Suppose that $T$ is a smooth complex projective algebraic surface, such that the finite cyclic group $\mathbb{Z_n}$ acts on it. Also, suppose that $S$ is another complex projective algebraic surface (not necessarily smooth) such that $S$ is birational to $T/\mathbb{Z}_{n}$. What can one say about the relation between the (sheaf theoretic) Euler characteristics of $T$ and $S$? in particular, can we claim that the Euler characteristic of $T$, $e(T)$ is greater than or equal to $ne(S)$?