2023/02/02 by Maxim, Laurenţiu, Saito, Morihiko, Yang, Ruijie
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2302.00970
openalex publication_date 2023/02/02 · openalex created_date 2023/02/04 · openalex updated_date 2026/07/28
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.