A note on holomorphic families of Abelian varieties

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