Jacobian rings and the infinitesimal Torelli Theorem

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 $\coker(κ_{\mathbb{P},f})$ (cokernel of the Kodaira-Spencer map) in dimensions $n \geq 4$.