Noether-Lefschetz locus for Beilinson-Hodge cycles. I
type: Item
instance of: scholarly article
title: Noether-Lefschetz locus for Beilinson-Hodge cycles. I
zbMATH Open document ID: 1088.14003
author: Masanori Asakura
author: Shuji Saito
published in: Mathematische Zeitschrift
publication date: 2006-01-23T00:00:00Z
review text: The classical Noether-Lefschetz theorem states that if \(X\) is a generic smooth surface of degree \(d\geq 4\) in \({\mathbb P}^3\) then every curve \(C\) on \(X\) is a complete intersection; or, equivalently, that every component of the Noether-Lefschetz locus has positive codimension. \textit{M. Green} [J. Differ. Geom. 27, 155--159 (1988; Zbl 0674.14005)] using infinitesimal variation of Hodge structure was able to strengthen this result by showing that for \(d\geq 3\) the codimension of any component of the Noether-Lefschetz locus must be \(\geq d-3\). In the paper under review, the authors consider schemes over a non-singular quasi-projective variety \(S\) of the form \[ {\mathbb P}_S^n\hookleftarrow {\mathcal X}\hookleftarrow {\mathcal Z}={\mathcal X}\cap (\bigcup_{1\leq j\leq s}{\mathcal Y}_j) \] where the fibers of \({\mathcal X}\) are smooth and those of \({\mathcal Z}\) are simple normal crossing divisors. They define the Noether-Lefschetz locus in degree \(q\) on the complement \({\mathcal U}={\mathcal X}\setminus {\mathcal Z}\) to be \[ S_{\text{NL}}^q=\{t\in S\: | \: H^q(U_t,{\mathbb Q}(q))\cap F^qH^q(U_t,{\mathbb C})\neq H^q(U_t,{\mathbb Q}(q))_{\text{triv}}\} \] where \(H^q(U_t,{\mathbb Q}(q))_{\text{triv}}=\text{Image}(H^q(V_t,{\mathbb Q}(q))\rightarrow H^q(U_t,{\mathbb Q}(q)))\) and \(V_t\) is the fiber of \({\mathcal V}={\mathbb P}_S^n\setminus (\cup_{1\leq j\leq s}{\mathcal Y}_j)\) over \(t\in S\). The main result of the paper concerns the case when the fibers of \({\mathcal X}\) are complete intersections, and then Lefschetz theory implies that \(S_{\text{NL}}^q=\phi\) unless \(q=m:=\text{dim}(U_t)\). Following Green, the authors use variation of mixed Hodge structure to get a lower bound on the codimension of a component of \(S_{\text{NL}}^m\) in terms of the multi-degree of the fibers of \(\mathcal X\), the degrees of the fibers of \({\mathcal Y}_j\) and an invariant \(c_S({\mathcal X},{\mathcal Z})\) which they introduce. They show that the bo…
reviewed by: Oliver Jones
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14C15
zbMATH DE Number: 5000055
zbMATH Keywords: Noether-Lefschetz locus
zbMATH Keywords: regulator map
MaRDI profile type: Publication
full work available at URL: s00209-005-0813-x
OpenAlex ID: W1988132138
cites work: Generalized Jacobian rings for open complete intersections
cites work: Maximal components of Noether-Lefschetz locus for Beilinson-Hodge cycles
cites work: The coniveau filtration and non-divisibility for algebraic cycles
cites work: Algebraic cycles and higher K-theory
cites work: On the Locus of Hodge Classes
cites work: Q3312351
cites work: Théorie de Hodge. II. (Hodge theory. II)
cites work: Q3731740
cites work: A new proof of the symmetrizer lemma and a stronger weak Torelli theorem for projective hypersurfaces
cites work: Q4320861
cites work: Q3715242
cites work: A new proof of the explicit Noether-Lefschetz theorem
cites work: Components of maximal dimension in the Noether-Lefschetz locus
cites work: On the periods of certain rational integrals. I, II
cites work: Q3809905
cites work: Continuous étale cohomology
cites work: Q3971596
cites work: Sur les séries $L$ d'une variété algébrique
cites work: Rational equivalence of O-cycles on surfaces
cites work: The Abel-Jacobi map for complete intersections
cites work: Algebraic cycles and Hodge theoretic connectivity
cites work: Mixed Hodge modules
cites work: Complete intersections with middle Picard number 1 defined over \({\mathbb{Q}}\)
cites work: Une précision concernant le théorème de Noether. (A precision concerning the theorem of Noether)
DOI: 10.1007/S00209-005-0813-X
description: scientific article; zbMATH DE number 5000055