Koszul complexes and spectra of projective hypersurfaces with isolated singularities

TL;DR

This paper extends classical results relating spectra, Jacobian rings, and Bernstein-Sato polynomials to projective hypersurfaces with isolated singularities, using Koszul complexes and spectral sequences to analyze their roots and cohomological properties.

Contribution

It generalizes known assertions to singular hypersurfaces, introduces new spectral sequence techniques, and establishes conditions for vanishing differentials and symmetry properties crucial for understanding Bernstein-Sato roots.

Findings

Roots of Bernstein-Sato polynomial with large absolute value are determined by the torsion part of the Jacobian ring.

Sufficient conditions for vanishing or non-vanishing of spectral sequence differentials are provided.

Symmetries of Koszul cohomology dimensions are proven, aiding in root computations.

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 Poincar\'e 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 Poincar\'e 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 $d_1$ 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 $d=\deg f$ and $n$ is the dimension of the ambient affine space.