Higher Chow groups and the Hodge-\({\mathcal D}\)-conjecture
type: Item
instance of: scholarly article
title: Higher Chow groups and the Hodge-\({\mathcal D}\)-conjecture
zbMATH Open document ID: 0899.14002
DOI: 10.1215/S0012-7094-96-08508-7
author: James D. Lewis
published in: Duke Mathematical Journal
publication date: 1998-11-11T00:00:00Z
review text: Let \(X\) be a complex quasi-projective variety, and let \(\text{CH}(X,m)\) denote Bloch's higher Chow groups of \(X\). In this paper, the author investigates a possible relationship between mixed Hodge structure of \(X\) and Chow groups. As a basic invariant of the latter, he introduces the notion of ``level'' analogous to that for the former. Then, under some assumptions including ``the hard Lefschetz conjecture'', he proves that \[ \text{Level} (N^{k-\ell} H^{2k- \ell-m} (X,\mathbb{Q}))= \ell-m\quad \Rightarrow\quad\text{Level(CH}^k (X,m)) \geq\ell-m. \] He points out that the inequality \[ \text{Level} (H^*(X)) \leq\text{Level(CH}^* (X,m)_\mathbb{Q}) \] holds under the same hypothesis plus the general Hodge conjecture, and conjectures \[ \text{Level} (H^*(X)) =\text{Level(CH}^* (X,m)_\mathbb{Q}) \] holds for all \(m\geq 0\). Moreover, he shows that this conjecture holds for any general hypersurfaces of degree \(d\) in \(\mathbb{P}^{n+1}\) satisfying the numerical condition \(k(n+2-k) +1- {k+d \choose d} \geq 0\) where \(k= [(n+1)/d]\).
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 949890
zbMATH Keywords: level
zbMATH Keywords: higher Chow groups
zbMATH Keywords: mixed Hodge structure
zbMATH Keywords: hypersurfaces
Wikidata QID: Q123233326
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 949890