The Levi phenomenon for holomorphic maps

Yusuke Dohi

We say that the q-Levi phenomenon occurs in dimension n for a complex manifold Y, if, for every (n − q + 1)-pseudoconvex hypersurface M ⊂ U ⊂ C^{n}, every holomorphic mapping f : U^{+} → Y extends holomorphically to a neighborhood of M.

In this paper, we shall prove that, if q-Levi phenomenon occurs in dimension q for a complex manifold Y, then q-Levi phenomenon occurs in dimension n for all n ≥ q.

In sections 1, we shall give basic definitions and a known fact needed in later sections. In section 2, we shall give a proof of the main result. In section 3, we shall give an example which indicates that, although our proof goes along the same line as Ivashkovich’s [3], our result is not a direct corollary of his. In section 4, we shall apply our theorem to studying an analytic property of divisors on a compact complex 3-manifold constructed in [1]. In section 5, as an appendix, we shall determine the algebraic dimension of the manifold considered in section 4.

1 Preparations

In [3], S. M. Ivashkovich proved that, if the 2-Levi phenomenon occurs in dimension 2 for a manifold Y, then 2-Levi phenomenon occurs in every dimension n ≥ 2. In this paper we will extend this fact for a general case.

The absolute value of a complex number z_{1} ∈ C will be denoted by |z_{1}|, and for z = (z_{1}, …, z_{n}) ∈ C^{n}, we define

|z| = \sqrt{|z_{1}|^{2} + ⋯ + |z_{n}|^{2}} .

For any compact subset K ⊂ D and any f ∈ O_{D} put

∥f∥ = sup_{z∈K} |f(z)| .

Let V_{1}, V_{2} ⊂ Y be nonempty compact subsets. Then the Hausdorff metric ν on the set of compact subsets in Y :

ν(V_{1}, V_{2}) = max [ sup{d(x, V_{2}) | x ∈ V_{1}} , sup{d(y, V_{1}) | y ∈ V_{2}}].

Let M be a real hypersurface in a domain U ⊂ C^{n} given by the equation M = {z ∈ U ; φ(z) = 0}, where φ is a real valued C^{∞} function with grad φ|_{M} ≠ 0 at any point on M. We write U^{+} = {z ∈ U ; φ(z) > 0} and U^{−} = {z ∈ U ; φ(z) < 0}. The surface M is said to be q-pseudoconvex at a point p ∈ M, if the Levi form

H_{φ}(p)(ξ, ξ̄) = ∑_{i,j} \frac{∂^{2}φ}{∂z^{i}∂z̄^{j}} (p)ξ^{i}ξ̄^{j}

has n − q positive eigenvalues on the complex tangent subspace T_{p}^{C}M of T_{p}U.

We can see the following facts immediately by this definition.

Remark 1 If M is q-pseudoconvex at a point p, then M is (q+1)-pseudoconvex at the same point for all q (1 ≤ q ≤ n − 1).

Remark 2 For any q (1 ≤ q ≤ n − 1), if the q-Levi phenomenon occurs in dimension n for a complex manifold Y, then the (q+1)-Levi phenomenon occurs in dimension n for Y.

Definition 1 Let (X, O_{X}) be an analytic space and K be a compact subset of X. Then the holomorphically convex hull of K is defined to be the set

K̂ = {z ∈ X ; |f(z)| ≤ ∥f∥_{K} for all f ∈ O_{X}}.

The complex space X is said to be holomorphically convex if for any compact subset K of X, K̂ is also compact.

Definition 2 Let (X, O_{X}) be a complex analytic space. Then X is a Stein space if

1. X has a countable topology.

2. X is holomorphically convex.

3. For z ∈ X, there are f_{1}, ⋯, f_{n} ∈ O_{X} such that rank_{z}(f_{1}, ⋯, f_{n}) = dim T_{z}X.

4. For any z, w ∈ X (z ≠ w), there is f ∈ O_{X} such that f(z) ≠ f(w)

It is well-known that any Stein manifold is biholomorphic to a closed submanifold of C^{n} for some n ∈ N.

Theorem 1 (Siu) Suppose X is a complex space and A is a subvariety of X. If A is Stein, then there exists an open neighborhood U of A in X such that U is Stein.

For a proof of this theorem, see [6]. The following theorem is our main result of this paper.

Theorem 2 Let q ≥ 2. If q-Levi phenomenon occurs in dimension q for a complex manifold Y, then q-Levi phenomenon occurs in dimension n for all n ≥ q.

2 Proof of the theorem

Lemma 1 Let M = {φ = 0} be a (n − q + 1)-pseudoconvex hypersurface in U ⊂ C^{n}, and let p ∈ M. Then, by a suitable choice of the defining equation φ of M, we can find a system of local coordinates (z^{1}, …, z^{q}; ξ) (ξ ∈ Δ^{n−q} ⊂ C^{n−q}) in a neighborhood V ∋ p with p = (0, ⋯, 0; 0) and, for every sufficiently small ξ_{0}, M_{ξ_{0}} = M ∩ V ∩ {ξ = ξ_{0}} is 1-pseudoconvex hypersurface in the domain {ξ = ξ_{0}} ∩ V.

Proof. By translating and rotating the coordinate system in C^{n}, we can arrange that p = 0 and T_{0}M = \{y^{q} = 0\}, where z^{j} = x^{j} + \sqrt{−1}y^{j}, and that the Hessian of the defining function φ is positive definite on the subspace {z^{q+1} = ⋯ = z^{n} = 0}.

To prove this, we develop φ as the Taylor series with respect to some system of local coordinates (z_{1}, …, z_{n}) centered at 0:

φ(z) = ∑_{i=1}^{n} \{\frac{∂φ}{∂z^{i}} (0)z^{i} + \frac{∂φ}{∂z̄^{i}} (0)z̄^{i}\} + Θ(z^{2})

where Θ(z^{2}) is the terms of order ≥ 2. Define a new system of local coordinates (w_{1}, …, w_{n}) as follows;

τ_{1} : \{ \begin{matrix}w^{1} & = & ∑_{i=1}^{n} \frac{∂φ}{∂z^{i}} (0)z^{i}, \\ w^{j} & = & z^{j}, & 2 ≤ j ≤ n. \end{matrix}

Put σ_{1} = τ_{1}^{−1} and ψ_{1}(w) = φ ∘ σ_{1}(w). Then we have

\frac{∂ψ}{∂w^{j}} (0) = ∑_{λ=1}^{n} \frac{∂φ}{∂z^{λ}} (0)\frac{∂z^{λ}}{∂w^{j}} (0) = ∑_{λ=1}^{n} \frac{∂w^{1}}{∂z^{λ}} (0)\frac{∂z^{λ}}{∂w^{j}} (0) = δ_{j}^{1}.

Hence the Taylor expansion of φ with respect to (w^{1}, …, w^{n}) at 0 is given by

ψ_{1}(w) = w^{1} + w̄^{1}

+ ∑_{i,j=1}^{n} \{\frac{1}{2} \frac{∂^{2}ψ_{1}}{∂w^{i}∂w^{j}} (0)w^{i}w^{j} + \frac{∂^{2}ψ_{1}}{∂w^{i}∂w̄^{j}} (0)w^{i}w̄^{j} + \frac{1}{2} \frac{∂^{2}ψ_{1}}{∂w̄^{i}∂w̄^{j}} (0)w̄^{i}w̄^{j}\}

+Θ(w^{3})

where Θ(w^{3}) is the terms of order ≥ 3. Next we put a_{ij} = \frac{∂^{2}ψ_{1}}{∂w^{i}∂w^{j}} (0) and define a new system of local coordinates (u_{1}, …, u_{n}) by

τ_{2} : \{ \begin{matrix}u^{1} & = & w^{1} + \frac{1}{2} ∑_{i,j=1}^{n} a_{ij}w^{i}w^{j}, \\ u^{j} & = & w^{j}, & 2 ≤ j ≤ n. \end{matrix}

Put σ_{2} = τ_{2}^{−1} and ψ_{2}(u) = ψ_{1} ∘ σ_{2}(u). Then at the point 0 we have

\frac{∂ψ_{2}}{∂u^{j}} = ∑_{μ} \frac{∂ψ_{1}}{∂w^{μ}} \frac{∂w^{μ}}{∂u^{j}} = \frac{∂ψ_{1}}{∂w^{j}} ,

\frac{∂ψ_{2}}{∂u^{j}} = ∑_{μ} \frac{∂ψ_{1}}{∂w^{μ}} \frac{∂w^{μ}}{∂u^{j}} = \frac{∂ψ_{1}}{∂w^{j}} ,

and

\frac{∂^{2}ψ_{2}}{∂u^{j}∂u^{k}} = ∑_{λμ} \frac{∂^{2}ψ_{1}}{∂w^{λ}∂w^{μ}} \frac{∂w^{λ}}{∂u^{j}} \frac{∂w^{μ}}{∂u^{k}} + ∑_{λ} \frac{∂ψ_{1}}{∂w^{λ}} \frac{∂^{2}w^{λ}}{∂u^{j}∂u^{k}}

= \frac{∂^{2}ψ_{1}}{∂w^{j}∂w^{k}} + ∑_{λ} \frac{∂ψ_{1}}{∂w^{λ}} \frac{∂^{2}w^{λ}}{∂u^{j}∂u^{k}} = \frac{∂^{2}u^{1}}{∂w^{j}∂w^{k}} + \frac{∂^{2}w^{1}}{∂u^{j}∂u^{k}} .

Differentiating the equality δ_{k}^{1} = ∑_{λ} \frac{∂u^{1}}{∂w^{λ}} \frac{∂w^{λ}}{∂u^{k}}, we have

0 = ∑_{λ} \frac{∂^{2}u^{1}}{∂w^{λ}∂w^{μ}} \frac{∂w^{λ}}{∂u^{l}} \frac{∂w^{μ}}{∂u^{k}} + ∑_{λ} \frac{∂u^{1}}{∂w^{λ}} \frac{∂^{2}w^{λ}}{∂u^{k}∂u^{l}} = \frac{∂^{2}u^{1}}{∂w^{k}∂w^{l}} + \frac{∂^{2}w^{1}}{∂u^{k}∂u^{l}}

at 0. Hence we have

\frac{∂^{2}ψ_{2}}{∂u^{j}∂u^{k}} (0) = 0.

Therefore the Taylor expansion of φ with respect to (u_{1}, …, u_{n}) at 0 is given by

ψ_{2}(u) = u^{1} + ū^{1} + ∑_{i,j≥1} a_{ij}u^{i}ū^{j} + Θ(u^{3}).

Replacing the original system of coordinates (z^{1}, …, z^{n}) with (u^{1}, …, u^{n}), we see that the Taylor expansion of φ with respect to (z^{1}, …, z^{n}) is of the form

φ(z) = z^{1} + z̄^{1} + ∑_{i,j≥1} a_{ij}z^{i}z̄^{j} + Θ(z^{3}),

where a_{ij} = \overline{a_{ji}}. Note that the complex tangent space T_{0}^{C}(M) is given by z_{1} = 0. The Hessian of φ at 0 is given by

Hesse(φ, 0) = ( \begin{matrix}a_{11} & a_{12} … & a_{1n} \\ a_{21} & a_{22} … & a_{2n} \\ ⋮ & ⋮ & ⋮ \\ a_{n1} & a_{n2} … & a_{nn} \end{matrix} ) , a_{ij} = \overline{a_{ji}}.

We set H_{1} = ( \begin{matrix}a_{22} … & a_{2n} \\ ⋮ & ⋮ \\ a_{n2} … & a_{nn} \end{matrix} ). Since H_{1} is a Hermitian matrix, we can

diagonalize H by a unitary matrix T_{1} of size n − 1 such that

T_{1}^{∗}H_{1}T_{1} = ( \begin{matrix}α_{2} \\ & ... & & & 0 \\ & & α_{q} \\ & & & α_{q+1} \\ & 0 & & & ... \\ & & & & & α_{n} \end{matrix} ) ,

where T_{1}^{∗} =^{t}T̄_{1}. Since M is (n−q +1)-pseudoconvex, at least q −1 of α_{2}, …, α_{n} are positive. Permuting the coordinates, we can assume that α_{j} > 0, 2 ≤ j ≤ q.

Furthermore we define T = ( \begin{matrix}1 & 0 \\ 0 & T_{1} \end{matrix} ) ∈ GL(n, C), and define a new system

of local coordinates u = (u^{1}, …, u^{n}) by w = T u. Put F (u) = ψ(T u) = ψ(w). Then we have

\frac{∂^{2}F}{∂u^{j}∂ū^{k}} = \frac{∂}{∂u^{j}} (\frac{∂ψ(T u)}{∂ū^{k}})
= \frac{∂}{∂u^{j}} (\frac{∂ψ}{∂w^{λ}} \frac{∂w^{λ}}{∂ū^{k}} + \frac{∂ψ}{∂w̄^{μ}} \frac{∂w̄^{μ}}{∂ū^{k}} )
= \frac{∂}{∂u^{j}} (\frac{∂ψ}{∂w̄^{μ}} \frac{∂w̄^{μ}}{∂ū^{k}} )
= \frac{∂}{∂u^{j}} (\frac{∂ψ}{∂w̄^{μ}} ) \frac{∂w̄^{μ}}{∂ū^{k}} + \frac{∂ψ}{∂w̄^{μ}} \frac{∂}{∂u^{j}} (\frac{∂w̄^{μ}}{∂ū^{k}} )
= \frac{∂}{∂w^{λ}} (\frac{∂ψ}{∂w̄^{μ}} ) \frac{∂w^{λ}}{∂u^{j}} \frac{∂w̄^{μ}}{∂ū^{k}}
= \frac{∂w̄^{μ}}{∂ū^{k}} \frac{∂^{2}ψ}{∂w^{λ}∂w̄^{μ}} \frac{∂w^{λ}}{∂u^{j}} .

Since \frac{∂w̄^{λ}}{∂u^{j}} = 0, this implies

Hesse(F, 0) = T^{∗}HT,

that is,

Hesse(F, 0) = ( \begin{matrix}1 & 0 \\ 0 & T̄_{1} \end{matrix} ) ( \begin{matrix}a_{11} & α \\ ᾱ^{t} & H_{1} \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & T_{1} \end{matrix} )
= ( \begin{matrix}a_{11} & αT_{1} \\ T̄_{1}^{t}ᾱ & T_{1}^{∗}H_{1}T_{1} \end{matrix} ) ,

where α = (a_{12}, …, a_{1n}). Since T_{0}^{C}M is given by u^{1} = 0, the restriction of Hesse(F, 0) to T_{0}^{C}M is given by T_{1}^{∗}H_{1}T_{1}.

Now we replace F with another defining equation E of M so that the Hessian of E has q positive eigen-values at p. Consider the function E(u) = e^{AF (u)}F (u), where A > 0 is sufficiently large. Then

\frac{∂^{2}E}{∂u^{i}∂ū^{j}} = \frac{∂}{∂u^{i}} \frac{∂}{∂ū^{j}} (e^{AF (u)}F (u))
= \frac{∂}{∂u^{i}} (Ae^{AF (u)} \frac{∂F}{∂ū^{j}} F (u) + e^{AF (u)} \frac{∂F}{∂ū^{j}} )
= A^{2}e^{AF (u)} \frac{∂F}{∂u^{i}} \frac{∂F}{∂ū^{j}} F (u) + Ae^{AF (u)} \frac{∂^{2}F}{∂u^{i}∂ū^{j}} F (u)

+ Ae^{AF (u)} \frac{∂F}{∂ū^{j}} \frac{∂F}{∂u^{i}} + Ae^{AF (u)} \frac{∂F}{∂u^{i}} \frac{∂F}{∂ū^{j}} + e^{AF (u)} \frac{∂^{2}F}{∂u^{i}∂ū^{j}}

= (e^{AF (u)} + Ae^{AF (u)}F (u))\frac{∂^{2}F}{∂u^{i}∂ū^{j}}

+ A(2e^{AF (u)} + Ae^{AF (u)}F (u))\frac{∂F}{∂u^{i}} \frac{∂F}{∂ū^{j}} .

Since F (0) = 0,

\frac{∂^{2}E}{∂u^{i}∂ū^{j}} (0) = \frac{∂^{2}F}{∂u^{i}∂ū^{j}} + 2A\frac{∂F}{∂u^{i}} (0)\frac{∂F}{∂ū^{j}} (0),

we have

\frac{∂^{2}E}{∂u^{1}∂ū^{1}} (0) = 2A + a_{11} > 0

for sufficiently large positive constant A.

In terms of the new system of local coordinates (u^{1}, …, u^{n}), the point p is given by (0, …, 0), the complex tangent subspace T_{0}^{C}(M) is given by u^{1} = 0, and the Hessian of E(u) is positive definite on the linear subspace {u^{q+1} = ⋯ = u^{n} = 0}. Thus we see that M ∩ {u^{q+1} = … = u^{n} = 0} is strictly convex in the polydisc {|u^{j}| < δ : j = 1, …, q} for some small constant δ > 0. Define a new system of local coordinates (z^{1}, …, z^{q}, ξ^{q+1}, …, ξ^{n}) by

σ \{ \begin{matrix}z^{j} & = & u^{j+1}, & j = 1, …, q − 1 \\ z^{q} & = & \sqrt{−1}u^{1}, \\ ξ^{j} & = & u^{j}, & j ≥ q + 1, \end{matrix}

and replace the original φ(z) with φ(z, ξ) = E(σ^{−1}(z, ξ)). Let

V = {z : |z^{j}| < δ, j = 1, …, q ; |ξ^{j}| < ε, j = q + 1, …, n}.

Then, for sufficiently small ε and δ, the intersection

M_{ξ_{0}} = M ∩ {z^{q+1} = ξ_{0}^{q+1}, …, z^{n} = ξ_{0}^{n}} ∩ V

is strictly convex in {ξ = ξ_{0}} ∩ V for all ξ_{0} = (ξ_{0}^{q+1}, …, ξ_{0}^{n}) with |ξ_{0}^{i}| < ε, i = q + 1, …, n. Thus the lemma is proved.

From now on, we identify U with the polydisc V = Δ^{q}(0, δ) × Δ^{n−q}(0, ε). Suppose now that M is a (n − q + 1)-pseudoconvex hypersurface given by the equation φ(z, ξ) = 0 in the polydisc V. The surface

M_{ξ} = {z ∈ Δ^{q}(0, δ); φ(z, ξ) = 0}

is 1-pseudoconvex in Δ^{q}(0, δ) × {ξ} for |ξ^{i}| < ε. We define a (q − 1)-dimensional ”analytic disc”

Δ_{z,ξ}^{s} = {w ∈ Δ^{q}(0, s^{−1}δ) × {ξ}; ⟨w − z, grad_{z} φ(z, ξ)⟩ = 0}

through each point (z; ξ) ∈ V^{+}. And at (0; 0), we define

Δ_{0}^{s} = lim_{z;ξ→0} Δ_{z;ξ}^{s} = {w ∈ Δ^{q}(0, s^{−1}δ) × {0}; ⟨w, grad_{z} φ(0, ξ)⟩ = 0}.

We fix a holomorphic mapping f : V^{+} → Y . The graph of f is written Γ_{f} ⊂ V × Y, and let p : V × Y → Y be a natural projection.

Lemma 2 Suppose that, for some k, the mapping f |_{Δ_{0}^{s}\{0}}: Δ_{0}^{s} \ {0} → Y extends to the origin 0 and that there exists a sequence {(z_{j}; ξ_{j})} of points in V^{+} converging to 0 such that the sequence of graphs {Γ_{f(Δ_{z_{j};ξ_{j}}^{s})}} converges in the Hausdorff metric to the graph Γ_{f(Δ_{0}^{s})}. Fix a large number r with r > s. Then, for any other sequence of discs {Δ_{ω_{p},η_{p}}^{r}} ⊂ V such that (ω_{p}, η_{p}) → 0 as p → ∞, the sequence {Γ_{f(Δ_{ωp,ηp}^{r})}} converges in the Hausdorff metric to Γ_{f(Δ_{0}^{r})}.

Proof. We shall prove the lemma by contradiction. We suppose the contrary. Then there exists a neighborhood W of Γ_{f|_{Δ^{r}_{0}}} and a sequence (ω_{p}, η_{p}) → 0 such that Γ_{f|_{Δ{r \atop ωp,ηp}}} ⊄ W. Since the graph Γ_{f|_{Δ^{r}_{0}}} is a Stein submanifold of V × Y, we can assume that W is a Stein domain by Siu [6]. Note that, for a large j > 0, we have

Γ_{f|∂Δ_{ω_{j} ,η_{j}}^{r}} ⊂ W and Γ_{f|Δ_{ω_{j} ,η_{j}}^{r}} ⊄ W.  (1)

Note also that Γ_{f|Δ_{z_{j} ,ξ_{j}}^{r}} ⊂ W for every j ≫ 0 by assumption. This contradicts that W is of Stein. Indeed, let h be any holomorphic function on W. Then, by the maximum modulus principle, we see that

∥h∥_{Γ_{f|∂Δ{r \atop (ω_{j} ,η_{j})}}} ≥ ∥h∥_{Γ_{f|Δ{r \atop (ω_{j} ,η_{j})}}} .  (2)

Since W is holomorphically convex, the inequality (2) implies that the holomorphically convex hull of the compact set Γ_{f|∂Δ_{(ω_{j} ,η_{j})}^{r}} contains Γ_{f|Δ_{(ω_{j} ,η_{j})}^{r}}, and is contained in W. Hence Γ_{f|Δ_{(ω_{j} ,η_{j})}^{l}} ⊂ W. But this contradicts (1). Thus the

lemma is proved.

Lemma 3 Under the hypotheses of Lemma 2, the mapping f extends to a neighborhood of 0 ∈ M.

Proof. The proof follows from the fact that for any neighborhood W of Γ_{f(Δ_{0}^{k})}, we can choose a neighborhood U of 0 such that Γ_{f(U^{+})} ⊂ W. We prove this by contradiction. Suppose that there exists a neighborhood W ⊃ Γ_{f(Δ_{0}^{s})} such that, for every neighborhood U of 0, Γ_{f(U^{+})}is not contained in W. That is, there exists a point (ω, η) ∈ U^{+} such that ((ω, η), f(ω, η)) ∉ W. Let U_{p}={z ∈ U ; |z| < \frac{1}{p}\}. Then, for all p, there exist a point (ω_{p}, η_{p}) ∈ U_{p}^{+} such that ((ω_{p}, η_{p}), f(ω_{p}, η_{p})) ∉ W. Since {(ω_{p}, η_{p})} is a sequence which converges to 0 with p → ∞, {Γ_{f(Δ_{ωp,ηp}^{r})}} with r > s converges to Γ_{f(Δ_{0}^{r})} by Lemma 2. But this contradicts Γ_{f(Δ_{0}^{r})} ⊂ Γ_{f(Δ_{0}^{s})} ⊂ W.

Taking s to be sufficiently large, we may assume that W is a Stein manifold as in Lemma 2. Then Lemma 3 now follows from the fact that a mapping into a Stein manifold is given by a set of holomorphic functions, and from a standard fact about extendability of holomorphic functions through the nonpseudoconvex point 0 of the hypersurface M.

Theorem 3 Fix an integer q with 2 ≤ q ≤ n. If the q-Levi phenomenon occurs in dimension q for a manifold Y then q-Levi phenomenon occurs for every dimension n ≥ q.

Proof. Let (V, (z, ξ)) be the system of local coordinates around of the point p ∈ M ⊂ U ⊂ C^{n} constructed in Lemma 1. The hypersurface M is assumed to be (n − q + 1)-pseudoconvex at p. By lemma 1, M_{0} = M ∩ {ξ = 0} is 1-pseudoconvex hypersurface. Since q-Levi phenomenon occurs in dimension q, f |_{ξ=0} extends to a neighborhood of M_{0} ⊂ U_{0} ⊂ C^{q} in the q-dimensional subspace {ξ = 0} in C^{n}, where U_{0} = U ∩ {ξ = 0}. That is, the restriction of f to Δ_{0}^{s} − {0} can extend to a neighborhood of 0 in U. Thus q-Levi phenomenon occurs in dimension n.

3 Example

In this section, we shall give an example which shows that our result does not follow from Ivashkovich’s directly.

Example. Let q ≥ 3 and consider

Y = T × (C^{q−1} − {0}/⟨g⟩),

where T is a complex torus and g is a linear transformation of C^{q−1} defined by z ↦ 2z. Put Ỹ = C×(C^{q−1}−{0}) and D^{k} = {z ∈ C^{k} ; |z| < 1}. Let π : Ỹ → Y be the natural projection. Note that since C^{q−1} − {0}/⟨g⟩ is a Hopf manifold, there is a holomorphic map h : D^{q−1} − {0} → Y which can not be extended to the whole D^{q−1} ;that is, a quotient map C^{q−1} − {0} → C^{q−1} − {0}/⟨g⟩ does not extend to zero.

We show that any holomorphic maps f : D^{q} −{0} → Y can extend holomorphically to the whole D^{q}. Since Ỹ is a covering of Y and simply-connected, Ỹ is a universal covering of Y. We consider a holomorphic map f̃: D^{q} − {0} → Ỹ which is a lift of f, f = π ∘ f̃. Then by a theorem of Hartogs, f̃ extends to a holomorphic map F̃ : D^{q} → C^{q} = Ỹ ∪W, where W = {(z_{1}, …, z_{q}) ∈ C^{q} ; z_{2} = ⋯ = z_{q} = 0}. We set F̃ = (F̃_{1}, …, F̃_{q}) with F̃_{i} : D^{q} → C, then F̃^{−1}(W) = {(z_{1}, …, z_{q}) ∈ D^{q} ; F̃_{2}(z_{1}, …, z_{q}) = ⋯ = F̃_{q}(z_{1}, …, z_{q}) = 0}. Therefore F̃^{−1}(W) is a complex analytic subset of D^{q} with dimension ≥ 1. Hence F̃^{−1}(W) − {0} is not empty. This implies that there is a point ω ∈ D^{q} − {0} which is mapped to W by f̃=F̃ |_{D^{q}−{0}}. This is a contradiction. Therefore F̃^{−1}(W)=∅, and consequently F̃(0) ∉ W. This shows that F̃ : D^{q} → C^{q}−W =Ỹ gives a holomorphic extension of f̃: D^{q} −{0} → Ỹ. Hence π ∘F̃ : D^{q} → Y gives a holomorphic extension of f : D^{q} −{0} → Y. Therefore, for any q ≥ 3, we have an example of manifolds for which the (q − 1)-Levi phenomenon in dimension q − 1 does not occur but the q-Levi phenomenon occurs in dimension q.

4 An application of the theorem

Let U be the following open subdomain in a complex 3-dimensional projective space P^{3} defined by

U = {[z_{0} : z_{1} : z_{2} : z_{3}] ∈ P^{3} ; |z_{0}|^{2} + |z_{1}|^{2} < |z_{2}|^{2} + |z_{3}|^{2}},

where [z_{0} : z_{1} : z_{2} : z_{3}] is a system of homogeneous coordinates on P^{3}. Then, by [2], there are torsion free discrete subgroups Γ ⊂ Sp(1, 1) such that the quotient spaces X = U/Γ are non-singular, where Sp(1, 1) is a Lie group which is defined by

{g ∈ M_{4}(C) ; ^{t}ḡ · H · g = H, J · g = ḡ · J}

where

H = ( \begin{matrix}−1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , J = ( \begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ) .

Here ℍ indicates the set of quaternions, that is,

ℍ = \{M ∈ M_{2}(C) : M = ( \begin{matrix}a & −b̄ \\ b & ā \end{matrix} ) , a, b ∈ C\}.

For later use, we shall also define the group

SL(2, ℍ) = \{g = ( \begin{matrix}A & C \\ B & D \end{matrix} ); det g = 1, A, B, C, D ∈ ℍ\} .

Then there are the natural inclusion relations

Γ ⊂ Sp(1, 1) ⊂ SL(2, ℍ) ⊂ SL(4, C).

We shall investigate some properties of these complex manifolds. Let π : U → X = U/Γ be the canonical projection. Suppose that there is a surface S in X and consider an irreducible component S̃_{j} of π^{−1}(S) ⊂ U. We begin by giving some definitions which are important for our purposes.

Definition 3 (Strongly plurisubharmonic function) :

1. Let G ⊂ C^{n} be a domain. Then a twice continuously differentiable function φ on G is plurisubharmonic if and only if,

(\frac{∂^{2}φ}{∂z^{i}∂z̄^{j}} )_{i,j=1,... ,n}

is positive definite on G.

2. Let X be a complex space and φ be a continuous function on X. Then φ is a strongly plurisubharmonic function, if and only if, for any x ∈ X, there is a neighborhood U of x in X and a map j : U → G ⊂ C^{n} such that j(U) is an analytic subset of G and U is isomorphic to j(U), and the following condition holds : there is a strongly plurisubharmonic function ψ on G such that φ|_{U} = ψ∘j.

Definition 4 (Strongly pseudoconcavity) : Let D be a domain on a complex space. We say that D is strongly pseudoconcave at x_{0} ∈ ∂D, if there is a neighborhood U of x_{0} and a strongly plurisubharmonic function φ in U such that

U ∩ D = {x ∈ U; φ(x) > 0}

Then we can show the following.

Proposition 1 The boundary ∂S̃_{j} has no strongly pseudoconcave boundary points for any locally closed complex analytic subvariety of dimention 2 which contains S̃_{j} as a subdomain.

Proof. We prove this by contradiction. It is known that π : U → X can’t extend holomorphically to the exterior of U at any point of the boundary by [1].

Suppose that there is a point p of ∂S̃_{j} such that S̃_{j} is pseudoconcave at p for a locally closed complex analytic subvariety Ŝ_{j} of dimension 2 which contains S̃_{j} as a subdomain. Let W be a neighborhood of p in Ŝ_{j} and k : W → C^{n} be a embedding map. There is a strongly plurisubharmonic function φ such that W ∩ S̃_{j} = {φ > 0}. By the definition, a property of strongly pseudoconcavity is invariant by a little translation of p. Therefore we can assume that p is a nonsingular point of Ŝ_{j}. By Lemma 1, we can choose a system of local coordinates (z_{1}, z_{2}) on Ŝ_{j} around p = (0, 0) such that

ψ = −2Imz_{1} + a_{1}|z_{1}|^{2} + a_{2}|z_{2}|^{2} + Θ(z^{3}) , (a_{1}, a_{2} > 0).

Therefore we can choose 0 < ε_{1} < ε_{2} and δ > 0 so that the Hartogs domain

{ε_{1} < |z_{2}| < ε_{2}, −δ < Imz_{1} < δ} ∪ {|z_{2}| < ε_{2}, −δ < Imz_{1} < 0}.

is contained in W ∩ S̃_{j}. Then, by theorem of [4], there exists a neighborhood W of p and a closed set A with the Hausdorff dimension less than 2 such that the restriction of π to S̃_{j} extends to a holomorphic map W − A → S. Hence there is a point q ∈ S̃_{j} ∩ ∂U and a neighborhood W′ of q such that π extends to W′ → S. But this is a contradiction.

5 The algebraic dimension of U/Γ

As an appendix, we shall prove the following theorem. Denote by X the compact complex manifold U/Γ constructed in section 4.

Theorem 4 There are no non-constant meromorphic functions on X.

Before proving the theorem, we shall make some preparations. Consider the unit 4-dimensional sphere

S^{4} = {x ∈ R^{5} : x_{1}^{2} + ⋯ + x_{5}^{2} = 1}

and define the differentiable surjective map

ϖ : P^{3} → S^{4}

by

[z, ź] ↦ (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}),

where

( \begin{matrix}x_{1} + ix_{2} & −x_{3} + ix_{4} \\ x_{3} + ix_{4} & x_{1} − ix_{2} \end{matrix} ) = \frac{2}{det Z + det Ź^{∗}} Z · Ź^{∗}, x_{5} = \frac{det Z − det Ź^{∗}}{det Z + det Ź^{∗}} ,

Z = ( \begin{matrix}z_{0} & −\overline{z_{1}} \\ z_{1} & \overline{z_{0}} \end{matrix} ) , Ź = ( \begin{matrix}z_{2} & −\overline{z_{3}} \\ z_{3} & \overline{z_{2}} \end{matrix} ) , Ź^{∗} = ^{t}¯Ź.

Then P^{3} is a differentiable fibre bundle over S^{4} with the projection ϖ and each fibre of ϖ is a line in P^{3}. The map ϖ sends the line ℓ_{∞} = {z_{2} = z_{3} = 0} to a single point N = (0, 0, 0, 0, 1) ∈ S^{4}. The stereographic projection

φ_{1} : S^{4} − {N} → ℍ = M_{2}(C) ≃ R^{4}

with the center N is given by

(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) ↦ \frac{1}{1 − x_{5}} ( \begin{matrix}x_{1} + ix_{2} & −x_{3} + ix_{4} \\ x_{3} + ix_{4} & x_{1} − ix_{2} \end{matrix} ) .

Put π_{1} = ϖ|(P^{3} − ℓ_{∞}) and ϖ_{1} = φ_{1} ∘ π_{1}. Then, since Ź^{∗} = (det Ź)Ź^{−1}, we have

ϖ_{1}([z, ź]) = Z · Ź^{−1}

for [z, ź] ∈ P^{3} − ℓ_{∞}. In the following, we identify S^{4} with the one point compactification ℍ ∪ {∞}. For any

g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ SL(2, ℍ),

we define a map σ_{g} : S^{4} → S^{4} by

σ_{g}(X) = (AX + B)(CX + D)^{−1}, X = ( \begin{matrix}u & −v̄ \\ v & ū \end{matrix} )

Then, we have the following commutative diagram.

\begin{matrix}P^{3} & ⟶^{g} & P^{3} \\ ϖ ↓ & & ↓ ϖ \\ S^{4} & ⟶^{σ_{g}} & S^{4} \end{matrix} .

Indeed, for any g ∈ Γ and (z, ź) ∈ P^{3}, we have the equality

( w ; ẃ ) = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ( z ; ź ) = ( \begin{matrix}Az + Bź \\ Cz + Dź \end{matrix} ) ,

π_{1}(w, ẃ) = W · Ẃ^{−1} = (AZ + BŹ)(CZ + DŹ)^{−1}
= (AZ · Ź^{−1} + B)(Ź){(Cz · ź^{−1} + D)ź}^{−1}
= (Aπ_{1}(z, ź) + B)(Cπ_{1}(z, ź) + D)^{−1}.

This implies the commutativity ϖ ∘ g = σ_{g} ∘ ϖ of the diagram above.

Proof of Theorem 4. We prove this by contradiction. Assume that X = U/Γ admits a non-constant meromorphic function f. Let a function f̃ = f ∘ π be the lift of f to U. Since f̃ is a meromorphic function on U, f̃ extends meromorphically to P^{3} by a theorem of Levi. Furthermore by a result in [2](in section 5), there is a plane H in P^{3} and a subgroup Γ_{1} ⊂ Γ with a finite index such that H is left invariant by Γ_{1} (that is, for all g ∈ Γ_{1}, g(H) = H). Replacing Γ with Γ_{1}, we assume the full group Γ leaves H invariant.

Since |z| < |ź|, z =^{t}(z_{0}, z_{1}), ź =^{t}(z_{2}, z_{3}), ϖ_{U} = ϖ|U : U → D^{4} is a differentiable sphere bundle over D^{4}, where D^{4} = {Z ∈ ℍ : det Z < 1}. Consider the group Γ̄ = {σ_{g} : g ∈ Γ}. The group Γ̄ acts on D^{4} as an isometric subgroup of the hyperbolic space D^{4}. Since Γ̄ has no fixed points in D^{4}, the fixed points of each element of Γ̄ are on the boundary ∂D^{4}. The number of fixed points are at most 2.

(a) Suppose that there is an element g ∈ Γ such that σ_{g} has two fixed points on ∂D^{4}. Let p =^{t}(p_{0}, p_{1}), q =^{t}(q_{0}, q_{1}) ∈ ∂D^{4} ⊂ ℍ be the fixed points of σ_{g}. Define σ_{k} by

k = c ( \begin{matrix}I & −P \\ I & −Q \end{matrix} ) ∈ SL(2, ℍ), c = det(P − Q)^{−1/4},

where

P = ( \begin{matrix}p_{0} & −\overline{p_{1}} \\ p_{1} & \overline{p_{0}} \end{matrix} ) , Q = ( \begin{matrix}q_{0} & −\overline{q_{1}} \\ q_{1} & \overline{q_{0}} \end{matrix} ) , I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) .

Put h = k ∘ g ∘ k^{−1} ∈ SL(2, ℍ). Then h leaves the lines ℓ_{0} = {z_{0} = z_{1} = 0} and ℓ_{∞} = {z_{2} = z_{3} = 0} invariant (and therefore σ_{h} = σ_{k} ∘ σ_{g} ∘ σ_{k}^{−1} leaves the points 0 and ∞ invariant). Since h(ℓ_{0}) = ℓ_{0}, there is λ ∈ C^{∗} = C − {0} such that

h ( \begin{matrix}0 \\ 0 \\ z_{2} \\ z_{3} \end{matrix} ) = λ ( \begin{matrix}0 \\ 0 \\ z_{2} \\ z_{3} \end{matrix} )

holds for all (z_{2}, z_{3}) ∈ C^{2}. Hence we have B = 0 and in the same way, C = 0 follows from h(ℓ_{∞}) = ℓ_{∞}. Hence we have

h = ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} )

for some A, D ∈ ℍ \ {0}. Note that h leaves k(H) invariant. Suppose that the plane k(H) is defined by

a_{0}z_{0} + a_{1}z_{1} + a_{2}z_{2} + a_{3}z_{3} = 0.  (3)

Put a = (a_{0}, a_{1}) and á = (a_{2}, a_{3}). Then (3) is written by a · z + á · ź = 0. Since k(H) is invariant by h, we get an equation aA · z + á · Dź = c(a · z + á · ź) for some constant c ∈ C^{∗}.

First, we consider the case where neither a nor á is zero. Then, since ca = aA and cá = áD for some constant c ∈ C^{∗}, both A and D have the same eigenvalues, say, λ and λ̄. Put

M = ( \begin{matrix}a_{0} & a_{1} \\ −ā_{1} & ā_{0} \end{matrix} ) and N = ( \begin{matrix}a_{2} & a_{3} \\ −ā_{3} & ā_{2} \end{matrix} )

then, we have

( \begin{matrix}M & 0 \\ 0 & N \end{matrix} ) ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ) = ( λ ; λ̄ ; λ ; λ̄ ) ( \begin{matrix}M & 0 \\ 0 & N \end{matrix} ) .

Since det h = 1, we can assume that |λ| = 1. Since h is of infinite order, we see that λ is not a root of unity. Therefore the action of h on k(U) is not properly discontinuous. This is a contradiction. Thus we conclude that, if neither a nor á is zero, then X has no non-constant meromorphic function.

Next, we consider the case where either a or á is zero. The argument above implies that, if σ_{g} (g ∈ Γ) has two fixed points p, q on ∂D, then one of the lines π_{1}^{−1}(p) and π_{1}^{−1}(q) is contained in H. Recall that k(H) is a projective plane. Since any two projective lines on a projective plane intersect each other, we see that, if both σ_{g_{1}} and σ_{g_{2}} have two fixed points in D^{4}, then σ_{g_{1}} and σ_{g_{2}} must have at least one common fixed point. Then, by [7, Theorem 5.5.4], if both σ_{g_{1}} and σ_{g_{2}} have two fixed points, then their sets of fixed points coincide, since Γ̄ is a discrete subgroup of the isometry group. Hence we can assume that any

g ∈ Γ is of the form ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ). Define a homomorphism r̄: Γ → R by

r̄(g) = log{det A det D^{−1}}.

Claim 1 Im r̄ ≅ Z.

Proof. It is enough to show that there are no sequences in ℑr̄ which converge to 0. We prove this by contradiction. Suppose that there is a se-

quence g_{n} = ( \begin{matrix}A_{n} & 0 \\ 0 & D_{n} \end{matrix} ) ∈ Γ such that \{det A_{n} det D_{n}^{−1}\}_{n} converges 1. By

det A_{n} det D_{n} = 1, we have det A_{n} det D_{n}^{−1} = det A_{n}^{2} → 1. Hence det A_{n} = |a_{n}|^{2} + |b_{n}|^{2} → 1. Since {A_{n}}_{n} is a sequence of matrix whose components are bounded, there is a subsequence of {g_{n}}_{n} such that lim_{n→∞} A_{n} define an element of SL(2, ℍ). This contradicts the discrete property of Γ.

Let K = ker r̄. Since K is a finite group and Γ is torsion free, we have K = {1}.

Hence Γ ≅ Z. Hence U/Γ cannot be compact. Thus, we have settled the case where there is an element g ∈ Γ such that σ_{g} has two fixed points.

(b) Now, we consider the case where, for every g ∈ Γ − {1}, σ_{g} ∈ Γ̄ has one fixed point in ∂D^{4}. Assume that g ∈ Γ leaves invariant the line ℓ_{∞} = {z_{2} = z_{3} = 0}. Let H be the plane which is left invariant by the full group Γ. Put

g = ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) , A, B, C, D ∈ ℍ.

Case 1. ℓ_{∞} ⊄ H. Put H = α_{0}z_{0} + ⋯ + α_{3}z_{3}. Then (α_{0}, α_{1}) ≠ (0, 0). Since g(H) = H,

(α_{0} ⋯ α_{3}) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) = λ(α_{0} ⋯ α_{3}), λ ∈ C^{∗}.  (4)

Since (α_{0}, α_{1})A = λ(α_{0}, α_{1}), λ is an eigenvalue of A. Put P = ( \begin{matrix}α_{0} & α_{1} \\ −ᾱ_{1} & ᾱ_{0} \end{matrix} )

then P AP^{−1} = ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ). Hence we can assume A = ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ). By (4), we

have (α_{0}, α_{1})B + (α_{2}, α_{3})D = λ(α_{2}, α_{3}).

Lemma 4 Let B, D ∈ ℍ and assume the equality

(α_{0}, α_{1})B + (α_{2}, α_{3})D = λ(α_{0}, α_{1}).

Then the equality

( \begin{matrix}α_{0} & α_{1} \\ −ᾱ_{1} & ᾱ_{0} \end{matrix} ) B + ( \begin{matrix}α_{2} & α_{3} \\ −ᾱ_{3} & ᾱ_{2} \end{matrix} ) D = ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ) ( \begin{matrix}α_{0} & α_{1} \\ −ᾱ_{1} & ᾱ_{0} \end{matrix} ) .

holds true.

Proof. Put B = ( \begin{matrix}b_{0} & b_{1} \\ −b̄_{1} & b̄_{0} \end{matrix} ) and D = ( \begin{matrix}d_{0} & d_{1} \\ −d̄_{1} & d̄_{0} \end{matrix} ). Then

(α_{0}, α_{1}) ( \begin{matrix}b_{0} & b_{1} \\ −b̄_{1} & b̄_{0} \end{matrix} ) + (α_{2}, α_{3}) ( \begin{matrix}d_{0} & d_{1} \\ −d̄_{1} & d̄_{0} \end{matrix} ) = λ(α_{0}, α_{1})

⇔ \{ \begin{matrix}α_{0}b_{0} + α_{1}b_{1} + α_{2}d_{0} + α_{3}d_{1} = λα_{0} \\ −α_{0}b̄_{1} + α_{1}b̄_{0} − α_{2}d̄_{1} + α_{3}d̄_{0} = λα_{0} \end{matrix}
⇔ \{ \begin{matrix}−ᾱ_{1}b_{0} + ᾱ_{0}b_{1} − ᾱ_{3}d_{0} + ᾱ_{2}d_{1} = −λᾱ_{1} \\ ᾱ_{1}b̄_{1} + ᾱ_{0}b̄_{0} + ᾱ_{3}d̄_{1} + ᾱ_{2}d̄_{0} = λ̄ᾱ_{0} \end{matrix}
⇔ (−ᾱ_{1}, ᾱ_{0})B + (−ᾱ_{3}, ᾱ_{2})D = λ̄(−ᾱ_{1}, ᾱ_{0}).

Therefore the equality (−ᾱ_{1}, ᾱ_{0})B + (−ᾱ_{3}, ᾱ_{2})D = λ̄(−ᾱ_{1}, ᾱ_{0}) also holds true. Hence the lemma follows. Put

F = ( \begin{matrix}α_{0} & α_{1} \\ −ᾱ_{1} & ᾱ_{0} \end{matrix} ) , M = ( \begin{matrix}α_{2} & α_{3} \\ −ᾱ_{3} & ᾱ_{2} \end{matrix} ) , A = ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ) ,

then

F B + MD = AM.

If λ ≠ λ̄, (α_{0}, α_{1}) ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ) = λ(α_{0}, α_{1}) by (4), and we have α_{1} = 0 and α_{0} ≠ 0. Hence we can assume F = I. Thus

( \begin{matrix}I & M \\ 0 & I \end{matrix} ) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ( \begin{matrix}I & −M \\ 0 & I \end{matrix} )

= ( \begin{matrix}A & −AM + B + MD \\ 0 & D \end{matrix} ) = ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ) .

Suppose λ = λ̄. Then, since A is a scalar matrix ( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} ), we have

( \begin{matrix}I & F^{−1}M \\ 0 & I \end{matrix} ) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ( \begin{matrix}I & −F^{−1}M \\ 0 & I \end{matrix} )

= ( \begin{matrix}A & −AF^{−1}M + F^{−1}MD + B \\ 0 & D \end{matrix} ) = ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ) .

Therefore, in both cases, we have

T · ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) · T^{−1} = ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} )

for some T ∈ SL(2, ℍ). This contradicts the fact that σ_{g} has one fixed point.

Case 2. ℓ_{∞} ⊂ H By the argument in Case 1, we can assume that for any g ∈ Γ, σ_{g} ∈ Γ̄ has only one fixed point on ∂D^{4} and that the line ℓ_{∞} left invariant by g is contained in H. Since any two lines in projective plane H intersect, all σ_{g} ∈ Γ̄ have the common fixed point. Therefore the full group Γ leaves ℓ_{∞} = {z_{2} = z_{3} = 0} invariant. Hence, we can write

Γ = \{( \begin{matrix}A & B \\ 0 & D \end{matrix} )\} .

Since ℓ_{∞} ⊂ H, H is written by

α_{2}z_{2} + α_{3}z_{3} = 0, (α_{2}, α_{3}) ≠ (0, 0).  (5)

By

(0, 0, α_{2}, α_{3}) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) = λ(0, 0, α_{2}, α_{3}), λ ∈ C^{∗},

we have that (α_{2}, α_{3})D = λ(α_{2}, α_{3}). Put M = ( \begin{matrix}α_{2} & α_{3} \\ −ᾱ_{3} & ᾱ_{2} \end{matrix} ), then

M · D = ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ) · M.

Hence we have

( \begin{matrix}I & 0 \\ 0 & M \end{matrix} ) ( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ( \begin{matrix}I & 0 \\ 0 & M^{−1} \end{matrix} ) = ( \begin{matrix}A & B · M^{−1} \\ 0 & M · D · M^{−1} \end{matrix} ) = ( \begin{matrix}A & B \\ 0 & Λ \end{matrix} ) ,

where Λ = ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ). Therefore, we can assume that

Γ = \{( \begin{matrix}A & B \\ 0 & Λ \end{matrix} )\} ,

where Λ is a diagonal matrix ( \begin{matrix}λ & 0 \\ 0 & λ̄ \end{matrix} ). Since the plane H_{3} := \{z_{3} = 0\} is

also left invariant by Γ, we consider the plane H_{3} in stead of H in the following. Take any g ∈ Γ and write as

g = ( \begin{matrix}a_{0} & −ā_{1} & b_{0} & −b̄_{1} \\ a_{1} & ā_{0} & b_{1} & b̄_{0} \\ 0 & 0 & λ & 0 \\ 0 & 0 & 0 & λ̄ \end{matrix} ) .

For simplicity, we denote also by U the transformed domain of the original domain U by an element of SL(2, ℍ). In our case, U is given by

U = {[z, ź] ∈ P^{3} : ϖ_{1}(z, ź) + ϖ_{1}(z, ź)^{∗} = rI, r > 0}.  (6)

Since ℓ_{∞} is in ∂U ∩ H_{3}, z_{2} ≠ 0 on U ∩ H_{3}. Hence we can define a system of coordinates by

x_{0} = \frac{z_{0}}{z_{2}} , x_{1} = \frac{z_{1}}{z_{2}} .

Then, g|H_{3} is given by

x′_{0} = λ̄^{−1} (a_{0}x_{0} − ā_{1}x_{1} + b_{0})
x′_{1} = λ̄^{−1} (a_{1}x_{0} + ā_{0}x_{1} + b_{1}) .

By (6), it is easy to check that

(x_{0}, x_{1}) ∈ H_{3} ∩ U ⇔ Re x_{0} > 0  (7)
(x_{0}, x_{1}) ∈ H_{3} ∩ ∂U ⇔ Re x_{0} = 0.  (8)

Put

\{ \begin{matrix}λ = λ_{0} + iλ_{1} \\ a_{j} = a_{j0} + ia_{j1} \\ x_{j} = x_{j0} + ix_{j1} \\ b_{0} = b_{00} + ib_{01} \end{matrix} ,

where λ_{0}, λ_{1}, a_{j0}, ia_{j1}, x_{j0}, x_{j1}, b_{00}, b_{01} are real numbers. Then we have

λ̄^{−1}(a_{0}x_{0} − ā_{1}x_{1}) = \frac{λ_{0} + iλ_{1}}{λ_{0}^{2} + λ_{1}^{2}} \{(a_{00} + ia_{01})(x_{00} + ix_{01} − (a_{10} − ia_{11})(x_{10} + ix_{11})\}
= \frac{λ_{0} + iλ_{1}}{λ_{0}^{2} + λ_{1}^{2}} \{(a_{00}x_{00} − a_{01}x_{01} − a_{10}x_{10} − a_{11}x_{11})
+ i(a_{00}x_{01} + a_{01}x_{00} − a_{10}x_{11} + a_{11}x_{10})}

Therefore, we obtain

Re x′_{0} = \frac{1}{λ_{0}^{2} + λ_{1}^{2}} \{λ_{0}(a_{00}x_{00} − a_{01}x_{01} − a_{10}x_{10} − a_{11}x_{11})
− λ_{1}(a_{00}x_{01} + a_{01}x_{00} − a_{10}x_{11} + a_{11}x_{10})}.

Since (x′_{0}, x′_{1}) ∈ H_{3} ∩ U if and only if (x_{0}, x_{1}) ∈ H_{3} ∩ U, and since (x′_{0}, x′_{1}) ∈ H_{3} ∩ ∂U if and only if (x_{0}, x_{1}) ∈ H_{3} ∩ ∂U, it follows from (7) and (8) that

λ_{0}a_{00} − λ_{1}a_{01} > 0
−λ_{0}a_{01} − λ_{1}a_{00} = 0
−λ_{0}a_{11} + λ_{1}a_{10} = 0
−λ_{0}a_{10} − λ_{1}a_{11} = 0
Re λ̄^{−1}b_{0} = 0.

Therefore we see that a_{1} = 0 and that g|H_{3} is given by

x′_{0} = λ̄^{−1}a_{0}x_{0} + λ̄^{−1}b_{0}
x′_{1} = λ̄^{−1}ā_{0}x_{1} + λ̄^{−1}b_{1},

where

λ̄^{−1}a_{0} ∈ R^{+} and λ̄^{−1}b_{0} ∈ iR.

Put r = λ̄^{−1}a_{0} > 0. Then, since g|H_{3} has no fixed point on H_{3} ∩ U, we have r = 1. Hence we have

x′_{0} = x_{0} + λ̄^{−1}b_{0}
x′_{1} = λ̄^{−1}λx_{1} + λ̄^{−1}b_{1}

with λ̄^{−1}b_{0} ∈ iR, or equivalently,

Re x′_{0} = Re x_{0}  (9)
Im x′_{0} = Im x_{0} − iλ̄^{−1}b_{0}
x′_{1} = λ̄^{−1}λx_{1} + λ̄^{−1}b_{1}

Note that the function Re x_{0} defines a surjective map of H_{3} ∩ U → R^{+}. By the relation (9), we see that Re x_{0} defines a continuous surjective map of a submanifold (H_{3} ∩ U)/Γ onto R^{+}. Hence (H_{3} ∩ U)/Γ is non-compact. On the other hand, since (H_{3} ∩ U)/Γ is a closed subset of U/Γ, it is compact. Thus we have a contradiction.

Thus Theorem 4 is proved completely.

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[5] Yum-Tong Siu, Techniques of extension of analytic objects, Marcel Dekker, Inc. New York, 1974.

[6] Yum-Tong Siu, Every Stein Subvariety Admits a Stein Neighborhood, Inventiones mathematicae.38(1976), pp.89-100

[7] John G.Ratcliffe, Foundations of Hyperbolic Manifolds, Springer-Verlag New York.Inc, 1994.