On algebraic 1-motives related to Hodge cycles

Abstract

The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.We also construct generalized cycle class maps from the K-cohomology groups H p+i (K p ) to the sub-quotiens W 2p H 2p+i /W 2p−2 given by the weight filtration. However, in general, the image of this cycle map (as well as the image of the canonical map from motivic cohomology) is strictly smaller than the rational part of the Hodge filtration F p on H 2p+i .