊂ Δ^{k} that

< R_{K,J̄}, ϕ >= ∫_{Δ^{k}} ϕ̄(z′, s(z′))dμ_{K,J̄}(z′).  (2.6.9)

If we choose ϕ not depending on z′′ := (z_{k+1}, ..., z_{k+l}) then (2.6.9) gives

< R_{K,J̄}, ϕ >= ∫_{Δ^{k}} ϕ̄(z′)dμ_{K,J̄}(z′).  (2.6.10)

From the closeness of R we obtain that

0 =< R, d[(\frac{i}{2})^{k}ϕ(z′)dz_{1}∧...dz_{p−1}∧dz_{p+1}∧...∧dz_{k}∧dz̄_{J}] >=< R_{1...(p−1)(p+1)...k,J̄}, \frac{∂ϕ}{∂z_{p}} >=

= ∫_{Δ^{k}} \frac{∂ϕ̄}{∂z̄_{p}} dμ_{K,J̄}.

So μ_{K,J̄}(z_{1}) = c_{K,J̄}(z_{1}) · (\frac{i}{2})^{k}dz′ ∧ dz̄′, where c_{K,J̄} are holomorphic for K = (1, ..., k) and all J. In particular c_{1...k,¯1...k̄} is constant. Now take the (k − 1, k)- forms ψ_{qp̄} = ϕ(z′) · z̄_{p} · (\frac{1}{2})^{k}dz_{1} ∧ ... ∧ dz_{q−1} ∧ dz_{q+1} ∧ ... ∧ dz_{k} ∧ dK̄. We have

0 =< R, dψ_{q,p̄} >=< R, \frac{∂ϕ}{∂z_{q}} · z̄_{p}(\frac{i}{2})^{k}dz^{K} ∧ dz̄^{K} >=< R_{K,K̄}, \frac{∂ϕ}{∂z_{q}} z̄p̄ >=

= c_{1¯1} ∫_{Δ} \frac{∂ϕ̄}{∂z̄_{q}} (z′) · s_{p}(z′)(\frac{i}{2})^{k}dz′ ∧ dz̄′,

i.e. s_{p} are holomorphic.

Thus we had proved that s is N-valued analytic map of Ω into Δ^{l}. Considering appropriate discriminants and using Rado’s Theorem, we obtain analyticity of s on the whole Δ^{k}.

If for some sphere S^{3} imbedded into Δ^{n+1} \ S^{0} f(S^{3}) is homologous to zero in X then one can extend f through one of the branches of S^{0} using the same arguments as in the proof of Step 4.

q.e.d.

2.5. Complex Plateau problem for three-dimensional contours. We shall apply the results of this paragraph to the complex Plateau problem. Namely, we shall prove part (b) of the Corollary 3 from Introduction.

Recall that complex Plateau problem for a real compact submanifold M of complex manifold X consists in finding a complex analytic subset A ⊂ X \ M such that ∂[A] = [M] in the sense of currents. The necessary condition on M for complex Plateau problem to have a solution is the maximal complexity of M, i.e. M should be a CR-submanifold of X of dim_{CR} M = p, where dim_{ℝ} M = 2p + 1. In the case when X is Stein this is also sufficient, see [H-L]. Already in the case X = ℂℙ^{3} the maximal complexity of the “contour” M is not sufficient any more, [Dl].

We suppose that M bounds an abstract Stein domain, i.e. there is a complex manifold D with boundary M such that D \ M is Stein, and the CR-embedding f : M → X is given. All that we need to prove is that f extends meromorphically onto D. Clearly we can suppose that f is already holomorphically extended to some neighborhood of M in D. Proposition 2.5.1. Let (D, M) be as above and suppose additionally that dim M = 3.

(a) Then any CR-map f : M → X, where X is a disk-convex complex space admitting

a pluriclosed Hermitian metric form, extends meromorphically onto D \ S. Here S is a finite subset of D.

(b) If f(M) is homologous to zero in X, or if X doesn’t contain spherical shells, then

S is empty. Proof. Let ρ : D → [0, 1] be a strictly plurisubharmonic (and thus Morse) exhausting function. Denote by D_{ε}^{+} = {z ∈ D : ρ(z) > ε}. Let E be the set of such ε that f can be meromorphically extended onto D_{ε}^{∗} \ S_{ε}, where S_{ε} is a discrete set. E is obviously closed and nonempty. All we need to prove is that E is open.

Let ε_{0} = inf{ε ∈ E}. If ε_{0} is a regular value of ρ then the needed result immediately follows from part (2) of Theorem 2.

Consider the case of not regular value ε_{0} of ρ. Denote by M_{ε_{0}} = {z : ρ(z) = ε_{0}}- the critical level set. Fix a critical point z_{0} ∈ M_{ε_{0}}. All we need to prove is that for any neighborhood W of z_{0} the envelope of holomorphy of W ∩ D_{ε}^{+} contains some neighborhood of z_{0}. For convenience we can suppose that z_{0} = 0 and ε_{0} = 0. Write

ρ(z) = Q(z)+ < z, z > +Q̄(z) + O(∥z∥^{3}),  (2.5.1)

where Q(z) is a holomorphic polynomial, < z, z > - Hermitian form - Levi form of ρ. By linear coordinate change we transform <, > to the sum of squares of absolute values. Then by unitary coordinate change we transform Q to the some of squares with real nonnegative coefficients. Now (2.5.1) has a form

ρ(z) = ∑_{j=1}^{p} a_{j}z_{j}^{2} + ∑_{j=1}^{p} a_{j}z̄_{j}^{2} + ∑_{j=1}^{n} |z_{j}|^{2} + O(∥z∥^{3}).  (2.5.2)

In coordinates z_{j} = x_{j} + iy_{j} we rewrite (2.5.2) as follows

ρ(z) = 2 ∑_{j=1}^{p} a_{j}(x_{j}^{2} − y_{j}^{2}) + ∑_{j=1}^{n}(x_{j}^{2} + y_{j}^{2}) + O(∥z∥^{3}) =

= ∑_{j=1}^{p}[(1 + 2a_{j})x_{j}^{2} + (1 − 2a_{j})y_{j}^{2}] + ∑_{j=p+1}^{n}(x_{j}^{2} + y_{j}^{2}) + O(∥z∥^{3}).  (2.5.3)

Renumerate the coordinates in such a way that a_{j} ≥ \frac{1}{2} for j = 1, ..., q and a_{j} < 1/2 for j = q + 1, ..., p. Then

ρ(z) ≥ ∑_{j=1}^{q}[(2a_{j} + 1)x_{j}^{2} − (2a_{j} − 1)y_{j}^{2}] + δ · ∑_{j=q+1}^{p} |z_{j}|^{2} + O(∥z∥^{3}) ≥

≥ ∑_{j=}^{q}[(2a_{j} − δ_{1} + 1)x_{j}^{2} − (2a_{j} + δ_{1} − 1)y_{j}^{2}] + δ · ∑_{j=q+1}^{p} |z_{j}|^{2} := ρ_{1}(z),  (2.5.4)

for some δ > 0 and δ_{1} can be chosen arbitrarily small for small ∥z∥. While obviously D^{+} := {z ∈ 𝔹^{n} : ρ_{1}(z) > 0} ⊂ D_{ε_{0}}^{+}, all we need is to prove the following Lemma 2.5.2. The envelope of holomorphy of D^{+} contains the origin. Proof. Consider two cases. Case 1: q ≤ n − 1. In this case D^{+} contains the following Hartogs figure:

H := {z ∈ 𝔹^{n} : ∑_{j=1}^{q}[(2a_{j} − δ_{1} + 1)x_{j}^{2} − (2a_{j} + δ_{1} − 1)y_{j}^{2}] > 0, δ · ∑_{j=q+1}^{n} |z_{j}|^{2} < 1

or

∑_{j=1}^{q}(2a_{j} − δ_{1} + 1)x_{j}^{2} − (2a_{j} + δ_{1} − 1)y_{j}^{2}] > −ε, δ · ∑_{j=q+1}^{n} |z_{j}|^{2} > ε}.

The envelope of holomorphy of H obviously contains the origin. Case 2: q = n. In this case

D^{+} = {z ∈ 𝔹^{n} : ∑_{j=1}^{n}[(2a_{j} − δ_{1} + 1)x_{j}^{2} − (2a_{j} + δ_{1} − 1)y_{j}^{2}] > 0}.  (2.5.5)

Put b_{j} = 2a_{j} − δ_{1} + 1, c_{j} = 2a_{j} + δ_{1} − 1, j = 1, ..., n. For small δ − 1, b_{j} > c_{j}. Write (2.5.5) in the form

D^{+} = {z ∈ 𝔹^{n} : ∑_{j=1}^{n} b_{j}x_{j}^{2} > ∑_{j=1}^{n} c_{j}y_{j}^{2}}.  (2.5.6)

In the new coordinates z_{j} → \sqrt{b_{j}}z_{j} (2.5.6) take a form

D^{+} = {z ∈ 𝔹^{n} : ∑_{j=1}^{n} x_{j}^{2} > ∑_{j=1}^{n} δ_{j}y_{j}^{2}},  (2.5.7)

where δ_{j} = \frac{c_{j}}{b_{j}} < 1, j = 1, ..., n. Put δ_{0} := max\{δ_{1}, ..., δ_{n}\} < 1. Then

D^{+} ⊃ D_{1}^{+} = {z ∈ 𝔹^{n} : ∥x∥^{2} > δ_{0} · ∥y∥^{2}}.  (2.5.8)

The set D_{1}^{+} contains clearly the following complete “tube torus”

T = {x + iy ∈ ℂ^{n} : ∥x∥ = 1, ∥y∥ ≤ 1/δ_{0}},  (2.5.9)

where 1/δ_{0} := η > 1. We shall prove that already the envelope of holomorphy of T contains the origin. For this consider the following continuous family of complex hypersurfaces

C_{t} = {z ∈ ℂ^{n} : z_{1}^{2} + ... + z_{n}^{2} = t}  (2.5.10)

or

C_{t} = {x + iy ∈ ℂ^{n} : ∥x∥^{2} − ∥y∥^{2} = t, (x, y) = 0},  (2.5.11)

where (x, y) = x_{1}y_{1} + ... + x_{n}y_{n}. Consider the intersections of C_{t} with a ball of radii 1 + η^{2}:

C̃_{t} = {x + iy ∈ 𝔹_{1+η^{2}}^{n} : ∥x∥ − ∥y∥ = t, (x, y) = 0}.  (2.5.12)

This is a continuous family of irreducible analytic hypersurfaces in 𝔹_{1+η}^{n} such that

C̃_{1+η^{2}} = {x + iy ∈ 𝔹_{1+η^{2}}^{n} : |x∥^{2} − ∥y∥^{2} = 1 + η^{2}, (x, y) = 0} =
= {x + iy ∈ 𝔹_{1+η^{2}}^{n} : ∥x∥^{2} + ∥y∥^{2} = 1 + η^{2} = ∥x∥^{2} − ∥∥^{2}, (x, y) = 0} =
{x + iy ∈ 𝔹_{1+η^{2}}^{n} : ∥x∥^{2} = 1 + η^{2}, y = 0} ⊂ T,

but M̃_{0} ∋ 0. By continuity principle the envelope of holomorphy of T contains the origin.

q.e.d

End of the proof of Corollary 3b.

So, as in the case of regular value, we can extend our map f meromorphically to the neighborhood of the critical level M_{ε_{0}} minus discrete set. As a result we obtain the extension f̂ of our map onto D̄ \S where S is a finite subset of D̄ not intersecting M = ∂D. If we put T := f^{∗}w then dd^{c}T̃ is nonpositive measure supported on S. We have

∫_{S} dd^{c}T̃ = ∫_{D} dd^{c}T̃ = ∫_{∂D} d^{c}T = ∫_{f(∂D)} d^{c}w = ∫_{M} d^{c}w = 0,

if M is homologous to zero in X, or if X doesn’t contain spherical shells. Corollary 3b is proved.

3. Generalisations, applications, open questions

In this last section of the paper we shall give some more applications of the results and (in fact more) techniques used here.

3.1. Multivalued mappings and singular domains of definition. Let D be a domain in complex space Ω and x_{0} ∈ ∂D be a boundary point. D is said to be q-concave at x_{0} if there is a neighborhood U ⊃ x_{0} and smooth function ρ : U → ℝ such that

1) D ∩ U = {x ∈ U : ρ(x) < 0};

2) Levi form of ρ at x_{0} has at least n − q + 1 negative eigenvalues. Here n = dim Ω. By the Projection Lemma of Siu, see [Si-T], if x_{0} is a q-concave boundary point of D, q ≤ n−1, one can find a neighborhoods U ∋ x_{0} and V ∋ 0 ∈ ℂ^{n} and a proper holomorphic map π : (U, x_{0}) → (V, 0) such that π(D ∩ V) will contain a Hartogs figure H, whose associated polydisk P contains the origin. Let d be the branching number of π.

Now suppose that a meromorphic map f : D → X is given, where X is another complex space. π^{−1} ∘ f defines a d-valued meromorphic correspondence between V and X. Definition 3.1.1. A d-valued meromorphic correspondence between complex spaces V and X is an irreducible analytic subset Z ⊂ V × X such that restriction p_{1} |_{Z} of the natural projection onto the first factor on Z is proper, surjective and generically d to one.

Thus the extension of f onto the neighborhood of x_{0} is equivalent to the extension of Z from H to P. Clear that if f was also a correspondence it will produce no additional complications. Thus we should discuss how far the problem of extending of correspondences go from the extension of mappings.

Let Z be a d-valued meromorphic correspondence between the Hartogs figure H and X. Z defines in a natural way a mapping f_{Z} : H → Sym^{d}(X) - symmetric power of X of order d. Clearly the extension of Z onto P is equivalent to the extension of f_{Z} onto P. If X was for example a Kähler manifold. Then by [V] Sym^{d}(X) is a Kähler space. So meromorphic correspondences with values in Kähler manifolds are extendable through pseudo-concave boundary points.

For the manifolds from class G_{1} this is no longer the case, if even they doesn’t contain spherical shells.

3.2. Construction of Example 2: appearance of shells in symmetric products. In this section we shall construct the following Example 2. There is a compact complex (elliptic) surface X such that:

(a) every meromorphic map f : H^{2}(r) → X extends meromorphically onto Δ^{2}, but (b) there exists a two-valued meromorphic correspondence Z between ℂ_{∗}^{2} and X which cannot be extended to origin. Consider a standard Hopf surface H = ℂ^{2} \ {0}/(z ∼ 2 · z). By π : H → ℂℙ^{1}

denote the standard projection. Let ϕ : C → ℂℙ^{1} be a nonconstant meromorphic function on the Riemann surface C of positive genus. ϕ will be a d-sheeted ramified covering of ℂℙ^{1} by C. If we take C to be a torus we can have such ϕ with d = 2. Following Kodaira we shall construct an elliptic surface over C in the following way. Put

X_{1} = {(z, y) ∈ C × H : ϕ(z) = π(y)}.  (3.2.1)

Elliptic structure on X_{1} is given by the restriction onto X_{1} of the natural projection p_{1} : C × H → C. Note that restriction onto X_{1} of the natural projection p_{2} : C × H → H gives us an d-sheeted covering p_{2} |_{X_{1}} of H by X_{1} which preserves the elliptic structure. Let n : X → X_{1} be a normalization of X_{1}. Then X is a smooth elliptic surface over C with elliptic fibration p := p_{1} |_{X_{1}} ∘n : X → C and F := p_{2} |_{X_{1}} ∘n : X → H will be a d- sheeted covering.

Z := F^{−1} ∘ π : ℂ_{∗}^{2} → X is d-valued meromorphic correspondence between ℂ_{∗}^{2} and X, which cannot be extended to origin, because the projection π : ℂ_{∗}^{2} → H cannot be extended meromorphically to zero.

On the other hand in [Iv-1] it was proved that meromorphic mappings from H^{2}(r) to an elliptic surface over a Riemann surface of positive genus are extendable onto Δ^{2}.

One can interpret this example in the way that Sym^{2}(X) could have a spherical shell if even X has not.

3.3. Extension theorems for meromorphic correspondences. In view of discussion in 3.1 and 3.2 we can restate our results for meromorphic correspondences. Definition 3.3.1. By a branched spherical shell of degree d in a complex space X we shall understand an image Σ of 𝕊^{3} ⊂ ℂ^{2} under the d-valued meromorphic correspondence between some neighborhood of 𝕊^{3} and X such that Σ ≁ 0 in X. Corollary 3.3.1. Let Z be a meromorphic correspondence from the domain D in complex space Ω into the disk-convex complex space X ∈ G_{1} and let x_{0} be a concave boundary point of D. Then Z extends onto some neighborhood of x_{0} in Ω minus (possibly empty) complex variety C of pure codimension two. If X doesn’t contain branched spherical shells then C = ∅. Remark. We would like to point out here that branched shell could be a much violate object that not branched one. For this purpose we consider a smooth complex curve C in the neighborhood of 𝔹̄^{2}(2) \ 𝔹^{2}(1/2) which doesn’t extend to 𝔹^{2}(1). Let π : W → 𝔹^{2}(2)\𝔹̄^{2}(1/2) be a covering branched along C. The images of Ŝ := π^{−1}(𝕊^{3}) in the complex spaces could be a branched shells which doesn’t bound an abstract Stein domains.

3.4. Complex Plateau problem for higher-dimensional contours. First let us prove part (a) of Corollary 3 from Introduction. Actually only in the end of the proof in 2.5 we used the fact that the dimension of D is two. The following proposition clearly enables us to finish the proof also of part (a) of Corollary 3 i.e for dim D ≥ 3.

Proposition 3.4.1. Every holomorphic map f from H_{1}^{n}(r) to disk-convex complex space X ∈ G_{1} extends meromorphically onto Δ^{n} provided n ≥ 2. Remark. In the Example 2 (see Introduction) the map F : 𝔹^{3} → X was not holomorphic ! We shall essentially use the condition of holomorphicity of f in the proof of this Proposition . Proof. It will be convenient for us simultaneously with the proof of the main statement of the Proposition to prove also the following weaker statement. Denote by A^{n}(a, b) := Δ^{n}(b) \ Δ^{n}(a), for 0 ≤ a < b.

Every holomorphic map f : A^{n}(\frac{1}{2}, 1) → X, where X from Proposition 5.1.1, extends meromorphically onto Δ^{n}, provided n ≥ 2 and f(∂Δ_{3/4}^{n}) is homologous to zero in X. We shall prove both statements by induction on n. For n = 2 the second statement

follows directly from Theorem 1. So it is sufficient to prove that for any n ≥ 2 from the second statement follows the statement of Proposition for this n.

So let a holomorphic mapping f : H_{1}^{n}(r) → X is given. For every z ∈ Δ restriction f_{z} of f onto Δ_{z}^{n} := {z} × Δ^{n} is holomorphic on A^{n}(r, 1). So, by the assumption f_{z} meromorphically extends onto Δ^{n}, because f(∂Δ_{z}^{n}) ∼ f(∂Δ_{0}^{n}) ∼ 0 in X! Lemma 2.3.2 immediately gives us (after shrinking Δ^{n+1} and taking different bends of z_{2}, ..., z_{n+1}) the meromorphic extension of f onto Δ^{n+1} \ S. Where S is zerodimensional pluripolar compact in Δ^{n+1}.

Because I(f) is an analytic set of positive dimension outside of zerodimensional set, \overline{I(f)} is analytic in Δ^{n+1} \ H_{1}^{n}, and thus empty. So the fundamental set of f is discrete in Δ^{n+1} \ S.

Put T := f^{∗}w, where w is pluriclosed metric form on X. T has locally summable coefficients in Δ^{n+1} and its trivial extension T̃ is plurinegative with dd^{c}T̃ supported on S. Observe that T̃ = T is pluriclosed outside of S.

Lemma 2.6.2 tells us that dd^{c}T̃ = 0 and moreover there is a (1,0)-current γ in any given ball W ⊂ Δ^{n+1}, ∂W ∩ S = ∅, smooth on W \ S, such that T̃ = i(∂γ̄ − ¯∂γ) . Remark that the conditions of Lemma 2.6.2 are satisfied, because S is zerodimensional and n+1 ≥ 3. All that remained is to repeat the arguments from the proof of Lemma 2.6.3 to estimate the volume of the graph of f in the neighborhood of S. Namely

Vol(Γ_{f|_{W \S}}) = ∫_{W \S}(T + dd^{c}∥z∥^{2})^{n+1} = ∑_{j=0}^{n+1} C_{n+1}^{j} ∫_{W \S} T^{j} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} ≤

≤ C · ∫_{W \S} T^{j} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} = C · lim_{ε↘0} ∫_{W \S} T̃_{ε}^{j} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} ≤

≤ C · lim_{ε↘0} ∫_{W} T̃_{ε}^{j} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} = C · lim_{ε↘0} ∫_{W} (∂γ̄_{ε}^{1,0} + ¯∂γ_{ε}^{1,0})^{j} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} =

= C · lim_{ε↘0} ∫_{W} (d(γ̄_{ε}^{1,0} + γ_{ε}^{1,0}))^{j} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} =

= C · lim_{ε↘0} ∫_{∂W} (γ̄_{ε}^{1,0} + γ_{ε}^{1,0}) ∧ d(γ̄_{ε}^{1,0} + γ_{ε}^{1,0})^{j−1} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} =

= C · ∫_{∂W} (γ̄^{1,0} + γ^{1,0}) ∧ d(γ̄^{1,0} + γ^{1,0})^{j−1} ∧ (dd^{c}∥z′∥^{2})^{n+1−j} < ∞

From Bishop theorem we get an extension of the graph of f onto Δ^{n+1}.

q.e.d.

3.5. Coverings of compact complex manifolds. Let X be a compact complex manifold and let D be a domain in complex manifold Ω which covers X. This means that there is some subgroup G of the group Aut(D) of biholomorphic automorphisms of D, acting properly and discontinuously on D without fixed points, such that D/G = X.

Before proving the Corollary 4 we want to recall that locally pseudoconvex domain in Stein manifold is Stein (Oka), and moreover locally pseudoconvex domain in ℂℙ^{n} is also Stein, provided it is different from ℂℙ^{n} itself, see [Ks] and [Hs-2].

Proof of Corollary 4. Let f : D → X be a covering map. Suppose that one can find a point p ∈ ∂D such that for any neighborhood V, biholomorphic to a ball, V ∩ D is not pseudoconvex. By Docquer-Grauert theorem,[D-G], this means that there exists an embedding ϕ : H_{n−1}^{1} → V such that ϕ(Δ^{n}) ∩ ∂D is nonempty, n = dim D. But f ∘ ϕ extends meromrophically onto Δ^{n} \ S, where S is:

(a) locally finite union of subvarieties of pure codimension two by Theorem 2 ;

(b) empty, by [Iv-3].

This means that locally ∂D ⊂ I(f) ∪ S.

Suppose we can find a point p ∈ (I(f) \ S) ∩ D. Take an analytic disk ψ : Δ → U through p, i.e.ψ(0) = p, which is not contained in I(f). Composition f ∘ ψ : Δ → X is holomorphic. Put x := (f ∘ ψ)(o). Choose a neighborhood G ∋ x such that every branch of f^{−1} is single valued on G. While f ∘ f^{−1} = id on G, the map f must be holomorphic in the neighborhood of p. Really, take p_{1} ∈ ψ(Δ) \ I(f) and close enough to p to have x_{1} = f(p_{1}) ∈ G. Then the branch of f^{−1} which sends x_{1} to p_{1} must send x to p. Contradiction.

Thus I(f) = ∅ and D ⊃ (U \ S).

q.e.d.

4. Open questions

In this paragraph we list some open problems which seem to us be of significant interest. Problem 1. Prove the conjecture stated in Itroduction. Problem 2. Let the complex manifold D is defined as two-sheeted cover of Δ^{2}\ℝ^{2}, i.e. D is “nonschlicht” domain over ℂ^{2}. Does there exist a compact compex manifold X and a holomorphic (meromorphic) mapping f : D → X which separates the points?

Note that the results of this paper imply that such X if exists cannot possed a plurinegative metrik form. Thus examples could occur starting from dim X ≥ 3.

In the following problems the space X is equipped with some Hermitian metrik. On the subsets of ℂ^{n} the metrik is allwayse dd^{c}∥z∥^{2}. Problem 3. Consider a class J_{R} of meromorphic mappings f : Δ^{k} → X, X beeing compact, such that

(a) ∥Df∥ ≥ R > 0. Here ∥Df∥ denotes the norme of the differential of f;

(b) vol(f_{s}(Δ^{k}) ≤ C_{1} for all s ∈ Δ^{k}.

Prove that there is a constant C_{2} = C_{2}(X, R, C_{1}), not depending on f, such that vol(Γ_{f_{s}}) ≤ C_{2}. Problem 4. Let f : Δ_{∗}^{k} → X be a meromorphic mapping from a punctured polydisk into a compact complex space X. Suppose that vol f(Δ_{∗}^{k}) < ∞. Prove that f meromorphically extends to zero. Problem 5. Let f : Δ_{∗}^{k+1} → X ∈ G_{k} be a meromorphic map from punctured (k + 1)-disk into a compact complex space from class G_{k}, see Introduction. Prove that vol(f(A^{k}(r, 1)) = O(log^{\frac{k+1}{k}} (\frac{1}{r})). In particular for equidimensional maps f : Δ_{∗}^{n} → X^{n} one allwayse should have vol(f(A^{n}(r, 1)) = O(log^{\frac{n+1}{n}} (\frac{1}{r})). Problem 6. Fix some 0 < r < 1 and some constant R. Fix also a compact complex space X. Consider the following class F_{R} of meromorphic mappings from f : Δ^{n} → X:

(a) vol_{2n}(Γ_{f} ∩ (A^{n}(r, 1) × X)) ≤ R;

(b) for every k-disk Δ_{z}^{k} which has sides parallel to the coordinate planes (i.e. Δ_{z}^{k} = {z} × Δ^{k}, where z ∈ Δ^{n−k}) vol_{2k}(Γ_{f|_{Δ^{k}∩A^{n}(r,1)}}) ≤ R.

6a. Prove that for any constant l there is a constant A such that for any f ∈ F_{R} satisfying vol_{2k}(Γ_{f_{z}}) ≤ l for all restrictions f_{z} of f onto the k-disks Δ_{z}^{k} one has vol_{2n}(Γ_{f}) ≤ A. 6b. Vice versa: for any constant a there is a constant L such that forany f ∈ F_{R} such that vol_{2n}(Γ_{f}) ≤ a one has vol_{2k}(Γ_{f_{s}}) ≤ L for all Δ_{z}^{k}. Problem 7. Let X is a compact complex manifold carrying a plurinegative metrik form, and let f : Δ^{3} \ S → X is a meromrohpic mapping. Suppose that S is a

42 S. IVASHKOVICH

minimal closed subset of Δ^{3} such that f extends onto Δ^{3} \ S. Prove that each connected component of S is a complex curve. Problem 8. Let K a complet pluripolar compact in pseudoconvex domain in ℂ^{n}, n ≥ 2. Let T a positive current on D \ K such that dd^{c}T ≤ O on D \ K.

(a) Prove that T has locally finite mass in the neighborhood of K.

(b) Suppose that T has locally finite mass in the neighborhood of K. Let T̃ its trivial extension. Prove that dd^{c}T̃ ≤ O.

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Current address: Université de Lille-I, U.F.R. de Mathématiques, Villeneuve d’Ascq Cedex, 59655 France

  E-mail address: ivachkov@gat.univ-lille1.fr

IAPMM Acad Sci. of Ukraine, Naukova 3/b, 290053 Lviv, Ukraine