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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  1. [B] E. BRIESKORN, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., 2 (1970), 103-161. Zbl0186.26101MR42 #2509
  2. [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. 
  3. [BM] J. BRIANCON and PH. MAISONOBE, Idéaux de germes d'opérateurs différentiels à une variable, Enseign. Math., 30 (1984), 7-38. Zbl0542.14008
  4. [Bo] L BOUTET DE MONVEL, D-modules holonômes réguliers en une variable, in Mathématique et Physique, Prog. in Math., Birkhäuser, 37, (1983), 281-288. Zbl0578.35080
  5. [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. Zbl0624.32006MR89k:32042
  6. [CN2] P. CASSOU-NOGUÈS, Calculs explicites sur les singularités isolées semi quasihomogènes. II, preprint. 
  7. [D1] P. DELIGNE, Équations différentielles à points singuliers réguliers, Lect. Notes in Math., 163, Springer, (1970). Zbl0244.14004MR54 #5232
  8. [D2] P. DELIGNE, Théorie de Hodge II, Publ. Math. IHES, 40, (1971), 5-58. Zbl0219.14007MR58 #16653a
  9. [K1] M. KASHIWARA, B-function and holonomic systems, Invent. Math., 38 (1976), 33-53. Zbl0354.35082MR55 #3309
  10. [K2] M. KASHIWARA, Introduction to Microlocal Analysis, Enseign. Math., 32 (1986), 227-259. Zbl0632.58030MR88e:58096
  11. [K3] M. KASHIWARA, Vanishing cycle sheaves and holonomic systems of differential equations, in Algebric Geometry, Lect. Notes in Math., Springer, 1016, (1983), 134-142. Zbl0566.32022MR85e:58137
  12. [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. Zbl0505.58033MR83e:58085
  13. [Ka] N.M. KATZ, The regularity theorem in Algebraic Geometry, Act. Congrès Intern. Math., (1970), 437-443. Zbl0235.14006MR57 #12512
  14. [Lo] E. LOOIJENGA, On the semi-universal deformation of a simple-elliptic hypersurface singularity, part II : The discrimiant, Topology, 17 (1978), 17-32. Zbl0392.57013MR58 #11503
  15. [M1] B. MALGRANGE, Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. Paris (4), 7 (1974), 405-430. Zbl0305.32008MR51 #8459
  16. [M2] B. MALGRANGE, Le polynôme de Bersnstein d'une singularité isolée, in Lect. Notes in Math., Springer, 459, (1975), 98-119. Zbl0308.32007MR54 #7845
  17. [M3] B. MALGRANGE, Déformations de systèmes différentiels et microdifférentiels, in Mathématique et Physique, Prog. in Math., Birkhäuser, 37, (1983), 353-379. Zbl0528.32016MR85h:58160
  18. [M4] B. MALGRANGE, Deformations of differentiel systems. II, Journal of the Ramanujan Math. Soc., 1 (1986), 3-15. Zbl0687.32019MR89j:58121
  19. [O] T. ODA, K. Saito's period map for holomorphic functions with isolated critical points, Advanced Studies in Pure Math., 10 (1987), 591-648. Zbl0643.32005MR89j:32030
  20. [Ph] F. PHAM, Singularités des Systèmes Différentiels de Gauss-Manin, Prog. in Math., Birkhäuser, 2, (1979). Zbl0524.32015MR81h:32015
  21. [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. Zbl0516.32012MR83f:14006
  22. [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. Zbl0549.32016
  23. [S3] M. SAITO, On the structure of Brieskorn lattices (preprint at Grenoble Sept. 1983). 
  24. [S4] M. SAITO, Hodge structure via filtered D-Modules, Astérisque, 130 (1985), 342-351. Zbl0621.14008MR87b:32019
  25. [S5] M. SAITO, Modules de Hodge polarisables, to appear in Publ. RIMS. Zbl0691.14007
  26. [S6] M. SAITO, Mixed Hodge Modules, preprint RIMS-585, July 1987. 
  27. [S7] M. SAITO, Exponents and Newton polyhedra for isolated hypersurface singularities, to appear in Math. Ann. Zbl0628.32038
  28. [Sk] K. SAITO, Period mapping associated to a primitive form, Publ. RIMS, Kyoto Univ., 19 (1983), 1231-1264. Zbl0539.58003MR85h:32034
  29. [SKK] M. SAITO, T. KAWAI and M. KASHIWARA, Microfunctions and pseudo-differential equations, Lect. Notes in Math., Springer, 287, (1973), 264-529. Zbl0277.46039MR54 #8747
  30. [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]). Zbl0618.14002MR87b:32014
  31. [Se] J.P. SERRE, Algèbre locale, multiplicités, Lect. Notes in Math., Springer 11, (1975). Zbl0296.13018
  32. [St] J. STEENBRINK, Mixed Hodge structure on the vanishing cohomology, in Real and Complex Singularities, Sijthoff & Noordhoff, Alphen aan den Rijn, (1977), 525-563. Zbl0373.14007MR58 #5670
  33. [V1] A. VARCHENKO, The asymptotics of holomorphic forms determine a mixed Hodge structure, Soviet Math. Dokl., 22 (1980), 772-775. Zbl0516.14007MR82g:14010
  34. [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. Zbl0497.32007
  35. [V3] A. VARCHENKO, The complex exponent of a singularity does not change along strata µ = const, Func. Anal. Appl., 16 (1982), 1-9. Zbl0498.32010MR83j:32005
  36. [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. Zbl0517.32004MR83j:10058

Citations in EuDML Documents

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

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