Beilinson's Hodge Conjecture for K_1 revisited

TL;DR

This paper refines Jannsen's counterexample to Beilinson's Hodge conjecture for K_1 and demonstrates that the cycle class map becomes surjective at the generic point of a smooth quasiprojective complex variety.

Contribution

It provides a refined counterexample to the conjecture and shows surjectivity of the cycle class map at the generic point.

Findings

Counterexample to Beilinson's Hodge conjecture for K_1 refined.

Cycle class map is surjective at the generic point.

Counterexample clarifies limitations of the conjecture.

Abstract

Let U be a smooth quasiprojective complex variety and CH^r(U,1) a special instance of Bloch's higher Chow groups. Jannsen was the first to show that the cycle class map cl_{r,1} from CH^r(U,1) (tensored with Q) to hom_{MHS}(Q(0), H^{2r-1}(U,Q(r)) is not in general surjective, contradicting an earlier conjecture of Beilinson. In this paper, we give a refinement of Jannsen's counterexample, and further show that the aforementioned cycle class map becomes surjective at the generic point.