In Atsushi Moriwaki's Arakelov Geometry, page 100, in a discussion about Greens functions, the proof of proposition 4.13 claims that $$dd^c(\log|\phi|)=0$$ for a non-zero rational function $\phi$ on a connected compact Riemann surface. The $dd^c$ operator is given by $$dd^c=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}$$ Why is it the case that $dd^c(\log|\phi|)=0$ ?
2026-03-25 22:25:12.1774477512
dd^c of log of a meromorphic function
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