  We prove an equivalence between the infinitesimal Torelli theorem for top
forms on a hypersurface contained inside a Grassmannian $\mathbb G$ and the
theory of adjoint volume forms presented in L. Rizzi, F. Zucconi, "Generalized
adjoint forms on algebraic varieties", Ann. Mat. Pura e Applicata, in press.
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.
