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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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