Pluripolar multifunctions are analytic

Nikolay Shcherbina

Department of Mathematics, University of Göteborg, 41296 Göteborg, Sweden

(e-mail: nikolay@math.chalmers.se)

Abstract. Let Ω be a domain in ℂ_{z}^{n} and let E be a subset of Ω × ℂ_{w} ⊂ ℂ_{z,w}^{n+1} defined by a Weierstrass pseudopolynomial, i.e. E = {(z, w) ∈ Ω × ℂ_{w} : w^{m} + a_{1}(z)w^{m−1}+...+a_{m}(z) = 0}, where a_{1}(z), a_{2}(z),..., a_{m}(z) are continuous complexvalued functions in Ω. We prove that the set E is a pluripolar subset of ℂ_{z,w}^{n+1} if and only if the functions a_{1}(z), a_{2}(z),..., a_{m}(z) are holomorphic in Ω.

1. Introduction

A function φ defined on a domain U ⊂ ℂ^{n} with values in [−∞, +∞) is called plurisubharmonic in U if φ is upper semicontinuous and its restriction to the components of the intersection of any complex line with U is subharmonic.

A set E ⊂ ℂ^{n} is called pluripolar if there is a neighbourhood U of E and a plurisubharmonic function φ on U such that E ⊂ {φ = −∞}. By a result of B. Josefson [J], the function φ in this definition can be chosen to be plurisubharmonic in the whole of ℂ^{n} (i.e. U = ℂ^{n}).

In 1963 T. Nishino raised the following question in connection to his paper [N1]:

Let Δ be the unit disk in ℂ_{z} and let f : Δ → ℂ_{w} be a continuous function such that its graph Γ(f) is a pluripolar subset of ℂ_{z,w}^{2}. Does it follow that f is holomorphic?

A positive answer to Nishino’s question in the case of an arbitrary continuous function f (when the mentioned above methods cannot be applied) was given in [S1]. In this paper we prove a more general result for multi-valued functions defined by a Weierstrass pseudopolynomial.

Theorem. Let Ω be a domain in ℂ_{z}^{n} and let E be a subset of Ω × ℂ_{w} ⊂ ℂ_{z,w}^{n+1} of the form

E = {(z, w) ∈ Ω × ℂ_{w} : w^{m} + a_{1}(z)w^{m−1} + ... + a_{m}(z) = 0},  (1)

where a_{1}(z), a_{2}(z), ..., a_{m}(z) are continuous complex-valued functions in Ω. Then E is a pluripolar subset of ℂ_{z,w}^{n+1} if and only if the functions a_{1}(z), a_{2}(z), ..., a_{m}(z) are holomorphic in Ω.

Note, that our assumption that the set E has the form (1) is essential for topological arguments used in the proof of Lemma 3, namely, for the fact that the first homology group H_{1}(Ω×ℂ_{w}\E, ℤ) is nontrivial. If we instead assume that the fibers E(z) = {w ∈ ℂ_{w} : (z, w) ∈ E} of E are finite and depend continuously on z ∈ Ω in the Hausdorff metric, then this homology group can be equal to zero. In the last section of the paper we construct an example of a compact subset E of Δ̄×ℂ_{w} ⊂ ℂ_{z,w}^{2} (Δ = {z : |z| < 1}) with finite fibers E(z) depending continuously on z ∈ Δ̄ in the Hausdorff metric such that H_{1}(Δ × ℂ_{w} \ E, ℤ) = 0. In particular, there is a neighbourhood U(E) of E which does not contain any subset of Δ̄ × ℂ_{w} defined by a Weierstrass pseudopolynomial.

Acknowledgements. Part of this work was done while the author was a visitor at the Max Planck Institute of Mathematics (Bonn). It is my pleasure to thank this institution for its hospitality and excellent working conditions. I would like to thank T. Ohsawa for informing me that the problem stated above was first raised in 1963 by T.Nishino and E.L.Stout for pointing me out reference on the paper [A].

2. Preliminaries

For a compact set K in ℂ^{n}, the polynomial hull K̂ of K is defined as

K̂ = {z ∈ ℂ^{n} : |P (z)| ≤ sup_{w∈K} |P (w)| for all holomorphic polynomials P in ℂ^{n}}.

The set K is called polynomially convex if K̂ = K.

The following statement proved in [S1] will be essential in the next section.

Lemma 1 Let K be a polynomially convex compact set in ℂ^{n} and let E be a pluripolar compact set in ℂ^{n}. Then the set ({ \atop K ∪ E}) \ K is pluripolar.

For bounded nonempty sets E_{1} and E_{2} in ℂ_{w}, the Hausdorff distance is defined as

d(E_{1}, E_{2}) = sup_{w_{2}∈E_{2}} inf_{w_{1}∈E_{1}} |w_{1} − w_{2}| + sup_{w_{2}∈E_{1}} inf_{w_{1}∈E_{2}} |w_{1} − w_{2}|.

A family of compact sets E(z) in ℂ_{w} parametrized by z ∈ Ω ⊂ ℂ_{z} is said to be continuously dependent on z in the Hausdorff metric if for each sequence {z_{n}} of points in Ω converging to a point z_{0} ∈ Ω one has d(E(z_{n}), E(z_{0})) → 0 as n → ∞. In particular, if Ω is a domain in ℂ_{z} and E is a nonempty closed subset of Ω × ℂ_{w} with bounded fibers E(z) = {w ∈ ℂ_{w} : (z, w) ∈ E} depending continuously on z ∈ Ω in the Hausdorff metric, then E(z) are nonempty for all z ∈ Ω.

Lemma 2 Let Ω be a domain in ℂ_{z} and let E be a closed subset of Ω×ℂ_{w} ⊂ ℂ_{z,w}^{2} such that the fibers E(z) = {w ∈ ℂ_{w} : (z, w) ∈ E} of E are bounded and depend continuously on z ∈ Ω in the Hausdorff metric. Suppose that the set E is not pseudoconcave. Then there is a point P = (z^{∗}, w^{∗}) ∈ E and a biholomorphic map 𝔽 : (z, w) → (z′, w′) of a neighbourhood V_{P} of P to a neighbourhood V_{O} of the origin O which has the form

z′ = z − z^{∗}, w′ = F (z, w)  (2)

such that

𝔽(E ∩ V_{P}) ⊂ {Re w′ ≥ 0}  (3)

and

𝔽(E ∩ V_{P}) ∩ {Re w′ = 0} = 𝔽(P) = O.  (4)

Proof. Consider the function ϕ(z, w) = − log R(z, w) on the domain Ω×ℂ_{w} \E, where R(·, ·) is the Hartogs radius of the domain Ω × ℂ_{w} \ E with respect to the axis w, i.e.

R(z_{0}, w_{0}) = inf_{(z_{0},w)∈E} |w − w_{0}|.

for each (z_{0}, w_{0}) ∈ Ω × ℂ_{w} \ E. From the assumption that the fibers E(z) depend continuously on z ∈ Ω in the Hausdorff metric we conclude that the function ϕ is continuous in Ω × ℂ_{w} \ E. The assumption that the domain Ω × ℂ_{w} \ E is not pseudoconvex implies that for some neighbourhood V of the set E the restriction of ϕ to V \ E is not plurisubharmonic. Since the restriction of ϕ to each complex line {z = z_{0}}

ϕ(z_{0}, w) = sup_{(z_{0},w̃)∈E}{− log |w − w̃|}

is subharmonic being the supremum of the subharmonic functions − log |w − w̃|, it follows that for some point (z_{1}, w_{1}) ∈ Ω × ℂ_{w} \ E and some complex line {w = w_{1} + a(z − z_{1})} the restriction ψ(z) of the function ϕ(z, w) to this line is not subharmonic near the point z = z_{1}. This means that there exists a small enough disk Δ_{r}(z_{1}) = {z ∈ ℂ_{z} : |z − z_{1}| < r} (so small that the closed disk {(z, w) ∈ Δ̄_{r}(z_{1}) × ℂ_{w} : w = w_{1} + a(z − z_{1})} is contained in Ω × ℂ_{w} \ E) and a function ψ̃ continuous on Δ̄_{r}(z_{1}) and harmonic on Δ_{r}(z_{1}) such that

ψ < ψ̃ on bΔ_{r}(z_{1})  (5)

and

ψ̃(z_{2}) < ψ(z_{2}) for some z_{2} ∈ Δ_{r}(z_{1}).  (6)

Consider the one-parameter family of functions

g_{t}(z) = t (1 − \frac{|z − z_{1}|^{2}}{r^{2}} )

strictly superharmonic for t > 0. It follows from (5) and continuity of ψ and ψ̃ that ψ(z) − ψ̃(z) < g_{t}(z) for all z ∈ Δ̄_{r}(z_{1}) if t is big enough. Since, by (6), for t = 0 one has ψ(z_{2}) − ψ̃(z_{2}) > g_{0}(z_{2}), we conclude that there exists t^{∗} > 0 such that

ψ(z) − ψ̃(z) ≤ g_{t^{∗}}(z) for all z ∈ Δ̄_{r}(z_{1})  (7)

and

ψ(z^{∗}) − ψ̃(z^{∗}) = g_{t^{∗}}(z^{∗}) for some z^{∗} ∈ Δ_{r}(z_{1}).  (8)

Consider the Hartogs domain

H = {(z, w) ∈ Δ_{r}(z_{1}) × ℂ_{w} : |w − w_{1} − a(z − z_{1})| < e^{−ψ̃(z)−g_{t^{∗}}(z)}}.

It follows from harmonicity of ψ̃ and strict superharmonicity of g_{t^{∗}} that the domain H is strictly pseudoconcave on the part bH∩(Δ_{r}(z_{1})×ℂ_{w}) of its boundary. Moreover, (7) and the definition of ψ imply that H ∩ E = ∅ and (8) implies that there exists a point (z^{∗}, w^{∗}) ∈ bH ∩ E ∩ (Δ_{r}(z_{1}) × ℂ_{w}). To exclude the other points of the set E ∩ {z = z^{∗}} from bH we deform H in the following way.

Denote by ω the complex number with the same argument as w^{∗} −w_{1} −a(z^{∗} − z_{1}) and such that |ω| = 1. Consider the Hartogs domain

H′ = {(z, w) ∈ Δ_{r_{1}}(z_{1}) × ℂ_{w} : |w − w_{1} − a(z − z_{1}) − εω| < e^{−ψ̃(z)−g_{t^{∗}}(z)} − ε},

where r_{1} < r is fixed and such that z^{∗} ∈ Δ_{r_{1}}(z_{1}). Since the property of a function to be strictly superharmonic is stable under a small perturbation in C^{2}-metric, the function h(z) = − log(e^{−ψ̃(z)−g_{t^{∗}}(z)} − ε) will be strictly superharmonic on Δ_{r_{1}}(z_{1}) ⊂⊂ Δ_{r}(z_{1}) for ε > 0 small enough. Therefore, for such ε the domain H′ is strictly pseudoconcave on the part bH′ ∩ (Δ_{r_{1}}(z_{1}) × ℂ_{w}) of its boundary. It follows from the definition of H′ that H′ ⊂ H and, hence, H′ ∩ E = ∅. Moreover, it also follows that

bH′ ∩ E ∩ {z = z^{∗}} = (z^{∗}, w^{∗}).  (9)

Strict superharmonicity of the function h implies that for some r^{∗} > 0 such that Δ̄_{r^{∗}}(z^{∗}) ⊂ Δ_{r_{1}}(z_{1}) there exists a harmonic function u defined in Δ̄_{r^{∗}}(z^{∗}) such that

h(z) < u(z) for z ∈ Δ̄_{r^{∗}}(z^{∗}) \ {z^{∗}}  (10)

and

h(z^{∗}) = u(z^{∗})  (11)

(one can actually choose u(z) to be a quadratic polynomial in Re z and Im z). Consider the Hartogs domain

H′′ = {(z, w) ∈ Δ_{r^{∗}}(z^{∗}) × ℂ_{w} : |w − w_{1} − a(z − z_{1}) − εω| < e^{−u(z)}}.

It follows from (10) that H′′ ⊂ H′ and, therefore,

H′′ ∩ E = ∅.  (12)

Moreover, from (9), (10) and (11) we conclude that

bH′′ ∩ E = (z^{∗}, w^{∗}).  (13)

Denote by f the function holomorphic in Δ_{r^{∗}}(z^{∗}) such that Re f(z) = u(z) and Im f(z^{∗}) = − arg(w^{∗}−w_{1}−a(z^{∗}−z_{1})−εω). Define the map 𝔽 as z′ = z−z^{∗}, w′ = Log(w − w_{1} − a(z − z_{1}) − εω) + f(z) choosing the branch of the function Log in such a way that 𝔽(z^{∗}, w^{∗}) = (0, 0) = O. Then 𝔽 is obviously biholomorphic in a small enough neighbourhood V_{P} of the point P = (z^{∗}, w^{∗}) and, by (12) and our choice of the function f, it satisfies the condition 𝔽(E ∩ V_{P}) ⊂ {Re w′ ≥ 0}, while by (13) it satisfies the condition 𝔽(E ∩ V_{P}) ∩ {Re w′ = 0} = 𝔽(P) = O. This completes the proof of Lemma 2. □

3. Proof of Theorem

If the set E is defined by a holomorphic Weierstrass pseudopolynomial, i.e.

E = {(z, w) ∈ Ω × ℂ_{w} : w^{m} + a_{1}(z)w^{m−1} + ... + a_{m}(z) = 0},

where the functions a_{1}(z), a_{2}(z),..., a_{m}(z) are holomorphic in Ω, then the function

φ(z, w) = ln |w^{m} + a_{1}(z)w^{m−1} + ... + a_{m}(z)|

is plurisubharmonic in Ω × ℂ_{w} and E = {φ = −∞}. Therefore, the set E is pluripolar in ℂ_{z,w}^{n+1}.

Suppose now that the set E is pluripolar. It follows from Oka’s theorem (see [N2], Th. 4.9, p.136) that if the set E is pseudoconcave, then all the functions a_{1}(z), a_{2}(z),..., a_{m}(z) are holomorphic in Ω. Hence, we only need to prove pseudoconcavity of E. For proving it we consider two cases.

1. The special case n = 1. We argue by contradiction and assume that the set E is pluripolar, has the form (1) and is not pseudoconcave. Then, in view of Lemma 2, the set E′ = 𝔽(E ∩V_{P}) satisfies properties (3) and (4). Using the special form (2) of 𝔽 and shrinking, if necessary, the neighbourhood V_{O} = 𝔽(V_{P}) of the origin O, we can assume that the set E′ has the same form (1) with, maybe, smaller value of m and some other continuous functions a_{1}(z), a_{2}(z), ..., a_{m}(z) and, moreover, that the domain Ω′ in the representation (1) of the set E′ is a disk with center at z = 0. Without loss of generality we can assume that Ω′ = Δ = {z ∈ ℂ_{z} : |z| < 1} is the unit disk. Since, by our assumptions, the set E is pluripolar, and since E′ is obtained from E ∩ V_{P} by a biholomorphic map 𝔽, it follows that the set E′ is also pluripolar. With a slight abuse of notations (like the use of E instead of E′) we can summarize our assumptions as follows:

(i) E is a subset of Δ̄ × ℂ_{w} ⊂ ℂ_{z,w}^{2} of the form

E = {(z, w) ∈ Δ̄ × ℂ_{w} : w^{m} + a_{1}(z)w^{m−1} + ... + a_{m}(z) = 0},

where a_{1}(z), a_{2}(z), ..., a_{m}(z) are continuous complex-valued functions in Δ̄.

(ii) E is pluripolar.

(iii) E satisfies the properties

E ⊂ {Re w ≥ 0}  (14)

and

E ∩ {Re w = 0} = (0, 0) = O.  (15)

Define the function F on Δ̄ by

F(z) = min{u : u = Re w, (z, w) ∈ E(z)}.

Then (i) implies that F is continuous on Δ̄, while (14) and (15) imply that F ≥ 0 on Δ̄ and F(z) = 0 if and only if z = 0. Therefore, one has

A =^{def} min_{z∈bΔ} F(z) > 0.

Let U be the connected component of the set

\{(z, u) ∈ Δ × ℝ_{u} : F(z) < u < \frac{A}{2}\}

such that the origin O of ℂ_{z} × ℝ_{u} is contained in bU. It follows from our choice of A that U is a nonempty domain in ℂ_{z} × ℝ_{u}.

Denote C = max{|w| : (z, w) ∈ E} and consider the following polynomially convex subset of ℂ_{z,w}^{2} :

K = \{(z, w) ∈ Δ̄ × ℂ_{w} : \frac{A}{2} ≤ u ≤ 3C, |v| ≤ C\}.

Let π : ℂ_{z,w}^{2} → ℂ_{z} × ℝ_{u} be the projection.

Lemma 3 U ⊂ π({ \atop K ∪ E} \ K).

Proof. Consider an approximation of the set E in the Hausdorff metric by a sequence {E_{n}} of smooth surfaces in Δ̄ × ℂ_{w} having the form (1), transversal to \{u = \frac{A}{2}\} and such that for the corresponding domains U_{n} ⊂ ℂ_{z} × ℝ_{u} one has U_{1} ⊂ U_{2} ⊂ U_{3} ⊂ ⋯ ⊂ U and ∪_{n=1}^{∞}U_{n} = U (one can get such approximation using suitable approximation of the functions a_{1}(z), a_{2}(z),..., a_{m}(z) defining the set E by smooth functions in general position). It follows then from the definition of polynomial hull that lim sup_{n→∞} { \atop K ∪ E_{n}} ⊂ { \atop K ∪ E}, where convergence is understood to be also in the Hausdorff metric. Hence, it is enough to prove the statement of Lemma 3 in the case when E is smooth and transversal to \{u = \frac{A}{2}\}.

Now we argue by contradiction and suppose that there is a point (z_{0}, u_{0}) ∈ U \ π({ \atop K ∪ E} \ K). Let Ẽ be the connected component of the set E ∩ \{u ≤ \frac{A}{2}\}

such that the origin O of ℂ_{z,w}^{2} is contained in Ẽ. Consider the image ˜ Ẽ of the set Ẽ under the reflection with respect to the plane \{u = \frac{A}{2}\}, i.e.

˜Ẽ = {(z, u + iv) ∈ ℂ_{z,w}^{2} : (z, (A − u) + iv) ∈ Ẽ}.

Then the set M_{E} = Ẽ ∪ ˜ Ẽ is an oriented compact closed piece-wise smooth manifold of dimension 2 in ℂ_{z,w}^{2} (with the orientation of Ẽ being the pull-back of the standard orientation of ℂ_{z} under the projection ℂ_{z,w}^{2} → ℂ_{z} and the orientation

of ˜ Ẽ being the inverse of the pull-back of the orientation of ℂ_{z} under the same projection). Consider the curve γ_{R} ⊂ ℂ_{z,w}^{2} consisting of the segment {z = z_{0}, u = u_{0}, −R ≤ v ≤ R\} and the half-circle \{z = z_{0}, w = (u_{0} + R cos θ) + iR sin θ, \frac{π}{2} ≤ θ ≤ \frac{3π}{2}\}. It follows from the construction of M_{E} and the choice of the point (z_{0}, u_{0}) that γ_{R} does not intersect M_{E} for R big enough. The curve γ_{R} bounds the half-disk D = \{z = z_{0}, w = (u_{0} + r cos θ) + ir sin θ, 0 ≤ r ≤ R, \frac{π}{2} ≤ θ ≤ \frac{3π}{2}\} which has a natural orientation induced from the standard orientation of ℂ_{w}. If we orient γ_{R} as the boundary of D, then the linking number l(γ_{R}, M_{E}) of γ_{R} and M_{E} will be positive, since D and M_{E} (more precisely, D and Ẽ) have a nonempty intersection by the definition of D, Ẽ and the choice of (z_{0}, u_{0}), and since all the intersection indexes of D and Ẽ are positive. From positivity of the linking number l(γ_{R}, M_{E}) we conclude that the curve γ_{R} represent a nontrivial element of the first homology group H_{1}(ℂ_{z,w}^{2} \ M_{E}, ℤ). Hence, by Alexander [A, Th. 1], M̂_{E} ∩ γ_{R} ≠ ∅. Since this argument can be applied to γ_{R} for all R big enough, it follows that M̂_{E} ∩{(z_{0}, u_{0}+iv) : v ∈ ℝ} ≠ ∅, i.e. (z_{0}, u_{0}) ∈ π(M̂_{E}). By the construction of M_{E} and the definition of K one has M_{E} ⊂ K ∪ E. Therefore,

M̂_{E} ⊂ { \atop K ∪ E} and, hence, (z_{0}, u_{0}) ∈ π(M̂_{E}) ⊂ π({ \atop K ∪ E}). Since (z_{0}, u_{0}) ∉ π(K),

we conclude that (z_{0}, u_{0}) ∈ π({ \atop K ∪ E} \ K). This contradicts the choice of (z_{0}, u_{0}) and the lemma follows. □

Since, by our assumption (ii), the set E is pluripolar, and since the compact set K is polynomially convex, we conclude from Lemma 1 that the set { \atop K ∪ E} \ K is pluripolar, i.e.

{ \atop K ∪ E} \ K ⊂ \{φ = −∞\}  (16)

for some plurisubharmonic function φ.

Let a be a point in ℂ_{z} such that U ∩ {z = a} ≠ ∅. For each a′ ∈ ℂ_{z} consider the complex line ℓ_{a′} = {(z, w) ∈ ℂ_{z,w}^{2} : z = a′} and the set

E_{a′} = ({ \atop K ∪ E} \ K) ∩ ℓ_{a′}.

Since the set U ⊂ ℂ_{z} × ℝ_{u} is open, it follows from Lemma 3 and the choice of the point a that for all a′ ∈ ℂ_{z} close enough to a the projection of E_{a′} on the real line ℓ_{a′} ∩{v = 0} contains an open segment. A polar set in ℂ has Hausdorff dimension zero (see, for example [T, Th.III.19, p.65]), hence it cannot be projected on an open segment in ℝ. Therefore, the set E_{a′} is not polar. It follows then from (16) that φ ≡ −∞ on ℓ_{a′}. Since this argument holds true for all a′ ∈ ℂ_{z} close enough to a, we conclude that φ ≡ −∞ on ℂ_{z,w}^{2}. This contradiction proves the Theorem in the case n = 1.

2. The general case. Consider, as in the proof of Lemma 2, the function ϕ(z, w) = − log R(z, w) on the domain Ω × ℂ_{w} \ E ⊂ ℂ_{z,w}^{n+1}, where R(·, ·) is the Hartogs radius of the domain Ω × ℂ_{w} \ E with respect to the axis w, i.e.

R(z_{0}, w_{0}) = inf_{(z_{0},w)∈E} |w − w_{0}|.

for each (z_{0}, w_{0}) ∈ Ω × ℂ_{w} \ E. Then ϕ(z, w) → +∞ as (z, w) → E and, in view of the form (1) of the set E, ϕ(z, w) is continuous on Ω × ℂ_{w} \ E.

Let k be one of the numbers 1, 2, …, n. For each a = (a_{1}, a_{2}, …, a_{n}) ∈ Ω consider the set

E_{k}^{a} = E ∩{z_{1} = a_{1}, …, z_{k−1} = a_{k−1}, z_{k+1} = a_{k+1}, …, z_{n} = a_{n}} ⊂ Ω_{k}^{a}×ℂ_{w} ⊂ ℂ_{z_{k},w}^{2},

where

Ω_{k}^{a} = Ω ∩ {z_{1} = a_{1}, …, z_{k−1} = a_{k−1}, z_{k+1} = a_{k+1}, …, z_{n} = a_{n}} ⊂ ℂ_{z_{k}}.

Since, by our assumptions, the set E is pluripolar, there is a function φ, plurisubharmonic in ℂ_{z,w}^{n+1}, such that E ⊂ {φ = −∞}. For all points a except for a pluripolar set in ℂ_{z}^{n} one obviously has that the function

φ_{k}^{a}(z_{k}, w) = φ(a_{1}, …, a_{k−1}, z_{k}, a_{k+1}, …, a_{n}, w)

is not identically equal to −∞ in ℂ_{z_{k},w}^{2}. For all such points a we can use the argument from case 1 and conclude from the inclusion E_{k}^{a} ⊂ {φ_{k}^{a} = −∞} that the set E_{k}^{a} is pseudoconcave. Therefore, the function

ϕ_{k}^{a}(z_{k}, w) = ϕ(a_{1}, …, a_{k−1}, z_{k}, a_{k+1}, …, a_{n}, w)

is plurisubharmonic in Ω_{k}^{a} × ℂ_{w} \ E_{k}^{a}. Since the complement of a pluripolar set is everywhere dense, it follows from continuity of ϕ that the functions ϕ_{k}^{a} are plurisubharmonic for all a ∈ Ω. This argument holds true for any k = 1, 2, …, n, hence the whole function ϕ is plurisubharmonic and, therefore, the set E is pseudoconcave. The proof of the Theorem is now completed. □

4. Example

We prove first the following simple lemma.

Lemma 4 Let f and g be holomorphic functions, defined in a neighbourhood U of a point a ∈ ℂ_{z}, such that f(a) = g(a) and f′(a) ≠ g′(a). Let r be a positive number such that Δ̄_{r}(a) = {z ∈ ℂ_{z} : |z − a| ≤ r} ⊂ U and f(z) ≠ g(z) for z ∈ Δ̄_{r}(a) \ {a}. Then for all sufficiently small ε > 0 the complex curve Σ ⊂ Δ_{r}(a) × ℂ_{w} defined by the equation

G(z, w) =^{def} (w − f(z))(w − g(z)) − ε = 0  (17)

is a branched covering over the disk Δ_{r}(a) with 2 branches and 2 branching points

b^{±} = a ± \frac{2i}{f′(a) − g′(a)}\sqrt{ε} + O(ε).  (18)

Proof. Equation (17) is quadratic with respect to w, hence Σ is a branched covering over Δ_{r}(a) with 2 branches. A point b is a branching point of Σ if for some w_{b} one has 0 = Gw^{′}(b, w_{b}) = 2w_{b}−f(b)−g(b). Therefore, w_{b} = \frac{1}{2}(f(b) + g(b)) and then (17) implies that −\frac{1}{4}(f(b) − g(b))^{2} − ε = 0, i.e.

f(b) − g(b) = ±2i\sqrt{ε}.  (19)

Hence, in view of our choice of r, b → a as ε → 0. Then, using Taylor expansions of f and g at the point a, we conclude from (19) and the assumption f(a) = g(a) that (f′(a)−g′(a))(b−a)+O(|b−a|^{2}) = ±2i\sqrt{ε}. Finilly, the assumption f′(a) ≠ g′(a) implies that

b − a = ±\frac{2i}{f′(a) − g′(a)}\sqrt{ε} + O(|b − a|^{2}) = ±\frac{2i}{f′(a) − g′(a)}\sqrt{ε} + O(ε).

□

Construction of the set E. Let ρ be a smooth real-valued function defined on the segment [0, 1] such that

ρ(t) = \{ \begin{matrix}1, & for 0 ≤ t ≤ \frac{1}{3}, \\ decreasing, & for \frac{1}{3} < t < \frac{2}{3}, \\ 0, & for \frac{2}{3} ≤ t ≤ 1. \end{matrix}

Consider the set

E_{1} = {(z, w) ∈ Δ̄ × ℂ_{w} : w^{2} = ρ(|z|)z},

where, as above, Δ = {z ∈ ℂ_{z} : |z| < 1} is the unit disk. This set has 2 branches over the disk Δ_{\frac{2}{3}} (0) with one branching point at z = 0. The branches are glued to each other along the circle A = \{|z| = \frac{2}{3}, w = 0\} and become one branch \{w = 0\} for \frac{2}{3} ≤ |z| ≤ 1. Consider some points A_{1} = (a_{1}, 0) and A_{3} = (a_{3}, \sqrt{a_{3}}) of E_{1} and a point A_{2} = (a_{2}, C) with a_{1}, a_{2}, a_{3} and C real and positive such that \frac{2}{3} < a_{1} < 1, 0 < a_{3} < \frac{1}{3} and a_{3} < a_{2} < a_{1}. Further, consider the complex line L′ passing through the points A_{2} and A_{1} and the complex line L′′ passing through the points A_{2} and A_{3}. Let a_{1}, a_{2}, a_{3} be already chosen and consider C so big that the line L′′ intersects E_{1} in two points A_{3} and A3^{′} = (a3^{′}, −\sqrt{a3^{′}}) with a3^{′} real such that 0 < a3^{′} < a_{3}, and the line L′ intersects E_{1} only at the point A_{1}. The set E will be constructed as a small deformation of the set E_{1} ∪ ((L′ ∪ L′′) ∩ (Δ̄ × ℂ_{w})) near the points A_{k}, k = 1, 2, 3, that creates, as in Lemma 4, 2 branching points instead of each selfintersection point.

Let r > 0 be so small that the disks Δ̄_{1} = Δ̄_{r}(a_{1}), Δ̄_{2} = Δ̄_{r}(a_{2}) and Δ̄_{3} = Δ̄_{r}(a_{3}) do not intersect each other and the circle \{|z| = \frac{2}{3}\} and, moreover, do not contain the point a3^{′}. Denote by E_{1} the set (E_{1} ∪ L′) ∩ (Δ_{1} × ℂ_{w}), by E_{2} the set (L′ ∪ L′′) ∩ (Δ_{2} × ℂ_{w}) and by E_{3} the connected component of the set (E_{1}∪L′′)∩(Δ_{3}×ℂ_{w}) containing the point A_{3}. Then each of the sets E_{k}, k = 1, 2, 3, is the union of the graphs of two holomorphic functions f_{k}^{j}, j = 1, 2, having the same value and different derivatives, both of them real (which is easy to check by direct calculation), at the center of the respective disk Δ_{k}. Therefore, we can apply Lemma 4 to each of these sets and, if ε is small enough, we will get branched coverings Σ_{1}, Σ_{2} and Σ_{3} over the disks Δ_{1}, Δ_{2} and Δ_{3}, respectively, with 2 branches and 2 branching points contained in the smaller disks Δ1^{′} = Δ\frac{r}{3} (a_{1}), Δ2^{′} = Δ\frac{r}{3} (a_{2}) and Δ3^{′} = Δ\frac{r}{3} (a_{3}). Moreover, since for each k = 1, 2, 3 the derivatives at the center of the disk Δ_{k} of the functions f_{k}^{j}, j = 1, 2, are real, we conclude from (18) that one point in the pair of the branching points contained in Δk^{′} is contained in the half-disk {z ∈ Δk^{′} : Im z > 0}, while the other point in this pair of branching points is contained in the half-disk {z ∈ Δk^{′} : Im z < 0}. Since both branching points of each set Σ_{k} are contained in the respective disk Δk^{′}, the set Σ_{k} ∩((Δ_{k} \Δk^{′})×ℂ_{w}) will be the union of the graphs of 2 holomorphic functions f̃_{k}^{j}, j = 1, 2, defined on Δ_{k} \Δk^{′} and, moreover, if ε is small enough, then each function f̃_{k}^{j} will be close enough to the corresponding function f_{k}^{j}. Define the functions

˜f̃_{k}^{j}(z) = ρ (\frac{|z − a_{k}|}{r}) f̃_{k}^{j}(z) + (1 − ρ (\frac{|z − a_{k}|}{r})) f_{k}^{j}(z),

for z ∈ Δ_{k} \ Δk^{′}, k = 1, 2, 3, j = 1, 2. Let Σ̃_{k} be the union of the graphs of ˜ f̃_{k}^{1} and ˜f̃_{k}^{2}. Now we can define the set E as

E = ((E_{1} ∪ ((L′ ∪ L′′) ∩ (Δ̄ × ℂ_{w}))) \ ⋃_{k=1}^{3} E_{k})∪ ⋃_{k=1}^{3} (Σ̃_{k} ∪ (Σ_{k} ∩ (Δ̄k^{′} × ℂ_{w}))) .

Define also the set E^{reg} as E with the circle A, the point A3^{′} of the transversal selfintersection of E and all the branching points of E being removed. Then, by our construction, E^{reg} is a smooth connected 2-dimensional surface transversal to the w-direction.

Note, that each fiber E(z) of the set E has at most 4 points and that the fibers E(z) depend continuously on z ∈ Δ̄ in the Hausdorff metric.

Claim 1. H_{1}(Δ × ℂ_{w} \ E, ℤ) = 0.

Proof. Let a be a real positive number such that a_{3} ≤ a < \frac{1}{3}. Consider the point A = (a, −\sqrt{a}) ∈ E and a disk D̄_{s} = \{(z, w) : z = a, |w + \sqrt{a}| ≤ s\} so small that it intersects the set E only at the point A. We prove first that the circle C_{s} = bD_{s} is homological to zero in Δ × ℂ_{w} \ E.

Consider the curve z(t) in ℂ_{z} defined as

z(t) = \{ \begin{matrix}a(1 − t) + (a_{1} + r)t, & for 0 ≤ t ≤ 1, \\ a_{1} + re^{πi(t−1)}, & for 1 < t ≤ 2, \\ (a_{1} − r)(3 − t) + (a_{3} + r)(t − 2), & for 2 < t ≤ 3, \\ a_{3} + re^{πi(t−3)}, & for 3 < t ≤ 4, \\ (a_{3} − r)(5 − t) + \frac{2}{3}(t − 4), & for 4 < t ≤ 5. \end{matrix}

If π_{z} : ℂ_{z,w}^{2} → ℂ_{z} is the projection, then the curve z(t) admits a uniquely defined lifting by π_{z}^{−1} to the piece-wise smooth curve γ in E with the initial point A.

The curve γ is transversal to the w-direction and has one point of selfintersection, namely, the endpoint (\frac{2}{3}, 0), where two smooth curves on the side \{|z| < \frac{2}{3}\} meet each other.

Geometric description of the curve γ looks as follows. We start from the point A = (a, −\sqrt{a}) and then, over the segment \{a ≤ Re z < \frac{2}{3}, Im z = 0\} the curve γ is contained in the “lower” branch of the set E_{1}, while over the segment \{\frac{2}{3} ≤ Re z ≤ a_{1} − r, Im z = 0\} γ is contained in the only branch \{w = 0\} of E_{1} for \{|z| > \frac{2}{3}\}. Since both branching points of Σ_{1} are contained in Δ_{1} = \{|z−a_{1}| < r\}, and since only one of them is contained in the half-disk {z ∈ Δ_{1} : Im z > 0}, we conclude that over the segment {a_{1} − r ≤ Re z ≤ a_{1} + r, Im z = 0} the curve γ will “change from the branch E_{1} to the branch L′”. Then, over the half-circle {|z − a_{1}| = r, Im z > 0} and the segment {a_{2} + r ≤ Re z ≤ a_{1} − r, Im z = 0} γ is contained in L′. After that, applying the same argument as we used for the segment {a_{1} − r ≤ Re z ≤ a_{1} + r, Im z = 0}, we conclude that over the segment {a_{2} −r ≤ Re z ≤ a_{2} +r, Im z = 0} the curve γ will “change from the branch L′ to the branch L′′”. Then, over the segment {a_{3} + r ≤ Re z ≤ a_{2} − r, Im z = 0} and the half-circle {|z − a_{3}| = r, Im z > 0} γ is contained in L′′. After that, the same argument as above shows that over the segment {a_{3}−r ≤ Re z ≤ a_{3}+r, Im z = 0} the curve γ will “change from the branch L′′ to the branch E_{1}”. And finally, over the segment \{a_{3}+r ≤ Re z ≤ \frac{2}{3}, Im z = 0\} the curve γ is contained in the “upper” branch of E_{1} up to the endpoint (\frac{2}{3}, 0), where we meet the first part of the curve γ which was (for |z| < \frac{2}{3}) contained in the “lower” branch of E_{1}.

For each z_{0} ∈ π_{z}(γ) and each s > 0 consider the sets

Γ_{s}(z_{0}) = {(z_{0}, w) : min_{(z_{0},w′)∈γ} |w − w′| = s}

and

Ω_{s}(z_{0}) = {(z_{0}, w) : min_{(z_{0},w′)∈γ} |w − w′| < s}.

Then, for s small enough, each set Ω_{s}(z_{0}) is the union of finitely many (at most 3) disks in {z_{0}} × ℂ_{w} which are disjoint, if z_{0} is far enough from the circle \{|z| = \frac{2}{3}\}, and is the union of two connected components, one of which is a disk and the other one is the union of two disks having nonempty intersection, if \{|z_{0}| < \frac{2}{3}\} and z_{0} is close enough to the circle \{|z| = \frac{2}{3}\}. As |z_{0}| → \frac{2}{3} from the side \{|z| < \frac{2}{3}\}, the centers of two disks constituting the second connected component of Ω_{s}(z_{0}) becomes closer to each other and for \{|z_{0}| ≥ \frac{2}{3}\} this component becomes just one disk. Each set Ω_{s}(z_{0}) has a natural orientation induced from ℂ_{w} and, hence, Γ_{s}(z_{0}) = bΩ_{s}(z_{0}) has also a natural orientation.

Consider the set

T_{s} = ⋃_{z_{0}∈π_{z}(γ)} Γ_{s}(z_{0}).

Since the curve γ is piece-wise smooth, it follows from the definition of Γ_{s}(z_{0}) that the set T_{s} is a piece-wise smooth surface of dimension 2 in Δ × ℂ_{w} with the boundary on the chosen above circle C_{s}. Moreover, since γ is oriented, and since each set Γ_{s}(z_{0}) is oriented, we can also orient the surface T_{s}. Topologically, T_{s} is a torus with a disk removed, C_{s} being the boundary of this disk. Since the curve γ ⊂ E is transversal to the w-direction, we conclude that T_{s} ⊂ Δ × ℂ_{w} \ E for s sufficiently small. This implies that the homology class [C_{s}] of the circle C_{s} in H_{1}(Δ × ℂ_{w} \ E, ℤ) is trivial.

Now we observe that for each point (z, w) ∈ E^{reg} the circle C_{s}(z, w) = {(z, w′) : |w−w′| = s} is homological to zero, if s > 0 is small enough. Indeed, since the set E^{reg} is connected, there is a smooth curve γ̃ ⊂ E^{reg} connecting the points A and (z, w). Then, for s > 0 small enough, the set M_{s} = {(z, w′) : |w−w′| = s, (z, w) ∈ γ̃} is a smooth “cylinder” of dimension 2 which is contained in Δ×ℂ_{w} \E and has the boundary on C_{s}(z, w) and C_{s}. Therefore, the circles C_{s}(z, w) and C_{s} represent the same homology class in H_{1}(Δ × ℂ_{w} \ E, ℤ). Since C_{s} is already proved to be homological to zero in Δ × ℂ_{w} \ E, it follows that C_{s}(z, w) is also homological to zero in Δ × ℂ_{w} \ E.

Finally, let C be any smooth closed curve in Δ × ℂ_{w} \ E. Then, there is a 2-dimensional disk D smoothly embedded into Δ × ℂ_{w} such that C = bD. We can assume that the disk D is in general position, in particular, that D intersects E in finitely many points {(z_{p}, w_{p})}_{p=1}^{n} which are contained in E^{reg}. Without restriction of generality, we can also assume that D is parallell to the w-direction in a neighbourhood of each point (z_{p}, w_{p}). Then the disks D_{s}(z_{p}, w_{p}) = {(z_{p}, w′) : |w_{p} − w′| ≤ s} are contained in D for s > 0 small enough. Therefore, C = bD is homological to ⋃_{p=1}^{n} bD_{s}(z_{p}, w_{p}) in Δ × ℂ_{w} \ E, D \ ⋃_{p=1}^{n} D_{s}(z_{p}, w_{p}) being the homology. Since each circle C_{s}(z, w) = bD_{s}(z_{p}, w_{p}) is already proved to be homological to zero in Δ × ℂ_{w} \ E, we conclude that C is also homological to zero. The proof of the claim is now completed. □

As an application of Claim 1 we show the following property of the set E.

Claim 2. There exists a neighbourhood U(E) of the set E which does not contain any subset of Δ̄ × ℂ_{w} defined by a Weierstrass pseudopolynomial.

Proof. Assume, to get a contradiction, that every neighbourhood U(E) of E contains a subset defined by a Weierstrass pseudopolynomial. For R big enough consider the circle C_{R} = {z = 0, |w| = R} ⊂ Δ × ℂ_{w} \ E oriented counterclockwise in w-variable. Then, in view of Claim 1, there is a 2-chain S such that bS = C_{R} and supp S ⊂ Δ × ℂ_{w} \ E. The last inclusion implies that there exists a neighbourhood U(E) of E such that supp S ∩U(E) = ∅. By our assumption, there is a subset Ẽ of U(E) which is defined by a Weierstrass pseudopolynomial, i.e. it has the form (1). Since supp S ∩Ẽ = ∅, the homology class [C_{R}] of the circle C_{R} in H_{1}(Δ × ℂ_{w} \ Ẽ, ℤ) is trivial. Consider the continuous map Φ : Δ × ℂ_{w} \ Ẽ → S^{1} defined by

Φ(z, w) = \frac{w^{m} + a_{1}(z)w^{m−1} + ... + a_{m}(z)}{|w^{m} + a_{1}(z)w^{m−1} + ... + a_{m}(z)|}.  (20)

Then, on one hand, [C_{R}] = 0 in H_{1}(Δ × ℂ_{w} \ Ẽ, ℤ) and, hence, Φ_{∗} ([C_{R}]) = 0 in H_{1}(S^{1}, ℤ). On the other hand, the term w^{m} in the nominator of formula (20) will be dominating for (z, w) ∈ C_{R}, if R is big enough. Therefore, the degree of the restriction of Φ to C_{R} (it is a map from S^{1} to S^{1}) is equal to m. Hence, Φ_{∗} ([C_{R}]) = m [S^{1}] ≠ 0 in H_{1}(S^{1}, ℤ). This gives the desired contradiction and proves the claim. □

References

[A] H. Alexander, Linking and holomorphic hulls, J. Diff. Geom. 38 (1993), 151-160.

[J] B. Josefson, On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on ℂ^{n}, Ark. Mat. 16 (1978), 109-115.

[N1] T. Nishino, Sur les valeurs exceptionnelles au sens de Picard d’une fonction entière, J. Math. Kyoto Univ. 2-3 (1963), 365-372.

[N2] T. Nishino, Function theory in several complex variables, Translations of Mathematical Monographs, 193., American Mathematical Society, Providence, RI, 2001.

[O] T. Ohsawa, Analyticity of complements of complete Kähler domains, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 484-487.

[S1] N. Shcherbina, Pluripolar graphs are holomorphic, submitted to Invent. Math.

[T] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959.