Hodge filtrations on Gauss-Manin systems. II
type: Item
instance of: scholarly article
title: Hodge filtrations on Gauss-Manin systems. II
zbMATH Open document ID: 0549.32017
author: Morihiko Saito
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: [For part I see the preceding review.] - Let \(f:X\to Y\) be a projective morphism of complex manifolds with dim Y\(=1\). The author asserts that the theory of limit mixed Hodge structures can be well combined with that of filtered D-modules, so that the hard Lefschetz and decomposition theorems for \({\mathbb{R}}f_*{\mathbb{C}}\) can be proved analytically in this case [they are proved by an \(\ell\)-adic method in the algebraic case, see \textit{A. A. Beilinson, J. Bernstein} and \textit{P. Deligne}, ''Faisceaux pervers'', Astérisque 100 (1982; Zbl 0536.14011]. But recently the author found a gap in Steenbrink's argument used in the proof; the decomposition was proved analytically only for the algebraic (or reduced normal crossing) case and the hard Lefschetz can not be reduced to the case of resolution. To prove the hard Lefschetz analytically for \({\mathbb{R}}f_*{\mathbb{C}}\) (or \(H^.(V,{\mathbb{C}})\) with V a projective V-manifold) we have to construct a new category (generalizable polarizable variations of Hodge structures) and prove the whole theory in the general case [cf. ''Hodge structures via filtered D-modules'' (to appear in Astérisque)].
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14D99
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 32K15
zbMATH DE Number: 3875683
zbMATH Keywords: hard Lefschetz theorems
zbMATH Keywords: limit mixed Hodge structures
zbMATH Keywords: filtered D-modules
MaRDI profile type: Publication
full work available at URL: pjaa.59.37
OpenAlex ID: W2019110407
cites work: Q3731741
cites work: Théoreme de Lefschetz et critères de dégénérescence de suites spectrales
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cites work: Singularités des systèmes différentiels de Gauss-Manin. Avec des contributions de Lo Kam Chan, Philippe Maisonobe et Jean-Etienne Rombaldi
cites work: Q3342878
cites work: Limits of Hodge structures
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DOI: 10.3792/PJAA.59.37
description: scientific article; zbMATH DE number 3875683