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 $f_h = ϕ$.

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I think that one importnt problem with Chow groups of compact Kähler manifolds is that a general compact Kähler manifold might have very few cycles. An example of a compact complex Kähler manifold is a non-algebraic torus $T$. If $T$ has no projective subtori, then $T$ contains no other closed analytic subvarieties than (non projective) subtori. So it is unlikely that a moving lemma can be proven with so few cycles. – Damian Rössler Nov 3 '12 at 20:44
About your question on morphisms of analytic spaces, I think you may apply the original GAGA statement to the graph of of the morphism $f_h$ to get what you want. – Damian Rössler Nov 3 '12 at 20:45
Dear Damian, thank you for this example! Yet closed analytic subvarieties of tori are analogues of smooth algebraic cycles. Below I propose an analytic analogue for arbitrary algebraic cycles: I consider images of compact Kahler manifolds with respect to morphisms of analytic spaces. Do you know a way to describe these more general 'cycles' for analytic tori? – Mikhail Bondarko Nov 4 '12 at 5:48
I think that the image of a compact Kähler manifold by a birational morphism has a natural structure of compact analytic subvariety. If $\phi:Z\to X$ is the morphism, then the ideal sheaf of the image is the kernel of the adjunction map ${\cal O}_X\to\phi_*{\cal O}_Z$. – Damian Rössler Nov 5 '12 at 8:58
Sorry, what do you mean by a subvariety? The image doesn't have to be smooth. – Mikhail Bondarko Nov 5 '12 at 9:11

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.

1. 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.

2. 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...

3. 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.

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