Let $X$ be a compact connected Riemann surface of genus $g>0$.

We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ for the space of holomorphic differentials $H^0(X,\Omega^1_X)$ with respect to the natural inner product $(\omega,\eta) \mapsto \frac{i}{2} \int_X \omega \wedge \overline{\eta}.$ Then $\mu = \frac{i}{2g} \sum_{k=1}^g \omega_k \wedge \overline{\omega_k}$. This is independent of the choice of orthonormal basis. Note that $\int_X \mu = 1$

Now, the following fact seems to be well-known.

Let $x\in X$ and let $z:U\longrightarrow B(0,1)$ be a chart at $x$, where $B(0,1)$ denotes the open unit disc in $\mathbf{C}$.

Write $\mu = i F dz d\overline{z}$ on $U\subset X$. Then, after shrinking $U$ if necessary, $F$ extends to a subharmonic function on the compactification/closure of $U$.

Does anybody know a nice reference for this?

This question is the higher genus analogue of my previous question Is it easy to see that this function is subharmonic? In that question it was made clear that the Fubini-Study metric has this property around $\infty$. The open $U$ being the complement of the open disc $B(0,1/2)$.

  • $\begingroup$ Farkas and Kra? $\endgroup$ – Braindead Mar 23 '13 at 18:33

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