On birational trivial families and Adjoint quadrics

Let $\pi \text{\hspace{0.17em}⁣}:\mathcal{X}\to B$ be a family whose general fiber $X_b$ gives a $(d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_a)$ 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.