COMPLEX STRUCTURES ON PRODUCT OF CIRCLE BUNDLES OVER COMPLEX MANIFOLDS

Let (L) over bar (i) X-i be a holomorphic line bundle over a compact complex manifold for i = 1, 2. Let S-i denote the associated principal circle-bundle with respect to some hermitian inner product on (L) over bar (i). We construct complex structures on S = S-1 x S-2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that Li are equivariant (C*)(ni)-bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X-i are (generalized) flag varieties and (L) over bar (i) negative ample line bundles over X-i. When H-1 (X-1; R) = 0 and c1 ((L) over bar (1)) is an element of H-2 (X-1; C) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kahler with respect to any complex structure. We obtain a vanishing theorem for H-q (S; O-S) when X-i are projective manifolds, (L) over bar (V)(i) are very ample and the cone over X-i with respect to the projective imbedding defined by (L) over bar (V)(i) are Cohen-Macaulay. We obtain applications to the Picard group of S. When X-i = G(i)/P-i where P-i are maximal parabolic subgroups and S is endowed with linear type complex structure with "vanishing unipotent part" we show that the field of meromorphic functions on S is purely transcendental over C.