  Let $\pi\colon\mathcal{X}\to B$ be a family over a smooth connected analytic
variety $B$, not necessarily compact, whose general fiber $X$ is smooth of
dimension $n$, with irregularity $\geq n+1$ and such that the image of the
canonical map of $X$ is not contained in any quadric of rank $\leq 2n+3$. We
prove that if the Albanese map of $X$ is of degree $1$ onto its image then the
fibers of $\pi\colon\mathcal{X}\to B$ are birational under the assumption that
all the $1$-forms and all the $n$-forms of a fiber are holomorphically liftable
to $\mathcal{X}$. Moreover we show that generic Torelli holds for such a family
$\pi\colon \mathcal{X}\to B$ if, in addition to the above hypothesis, we assume
that the fibers are minimal and their minimal model is unique. There are
counterexamples to the above statements if the canonical image is contained
inside quadrics of rank $\leq 2n+3$. We also solve the infinitesimal Torelli
problem for an $n$-dimensional variety $X$ of general type with irregularity
$\geq n+1$ and such that its cotangent sheaf is generated and the canonical map
is a rational map whose image is not contained in a quadric of rank less or
equal to $2n+3$.
