Abstract Let $X/{\mathbb C}$ be a smooth projective variety. We consider two integral invariants, one of which is the level Hodge cohomology algebra $H^*(X,{\mathbb C})$ and other involving complexity higher Chow groups ${\mathrm {CH}}^*(X,m;{\mathbb Q})$ for $m\geq 0$ . conjecture that these invariants are same accordingly provide some strong evidence in support this.

In this paper, we investigate Murre’s conjectures on the structure of rational Chow groups for a rational homogeneous bundle Z → S over a smooth variety. Absolute Chow-Künneth projectors are exhibited for Z whenever S has a Chow–Künneth decomposition.

Let W be a vector space of dimension n+ 1 over a field K. The Chow divisor of a k-dimensional variety X in P = P(W ) is the hypersurface, in the Grassmannian Gk+1 of planes of codimension k+1 in P, whose points (over the algebraic closure of K) are the planes that meet X . The Chow form of X is the defining equation of the Chow divisor. For example, the resultant of k+ 1 forms of degree e in k ...

In this paper we investigate Murre’s conjecture on the Chow–Künneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space Mg, in genus at most 8 and show existence of a Chow–Künneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. T...