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We extend the Hirzebruch-Milnor class of a hypersurface $X$ to the case wherethe normal bundle is nontrivial and $X$ cannot be defined by a global function,using the associated line bundle and the graded quotients of the monodromyfiltration. The earlier definition requiring a global defining function of $X$can be applied rarely to projective hypersurfaces with non-isolatedsingularities. Indeed, it is surprisingly difficult to get a one-parametersmoothing with total space smooth without destroying the singularities byblowing-ups (except certain quite special cases). As an application, assumingthe singular locus is a projective variety, we show that the minimal exponentof a hypersurface can be captured by the spectral Hirzebruch-Milnor class, andhigher du~Bois and rational singularities of a hypersurface are detectable bythe unnormalized Hirzebruch-Milnor class. Here the unnormalized class can bereplaced by the normalized one in the higher du~Bois case, but for the higherrational case, we must use also the decomposition of the Hirzebruch-Milnorclass by the action of the semisimple part of the monodromy (which isequivalent to the spectral Hirzebruch-Milnor class). We cannot extend thesearguments to the non-projective compact case by Hironaka's example.

Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities