Hirzebruch–Milnor Classes of Hypersurfaces with Nontrivial Normal Bundles and Applications to Higher du Bois and Rational Singularities

We extend the Hirzebruch–Milnor class of a hypersurface X in an ambient complex algebraic manifold 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.