Suppose N is a complex manifold with a holomorphic C* action $\sigma (re^{ia})$. Assume $\sigma (r)$ is globally free, $\sigma (e^{ia})$ is locally free and the orbit space M=N/$\sigma$ is compact. Then we have a local index theorem for N through a heat kernel approach. As an application, the singular strata in Kawasaki's Riemann-Roch theorem for the compact complex orbifold M have no contribution.

This is joint work with Chin-Yu Hsiao and I-Hsun Tsai.