Standard canonical support loci

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 $q(X)=\dim X$.