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On the structure of Brieskorn lattice. (English) Zbl 0644.32005
Let f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a holomorphic function with an isolated singularity. We have the filtered Gauss-Manin system \((M,F)=(\int ^{0}_{f}\underline O_ X,F)_ 0\) with \(f: X\to S\) a good represent of f (i.e. Milnor fibration) such that \(F_{-n}M\) is the Brieskorn lattice \(\Omega ^{n+1}_{X,0}/df\wedge d\Omega ^{n- 1}_{X,0}\). We study the structure of the filtered \b{E}-module (M,F) using the filtration V of M, and get a free basis \(v=(v_ 1,...,v_{\mu})\) of \(F_{-n}M\) over \({\mathbb{C}}\{\{\partial _ t^{- 1}\}\}\) such that the action of t is expressed by \(tv=A_ 0+A_ 1\partial _ t^{-1}v\) for two matrices \(A_ 0\), \(A_ 1\) with \(A_ 1\) semisimple. For the proof we use essentially the theory of V- filtration and mixed Hodge structures. The relation between the filtered microlocal Poincaré duality of (M,F) and the Grothendieck residue pairings is also explained. As an application, the b-function of f is calculated in some cases.
Reviewer: M.Saito

MSC:
32Sxx Complex singularities
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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References:
[1] E. BRIESKORN, Die monodromie der isolierten singularitäten von hyperflächen, Manuscripta Math., 2 (1970), 103-161. · Zbl 0186.26101
[2] 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] J. BRIANCON and PH. MAISONOBE, Idéaux de germes d’opérateurs différentiels à une variable, Enseign. Math., 30 (1984), 7-38. · Zbl 0542.14008
[4] L BOUTET DE MONVEL, D-modules holonômes réguliers en une variable, in Mathématique et Physique, Prog. in Math., Birkhäuser, 37, (1983), 281-288. · Zbl 0578.35080
[5] 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. · Zbl 0624.32006
[6] P. CASSOU-NOGUÈS, Calculs explicites sur LES singularités isolées semi quasihomogènes. II, preprint.
[7] P. DELIGNE, Équations différentielles à points singuliers réguliers, Lect. Notes in Math., 163, Springer, (1970). · Zbl 0244.14004
[8] P. DELIGNE, Théorie de Hodge II, Publ. Math. IHES, 40, (1971), 5-58. · Zbl 0219.14007
[9] M. KASHIWARA, B-function and holonomic systems, Invent. Math., 38 (1976), 33-53. · Zbl 0354.35082
[10] M. KASHIWARA, Introduction to microlocal analysis, Enseign. Math., 32 (1986), 227-259. · Zbl 0632.58030
[11] M. KASHIWARA, Vanishing cycle sheaves and holonomic systems of differential equations, in Algebric Geometry, Lect. Notes in Math., Springer, 1016, (1983), 134-142. · Zbl 0566.32022
[12] 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. · Zbl 0505.58033
[13] N.M. KATZ, The regularity theorem in algebraic geometry, Act. Congrès Intern. Math., (1970), 437-443. · Zbl 0235.14006
[14] E. LOOIJENGA, On the semi-universal deformation of a simple-elliptic hypersurface singularity, part II : The discrimiant, Topology, 17 (1978), 17-32. · Zbl 0392.57013
[15] B. MALGRANGE, Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. Paris (4), 7 (1974), 405-430. · Zbl 0305.32008
[16] B. MALGRANGE, Le polynôme de bersnstein d’une singularité isolée, in Lect. Notes in Math., Springer, 459, (1975), 98-119. · Zbl 0308.32007
[17] B. MALGRANGE, Déformations de systèmes différentiels et microdifférentiels, in Mathématique et Physique, Prog. in Math., Birkhäuser, 37, (1983), 353-379. · Zbl 0528.32016
[18] B. MALGRANGE, Deformations of differentiel systems. II, Journal of the Ramanujan Math. Soc., 1 (1986), 3-15. · Zbl 0687.32019
[19] T. ODA, K. Saito’s period map for holomorphic functions with isolated critical points, Advanced Studies in Pure Math., 10 (1987), 591-648. · Zbl 0643.32005
[20] F. PHAM, Singularités des systèmes différentiels de Gauss-Manin, Prog. in Math., Birkhäuser, 2, (1979). · Zbl 0524.32015
[21] 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. · Zbl 0516.32012
[22] 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 0549.32016
[23] M. SAITO, On the structure of Brieskorn lattices (preprint at Grenoble Sept. 1983).
[24] M. SAITO, Hodge structure via filtered D-modules, Astérisque, 130 (1985), 342-351. · Zbl 0621.14008
[25] M. SAITO, Modules de Hodge polarisables, to appear in Publ. RIMS. · Zbl 0691.14007
[26] M. SAITO, Mixed Hodge modules, preprint RIMS-585, July 1987.
[27] M. SAITO, Exponents and Newton polyhedra for isolated hypersurface singularities, to appear in Math. Ann. · Zbl 0628.32038
[28] K. SAITO, Period mapping associated to a primitive form, Publ. RIMS, Kyoto Univ., 19 (1983), 1231-1264. · Zbl 0539.58003
[29] M. SAITO, T. KAWAI and M. KASHIWARA, Microfunctions and pseudo-differential equations, Lect. Notes in Math., Springer, 287, (1973), 264-529. · Zbl 0277.46039
[30] 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]). · Zbl 0618.14002
[31] J.P. SERRE, Algèbre locale, multiplicités, Lect. Notes in Math., Springer 11, (1975). · Zbl 0296.13018
[32] J. STEENBRINK, Mixed Hodge structure on the vanishing cohomology, in Real and Complex Singularities, Sijthoff & Noordhoff, Alphen aan den Rijn, (1977), 525-563. · Zbl 0373.14007
[33] A. VARCHENKO, The asymptotics of holomorphic forms determine a mixed Hodge structure, Soviet Math. Dokl., 22 (1980), 772-775. · Zbl 0516.14007
[34] 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 0497.32007
[35] A. VARCHENKO, The complex exponent of a singularity does not change along strata µ = const, Func. Anal. Appl., 16 (1982), 1-9. · Zbl 0498.32010
[36] 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. · Zbl 0517.32004
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