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Polynomial convexity, rational convexity, and currents. (English) Zbl 0838.32006
Dans les énoncés suivants, extraits de l’article, \(X\) est toujours un compact de \(\mathbb{C}^k\), \(S\) une variété compacte totalement réelle dans \(\mathbb{C}^k\).
(1) Soit \(\varphi\) une fonction p.s.h. sur un domain pseudoconvexe \(\Omega\), telle que \(\Omega \backslash \text{Supp} (dd^c \varphi)\) soit relativement compact dans \(\Omega\); alors, \(\forall s > 0\), le compact \(K_s = \{z \in \Omega : \text{dist} [z, \text{Supp} (dd^c \varphi)] \geq s\}\) est méromorphiquement convexe dans \(\Omega\), i.e. \(\Omega \backslash K_s\) réunion des ensembles de zéros des fonctions holomorphes sur \(\Omega\) qui ne s’annullent pas sur \(K_s\).
(2) L’enveloppe rationnelle \(r(X)\), définie par \(\mathbb{C}^k \backslash r(X)\) réunion des ensembles de zéros des polynomes en \(z_1, \dots, z_k\) qui ne s’annullent pas sur \(X\), peut aussi être caractérisée comme suit à l’aide de supports de courants positifs fermés de bidegré (1,1): si le support d’un tel courant est disjoint à \(X\), il est aussi disjoint à \(r(X)\); réciproquement, \(\forall x \notin r(X)\) il existe une (1,1)-forme positive fermée, de classe \({\mathcal C}^\infty\), strictement positive en \(x\) et nulle sur un voisinage de \(r(X)\).
(3) Si \(H^1 (X, \mathbb{Z}) = 0\) et si \(T\) est un courant positif, de bidegré \((k - 1, k - 1)\), à support compact: \(\text{Supp} (dT) \subset X\) entraîne \(\text{Supp} T \subset r (X)\).
(4) \(S\) est rationnellement convexe, i.e. \(S = r(S)\), si et seulement s’il existe \(\varphi \in {\mathcal C}^\infty (\mathbb{C}^k) \), strictement p.s.h., telle que \(j^* dd^c \varphi = 0\) \((j\) étant l’injection \(S \to \mathbb{C}^k)\), résultat obtenu auparavant par le \(1^{\text{er}}\) auteur [Acta Math. 172, No. 1, 77-89 (1994; Zbl 0810.32008)] dans le cas \(\dim S = 2\).
(5) \(x \in \widehat X\) si et seulement s’il existe un courant positif \(T\), be bidegré \((k - 1, k - 1)\), à support compact, tel que \(dd^c T = \mu - \delta_x\), où \(\mu\) est une mesure positive de masse 1 sur \(X\).
(6) \(\widehat S \neq S\) si et seulement s’il existe un courant positif \(T\), de bidegré \((k - 1, k - 1)\), à support compact non contenu dans \(S\), tel que \(\text{Supp} (dd^cT) \subset S\).
Reviewer: M.Hervé (Paris)

MSC:
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32C30 Integration on analytic sets and spaces, currents
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