PSEUDOCONVEXITY OF RIEMANN

DOMAINS OVER PRODUCT SPACES

Kwang Ho Shon

Department of Mathematics, College of Natural Sciences, Pusan National University, Pusan 609-735, Korea

1. Introduction.

Let F be the complexification of a locally convex space F_{R} over R. Let ω be an open subset of the space F_{R} and let (Ω, φ) be a Riemann domain, which contains the product set E ×ω, over the product space E ×F of locally convex spaces E and F. Then we show that Ω is pseudoconvex if and only if there exist a pseudoconvex Riemann domain (V, φ|_{V}) over F and a biholomorphic mapping σ of E × V onto Ω such that φ ∘ σ = (id_{E}, φ|_{V}).

Let M be a complex manifold of dimension n and (Ω, φ) be a Riemann domain over C^{m} × M. We assume that there exist a chart (U, α) of M, a nonempty subset A of U such that A = α^{−1}(R^{n}) and that there exists a section s : C^{m} × A ⟶ Ω of (Ω, φ). Then Ω is pseudoconvex if and only if there exist a pseudoconvex Riemann domain (V, φ|_{V}) over M and a biholomorphic mapping σ : C^{m} ×V ⟶ Ω such that φ ∘ σ = (id_{C^{m}}, φ|_{V}). Let cs(E) be the set of all nontrivial continuous seminorms on E. Let M and N be complex manifolds modeled on locally convex spaces over C. Then a holomorphic mapping f of M into N can be defined by the same way as in case the dimensions of both M and N are finite. We denote by H(M) the vector space of all holomorphic functions in M. Let 𝔉 = {(U_{i}, φ_{i}) ; i ∈ I} be the atlas of M. A function p : M → [−∞, ∞) is said to be plurisubharmonic if for each i ∈ I, the function p∘φ_{i}^{−1} is plurisubharmonic. If there exists a local biholomorphic mapping φ of a complex manifold Ω into the complex manifold M, (Ω, φ) is called a Riemann domain over M. A section of Ω is a continuous mapping s : A → Ω, with A ⊂ M, such that φ ∘ s = id on A. Then the Riemann domain (Ω, φ) is said to be contain the set A. In this paper, we assume that the Riemann domain is connected.

We shall collected some properties of Riemann domains over a locally convex space, for which we have use afterwards. We define the distance functions d_{Ω}^{α} : Ω → [0, +∞], for α ∈ cs(E), and δ_{Ω} : Ω × E → (0, +∞], as follows:

d_{Ω}^{α}(x) = sup({r > 0 ; there is a section σ : B_{E}^{α}(φ(x), r) ⟶ Ω
with σ ∘ φ(x) = x} ∪ {0}),

Typeset by AMS-TEX

Key words and phrases. Riemann domains, pseudoconvex domain, locally convex space, complex manifold, plurisubharmonic function.

Received December 22, 2000.

The author was supported by Brain Korea 21 Project, 2000

δ_{Ω}(x, a) = sup{r > 0 ; there is a section σ : D_{E}(φ(x), a, r) ⟶ Ω
with σ ∘ φ(x) = x},

where for ξ, a ∈ E and r > 0 we write

B_{E}^{α}(ξ, r) = {ξ + b ; b ∈ E, α(b) < r},
D_{E}(ξ, a, r) = {ξ + λa ; λ ∈ C, |λ| < r}.

The domain Ω is said to be pseudoconvex if the function − log δ_{Ω} is plurisubharmonic on Ω × E.

2. Pseudoconvexity of Riemann domains.

Let E_{R} be a real locally convex space, let E(C) be the complexification of E_{R} and let D be a connected open subset of E with D ∩ E_{R} ≠ ∅. In this section we shall show that the non-pluripolarity of the set D ∩ E_{R} in the open set D.

A set P in C is polar if and only if, on some (or every) open set U ⊃ P, there is u subharmonic (which may be chosen ≦ 0 if U is bounded) such that P ⊂ u^{−1}(−∞). Polar subsets of an open set Ω in a locally convex space E have several possible definitions (cf. M. Hervé [3, 4], P. Noverraz [16]).

Definition 2.1. A subset S of an open set U in a locally convex space is said to be pluripolar in U if there is a plurisubharmonic function p on U such that p ≢ −∞ on each connected component of U and that S ⊂ {x ∈ U ; p(x) = −∞}.

Lemma 2.2. Let U be a connected open suset of a locally convex space over C and let S be a subset of U. Then the set S is not a pluripolar set of U if and only if all plrisubharmonic functions p on U with p ≡ −∞ on S satisfy p ≡ −∞ on U.

The proof of the following proposition 2.3 is on M. Hervé [4, Corollary 4. 5. 17].

Proposition 2.3. Let U be a connected open subset of E and let S be a pluripolar subset of U. Then the set U \ S is connected too.

Lemma 2.4. Let Δ^{n} be the open unit polydisk of C^{n} and let S be a subset of Δ^{n} defined by S = Δ^{n} ∩ R^{n}. Then S is not a pluripolar subset of Δ^{n}.

Lemma 2.5. Let U be a connected open subset of C^{n}. Then the set U ∩ R^{n} is not a pluripolar set of U if U ∩ R^{n} ≠ ∅.

Definition 2.6. Let Ω be an open set in a locally convex space E. A set A ⊂ Ω is unipolar in Ω if :

(1) for every complex line L in E and every connected component U of L ∩ Ω, A ∩ U is either polar or A ∩ U = U,

(2) Å = ∅ or, equivalently, A does not include any connected component of Ω.

Lemma 2.7. R is not a polar set in C.

Let E_{R} be any locally convex Hausdorff space over R. For any (a, b), (c, d) ∈ E_{R} × E_{R} and any complex number λ = α + \sqrt{−1}β, we define

(a, b) + (c, d) = (a + c, b + d)
λ · (a, b) = (αa − βb, βa + αb).

Then E_{R} × E_{R} can be turned into a vector space over C. If B is a fundamental system of 0-neighborhoods of E_{R} consisting of convex balanced sets, then the family

B(C) = {λ(x, y); x ∈ V_{1}, y ∈ V_{2}, |λ| ≦ 1}_{V_{1},V_{2}∈B}

forms a fundamental system of 0-neighborhoods of E_{R} × E_{R} consisting of convex balanced sets. We denote by E(C) the vector space E_{R} × E_{R} equipped with the topology defined by B(C). Then E(C) is also a locally convex Hausdorff space over C, and E(C) is called the complexification of E_{R}.

Proposition 2.8. Let E be a locally convex space over R, Ω be on open subset of E and E(C) be the complexification of E. If Ω is an open subset of E(C) with G ⊂ Ω, then G is not a unipolar set in Ω.

Theorem 2.9. Let E(C) be the complexification of a locally convex space E_{R} over R. Let U be a connected open subset of E with U ∩ E_{R} ≠ ∅. Then the set U ∩ E_{R} is not pluripolar in U.

Proof. By Lemma 2.2 we have only to prove that all plurisubharmonic functions p on U with p ≡ −∞ on U ∩E_{R} satisfy p ≡ −∞ on U. Let p be any plurisubharmonic function on U with p ≡ −∞ on U ∩ E_{R}. Let a be any point of U. Then since U is connected, there exists a finite dimensional complex linear subspace L of E(C) such that there exists a connected component U_{L} of U ∩ L with a ∈ U_{L} and with U_{L} ∩ E_{R} ≠ ∅. Let {(x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n})} be a basis of the compex vector space L. We denote by L̃ the real linear span of the set {x_{1}, x_{2}, …, x_{n}, y_{1}, y_{2}, …, y_{n}}. Let {c_{1}, c_{2}, …, c_{m}} be a real basis of the real vector space L̃. We denote by L̃_{C} the complexfication of the real vector space L̃. Then we have L ⊂ L̃_{C}. We denote by D be a connected component of U ∩ L̃_{C} with U_{L} ⊂ D. Then the connected open subset D of L̃_{C} satisfies the properties: a ∈ D and D ∩ L̃ ≠ ∅. Since p ≡ −∞ on U ∩ E_{R} and L̃ ⊂ E_{R}, we have p ≡ −∞ on D ∩ L̃. Thus by Lemma 2.5 we have p ≡ −∞ on D. Especially p(a) = −∞. Since a is chosen arbitrariy, it is valid that p ≡ −∞ on U. This completes the proof.

Theorem 2.10. Let E and F be locally convex spaces over C. Moreover assume that the locally convex space F is the complexification of a locally convex space F_{R} over R. Let ω be an open subset of the space F_{R}. Let (Ω, φ) be a Riemann domain over E × F. Suppose that there exists a section s : E × ω ⟶ Ω of the Riemann domain (Ω, φ). Then Ω is a pseudoconvex domain if and only if there exist a pseudoconvex Riemann domain (V, φ|_{V}) over F and a biholomorphic mapping σ : E × V ⟶ Ω such that (id_{E}, φ|_{V}) = φ ∘ σ.

Proof. We denote by p_{E} and p_{F} projections of E × F onto E and F, respectively. We set Γ = s(E × ω). Then by Theorem 2.6, Γ is not a pluripolar subset of Ω. For any x ∈ Γ, any a ∈ E and any λ ∈ C, we have

φ(x) + λ(a, 0) = (p_{E} ∘ φ(x), p_{F} ∘ φ(x)) + λ(a, 0)
= (p_{E} ∘ φ(x) + λa, p_{F} ∘ φ(x)) ∈ E × ω.

Therefore φ(x) + λ(a, 0) ∈ E × ω and hence

D_{E×F} (φ(x), (a, 0), r) ⊂ E × ω

for every x ∈ Γ and r > 0. That is, there exists a section

s : D_{E×F} (φ(x), (a, 0), r) ⟶ Ω

satisfying s ∘ φ(x) = x for any x ∈ Γ, a ∈ E and r > 0. Since r is arbitrary positive number, we have δ_{Ω}(x, (a, 0)) = ∞. Since x ∈ Γ and a ∈ E are arbitrary points, we have δ_{Ω}|_{Γ×(E×{0})} ≡ ∞ and hence

− log δ_{Ω}|_{Γ×(E×{0})} ≡ −∞.

Since Ω is a pseudoconvex domain, the function − log δ_{Ω} is plurisubharmonic on Ω × (E × F). Since Γ × (E × {0}) is not a pluripolar subset of Ω × (E × {0}) and Ω × (E × {0}) is connected, we have

− log δ_{Ω}|_{Ω×(E×{0})} ≡ −∞,

that is, δ_{Ω}|_{Ω×(E×{0})} ≡ ∞. We denote by (V, φ|_{V}) a Riemann domain over F defined by V = φ^{−1}({0} × F) and φ_{V} = p_{F} ∘ φ|_{V}. Since δ_{Ω}|_{Ω×(E×{0})} ≡ ∞, there exists a section s_{x,a} : φ(x) + C · (a, 0) ⟶ Ω satisfying s_{x,a} ∘ φ(x) = x for each a ∈ E and x ∈ V where C · (a, 0) means the linear space {λa; λ ∈ C}. We define a mapping σ : E × V ⟶ Ω by σ(a, x) = s_{x,a}(φ(x) + (a, 0)) for a ∈ E and x ∈ V. We shall show that the mapping σ satisfies the required conditions. We have

φ ∘ σ(a, x) = φ ∘ s_{x,a} (φ(x) + (a, 0))
= (id_{E}(a), φ_{V} (x)) .

In order to show that σ is injective, we assume that σ(a, x) = σ(a′, x′) for some (a, x), (a′, x′) ∈ E × V. Then we have

φ(x) + (a, 0) = φ ∘ s_{x,a} (φ(x) + (a, 0)) = φ ∘ s_{x′,a′} (φ(x′) + (a′, 0)) = φ(x′) + (a′, 0).

Since a′ − a ∈ E and φ(x) − φ(x′) ∈ {0} × F we have φ(x) = φ(x′) and a′ = a. Assume that x ≠ x′. Since s_{x,a} ∘ φ(x) = x and s_{x′,a′} ∘ φ(x′) = x′, we have s_{x,a} ∘ φ(x) ≠ s_{x′,a′} ∘ φ(x′). From the definition of the section, s_{x,a} (φ(x) + (a, 0)) ≠ s_{x′,a′} (φ(x′) + (a′, 0)). This is a contradiction. Thus we have x = x′ and σ is injective. Next we shall show that σ is surjective. Let z be any point of Ω. Then we have φ(z) = (p_{E} ∘ φ(z), p_{F} ∘ φ(z)) ∈ E × F. There exists a section

s : φ(z) + C · (p_{E} ∘ φ(z), 0) ⟶ Ω

satisfying s ∘ φ(z) = z. Since φ(z) − (p_{E} ∘ φ(z), 0) ∈ {0} × F, we have

s ((0, p_{F} ∘ φ(z))) ∈ V.

And we have

σ (p_{E} ∘ φ(z), s(0, p_{F} ∘ φ(z)))
= s_{s(0, p_{F} ∘φ(z)), p_{E} ∘φ(z)} (φ(z) − (p_{E} ∘ φ(z), 0)) + (p_{E} ∘ φ(z), 0))
= z.

Thus the mapping σ is surjective. Finally we shall show that σ is biholomorphic. For x ∈ Ω, there exist continuous seminorms α and β on E and F, respectively, and a section

s : B_{E}^{α}(p_{E} ∘ φ(x), 1) × B_{F}^{β} (p_{F} ∘ φ(x), 1) ⟶ Ω

satisfying s ∘ φ(x) = x. We put

W = s({0} × B_{F}^{β} (p_{F} ∘ φ(x), 1)) ⊂ V,
D = s(B_{E}^{α}(p_{E} ∘ φ(x), 1) × B_{F}^{β} (p_{F} ∘ φ(x), 1))

Then we have

σ|_{B_{E}^{α}(p_{E} ∘φ(x), 1)×W} = (φ|_{D})^{−1} ∘ ((id_{E}, 0) + φ|_{W}) |_{B_{E}^{α}(p_{E} ∘φ(x), 1)×W} ,
σ^{−1}|_{D} = (p_{E} ∘ φ|_{D}, s ∘ p_{F} ∘ φ|_{D}) ⊂ E × V,

and hence σ is a biholomorphic mapping. The ”if part” is clear.

Let (Ω, φ) be a Riemann domain over C^{N} such that H(Ω) separated the points of Ω. M. Matos[10] proved that (Ω, φ) is a domain of holomorphy if and only if there exists a positive integer n such that (Ω, φ) is of order n in Ω and (Ω_{n}, φ_{n}) is a manifold of holomorphy spread over C^{n}, if φ_{n} = φ|_{Ω_{n}} and Ω_{n} = φ^{−1}[π_{n} ∘ φ(Ω)] where π_{n} denotes the projection mapping from C^{N} onto the space of the first n variables. A. Hirschowitz[5] proved it for the case in which (Ω, φ) is an open subset of C^{N}.

Theorm 2.11. Let (Ω, φ) be a Schlicht Riemann domain over C^{N}. If Ω is a pseudoconvex domain, then there exists a number n ∈ N and a psedoconvex Riemann domain (V, φ|_{V}) over C^{n} such that

Ω = C^{N−{1,2,··· ,n}} × V.

Proof. For x ∈ Ω, there is α ∈ cs(C^{N}) with d_{Ω}^{α}(x) ≧ 1. Thus for z = (z_{i}), there exist n ∈ N and c > 0 such that

c( sup_{1≦i≦n}|z_{i}|) ≧ α(z).

Hence there exists a section

𝔰 : B_{C^{N}}^{α}(φ(x), 1) ⟶ Ω

satisfying 𝔰 ∘ φ(x) = x. From the theorem 2.10, we have the result.

Lemma 2.12. Let M be a Stein manifold of dimension finite and let (Ω, φ) be a Riemann domain over M. We assume that Ω is a Stein manifold. Then, for any Stein open subset D of M, φ^{−1}(D) is also a Stein manifold.

Let M be a connected complex manifold of dimension finite and (Ω, φ) be a Riemann domain over M. Ω is said to be pseudoconvex if for every x ∈ M there exists an open neighborhood U of x in M such that φ^{−1}(U) is a Stein manifold. Let (Ω, φ) is a Riemann domain over the product space C^{m} × M of the complex vector space C^{m} of dimension m and M. For any p = (x, y) ∈ C^{m} × M, a ∈ C^{m} and r > 0 we set

K_{C^{m}×M}(p, a, r) = {(x + λa, y) ; λnC, |λ| < r}.

Theorem 2.13. Let M be a complex manifold of dimension n and (Ω, φ) be a Riemann domain over C^{m} × M. We assume that there exist a chart (U, α) of M, a nonempty subset A of U such that A = α^{−1}(R^{n}) and that there exists a section s : C^{m} × A ⟶ Ω of (Ω, φ). Then Ω is pseudoconvex if and only if there exist a pseudoconvex Riemann domain (V, φ|_{V}) over M and a biholomorphic mapping σ : C^{m} × V ⟶ Ω such that φ ∘ σ = (id_{C^{m}}, φ|_{V}). Especially if M is a Stein manifo , V is also a Stein manifold.

Proof. We define a Riemann domain (V, φ|_{V}) over M by

V = φ^{−1}({0} × M), φ_{V} = p_{M} ∘ φ|_{V}

where p_{M} is the projection of the product space C^{m} × M onto M. Since V is a closed complex submanifold of Ω and a closed submanifold of a Stein manifold is also a Stein manifold, the Riemann domain (V, φ_{V}) is pseudoconvex. Especially if M is a Stein manifold, it follows from F. Docquier-H. Grauert [1] that V is also a Stein manifold. We define the function Δ_{Ω} on Ω × C^{m} by

Δ_{Ω}(x, a) = sup{r > 0 ; there is a section s : K_{C^{m}×M}(φ(x), a, r) → Ω
with s ∘ φ(x) = x}.

Let (W, β) be a chart of M such that W is a Stein manifold. Then (C^{m} × W, β̃) is a chart of the complex manifold C^{m} × M. Since Ω is pseudoconvex and C^{m} × W is a Stein manifold, φ^{−1}(C^{m} × W) is also a Stein manifold by Lemma 2.12 and F. Docquier-H. Grauert [1]. Therefore (φ^{−1}(C^{m} × W), β̃ ∘ φ|_{φ^{−1}(C^{m}×W)}) is a pseudoconvex Riemann domain over C^{m} × C^{n}. Moreover we have

Δ_{Ω}(x, a) = δ_{φ^{−1}(C^{m}×W)} (x, (a, 0))

for every (x, a) ∈ φ^{−1}(C^{m} ×W)×C^{m} where δ_{φ^{−1}(C^{m}×W)} is the distance function of the Riemann domain (φ^{−1}(C^{m} × W), β̃ ∘ φ|_{φ^{−1}(C^{m}×W)}) defined in Section 1. Since (φ^{−1}(C^{m} × W), β̃ ∘ φ|_{φ^{−1}(C^{m}×W)}) is a pseudoconvex Riemann domain over C^{m} × C^{n}, − log δ_{φ^{−1}(C^{m}×W)} is plurisubharmonic on φ^{−1}(C^{m} × W) × C^{m} × {0}. Thus − log Δ_{Ω} is plurisubharmonic on φ^{−1}(C^{m}×W)×C^{m} × {0}. Since W is chosen arbitralily, − log Δ_{Ω} is plurisubharmonic on Ω × C^{m}. By the assumption we have Δ_{Ω} ≡ ∞ on A × C^{m}. Since − log Δ_{Ω} is plurisubharmonic on φ^{−1}(C^{m} × U) × C^{m} and since the set C^{m}×A is not pluripolar from Lemma 2.5, we have − log Δ_{Ω} ≡ −∞ on φ^{−1}(C^{m} × U) × C^{m}. Since C^{m} × U is an open subset of the connected complex manifold C^{m} × M and since − log Δ_{Ω} is plurisubharmonic on Ω × C^{m}, we have − log Δ_{Ω} ≡ −∞ on Ω × C^{m}, that is, Δ_{Ω} ≡ ∞ on Ω × C^{m}. Thus by the same way as the proof of Theorem 2.10 the proof of this theorem can be completed.

Proposition 2.14. Let (Ω, φ) be a Riemann domain over C^{n}, p be plurisubharmonic on Ω and p ≢ −∞. Suppose that φ^{−1}(R^{n}) ≠ ∅. Then the set {x ∈ Ω : p(x) = −∞} does not contain the set φ^{−1}(R^{n}).

References

1. F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94–123.

2. H. Hamada and J. Kajiwara, Ensembles totalment réels et domains pseudoconvexes, Mem. Fac. Sci. Kyushu Univ., Ser. A 39-2 (1985), 243–247.

3. M. Hervé, Analytic and plurisubharmonic functions in finite and infinite dimensional spaces, Lecture notes in Math., Springer-verlag, 198, 1971.

4. M. Hervé, Analyticity in infinite dimensional spaces, De Gruyter Studies in Mathematics 10, Walter de Gruyter, Berlin, New York, 1989.

5. A. Hirschowitz, Remarques sur les ouverts d’holomorphie d’un produit denombable de droices, Ann. Inst. Fourier, Grenoble 19-1 (1969), 219–229.

6. M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis, Walter de Gruyter, Berlin, New York, 1993.

7. J. Kajiwara, S. Ohgai and K. H. Shon, Groups de Lie complexes et fonctions holomorphes, Mem. Fac. Sci. Kyushu Univ., Ser. A 39-1 (1985), 127–132.

8. H. Kazama, Note on Stein neighborhoods of C^{k} ×R^{l}, Math. Rep. College Gen. Educ. Kyushu Univ. 14-1 (1983), 47–55.

9. P. Lelong, Fonctionnelles analytiques et fonctions entieres (n variables), Sémina. de Math. Supérieures-été 1967, Les Presses de L’Université de Montreál, Montreál, 1968.

10. M. Matos, The envelope of holomorphy of Riemann domains over a countable product of complex planes, Tran. Amer. Math. Soc. 167 (1972), 379–387.

11. J. Mujica, Holomorphic approximation in infinite-dimensional Riemann domains, Studia Mathematica T. 82 (1985), 107–134.

12. J. Mujica, Complex analysis in Banach spaces, North-Holland Math. Studies 120,, 1986.

13. M, Nishihara, On the indicator of growth of entire functions of exponential type in infinite dimensional spaces and the Levi problem in infinite dimensional projective spaces, Portugal Math. J 52 (1995), 61–94.

14. M. Nishihara, K. H. Shon and N. Sugawara, On pseudoconvex neighborhoods of real subspaces in some locally convex spaces over C, Mem. Fac. Sci. Kyushu Univ., Ser. A 39-1 (1985), 85–93.

15. P. Noverraz, Pseudo-Convexité, Convexité Polynomiale et Domaines d’Holomorphie en Dimension Infinite, North-Holland Math. Studies 3, North-Holland Publ. Co., New York, 1973.

16. , Fonctions Plurisousfarmoniques et Analytiues dans les Espaces Vectoriels Topologiques Complexes, Ann Inst. Fourie, Grenoble, 19-2 (1969), 419–493.

17. S. Ohgai and K. H. Shon, Stein neighborhoods of real axis, Mem. Fac. Sci. Kyushu Univ., Ser. A 38-2 (1984), 199–202.

18. M. Schottenloher, Das Levi problem in unendlichdimensionalen Räumen mit Schauder zerlegung, Habilitations Schrift, Universität, München, 1974.