Complex structures on products of circle bundles over complex manifolds

Abstract

We propose, in this Note, a construction of complex structures on the product of two circle bundles associated to negative ample line bundles over flag varieties X i : = G i / P i , i = 1 , 2 , where the G i are complex semisimple linear Lie groups and the P i ⊂ G i are parabolic subgroups. The resulting manifold S is non-symplectic and hence non-Kählerian. We show that the group Pic 0 ( S ) of topologically trivial holomorphic line bundles on S is isomorphic to C .