  Let $\pi\colon \mathcal{X}\to B$ be a family whose general fiber $X_b$ gives
a $(d_1,...,d_a)$ polarisation of a general Abelian variety where $1\leq
d_i\leq 2$, $i=1,...,a$ and $a\geq 4$. We show that the fibers are in the same
birational class if all the $(m,0)$ forms on $X_b$ are liftable to $(m,0)$
forms on $\mathcal{X}$ where $m=1$ and $m=a-1$. Actually we show general
criteria to find families with fibers in the same birational class, which leads
together with a famous theorem of Nori to some interesting applications.
