Hodge theory, graded rings and Grassmannians

One of the most classical results in Hodge theory is Griffiths' description of the Hodge filtration of a smooth projective hypersurface in terms of a very explicit polynomial algebra, the so-called Jacobian ring. This turns to be extremely useful in solving Torelli-type problems, amongst others. Griffiths' result has been generalised to the smooth projective complete intersection case by Dimca et al., but not much other progress has been made so far. In this talk we present two different generalisations of Griffiths' theory. First we show how to attach to a smooth projective variety (with no hypotheses on the codimension) a graded module that controls (part of) its Hodge theory and deformation theory (joint work with Carmelo Di Natale/Domenico Fiorenza). Then we analyze the case of smooth hypersurfaces in Grassmannians, and show how to construct an explicit analogue of the Jacobian ring in this case.