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The present paper is the first part of our study of a class of domains of holomorphy which includes certain complex manifolds with nonStein algebras, i.e., the algebra of holomorphic functions on the complex manifold is not a Stein algebra. Following H. Kerner [7], we can define the concept of domains of holomorphy for holomorphically separable manifolds. In what follows X is assumed to be a holomorphically separable manifold which is a domain of holomorphy (see Definition (2.5)). If X admits a fibre discrete holomorphic mapping with empty branched locus 0: X-+C9 then the classical fundamental Oka Theorem states that X is a Stein manifold. Unfortunately X does not always have such a fibre discrete mapping. In this situation we encounter with tremendous difficulties. For example, there exists a non-pseudoconvex domain with a non-Stein algebra (for the definition of pseudoconvex domains, see Definition (2.8)). Then we have the following problems: What are the necessary conditions of domains of holomorphy? and what are the good sufficient conditions of domains of holomorphy? The purpose of the present paper is to give a class of domains of holomorphy including (1) non-holomorphically convex manifolds, (2) non-pseudoconvex domains and (3) manifolds with non-Stein algebras.

A new class of domains of holomorphy. I. The concepts of boundary resolutions and {$L$}-manifolds