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VOL. 58 | 2010 Fermat varieties and the periods of some hypersurfaces


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. In particular, we thus recover the results of Allcock–Carlson–Toledo on the cubic surface case.


Published: 1 January 2010
First available in Project Euclid: 24 November 2018

zbMATH: 1225.14032
MathSciNet: MR2676157

Digital Object Identifier: 10.2969/aspm/05810047

Rights: Copyright © 2010 Mathematical Society of Japan


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