Beilinson's Hodge conjecture for smooth varieties

TL;DR

This paper investigates Beilinson's Hodge conjecture for smooth varieties by analyzing the cycle class map and its image, revealing that at the generic point, the cokernel is consistent for both integral and rational coefficients.

Contribution

It establishes a connection between the cycle class map's cokernel and Abel-Jacobi maps, especially when r=m, using the Bloch-Kato theorem.

Findings

Cokernel of cycle class map is the same for integral and rational coefficients at the generic point.

The study links the image of the cycle class map to kernels of Abel-Jacobi maps.

Results support Beilinson's Hodge conjecture for specific cases.

Abstract

Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to \Gamma H^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r)) is singular cohomology. We study the image of cl_{r,m} in terms of kernels of Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the cokernel of cl_{r,m} at the generic point of U is the same for integral or rational coefficients.