Complex structures on product of circle bundles over compact complex manifolds

Let Li → Xi be a holomorphic line bundle over a compact complex manifold, for i = 1, 2. With respect to any hermitian inner product over Li, we denote the associated circle bundle by S(Li). The aim of this talk is to describe a family of complex structures on S(L1) × S(L2). As a special case when Xi are projective space Pni and the line bundles are tautological line bundles, Calabi-Eckmann obtained a family of complex structures on the product of odd dimensional spheres, S2n1+1×S2n2+1. Later Loeb and Nicolau constructed a more general family of complex structures on S2n1+1×S2n2+1. We generalize the Loeb-Nicolau construction to obtain the complex structures on S(L1)× S(L2). This is joint work with Prof. P Sankaran.