Motivation for the definition of orbifold euler characteristic

Question

Let $S$ be a normal projective surface with quotient singularities only. For each singular point $p\in S$, $(S,p)$ is locally analytically isomorphic to $(\Bbb C^2/G_p,0)$ for some finite subgroup $G_p\subset \text{GL}(2,\Bbb C)$, by the definition of quotient singularity. In this case, $S$ is a orbifold and the orbifold Euler characteristic of $S$ is defined by $e_{orb}(S)=e(S)-\sum_{p \in \text{Sing}(S)} (1-1/|G_p|)$ where $e(S)$ is the (ordinary) Euler charateristic of $S$ (cf. p.7 of https://arxiv.org/pdf/0801.3021.pdf). Is there a motivation for this definition? Why is $e_{orb}$ defined in this way? By the way, it seems this definition is different (not sure) with the one given in https://en.wikipedia.org/wiki/Euler_characteristic_of_an_orbifold.

Answer 1

Ordinary Euler characteristic is multiplicative with respect to taking finite covers of manifolds. The orbifold Euler characteristic (as defined) has the same property, for orbifold covers of orbifolds. Finally, the two concepts play nicely with each other - if $\rho : M \to \mathcal{O}$ is an orbifold cover, where $M$ is a manifold and $\mathcal{O}$ is an orbifold, then $\chi(M) = \mathrm{deg}(\rho) \cdot \chi_\mathrm{orb}(\mathcal{O})$.