Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures
type: Item
instance of: scholarly article
title: Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures
zbMATH Open document ID: 0814.14010
DOI: 10.1007/BF01446621
author: Jozef Steenbrink
published in: Mathematische Annalen
publication date: 1995-06-20T00:00:00Z
full work available at URL: https://eudml.org/doc/165283
review text: Recently, the theory of logarithmic structures on algebraic varieties has been developed by \textit{I. Illusie} [``Logarithmic spaces (according to K. Kato)'' in Barsotti Sympos. Algebraic Geometry, Abano Terme 1991, Perspect. Math. 15, 183-203 (1994)] and \textit{K. Kato} [in Algebraic Analysis, Geometry and Number Theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 191-224 (1989; Zbl 0776.14004)]. Each embedding of a variety \(D\) as a divisor in a smooth ambient space \(X\) gives rise to logarithmic structures on \(D\) and \(X\). In this paper, the author considers logarithmic embeddings of varieties over \(\mathbb{C}\) with normal crossings; these are logarithmic structures which are locally isomorphic to the standard one obtained from an embedding of the variety as a divisor in a smooth space. To each logarithmic embedding of \(D\) is associated a filtered logarithmic de Rham complex and a counterpart of this with integral coefficients, which is a constructible complex. If \(D\) is compact, the hypercohomology groups of these form mixed Hodge structures, which in the case of genuine embeddings reduces to the mixed Hodge structure of a deleted neighborhood of \(D\) [cf. \textit{A. Durfee}, Duke Math. J. 50, 1017-1040 (1983; Zbl 0545.14005) and \textit{F. Elzein}, Trans. Am. Math. Soc. 275, 71-106 (1983; Zbl 0511.14003)]. A logarithmic embedding provided with a suitable global section of its logarithmic sheaf is called a logarithmic deformation. Such a structure exists exactly for \(d\)-semistable varieties in the sense of \textit{R. Friedman} [Ann. Math., II. Ser. 118, 75-114 (1983; Zbl 0569.14002)]. The previous result is used for the construction of a ``limit mixed Hodge structure'' associated to a logarithmic deformation, which reduces to the construction by the reviewer [Invent. Math. 31, 229-257 (1976; Zbl 0305.14002)] in the case of a one-parameter smoothing with smooth total space.
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14E25
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 734581
zbMATH Keywords: logarithmic embeddings
zbMATH Keywords: logarithmic de Rham complex
zbMATH Keywords: hypercohomology groups
zbMATH Keywords: mixed Hodge structures
zbMATH Keywords: logarithmic deformation
MaRDI profile type: Publication
cites work: Théorie de Hodge. II. (Hodge theory. II)
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cites work: Mixed Hodge Structures
cites work: Global smoothings of varieties with normal crossings
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cites work: Logarithmic structures of Fontaine-Illusie. II
cites work: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties
cites work: Modules de Hodge polarisables. (Polarisable Hodge modules)
OpenAlex ID: W2078935957
description: scientific article; zbMATH DE number 734581