Koszul complexes and spectra of projective hypersurfaces with isolated singularities

For a projective hypersurface with isolated singularities, we generalize some well-known results in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others. They showed, for instance, a relation between the mixed Hodge structure on the vanishing cohomology and the Gauss-Manin system filtered by shifted Brieskorn lattices of a defining homogeneous polynomial by using the V-filtration of Kashiwara and Malgrange. Numerically this implied an identity between the Steenbrink spectrum and the Poincare polynomial of the Milnor algebra. In our case, however, we have to replace these with the pole order spectrum and the alternating sum of the Poincare series of certain subquotients of the Koszul cohomologies respectively, and then study the pole order spectral sequence which does not necessarily degenerate at E_2. This non-degeneration is closely related with the torsion of the Brieskorn module which vanished in the classical case.