Jacobian rings and the infinitesimal Torelli Theorem

TL;DR

This paper explores Jacobian rings and their relation to the infinitesimal Torelli theorem for hypersurfaces in tori, providing explicit computations of the period map's differential and its kernel.

Contribution

It introduces a period map for hypersurfaces in tori, explicitly computes its differential and kernel, and advances the understanding of the infinitesimal Torelli theorem in higher dimensions.

Findings

Explicit description of the kernel of the differential of the period map.

Complete proof of the infinitesimal Torelli theorem for certain hypersurfaces in dimensions n ≥ 4.

Identification of a mixed Hodge component with a lattice geometric quotient.

Abstract

In this article we deal with jacobian rings and identify a mixed Hodge component of a nondegenerate hypersurface in the torus with a lattice geometric quotient vector space. We introduce a period map, study its differential and compute the kernel of the differential much explicitly via certain Laurent polynomials. As a main application we deal with the infinitesimal Torelli theorem (ITT) for such explicit deformations. We study the kernel of the cohomological map for explicit deformations and complete the ITT by dealing with the remaining part $\coker(\kappa_{\mathbb{P},f})$ (cokernel of the Kodaira-Spencer map) in dimensions $n \geq 4$.