  Let $\bar{L}_i\lr X_i$ 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 $\bar{L}_i$. We construct
complex structures on $S=S_1\times S_2$ which we refer to as {\em scalar,
diagonal, and linear types}. While scalar type structures always exist, the
more general diagonal but non-scalar type structures are constructed assuming
that $\bar{L}_i$ are equivariant $(\bc^*)^{n_i}$-bundles satisfying some
additional conditions. The linear type complex structures are constructed
assuming $X_i$ are (generalized) flag varieties and $\bar{L}_i$ negative ample
line bundles over $X_i$. When $H^1(X_1;\br)=0$ and $c_1(\bar{L}_1)\in
H^2(X_1;\br)$ is non-zero, the compact manifold $S$ does not admit any
symplectic structure and hence it is non-K\"ahler with respect to {\em any}
complex structure.
  We obtain a vanishing theorem for $H^q(S;\mathcal{O}_S)$ when $X_i$ are
projective manifolds, $\bar{L}_i^\vee$ are very ample and the cone over $X_i$
with respect to the projective imbedding defined by $\bar{L}_i^\vee$ 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 $\bc$.
