By a Calabi-Yau variety I mean a smooth projective variety over the complex numbers with numerically trivial canonical divisor.
For each postive integer $n$ does there exist a finite group $G$ (possibly depending on $n$) which is not the fundamental group of a Calabi-Yau variety of dimension $n$?
For $n=1,2$ this follows easily from the classification of Calabi-Yau varieties of these dimensions, the only non-trivial finite fundamental group being $\mathbb{Z}/2\mathbb{Z}$ (for Enriques surfaces). For $n=3$ some finite non-abelian groups are known to occur as fundamental groups but I do not know of any non-existence results.
If there are only finitely many families of Calabi-Yau varieties of a given dimension then the question would clearly have a positive answer. However, this is far from being known so I am interested in other possible approaches.
Best Answer
If n is even, then I believe you can use the Atiyah-Bott fixed point formula to rule out many cases. For instance, let G be a simple, non-cyclic group. Consider the action of G on the Hodge group $H^{0,n}(X)$. Since G has only the trivial character, this action must be trivial. Then for every element g in G, the holomorphic Lefschetz number is 2 (if n is odd, the number is 0, which doesn't help). Therefore g has a fixed point.