Domains of holomorphy in Segre cones (Q1821910)

Consider the space \({\mathbb{C}}^{rs}\) as the set of all rs-matrices with complex components. Let D be a domain in the Segre cone \(Z=\{z\in {\mathbb{C}}^{rs}:\) rank \(z\leq 1\}\). The author's main result says that if D is locally Stein at every non-singular boundary point in \(Z\setminus \{0\}\), then either D is a domain of holomorphy or the origin is an isolated boundary point of D, D is not a domain of holomorphy and \(D\cup \{0\}\) is a Stein domain. The method of the proof is similar to that in a previous author's paper [J. Math. Kyoto Univ. 20, 391-394 (1980; Zbl 0456.32011)], namely to go over a vector space and to apply Oka's theorem. As a consequence, D is Stein if it is locally Stein at every boundary point.