On the Hodge spectral sequence for some classes of non-complete algebraic manifolds
type: Item
instance of: scholarly article
title: On the Hodge spectral sequence for some classes of non-complete algebraic manifolds
zbMATH Open document ID: 0707.14005
DOI: 10.1007/BF01443352
author: Siegmund Kosarew
author: Ingrid Bauer
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164573
review text: The following two theorems concerning the degeneration of the Hodge spectral sequence for non-complete algebraic manifolds are proved via Deligne's and Illusie's purely algebraic proof of the \(E_ 1\)- degeneration [\textit{P. Deligne} and \textit{L. Illusie}, Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)]: Theorem A. Let K be a field of characteristic zero, X a scheme, integral and proper over Spec(K) and \(Y\subset X\) a closed subscheme such that \(U:=X\setminus Y\) is smooth. Then (1) the K-vector spaces \(H^ n_{DR}(U/K)\) and \(H^ j(U,\Omega^ i_{U/K})\) are finite dimensional for n, j\(<co\dim (Y,X)-1\) and any i, (2) the differentials \(d_ n^{i,j}\) of the Hodge spectral sequence \(E_ 1^{i,j}=H^ j(U,\Omega^ i_{U/K})\Rightarrow H_{DR}^{i+j}(U/K)\) are zero for \(i+j<co\dim (Y,X)-1\) and \(r\geq 1.\) Theorem B. Let K be a field of characteristic zero, X an integral scheme, smooth and proper over Spec(K), \(Y\subset X\) a divisor and \(U:=X\setminus Y\). If the normal bundle \(N_{Y/X}\) of Y in X is k-ample, then (1) the K-vector spaces \(H^ n_{DR}(U/K)\) and \(H^ j(U,\Omega^ i_{U/K})\) are finite dimensional for \(n\geq \dim (X)+k+1\) and \(j\geq k+1\), respectively, (2) the differentials \(d_ r^{i,j}\) of the Hodge spectral sequence \(E_ 1^{i,j}=H^ j(U,\Omega^ i_{U/K})\Rightarrow H_{DR}^{i+j}(U/K)\) are zero for \(i+j\geq \dim (X)+k+1\) and \(r\geq 1.\)
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14D07
zbMATH DE Number: 4160908
zbMATH Keywords: de Rham complex
zbMATH Keywords: degeneration of the Hodge spectral sequence for non- complete algebraic manifolds
reviewed by: M. Miyanishi
MaRDI profile type: Publication
cites work: Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham. (Lifting modulo \(p^ 2\) and decomposition of the de Rham complex)
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili
cites work: Q4143433
cites work: Q3491899
cites work: Hodge spectral sequence and symmetry on compact Kähler spaces
cites work: A reduction theorem for cohomology groups of very strongly q-convex Kähler manifolds
cites work: A comparison theorem
cites work: Q3701753
cites work: Q5598244
cites work: Q5596003
cites work: Submanifolds of Abelian varieties
OpenAlex ID: W2082942424
Recommended article: Decomposition of the de Rham complex
Recommended article: On infinitesimal liftings and degenerations of Hodge-de Rham spectral sequences
Recommended article: Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie
Recommended article: Cartier isomorphism and Hodge Theory in the non-commutative case
Recommended article: Q4023663
Recommended article: Some remarks on Beilinson adeles
Recommended article: Spectral algebras and non-commutative Hodge-to-de Rham degeneration
Recommended article: Hodge cohomology on blow‐ups along subvarieties
Recommended article: Hodge theorem for the logarithmic de Rham complex via derived intersections
Recommended article: Limits of Hodge structures in several variables, II
description: scientific article; zbMATH DE number 4160908