  For a projective variety $X$ we have the intersection complex $L$-class
defined by using the self-duality of the intersection complex and also the
constant coefficient $L$-class which is the specialization at $y=1$ of the
Hirzebruch characteristic class defined by taking a cubic hyperresolution. We
show that these two $L$-classes differ if they do for an intersection of
general hyperplane sections which has only cohomologically isolated
singularities. So the study of a sufficient condition for non-coincidence is
reduced to the latter case, where a necessary and sufficient condition has been
obtained in terms of mixed Hodge structures on the stalks of the intersection
complex in our previous paper. We also construct examples of projective
varieties where the two $L$-classes differ although the constant coefficient
and intersection cohomologies coincide.
