## 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.

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.

Original language | English |
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Title of host publication | Algebraic and arithmetic structures of moduli spaces (Sapporo 2007) |

Editors | Iku Nakamura, Lin Weng |

Place of Publication | Tokyo |

Publisher | Mathematical Society of Japan |

Pages | 47–67 |

Number of pages | 20 |

ISBN (Print) | 978-4-931469-59-4 |

Publication status | Published - 2010 |

### Publication series

Name | Advanced Studies in Pure Mathematics |
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Publisher | Mathematical Society of Japan |

Volume | 58 |

## Keywords

- Fermat hypersurface
- period map