# Atiyah-Singer for Riemannian and Kaehler manifolds

I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special case of

1. A Riemannian manifold and its associated Dirac operator $$d+d^*: \Omega^\bullet \to \Omega^\bullet,$$
2. a Kaehler manifold $$(M,g)$$ and its Dirac operator $$(\overline{\partial} + \overline{\partial}^*): \Omega^{0,\bullet} \to \Omega^{0,\bullet}.$$
• I revised my answer to perhaps address the question in your post more directly. – Danny Ruberman Oct 29 at 17:13