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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?
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