  We consider the union of certain irreducible components of cohomological
support loci of the canonical bundle, which we call standard. We prove a
structure theorem about them and single out some particular cases, recovering
and improving results of Beauville and Chen-Jiang. Finally, as an example of
application, we extend to compact Kahler manifolds the classification of smooth
complex projective varieties with $p_1(X)=1$, $p_3(X)=2$ and $q(X)=\dim X$.
