# On Green's proof of infinitesimal Torelli theorem for hypersurfaces

## TL;DR

This paper establishes a link between the infinitesimal Torelli theorem for hypersurfaces in Grassmannians and the theory of adjoint volume forms, providing a criterion involving twisted volume forms and the Jacobi ideal.

## Contribution

It generalizes Macaulay's theorem and connects the infinitesimal Torelli theorem with adjoint volume forms in a new, explicit way.

## Key findings

- Differential of the period map vanishes iff certain twisted volume forms lie in the generalized Jacobi ideal.
- Provides an explicit criterion for infinitesimal Torelli in terms of volume forms and the Jacobi ideal.
- Links the infinitesimal Torelli theorem to the theory of adjoint volume forms in hypersurfaces.

## Abstract

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.