On the extended tube conjecture. (English) Zbl 0855.32006

Etant donné les entiers \(n \geq 2\), \(N \geq 1\), on note: d’abord \(L (\mathbb{C}^n)\) le groupe des endomorphismes de \(\mathbb{C}^n\) qui conservent la forme quadratique \(z = (z_1, \dots, z_n) \mapsto z^2_1 - \sum^n_{j = 2} z^2_j\) et \(T(n)\) le cône supérieur de lumière \(\{z = x + iy \in \mathbb{C}^n : y_1 > (\sum^n_{j = 2} y^2_j)^{1/2}\}\); puis \(T(n;N)\) le domaine dans \((\mathbb{C}^n)^N\) formé des points \((Az^1, \dots, Az^N)\), où \(A \in L (\mathbb{C}^n)\), \(\text{det} A = 1\) et \(z^1, \dots, z^N \in T(n)\). Le tube \(T(n;1)\) est domaine d’holomorphie, car \(T(n,1) = \{z \in \mathbb{C}^n\): \(z^2_1 - \sum^n_{j = 2} z^2_j \in \mathbb{C} \backslash [0, \infty[\}\); mais le même question pour le tube \(T(n;N)\) est ouverte si \(N \geq 2\). Les Auteurs considèrent donc le tube \(T_0 (n;N)\) obtenu en ajoutant \(A\) unitaire aux 2 conditions imposées ci-dessus à \(A\), et montrent: d’une part l’équivalence des 3 propriétés \(T_0 (n;N) = T(n;N)\), \(n = 2\) ou \(N = 1\), \(T_0 (n; N)\) domaine d’holomorphie; d’autre part que \(T(n;2)\) est domaine d’holomorphie \(\forall n \geq 2\).
Reviewer: M.Hervé (Paris)


32D05 Domains of holomorphy
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