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I'm looking at the definition of Green's function $g_\mu$ for the Laplacian $\Delta_\mu$ associated to a positive $(1,1)$-form $\mu$ on a Riemann Surface $X$.

In specific the main request that the function $g_\mu:X \times X \rightarrow \mathbb{R}\cup \{\infty\}$ must satisfy is the following equation of $(1,1)$-forms:

$$dd^c g_\mu(w,z) + \delta_w(z)=\frac{\mu(z)}{Vol_\mu(X)}.$$

I'm confused about the meaning of the first term (also of the second, but it is another question). Googling around I found post referrings to the $dd^c$-Lemma (For example on Math.Overflow there are many), but I didn't find neither a statement nor a proof. Since I think it could be useful to understand (and to use in relation to) the formula above I would like to have a precise reference for it.

Thank you for your time!

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up vote 2 down vote accepted

P. Deligne, P. Griffiths, J. Morgan, D. Sullivan: Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274.

See page 246.

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Thank you very much! – Giovanni De Gaetano Dec 1 '11 at 14:49

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