I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for a "simple" example. Could somebody direct me to a proof for the special case of a Kaehler manifold $(M,g)$ and its Dirac operator $$ (\overline{\partial} + \overline{\partial}^*): \Omega^{0,\bullet} \to \Omega^{0,\bullet}. $$

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 $$ (\overline{\partial} + \overline{\partial}^*): \Omega^{0,\bullet} \to \Omega^{0,\bullet}. $$