I'm working with Kahler manifolds at the moment and looking at their spin$^c$ structure. I really don't know much about spin$^c$ structures in general and don't have enough time to learn it all at present, so I was hoping someone might be able to show me a shortcut in this case.

As far as I understand, for the usual spin$^c$ structure on an $N$-dimensional Kahler manifold $M$, the spinor bundle $S$ is given by the the direct sum of all the anti-holomorphic, that is $S = \bigoplus_{i=1}^N \Omega^{(0,i)}(M)$. Let $$ \nabla^s:S \to S \otimes \Omega^{(1,1)}, ~~~~~~~~~ s \mapsto \sum s_i \otimes \omega_i $$ be the spin$^c$ connection.

Is there a simple direct algebraic description of the Clifford action $$ c:S \otimes \Omega^{(1,1)} \to S $$ in this case? For example, an initial stupid guess might be, for $\omega_i = \omega_i^h + \omega_i^{ah}$, with $\omega_i^{h} \in \Omega^{(1,0)}$ and $\omega_i^{ah} \in \Omega^{(0,1)}$, we would have $$ c(\sum_i s_i \otimes \omega_i) = \sum_i s_i\omega^{ah}_i. $$ I am quite sure this is complete rubbish, but it illustrates the kind of result-from-heaven I'm hoping exists