On the structure of Brieskorn lattice

Morihiko Saito

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 1, page 27-72
  • ISSN: 0373-0956

Abstract

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We study the structure of the filtered Gauss-Manin system associated to a holomorphic function with an isolated singularity, and get a basis of the Brieskorn lattice Ω X , 0 n + 1 / d f d Ω X , 0 n + 1 over { { t - 1 } } such that the action of t is expressed by t v = A 0 + A 1 t - 1 v for two matrices A 0 , A 1 with A 1 semi-simple, where v = t ( v 1 ... v μ ) is the basis. As an application, we calculate the b -function of f in the case of two variables.

How to cite

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Saito, Morihiko. "On the structure of Brieskorn lattice." Annales de l'institut Fourier 39.1 (1989): 27-72. <http://eudml.org/doc/74829>.

@article{Saito1989,
abstract = {We study the structure of the filtered Gauss-Manin system associated to a holomorphic function with an isolated singularity, and get a basis of the Brieskorn lattice $\Omega ^\{n+1\}_\{X,0\}/df\wedge d\Omega ^\{n+1\}_\{X,0\}$ over $\{\Bbb C\}\lbrace \lbrace \partial ^\{-1\}_t\rbrace \rbrace $ such that the action of $t$ is expressed by\begin\{\}tv=A\_0+A\_1\partial ^\{-1\}\_tv\end\{\}for two matrices $A_0,A_1$ with $A_1$ semi-simple, where $v=\{\}^t(v_1\ldots v_\mu )$ is the basis. As an application, we calculate the $b$-function of $f$ in the case of two variables.},
author = {Saito, Morihiko},
journal = {Annales de l'institut Fourier},
keywords = {mixed Hodge structures; Gauss-Manin system; Brieskorn lattice; b- function; microlocalization; vanishing cycles},
language = {eng},
number = {1},
pages = {27-72},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the structure of Brieskorn lattice},
url = {http://eudml.org/doc/74829},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Saito, Morihiko
TI - On the structure of Brieskorn lattice
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 1
SP - 27
EP - 72
AB - We study the structure of the filtered Gauss-Manin system associated to a holomorphic function with an isolated singularity, and get a basis of the Brieskorn lattice $\Omega ^{n+1}_{X,0}/df\wedge d\Omega ^{n+1}_{X,0}$ over ${\Bbb C}\lbrace \lbrace \partial ^{-1}_t\rbrace \rbrace $ such that the action of $t$ is expressed by\begin{}tv=A_0+A_1\partial ^{-1}_tv\end{}for two matrices $A_0,A_1$ with $A_1$ semi-simple, where $v={}^t(v_1\ldots v_\mu )$ is the basis. As an application, we calculate the $b$-function of $f$ in the case of two variables.
LA - eng
KW - mixed Hodge structures; Gauss-Manin system; Brieskorn lattice; b- function; microlocalization; vanishing cycles
UR - http://eudml.org/doc/74829
ER -

References

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Citations in EuDML Documents

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  1. Antoine Douai, Équations aux différences finies, intégrales de fonctions multiformes et polyèdres de Newton
  2. Alexandru Dimca, Morihiko Saito, On the cohomology of a general fiber of a polynomial map
  3. Daniel Barlet, Sur les fonctions à lieu singulier de dimension 1
  4. Claude Sabbah, Non-commutative Hodge structures
  5. Morihiko Saito, On microlocal b -function
  6. Antoine Douai, Très bonnes bases du réseau de Brieskorn d'un polynôme modéré
  7. Andréa G. Guimarães, Abramo Hefez, Bernstein-Sato Polynomials and Spectral Numbers
  8. Claus Hertling, μ -constant monodromy groups and marked singularities
  9. Morihiko Saito, Period mapping via Brieskorn modules
  10. Antoine Douai, Claude Sabbah, Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I)

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