A note on holomorphic families of Abelian varieties

Let \pi :X\to \Delta be a smooth family of complex manifolds. Suppose that X is Kähler and the fibers X_t are biholomorphic to S for all t\in \Delta ^*≔\Delta \setminus 0, where S is a fixed complex manifold. We prove that the central fiber X_0 is biholomorphic to S when S is an Abelian variety or a holomorphic principal bundle of Abelian varieties over a smooth curve T with genus g(T)>1.