I'm trying to understand AtiyahSinger by looking at the usual starting point of $CP^1$ and the DiracDolbeault operator. If I've reduced everything down correctly, then in this case the theorem gives $$ \text{Index}(\overline{\partial} + \overline{\partial}^*) = \frac{1}{2}\int_{CP^1} \text{ch}_1(T(CP^1)), $$ where ch$_1(T(CP^1))$ is the first Chern class of the tangent bundle $CP^1$. I would like a direct explanation of why this is true.
