Let $X$ be a closed oriented orbifold with singularity $\Sigma X$. The singularity is defined as
$$ \Sigma X=\{(x)|~x\in X,~G_x\neq1\}, $$
where $G_x$ is the isotropy group.
For $u\in K_v(TX)$, as stated in Kawasaki's paper https://projecteuclid.org/euclid.nmj/1118786571 we have $$ Ind(u)=(-1)^{\dim(X)}<ch(u)\hat A^2(X),[TX]>+\sum^n_{i=1}\frac{(-1)^{\dim(\Sigma_i)}}{m_i}<ch^\Sigma(u)\hat A^{2,\Sigma}(\Sigma_iX),[T\Sigma_i X]>,$$ where $\{\Sigma_iX\}$ denotes all the components of the singularity set.
Q
- I do not follow how to choose the orientations of the singularity set $\Sigma X$.
- It seems that if we reverse the orientation of the singularity, then the formula changes?