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Is there a discrete group G which is the fundamental group of a compact Kahler manifold but which is not the fundamental group of any smooth projective complex algebraic variety?

It is known that there are cohomology rings of compact Kahler manifolds not realisable by smooth projective complex algebraic varieties (some of the recent results are due to Voisin).

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This is a well-known open problem. –  Andy Putman Apr 14 '12 at 19:41
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The best positive result (as far as I know) is an old theorem of Campana stating that Malcev completion of every Kahler group $\pi_1(X)$ is isomorphic to the Malcev completion of the fundamental group of a smooth projective variety $Y$, which is a desingularization of the image of $X$ under the Albanese map. –  Misha Apr 14 '12 at 22:41
    
Thanks to both of you! I did not know any of this. @Andy: could you please let me know if this is mentioned in print anywhere? Thanks! –  SGP Apr 22 '12 at 16:34
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As far as I know, still open. In the 2011 paper HOMOMORPHISMS BETWEEN FUNDAMENTAL GROUPS OF KA ̈HLER MANIFOLDS, one Botong Wang shows that there is a homomorphism between Kahler groups not realized by a holomorphic map between projective varieties, but this seems to be closest to a counterexample.

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Thanks! That is very inveresting! –  SGP Apr 22 '12 at 16:35
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