Noether-Lefschetz locus for Beilinson-Hodge cycles. I
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 bound is optimal in the case when the fibers of \({\mathcal X}\) are plane curves but point out that in general it will be far from such. As an application of their work, the authors prove some results concerning the regulator map \(\text{reg}_U^q:\text{CH}^q(U,q)\otimes {\mathbb Q}\rightarrow H^q(U,{\mathbb Q}(q))\cap F^qH^q(U,{\mathbb C})\) and its étale variant \(\text{reg}_{\text{ét},U}^q:\text{CH}^q(U,q)\otimes {\mathbb Q}_l\rightarrow H^q_{\text{ét}}(U_{\bar k},{\mathbb Q}_l(j))^{\text{Gal}(\bar k/k)}\), where \(\text{CH}^q(U,q)\) denotes Bloch's higher Chow group, which have been conjectured by Beilinson to be surjective.
scientific article; zbMATH DE number 5000055
scientific article