Computing Cohomology with Griffiths--Dwork Reduction

Location: 11th Floor Lecture Hall Speaker Andrew Huchala, University of Oregon Abstract The Hodge structure of a smooth hypersurface Z in P^n may be understood by Griffiths' theory of residues. To each associated graded Hodge piece of the primitive cohomology of Z we identify a graded part of the so-called Jacobian ring, given in terms of the defining equation for Z. Griffiths--Dwork reduction is a technique that makes use of Griffiths' construction to identify primitive cohomology classes with meromorphic forms by lowering pole orders modulo exact forms. This has applications in Picard--Fuchs equations and zeta function computations via p-adic cohomology. I will explain the classical construction and discuss its extension to smooth hypersurfaces in Grassmannians, where the ambient geometry introduces additional cohomology and new conditions on the residue description.