Une nouvelle version du théorème d'extension de Hartogs pour les applications séparément holomorphes entre espaces analytiques

Abstract

Abstract. This paper is concerned with the problem of extension of separately holomorphic mappings defined on a "generalized cross" of a product of complex analytic spaces with values in a complex analytic space.The crosses considered here are inscribed in Borel rectangles (of a product of two complex analytic spaces) which are not necessarily open but are non-pluripolar and can be quite small from the topological point of view.Our first main result says that the singular set of a given separately holomorphic mapping defined on such a cross is quite small from the pluripotential point of view in the product space in the sense that each of its projections is pluripolar.Then for some special crosses, we deduce more precise results on the extension of separately holomorphic mappings on such crosses, giving generalizations of all the main results obtained earlier by various authors in this direction.

A new cross theorem for separately holomorphic functions

“…Theorem 1 ([12], [13], [15], [14], [8], [9], [10], [7], [1], [16]). For each f ∈ O s (X) there exists exactly one f ∈ O( X) such that f = f on X and sup X | f | = sup X |f | ≤ +∞.…”

An extension theorem for separately holomorphic functions with pluripolar singularities

“…[17], [20], [18], [16], [12], [10], [1] (for N = 2), and [18], [13], [8] (for arbitrary N ). The case where M is analytic was studied in [14], [15], [19], [6].…”

A boundary cross theorem for separately holomorphic functions

“…Put M := {(z 1 , w 1 ), (z 2 , w 2 )}. By (1) there are open sets T ⊂ D, S ⊂ G and 0 < ε < 1 such that…”