On the structure of Brieskorn lattice

Saito, Morihiko

doi:10.5802/aif.1157

Saito, Morihiko

Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 27-72.

Résumé
Abstract

On étudie la structure du système de Gauss-Manin filtré associé à une fonction holomorphe à singularité isolée, et on obtient une base du réseau de Brieskorn $Ω_{X, 0}^{n + 1} / d f \land d Ω_{X, 0}^{n + 1}$ sur $ℂ {{\partial_{t}^{- 1}}}$ telle que l’action de $t$ s’écrit

t v = A_{0} = A_{0} + A_{1} \partial_{t}^{- 1} v

pour deux matrices $A_{0}, A_{1}$ avec $A_{1}$ semi-simple, où $v =^{t} (v_{1}, ..., v_{μ})$ est la base. Comme application, on calcule la $b$ -fonction de $f$ dans le cas en deux variables.

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 $Ω_{X, 0}^{n + 1} / d f \land d Ω_{X, 0}^{n + 1}$ over $ℂ {{\partial_{t}^{- 1}}}$ such that the action of $t$ is expressed by

t v = A_{0} + A_{1} \partial_{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.

MR Zbl

DOI : 10.5802/aif.1157

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     author = {Saito, Morihiko},
     title = {On the structure of {Brieskorn} lattice},
     journal = {Annales de l'Institut Fourier},
     pages = {27--72},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {1},
     year = {1989},
     doi = {10.5802/aif.1157},
     zbl = {0644.32005},
     mrnumber = {91i:32035},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1157/}
}

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Saito, Morihiko. On the structure of Brieskorn lattice. Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 27-72. doi : 10.5802/aif.1157. https://aif.centre-mersenne.org/articles/10.5802/aif.1157/

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