Equivalence of two kinds of orbifold Euler characteristic (Q652221)

The paper describes two definitions of Euler characteristic for orbifolds and shows their equivalence. One was given by \textit{I. Satake} in his original paper [J. Math. Soc. Japan 9, 464--492 (1957; Zbl 0080.37403)]), in terms of indices of vector fields with isolated singularities; and the other one was given by \textit{K. Fukaya} and \textit{K. Ono} [Topology 38, No. 5, 933--1048 (1999; Zbl 0946.53047)], in terms of multiplicities of zeros of multi-sections. It is made explicit that vector fields can be perturbed locally at the singularities using multi-sections of the orbifold tangent bundle. Local computations identify the contributions from indices of vector fields to that of the multi-sections.