Complex Structures on Product of Circle Bundles over Compact Complex Manifolds.

Abstract:
Let $L_i \rightarrow X_i$ be  holomorphic line bundle over 
compact complex manifold $X_i$, for $i = 1,2$. With respect to any
hermitian inner product over $L_i$, we denote the associated circle bundle
by $S(L_i)$. The aim of this talk is to describe a family of complex
structures on $S(L_1) \times S(L_2)$. As a special case when $X_i$ are
projective space $\mathbb P^{n_i}$ and the line bundles are tautological
line bundles, Calabi-Eckmann  obtained a family of  complex structures on
the product of odd dimensional spheres, $S^{2n_1 +1} \times S^{2n_2 +1}$.
Later Loeb and Nicolau constructed a more general family of complex
structures on  $S^{2n_1+1} \times S^{2n_2 +1}$.  We generalize the
Loeb-Nicolau construction to obtain the complex structures on  $S(L_1)
\times S(L_2)$. These complex manifolds will be non-K\"{a}hler. This is
joint work with Prof P.Sankaran.