Murre's conjecture for certain product varieties (Q932939)

Let \(X\) be a projective algebraic manifold of dimension \(d\) and \(\Delta\in \text{ CH}^d(X\times X)\otimes{\mathbb{Q}}\) the diagonal class. Jacob Murre's conjectures concern a conjectured lifting of the Künneth decomposition of the diagonal class \([\Delta]\) (on the level of cohomology) to the Chow group. Briefly, his conjectures are the following: \[ \sum_{i=0}^{2d} \pi_i = \Delta \in \text{ CH}^d(X\times X)\otimes{\mathbb{Q}}, \tag{A} \] where \(\{\pi_i\}\) are orthogonal projectors in \(\text{ CH}^d(X\times X)\otimes{\mathbb{Q}}\). (B) \(\{\pi_0,\dots,\pi_{j-1},\pi_{2j+1},\dots,\pi_{2d}\}\) act as zero on \(\text{ CH}^j(X)\otimes{\mathbb{Q}}\). (C) The filtration \[ \big\{F^{\nu}\text{ CH}^j(X)\otimes{\mathbb{Q}} := \ker \pi_{2j}\cap \ker \pi_{2j-1}\cap\cdots \cap \ker \pi_{2j-\nu+1}\big\}, \] is independent of the choice of \(\pi_i\). (D) \(F^1\text{ CH}^j(X)\otimes{\mathbb{Q}} = \text{ CH}^j_{\hom}(X)\otimes{\mathbb{Q}}\). The author provides partial results towards Conjecture (D) in the case that \(X\) is the fourfold made up of a product of two curves with a surface. In particular he describes a certain class of codimension two cycles on \(X\) killed by a suitable projector, whose kernel should (i.e. conjecturally) contain \(\text{ CH}^2_{\hom}(X)\otimes{\mathbb{Q}}\). He also proves Conjecture (B) for a product of two surfaces.