Can there exist Chow motives/motivic cohomology for compact Kähler manifolds? 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 $f_h = ϕ$.
 A: Hi Mikhail,
I honestly don't have a good answer, but I'll share my thoughts on this and related things, since this is potentially quite interesting.


*

*I don't see a problem in formally defining the Chow group of a compact Kähler manifold
as the group of cycles modulo rational equivalence.
However, if you want to compose correspondences, then you would need a ring structure,
and this might be difficult. Cycles on a Kähler manifold are probably more rigid than on an algebraic variety, so proving a moving lemma could be problematic.

*In the algebraic case, $CH^*(X)\otimes \mathbb{Q}$ is the same as the associated graded
of the Grothendieck group of algebraic vector bundles $K^0(X)\otimes\mathbb{Q}$. I doubt
that this isomorphism would work in the analytic case because of the failure to have
global resolutions, but you could simply use $K^0(X)$ directly. This would give a ring, and therefore a category of $K^0$-correspondences...

*You could also consider cycles modulo homological equivalence, and I don't see a problem in getting everything to work. So that you should be able construct a category of pure
homological motives for Kähler manifolds. However, unlike that algebraic case, I don't
think that one should expect this to be semisimple and abelian. Part of my pessimism stems from the fact the Hodge conjecture is known to fail in this setting (Voisin).
I don't really want to say much more here, but I think you have my email in case you want to discuss this further.
