Hodge filtrations on Gauss-Manin systems. II
[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)].
scientific article; zbMATH DE number 3875683