Mixed Hodge structure of affine hypersurfaces

Hossein Movasati[1]

  • [1] Instituto de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina, 110 22460-320, Rio de Janeiro (Brazil)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 3, page 775-801
  • ISSN: 0373-0956

Abstract

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In this article we give an algorithm which produces a basis of the n -th de Rham cohomology of the affine smooth hypersurface f - 1 ( t ) compatible with the mixed Hodge structure, where f is a polynomial in n + 1 variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of f is given in terms of the vanishing of integrals of certain polynomial n -forms in ℂ n + 1 over topological n -cycles on the fibers of f . Since the n -th homology of a regular fiber is generated by vanishing cycles, this leads us to study Abelian integrals over them. Our result generalizes and uses the arguments of J. Steenbrink for quasi-homogeneous polynomials.

How to cite

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Movasati, Hossein. "Mixed Hodge structure of affine hypersurfaces." Annales de l’institut Fourier 57.3 (2007): 775-801. <http://eudml.org/doc/10241>.

@article{Movasati2007,
abstract = {In this article we give an algorithm which produces a basis of the $n$-th de Rham cohomology of the affine smooth hypersurface $f^\{-1\}(t)$ compatible with the mixed Hodge structure, where $f$ is a polynomial in $n+1$ variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\mathbb\{C\}^\{n+1\}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th homology of a regular fiber is generated by vanishing cycles, this leads us to study Abelian integrals over them. Our result generalizes and uses the arguments of J. Steenbrink for quasi-homogeneous polynomials.},
affiliation = {Instituto de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina, 110 22460-320, Rio de Janeiro (Brazil)},
author = {Movasati, Hossein},
journal = {Annales de l’institut Fourier},
keywords = {Mixed Hodge structures of affine varieties; Gauss-Manin connection; mixed Hodge structures},
language = {eng},
number = {3},
pages = {775-801},
publisher = {Association des Annales de l’institut Fourier},
title = {Mixed Hodge structure of affine hypersurfaces},
url = {http://eudml.org/doc/10241},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Movasati, Hossein
TI - Mixed Hodge structure of affine hypersurfaces
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 775
EP - 801
AB - In this article we give an algorithm which produces a basis of the $n$-th de Rham cohomology of the affine smooth hypersurface $f^{-1}(t)$ compatible with the mixed Hodge structure, where $f$ is a polynomial in $n+1$ variables and satisfies a certain regularity condition at infinity (and hence has isolated singularities). As an application we show that the notion of a Hodge cycle in regular fibers of $f$ is given in terms of the vanishing of integrals of certain polynomial $n$-forms in $\mathbb{C}^{n+1}$ over topological $n$-cycles on the fibers of $f$. Since the $n$-th homology of a regular fiber is generated by vanishing cycles, this leads us to study Abelian integrals over them. Our result generalizes and uses the arguments of J. Steenbrink for quasi-homogeneous polynomials.
LA - eng
KW - Mixed Hodge structures of affine varieties; Gauss-Manin connection; mixed Hodge structures
UR - http://eudml.org/doc/10241
ER -

References

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