Estimate of dimension of Noether-Lefschetz locus for Beilinson-Hodge cycles on open complete intersections

In his lectures in [G1], M. Green gives a lucid explanation how fruitful the infinitesimal method in Hodge theory is in various aspects of algebraic geometry. A significant idea is to use Koszul cohomology for Hodge-theoretic computations. The idea originates from Griffiths work [Gri] where the Poincaré residue representation of the cohomology of a hypersurface played a crucial role in proving the infinitesimal Torelli theorem for hypersurfaces. Since then many important applications of the idea have been made in different geometric problems such as the generic Torelli problem and the Noether-Lefschetz theorem for Hodge cycles and the study of algebraic cycles (see [G1, Lectures 7 and 8]). In this paper we apply the method to study an analog of the Noether-Lefschetz theorem in the context of Beilinson’s Hodge conjecture. Beilinson’s Hodge conjecture and its Tate variant concern the regulator maps for open varieties (cf. [J1, Conjecture 8.5 and 8.6]). To be more precise we let U be a smooth variety over a field k of characteristic zero.