On the structure of Brieskorn lattice
Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 27-72.

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 /dfdΩ X,0 n+1 sur {{ t -1 }} telle que l’action de t s’écrit

tv=A0=A0+A1t-1v

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 /dfdΩ X,0 n+1 over {{ t -1 }} such that the action of t is expressed by

tv=A0+A1t-1v

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.

@article{AIF_1989__39_1_27_0,
     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/}
}
TY  - JOUR
AU  - Saito, Morihiko
TI  - On the structure of Brieskorn lattice
JO  - Annales de l'Institut Fourier
PY  - 1989
SP  - 27
EP  - 72
VL  - 39
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1157/
DO  - 10.5802/aif.1157
LA  - en
ID  - AIF_1989__39_1_27_0
ER  - 
%0 Journal Article
%A Saito, Morihiko
%T On the structure of Brieskorn lattice
%J Annales de l'Institut Fourier
%D 1989
%P 27-72
%V 39
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1157/
%R 10.5802/aif.1157
%G en
%F AIF_1989__39_1_27_0
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/

[B] E. Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., 2 (1970), 103-161. | MR | Zbl

[BGM] J. Briancon, M. Granger and Ph. Maisonobe, Sur le polynôme de Bernstein des singularités semi quasihomogènes, Prépublication de l'Université de Nice, n° 138, novembre 1986.

[BM] J. Briancon and Ph. Maisonobe, Idéaux de germes d'opérateurs différentiels à une variable, Enseign. Math., 30 (1984), 7-38. | Zbl

[Bo] L Boutet De Monvel, D-modules holonômes réguliers en une variable, in Mathématique et Physique, Prog. in Math., Birkhäuser, 37, (1983), 281-288.

[CN1] P. Cassou-Noguès, Étude du comportement du polynôme de Bernstein lors d'une déformation à µ constant de Xa + Yb avec (a, b) = 1, Compositio Math., 63 (1987), 291-313. | Numdam | MR | Zbl

[CN2] P. Cassou-Noguès, Calculs explicites sur les singularités isolées semi quasihomogènes. II, preprint.

[D1] P. Deligne, Équations différentielles à points singuliers réguliers, Lect. Notes in Math., 163, Springer, (1970). | MR | Zbl

[D2] P. Deligne, Théorie de Hodge II, Publ. Math. IHES, 40, (1971), 5-58. | Numdam | MR | Zbl

[K1] M. Kashiwara, B-function and holonomic systems, Invent. Math., 38 (1976), 33-53. | MR | Zbl

[K2] M. Kashiwara, Introduction to Microlocal Analysis, Enseign. Math., 32 (1986), 227-259. | MR | Zbl

[K3] M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, in Algebric Geometry, Lect. Notes in Math., Springer, 1016, (1983), 134-142. | MR | Zbl

[KK] M. Kashiwara and T. Kawai, On the holomic systems of microdifferential equations III, Systems with regular singularites, Publ. RIMS, Kyoto Univ., 17 (1981), 813-979. | MR | Zbl

[Ka] N.M. Katz, The regularity theorem in Algebraic Geometry, Act. Congrès Intern. Math., (1970), 437-443. | MR | Zbl

[Lo] E. Looijenga, On the semi-universal deformation of a simple-elliptic hypersurface singularity, part II : The discrimiant, Topology, 17 (1978), 17-32. | MR | Zbl

[M1] B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. Paris (4), 7 (1974), 405-430. | Numdam | MR | Zbl

[M2] B. Malgrange, Le polynôme de Bersnstein d'une singularité isolée, in Lect. Notes in Math., Springer, 459, (1975), 98-119. | MR | Zbl

[M3] B. Malgrange, Déformations de systèmes différentiels et microdifférentiels, in Mathématique et Physique, Prog. in Math., Birkhäuser, 37, (1983), 353-379. | MR | Zbl

[M4] B. Malgrange, Deformations of differentiel systems. II, Journal of the Ramanujan Math. Soc., 1 (1986), 3-15. | MR | Zbl

[O] T. Oda, K. Saito's period map for holomorphic functions with isolated critical points, Advanced Studies in Pure Math., 10 (1987), 591-648. | MR | Zbl

[Ph] F. Pham, Singularités des Systèmes Différentiels de Gauss-Manin, Prog. in Math., Birkhäuser, 2, (1979). | MR | Zbl

[S1] M. Saito, Gauss-Manin system and mixed Hodge structure, Proc. Japan Acad., 58 (A) (1982), 29-32 ; Supplement, in Astérisque, 101/102 (1983), 320-331. | MR | Zbl

[S2] M. Saito, Hodge filtrations on Gauss-Manin systems I, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math., 30 (1984), 489-498 ; II, Proc. Japan Acad., 59 (A) (1983), 37-40. | Zbl

[S3] M. Saito, On the structure of Brieskorn lattices (preprint at Grenoble Sept. 1983).

[S4] M. Saito, Hodge structure via filtered D-Modules, Astérisque, 130 (1985), 342-351. | MR | Zbl

[S5] M. Saito, Modules de Hodge polarisables, to appear in Publ. RIMS. | Zbl

[S6] M. Saito, Mixed Hodge Modules, preprint RIMS-585, July 1987.

[S7] M. Saito, Exponents and Newton polyhedra for isolated hypersurface singularities, to appear in Math. Ann. | Zbl

[Sk] K. Saito, Period mapping associated to a primitive form, Publ. RIMS, Kyoto Univ., 19 (1983), 1231-1264. | MR | Zbl

[SKK] M. Saito, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, Lect. Notes in Math., Springer, 287, (1973), 264-529. | MR | Zbl

[SS] J. Scherk and J. Steenbrink, On the mixed Hodge structure on the cohomology of the Milnor fiber, Math. Ann., 271 (1985), 641-665. (This is the corrected version of the preprint quoted in [S1][S2]). | MR | Zbl

[Se] J.P. Serre, Algèbre locale, multiplicités, Lect. Notes in Math., Springer 11, (1975). | Zbl

[St] J. Steenbrink, Mixed Hodge structure on the vanishing cohomology, in Real and Complex Singularities, Sijthoff & Noordhoff, Alphen aan den Rijn, (1977), 525-563. | MR | Zbl

[V1] A. Varchenko, The asymptotics of holomorphic forms determine a mixed Hodge structure, Soviet Math. Dokl., 22 (1980), 772-775. | MR | Zbl

[V2] A. Varchenko, On the monodromy operator in vanishing cohomology and the operator of multiplication by f in the local ring, Soviet Math. Dokl., 24 (1981), 248-252. | Zbl

[V3] A. Varchenko, The complex exponent of a singularity does not change along strata µ = const, Func. Anal. Appl., 16 (1982), 1-9. | MR | Zbl

[V4] A. Varchenko, A lower bound for the codimension of the strata µ = const in terms of the mixed Hodge structure, Moscow Univ. Math. Bull., 37-6 (1982), 30-33. | MR | Zbl

Cité par Sources :