On Green’s proof of the infinitesimal Torelli theorem for hypersurfaces

Abstract

We prove an equivalence between the infinitesimal Torelli theorem for top forms on a hypersurface $X$ contained inside a Grassmannian and the theory of adjoint volume forms. More precisely, via this theory and a suitable generalization of Macaulay’s theorem we show that the differential of the period map vanishes on an infinitesimal deformation if and only if certain explicitly given twisted volume forms go in the generalized Jacobi ideal of $X$ via the cup product homomorphism.