Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?



A complex variety with a finite non-abelian simple fundamental group
Yes. In fact, Serre proved that any finite group is the fundamental group of a smooth projective complex variety. See Proposition 15 of:


J.-P. Serre, Sur la topologie des variétés algébriques en charactéristique $p$, Symposium Internacional de Topologia Algebraica, Universidad Nacional Autonoma de Mexico, 1958, pp. 24–53 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=98097)).


