Domains of Holomorphy In Segre Cones

Abstract

Let X be a normal Stein space and D a domain (open set) In X, If D is Stein, then it is a domain of holomorphy. The converse is valid when X is a manifold (Docquier-Grauert [2]). However, this is not the case in general, as was pointed out by GrauertRemmert [6], [7], They gave an example of a non-Stein domain of holomorphy in a Stein space (Segre cone). This problem is naturally related with the Levi problem, which asks whether a domain in X is Stein if it is locally Stein (at all boundary points). Concerning some results on the Levi problem for Stein spaces, see Andreotti-Narasimhan [1], Fornaess-Narasimhan [4], and Fornaess [3] particularly for Segre cones. A domain of holomorphy is locally Stein at the boundary points which are non-singular points of X. So we pose the problem : Suppose that D is locally Stein at the boundary points which are nonsingular points of X. Under what additional condition Is D a domain of holomorphy, or a Stein domain? In the present note we will give an answer to this problem for the case where X is a Segre cone. The method used here is the same as the one in the previous note of the author [11], i.e., to go over to a domain in an affine space and to apply Oka's theorem.