Fermat varieties and the periods of some hypersurfaces

TL;DR

This paper investigates the Hodge structures of Fermat hypersurfaces and analyzes the period map for cubic hypersurfaces, providing new insights into their geometric and algebraic properties.

Contribution

It determines the integral polarized Hodge structure of Fermat hypersurfaces and extends the understanding of the period map for cubic hypersurfaces of various dimensions.

Findings

Computed the Hodge structure of Fermat hypersurfaces.

Analyzed the period map for cubic fourfolds and lower dimensions.

Reproduced known results for cubic surfaces.

Abstract

The variety of all smooth hypersurfaces of given degree and dimension has the Fermat hypersurface as a natural base point. In order to study the period map for such varieties, we first determine the integral polarized Hodge structure of the primitive cohomology of a Fermat hypersurface (as a module over the automorphism group of the hypersurface). We then focus on the degree 3 case and show that the period map for cubic fourfolds as analyzed by R. Laza and the author gives complete information about the period map for cubic hypersurfaces of lower dimension dimension. In particular, we thus recover the results of Allcock-Carlson-Toledo on the cubic surface case.