Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any obstructions for the existence of such a ('nice') extension? In particular, could there exist some 'Chow motives for compact Kähler manifolds'?
If one tries to mimick the usual 'algebraic' definitions, then one should define an analogue of algebraic cycles for compact Kähler manifolds. Is it reasonable to consider subsets that are images of compact Kähler manifolds with respect to birational morphisms?
Upd. Which (GAGA?) statements could help here? I would be deeply grateful for any references! In particular, does there exist a good exposition of GAGA that includes the following statement: let X and Y be projective complex varieties and let $ϕ: X_h \to Y_h$ be a morphism of analytic spaces, then there is a unique morphism $f : X \to Y$ such that $f_h = ϕ$.