Introduction

The Hopf manifolds afford a quite elementary, though quite typical example of non-Kähler compact complex manifolds. Those were defined by H. Hopf in 1948, and investigated completely by K. Kodaira [3], [4] in the case of dimension 2. In this paper, we intend to study higher dimensional Hopf manifolds and their subvarieties.

A Hopf manifold of dimension n≥2 is defined to be a compact complex manifold of which the universal covering manifold is biholomorphic to the domain Cn — {0} ([3]). Any Hopf manifold contains nowhere discrete subvarieties. These subvarieties have rather special properties (§ 3, see also [2]), It is our aim to give a complete characterization of submanifolds of Hopf manifolds. In the case of dimension 2, the following result is known:

Theorem 1 ([1], [2]). A compact complex manifold S of dimension 2 is biholomorphic to a submanifold of a Hopf manifold if and only if S is of class VI0, VII0-elliptic or a Hopf surface.

The proof of the theorem depends on Kodaira's classification theory of surfaces. The purpose of this paper is to give a sufficient condition for a compact complex manifold of dimension ≥4 to dominate bimeromorphically a subvariety of a Hopf manifold (Main Theorem § 1).

In § 1, we give basic definitions and the statement of our main theorem.

In § 2, we recall some recent results due to Y-T. Siu [8], [9] and H-S. Ling [5], which play an essential role in the proof of our main theorem.