Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern classes of $X/G$ in terms of $X$ and $G$.
The top Chern class is simply the topological Euler number and thus we have $e(X/G)=e(X)/|G|$. What about others? I am particularly interested in the case $\dim X=3,4$.