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This turned out to be a highly non-trivial topic. For a detailed analysis, see Jorgenson and Kramer's paper Bounds on canonical Green functions. Their result can be summarized as $$ g_{\textrm{can},X …

Let $E\rightarrow X$ be a line bundle over a compact Riemann surface. Let $\Delta_{E}$ be the Dolbeault Laplacian $\Delta_{\overline{\partial}}$ extended to sections of $E$. Let $g_1, g_2$ be two metr …

Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation $$ (\partial_{ …

Here is a rather low-brow way of tracing through Arakelov's original ideas. Recall that the intersection of two ordinary divisors $D,E$ can be written as $$ (D.E)_{v}=\sum^{r}_{i=1}-\log \lVert (f| …

Here is another low brow way to look at it by tracing Arakelov's ideas in his paper. Let $X$ be a curve over $K$, where $K$ is a number field. Let $\infty$ denoting the archimedean valuations of $K$ …

A very short answer I learned from summer school - because they want to construct local-global correspondence. A Neron function correspond exactly to the local part and Riemann-Roch is a local-global …

I do not know if this is really the end of the story. You may be interested in the following paper by Jay Jorgenson. Basically he extended the work by Ray-Singer by fixing all unknown invariants, bu …