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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!

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  • $\begingroup$ Could you please define what you mean by Green's function? $\endgroup$ – Docteur Cottard Feb 4 '18 at 22:30
  • $\begingroup$ @DocteurCottard: I would define the equation as $\partial_{\bar z} G(z, \bar z) = \delta(z, \bar z)$, where $\delta(z, \bar z)$ is the delta function on the Riemann surface. I hope there is no ambiguity in such definition. $\endgroup$ – Lelouch Feb 5 '18 at 8:54

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