Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points

TL;DR

This paper extends classical theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points, providing algebraic descriptions of cohomology filtrations and spectra, and simplifying their computation in this singular case.

Contribution

It generalizes Griffiths and Steenbrink's theorems to hypersurfaces with ordinary double points, offering new formulas for spectral sequences and cohomology structures.

Findings

Derived algebraic descriptions of Hodge filtration quotients

Extended theorems to Milnor cohomology for certain degrees

Provided simple formulas for Steenbrink and pole order spectra

Abstract

Let Y be a hypersurface in projective space having only ordinary double points as singularities. We prove a variant of a conjecture of L. Wotzlaw on an algebraic description of the graded quotients of the Hodge filtration on the top cohomology of the complement of Y except for certain degrees of the graded quotients, as well as its extension to the Milnor cohomology of a defining polynomial of Y for degrees a little bit lower than the middle. These partially generalize theorems of Griffiths and Steenbrink in the Y smooth case, and enable us to determine the structure of the pole order spectral sequence. We then get quite simple formulas for the Steenbrink and pole order spectra in this case, which cannot be extended even to the simple singularity case easily.