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Is there a known example of a Calabi-Yau manifold (say, a Kähler compact manifold with $c_1$ torsion) with finite simple (non cyclic) fundamental group, for instance $\mathfrak{A}_5$? I am pretty sure such an example is not known in dimension 3 (see this paper1), but perhaps it is easier to find it in higher dimension (?).

1Davies, Rhys, The expanding zoo of Calabi-Yau threefolds, Adv. High Energy Phys. 2011, Article ID 901898, 18 p. (2011). ZBL1234.81110, MR2821564.

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  • $\begingroup$ Please see my answer to the following MO question (which rules out all Calabi-Yau manifolds of even dimension): mathoverflow.net/questions/65208/… $\endgroup$ Commented Sep 7, 2019 at 19:03
  • $\begingroup$ @Jason Starr: Well, I didn't rule out holomorphic symplectic manifolds. But I agree that finding a IHS manifold with a free action of $\mathfrak{A}_5$, hence of dimension $2(60k-1)$, is extremely unlikely. I had in mind the odd-dimensional case. $\endgroup$
    – abx
    Commented Sep 7, 2019 at 19:33

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