  For a subvariety of a smooth projective variety, consider the family of
smooth hypersurfaces of sufficiently large degree containing it, and take the
quotient of the middle cohomology of the hypersurfaces by the cohomology of the
ambient variety and also by the cycle classes of the irreducible components of
the subvariety. Using Deligne's semisimplicity theorem together with
Steenbrink's theory for semistable degenerations, we give a simpler proof of
the first author's theorem (with a better bound of the degree of hypersurfaces)
that this monodromy representation is irreducible.
