From what I learned in early math class, the Green's function of $\partial_{\bar z}$ on the plane $\mathbb{C}$ is given by $1/z$, up to some constants.
I wonder what is the Green's function $\partial_{\bar z}G(z, \bar z) = \delta_\Sigma(z, \bar z)$ if we look at a torus with complex structure $\tau$, or a more general Riemann surface $\Sigma$? Let me also normalize $\delta_{\Sigma}(z, \bar z)$ using $\int_\Sigma \delta_{\Sigma}(z, \bar z) \sqrt{g}dz \wedge d\bar z = 1$.
On a torus, the answer looks like some $1/\vartheta(z;\tau)$, but I'm not able to fix the periodicity.
Reference or comments are welcome!