Mixed Hodge structures on the homotopy of links
type: Item
instance of: scholarly article
title: Mixed Hodge structures on the homotopy of links
zbMATH Open document ID: 0617.14012
DOI: 10.1007/BF01474182
author: Alan H. Durfee
author: Richard Hain
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164351
review text: This paper uses mixed Hodge theory to study the homotopy of a complex algebraic variety in the neighborhood of a singular point, or more generally, in the neighborhood of a subvariety. An important application is finding topological restrictions on the links of isolated singular points. One such application is the following theorem: Let L be the link of an isolated singularity of an n-dimensional variety. If s, t\(<n\) and \(s+t\geq n\), then the cup product \(H^ s(L;{\mathbb{Q}})\otimes H^ t(L;{\mathbb{Q}})\to H^{s+t}(L;{\mathbb{Q}})\quad vanishes\). (Recall that the link of an isolated singularity in an n-dimensional variety is a real (2n-1)-manifold.) This theorem is a direct consequence of the following three facts: (1) The cohomology \(H^ k(L)\) (which is isomorphic to \(H_ x^{k+1}(X))\) has a natural mixed Hodge structure. (2) The weights of \(H^ k(L)\) are less than or equal to k for \(k<n\), and greater than or equal to \(k+1\) for \(k\geq n.\) (3) The cup product in the cohomology of L is a morphism of mixed Hodge structures and therefore preserves weights. The first assertion is due to Deligne. The second is a well-known consequence of Gabber's purity theorem and the decomposition theorem of intersection homology, both deep theorems. This paper establishes fact 3. It is more convenient to work not just with isolated singularities, but in the following more general situation: Suppose that X is a projective variety and that Z and Z' are closed subvarieties. Set \(Y=Z\cup Z'\) and suppose that the singular locus of X is contained in Y. Let T be a sufficiently nice neighborhood of Z in X. Define the link L(X,Y,Z) of Z in X with Y removed by \(L(X,Y,Z)=T-Y\). This is well defined up to homotopy. A de Rham mixed Hodge complex is constructed for L(X,Y,Z). The main result of this paper then follows from work of \textit{R. M. Hain} [''The de Rham homotopy theory of complex algebraic varieties. I'', J. K-Theory (to appear)] or \textit{J. W. Morgan} [Publ. Math., Ins…
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 57R19
zbMATH DE Number: 4000099
zbMATH Keywords: mixed Hodge theory
zbMATH Keywords: homotopy of a complex algebraic variety
zbMATH Keywords: neighborhood of a subvariety
zbMATH Keywords: links of isolated singular points
zbMATH Keywords: cup product
zbMATH Keywords: decomposition theorem of intersection homology
MaRDI profile type: Publication
cites work: Théorie de Hodge. II. (Hodge theory. II)
cites work: Neighborhoods of Algebraic Sets
cites work: Mixed Hodges structures on punctured neighborhoods
cites work: Q3663424
cites work: Duality of mixed Hodge structures of algebraic varieties
cites work: Iterated integrals and homotopy periods
cites work: Mixed Hodge structures on homotopy groups
cites work: The de Rham homotopy theory of complex algebraic varieties. I
cites work: Unipotent variations of mixed Hodge structure
cites work: Lectures on minimal models
cites work: The algebraic topology of smooth algebraic varieties
cites work: Sur la théorie de Hodge-Deligne. (On the Hodge-Deligne theory)
cites work: Q5522742
cites work: Q3661709
cites work: Variation of mixed Hodge structure. I
cites work: On the intersection ring of compact three manifolds
cites work: Infinitesimal computations in topology
cites work: Variation of mixed Hodge structure. II
OpenAlex ID: W2077599582
description: scientific article; zbMATH DE number 4000099