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May 16, 2017 at 21:54 comment added David E Speyer Also note, there are compact non-Kahler complex manifolds with $\pi_1 = \mathbb{Z}$, namely, the Hopf surface en.wikipedia.org/wiki/Hopf_surface
May 16, 2017 at 21:07 vote accept Nick L
May 16, 2017 at 20:53 history answered Michael Albanese CC BY-SA 3.0