Algebraicity of Künneth projectors (Grothendieck’s standard conjecture)

Determine whether, for every smooth projective variety X over a field F, the Chow motive M(X) decomposes as a direct sum ⊕_n M^n(X) that lifts the individual cohomology groups H^n_?(X); equivalently, prove that the Künneth projectors H^•_?(X) → H^n_?(X) → H^•_?(X) are induced by algebraic correspondences in CH^{dim X}(X × X)_Q.

Background

Grothendieck’s category of Chow motives CHM(F) encodes correspondences as morphisms of motives. A fundamental open issue is whether motives admit a canonical splitting mirroring the cohomological Künneth decomposition.

This is one of Grothendieck’s standard conjectures, asserting that Künneth projectors are algebraic (i.e., represented by cycles). Its resolution would provide functorial pieces M^n(X) lifting cohomology and would clarify the structure of pure motives.