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

Fermat varieties and the periods of some hypersurfaces