Fermat varieties and the periods of some hypersurfaces

Eduard Looijenga

Fermat varieties and the periods of some hypersurfaces

Eduard Looijenga

Sub Fundamental Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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.

Original language	English
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
Publisher	Mathematical Society of Japan
Volume	58

Keywords

Fermat hypersurface
period map

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Cite this

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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.",

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Fermat varieties and the periods of some hypersurfaces. / Looijenga, Eduard.
Algebraic and arithmetic structures of moduli spaces (Sapporo 2007). ed. / Iku Nakamura; Lin Weng. Tokyo: Mathematical Society of Japan, 2010. p. 47–67 (Advanced Studies in Pure Mathematics; Vol. 58).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

TY - CHAP

T1 - Fermat varieties and the periods of some hypersurfaces

AU - Looijenga, Eduard

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Fermat hypersurface

KW - period map

M3 - Chapter

SN - 978-4-931469-59-4

T3 - Advanced Studies in Pure Mathematics

SP - 47

EP - 67

BT - Algebraic and arithmetic structures of moduli spaces (Sapporo 2007)

A2 - Nakamura, Iku

A2 - Weng, Lin

PB - Mathematical Society of Japan

CY - Tokyo

ER -

Fermat varieties and the periods of some hypersurfaces

Abstract

Publication series

Keywords

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