  We extend the Hirzebruch-Milnor class of a hypersurface $X$ to the case where
the normal bundle is nontrivial and $X$ cannot be defined by a global function,
using the associated line bundle and the graded quotients of the monodromy
filtration. The earlier definition requiring a global defining function of $X$
can be applied rarely to projective hypersurfaces with non-isolated
singularities. Indeed, it is surprisingly difficult to get a one-parameter
smoothing with total space smooth without destroying the singularities by
blowing-ups (except certain quite special cases). As an application, assuming
the singular locus is a projective variety, we show that the minimal exponent
of a hypersurface can be captured by the spectral Hirzebruch-Milnor class, and
higher du~Bois and rational singularities of a hypersurface are detectable by
the unnormalized Hirzebruch-Milnor class. Here the unnormalized class can be
replaced by the normalized one in the higher du~Bois case, but for the higher
rational case, we must use also the decomposition of the Hirzebruch-Milnor
class by the action of the semisimple part of the monodromy (which is
equivalent to the spectral Hirzebruch-Milnor class). We cannot extend these
arguments to the non-projective compact case by Hironaka's example.

Algebraic Geometry