Koszul complexes and spectra of projective hypersurfaces with isolated singularities

2012/12/05 by Dimca, Alexandru, Saito, Morihiko
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1212.1081

Abstract

For a projective hypersurface Z with isolated singularities, we generalize some well-known assertions in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others about the relations between the Steenbrink spectrum, the Poincaré polynomial of the Jacobian ring, and the roots of Bernstein-Sato polynomial for a defining polynomial f up to sign forgetting the multiplicities. We have to use the pole order spectrum and the alternating sum of the Poincaré series of certain subquotients of the Koszul cohomologies, and study the pole order spectral sequence. We show sufficient conditions for vanishing or non-vanishing of the differential d₁ of the spectral sequence, which are useful in many applications. We prove also symmetries of the dimensions of the subquotients of Koszul cohomologies, which are crucial for computing the roots of BS polynomials. We can deduce that the roots of BS polynomial whose absolute values are larger than n-1-n/d are determined by the ``torsion part" of the Jacobian ring (modulo the roots of BS polynomial for Z) if all the singularities of Z are weighted homogeneous. Here d=°f and n is the dimension of the ambient affine space.

Related

• Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities
• Koszul complexes and pole order filtrations
• Multiplier ideals, b-function, and spectrum of a hypersurface singularity
• Roots of Bernstein-Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci
• Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points