Let $X,Y$ be smooth projective varieties over $\mathbb C$ where $X$ is the universal cover of $Y$. Assume that the fundamental group of $Y$ is finite and has order $d$. Then we want to show that $\chi (X)=d \chi(Y)$ where $\chi(\cdot)$ means Euler characteristic with respect to the sheaf of regular functions.

Any ideas here? I guess it might be an easy application, but I don't see the connection between the chern class and the Todd class of $X$ and $Y$ in order to apply the Riemann Roch.