Double zeros of the szego kernel

When we study complex analysis, we often encounter multi-valued functions. Provably the best known example would be $\sqrt{z}$, which has a singularity at the oritgin. In general, let $\Omega_1 and \Omega_2$ be domains in C, and let $\Gamma$ be a subvariety of $\Omega_1 \times \Omega_2$.