Since you are looking for an official source, I recommend the book Symplectic Fibrations and Multiplicity Diagrams by Guillemin, Lerman and Sternberg. It has a lot to say about symplectic reduction on homogeneous manifolds. However, I don't have it with me right now and cannot point you directly to relevant examples there.
For your example at hand, here is number of claims that you should check step by step.
Your $S^1$ action has a moment map that is a function of $|z_1|/|(z_1,z_2,z_3)|$. The regular level sets are diffeomorphic to $S^3$, and the quotient by the $S^1$ action is $\mathbb P^1$. In fact, you can fix $z_1\in\mathbb R_+\subset\mathbb C$, then your level set can be represented by $\{(z_1,z_2,z_3)\mid(z_2,z_3)\in S^3\}$, where $\theta\in S^1$ acts by multiplying $(z_2,z_3)$ with $\bar\theta$. For $z_1\to 0$, the level sets collapse to a copy of $\mathbb P^1$ on which $S^1$ acts trivially.
Your bundle $\mathcal O(1)$ is dual to the tautological bundle $L^*$, which I find easier to handle. The group $S^1$ acts on it, and $(L^*)^{S^1}$ is the quotient of $L^*|_{\mu^{-1}(r)}$ by this action. Hence it is the tautological bundle on $\mathbb P^1$. Dual to this, $L^{S^1}\cong\mathcal O(1)$.
Finally, compute the indices. $D$ is the Dolbeault operator. Holomorphic sections of $\mathcal O(1)\to\mathbb P^k$ correspond to linear functions on $\mathbb C^{k+1}$, and to the best of my knowledge, there is no higher cohomology, so $\mathrm{ind}(D_{\mathbb P^1}^{\mathcal O(1)})$ would be $2$. On the other hand, the index on $\mathbb P^2$ would be the vector space $\mathbb C^3$, where $S^1$ fixes a two-dimensional subspace, so again, $\mathrm{ind}(D_{\mathbb P^2}^{\mathcal O(1)})^{S^1}=2$.