Saito, Morihiko On the structure of Brieskorn lattice. (English) Zbl 0644.32005 Ann. Inst. Fourier 39, No. 1, 27-72 (1989). 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 Cited in 7 ReviewsCited in 80 Documents MSC: 32Sxx Complex singularities 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) Keywords:mixed Hodge structures; Gauss-Manin system; Brieskorn lattice; b- function; microlocalization; vanishing cycles PDFBibTeX XMLCite \textit{M. Saito}, Ann. Inst. Fourier 39, No. 1, 27--72 (1989; Zbl 0644.32005) Full Text: DOI Numdam EuDML References: [1] E. BRIESKORN, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math., 2 (1970), 103-161.0186.2610142 #2509 · Zbl 0186.26101 [2] [BGM] , and , Sur le polynôme de Bernstein des singularités semi quasihomogènes, Prépublication de l’Université de Nice, n° 138, novembre 1986. 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