JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. I. Introduction. There is an example due to Thullen [14, p. 76; 2, pp. 428-429] of a domain D in C2 whose envelope of holomorphy, D, projects, under the canonical map D-4C2, onto a domain that is not holomorphically convex. Granted this example, it is natural to ask whether there are any conditions whatsoever that must be satisfied by a domain in CN before it can be the projection of a holomorphically convex Riemann domain. For example, is
CN\(O) the projection of a holomorphically convex Riemann domain?Our main result provides a definitive answer to this question-and to its generalization to complex manifolds. (Although we shall not mention it again, we shall consider only connected, Hausdorff, separable complex manifolds.) The following theorem is the principal result of this paper: MAIN THEOREM.
If 9T is an m-dimensional complex manifold, then there exists a locally biholomorphic.map 1D from the open unit polydisc in Cmonto 1Z with the property that for each z E 91, the fiber D-'(z) consists of not more than (2m + 1)4m + 2 points.
In particular then, every domain in CN is the projection of a holomorphically convex Riemann domain.We do not know whether every complex manifold is the finite regularl image of the unit ball, rather than the unit polydisc. In light of the main theorem, this question amounts to the problem of constructing a finite regular map from the ball onto the polydisc.The classical uniformization theorem implies that every Riemann surface is the image of the unit disc under a locally biholomorphic map, and the uniformization theorem of Griffiths [4] shows that certain Zariski open subsets of affine algebraic manifolds can be realized as images of bounded domains of holomorphy that are topological cells under regular holomorphic maps. These maps are not finite-to-one.The main theorem is proved in Sections II and III. In Section III, we also prove that an irreducible, locally irreducible complex space is the locally biholomorphic image of a Stein space under a map whose multiplicity depends only on the dimension.Section IV treats the case of real analytic manifolds, and in Section V, we prove a version of the main theorem that works simultaneously for a complex manifold and a complex submanifold. The last section of the paper is devoted to the proof of the fact that a complex manifold can be covered by a finite number of polydiscs.II. Proof of the Main Theorem. We begin with a geometric fact. LEMMA II.1.
If GT is an m-dimensional complex manifold, then DT has open covers of order not more than (2m + 1)4m consisting of relative...
