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

top
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 ℂ { { ∂ 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

top

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

top
  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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.