  The goal of this article is three-fold; first, we establish an interesting
link between some sets related to cohomologies of complex local systems and the
stratification of the Albanese morphism of smooth irregular complex projective
varieties. This gives a different interpretation of a result due to Green and
Lazarsfeld. This observation originated out of our interest in studying zeros
of holomorphic 1-forms. Secondly, we show that the set of 1-forms that admit
codimension 1 zeros are in fact a linear subvariety of $H^0(X,\Omega_X^1)$. We
ask the same question for all codimensions, in particular for the set of all
1-forms that admit zeros. Thirdly, in a somewhat different direction, we deduce
some geometric consequences. The key ingredient in the third part is a result
of Kashiwara which relates the singular support produced by Albanese morphism
to 1-forms that admit zeros. We show that when a smooth projective variety with
simple Albanese admits a $\mathscr{C}^{\infty}$-fibre bundle structure over the
circle, its Albanese morphism is cohomologically trivial, i.e., the
decomposition theorem behaves like that of a smooth morphism. Such a
$\mathscr{C}^{\infty}$-fibre bundle structure is conjecturally equivalent to
the existence of holomorphic 1-forms without zeros.
