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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.

Bernard
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    This was crossposted to MO. In the future, please wait some time before posting your question in multiple fora, and when you do, provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to see hear from you that you'd already gotten an answer elsewhere. – Zev Chonoles Mar 10 '13 at 06:39
  • Ok, sorry I'll keep that in mind. – Bernard Mar 10 '13 at 06:47

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