  Let $f\colon X\to B$ be a semistable fibration where $X$ is a smooth variety
of dimension $n\geq 2$ and $B$ is a smooth curve. We give the structure theorem
for the local system of the relative $1$-forms and of the relative top forms.
This gives a neat interpretation of the second Fujita decomposition of
$f_*\omega_{X/B}$. We apply our interpretation to show the existence, up to
base change, of higher irrational pencils and on the finiteness of the
associated monodromy representations under natural Castelnuovo-type hypothesis
on local subsystems. Finally we give a criterion to have that $X$ is not of
Albanese general type if $B=\mathbb{P}^1$.

Algebraic Geometry