Introduction

In the present paper we study certain complex structures of a compact complex manifold X of dimension n which is homeomorphic to the product of two odddimensional spheres S2p+1 × S2q+1 with p + q≥0. Since the second Betti number of X vanishes, the transcendence degree over C of the field of all meromorphic functions on X does not exceed n−1. In the following we restrict ourselves to the case where X has exactly (n−1) algebraically independent meromorphic functions. A so-called Hopf manifold is an example of such a manifold with p = 0. E. Brieskorn and A. Van de Ven [2] have constructed a somewhat different kind of complex structure on S1 × S2p+1 which also has p algebraically independent meromorphic functions. A complex structure on S2p+1 × S2q+1 with p≥l and q≥l was first constructed by E. Calabi and B. Eckmann [3]. It also satisfies the above condition. (See § 2 below.) Recently Ma. Kato [8] [9] has studied complex structures on S1 × S5 with algebraic dimension 2 which satisfy some additional conditions. Our results are generalizations of his to higher dimensional cases. Now we summarize our main results. First in § 1 we study the structure of a compact complex manifold X of dimension n such that π(X) ≃ {1} or Z, b2(X)=0 and such that a(X) = n −1. For such an X, we have the following.