Extending holomorphic mappings from subvarieties in Stein manifolds

Forstneric, Franc

Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 733-751.

Soit $Y$ une variété analytique complexe telle que toute application holomorphe d’une partie convexe compacte de l’espace euclidien ${ℂ}^n$ à valeurs dans $Y$ est limite uniforme d’applications entières à valeurs dans $Y$. On prouve que toute application holomorphe d’un sous ensemble analytique complexe fermé $X_0$ d’une variété de Stein $X$ à valeurs dans $Y$ possède un prolongement holomorphe à $X$ à condition qu’elle admette un prolongement continu. On établit ensuite l’équivalence entre quatre propriétés de type Oka pour une variété analytique complexe.

Suppose that $Y$ is a complex manifold such that any holomorphic map from a compact convex set in a Euclidean space ${ℂ}^n$ to $Y$ is a uniform limit of entire maps ${ℂ}^n\to Y$. We prove that a holomorphic map $X_0\to Y$ from a closed complex subvariety $X_0$ in a Stein manifold $X$ admits a holomorphic extension $X\to Y$ provided that it admits a continuous extension. We then establish the equivalence of four Oka-type properties of a complex manifold.

Classification : 32E10, 32E30, 32H02
Keywords: Stein manifold, holomorphic mappings, Oka property
Mot clés : variétés de Stein, applications holomorphes, propriété d'Oka

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