I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture.




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**Summary:**


Let $X$ be an arithmetic surface over $\operatorname{Spec } O\_K$ where $K$ is a number field (we put on $X$ the good properties: regular, projective,...). Moreover suppose that $\{X\_\sigma\}$ are the "archimedean fibers" of $X$, where each $\sigma$ is a field embedding $\sigma:K\to\mathbb C$ (up to conjugation). Each $X\_\sigma$ is a smooth Riemann surface. We accomplished to "compactify" our arithmetic surface $X$, so now we want a reasonable intersection theory.


An Arakelov divisor $\widehat D$ is a formal sum 
$$\widehat D=D+\sum\_\sigma g\_\sigma\sigma$$
where $D$ is a usual divisor of $X$ and $g\_\sigma$ is a Green function on $X\_\sigma$. 




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**Question:**


Why on the archimedian fibers $X\_\sigma$ do we need Green functions? A Green function on a Riemann surface is simply a function $g:X\_\sigma\to\mathbb R$ which is $C^\infty$ on all but finitely points and at the "non-smooth" points is locally represented by
$$a\log|z|^2+\text{smooth function}$$
in other words a Green function is a decent real valued smooth function which "explodes at infinity" in a finite set of points. Simply I don't understand why in order to define a reasonable intersection theory we need an object with such properties. The key point should be that Green functions satisfy a Poincare-Lelong formula which can be expressed in terms of currents as:
$$dd^c[g]=[\operatorname{div}^G(g)]+[dd^c(g)]$$
but still I don't get the meaning of this formula.


Somewhere I've read something like:



> 
> Green function are useful in order to "measure distances" on a Riemann
>  surface. (This is not a quote, but simply what I remember).
> 
> 
> 


I really don't have any idea on the meaning of this sentence.


**Addendum:**


Moreover if one wants to calculate the intersection between two Arakelov divisors, at the archimedian places one has to take the $\ast$-product between Green functions. I don't understand the geometry of this product, probably because I don't understand well the geometry behind distribution and currents on Riemann surfaces.



Why are Green functions involved in intersection theory?
The Green's function is used not to measure distances in the surface but to measure distances in the line bundle. A Green's function on $X\_{\mathbb C}$ that blows up at $D$ can be used to measure sections of $\mathcal O(D)$. Indeed, if we represent a section of $\mathcal O(D)$ as a holomorphic function $f$ on $X\_{\mathbb C}$ with poles at the points of $D$, and $g$ is a Green's function on $X\_{\mathbb C}$ with poles at $D$, then $|f(x)|^2 e^{- |g(x)|}$ is a smooth nonnegative function that vanishes only where $f$ vanishes as a section of $\mathcal O(D)$ - i.e. where it lies in the image of the natural map $\mathcal O(D-P) \to \mathcal O(D)$. In fact it is easy to see that this is a Hermitian form on the space of sections. So the Green's function gives a metric on the line bundle.


We need a metric at $\infty$ in an Arakelov line bundle because we already have a metric (in the form of a $p$-adic valuation) at every other place - any section of $\mathcal O(D)$ on $X\_{\mathbb Q\_p}$ has a well-defined $p$-adic valuation at each point of $x\in X\_{\mathbb Q\_p}$, where a local generator of the line bundle at the reduction mod $p$ $\overline{x}\in X\_{\mathbb F\_p}$ is given absolute value $0$ - because such a generator is also a generator of the fiber of $\mathcal O(D)$ at $x$, this determines the valuation on every section.


