Complex structures on products of circle bundles over complex manifolds Structures

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-Kahlerian. We show that the group Pic⁰(S) of topologically trivial holomorphic line bundles on S is isomorphic to C.