TITLE: Why are Green functions involved in intersection theory?
QUESTION [40 upvotes]: 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.

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 $X_\sigma$. 

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.

REPLY [3 votes]: Just as comment to previous answers: Concrete example for Green function for $X=\mathbb P^d_\mathbb Z= \text{Proj}\mathbb Z[x_0,...,x_d]$ to undrestand the meaning of "measure distances"
For an integer $p\geq 0$ let $Z^p(\mathcal X)$ be the group of cycles $Z$ in $\mathcal X$ of codimension
$p$. Any cycle $Z =\sum n_iZ_i$, where $n_i \in \mathbb Z$, is a formal sum of irreducible cycles $Z_i$,
i.e. irreducible closed subschemes of $\mathcal X$. An irreducible cycle $Z \in Z^p(\mathcal X)$ defines a real current $\delta_{Z(\mathbb C)}\in D^{(p,p)}(\mathcal X)$ by integration
along the smooth part of $Z(\mathbb C)$. More explicitly, the current $Z(\mathbb C)$ is defined by
$$\delta_{Z(\mathbb C)}(\omega)=\int_{Z(\mathbb C)_{reg}}\vartheta^*(\omega)$$
where $\vartheta:Z(\mathbb C)_{reg}\to Z(\mathbb C)$ is a desingularization of $Z(\mathbb C)$ along the set of singular points of $Z(\mathbb C)$. Such desingularization exists due to theorem of Hironaka.
Now we define Green current.
Suppose $Z\in Z^p(\mathcal X)$. A Green's current for $Z$ is a current $g_Z \in D^{(p-1,p-1)}(\mathcal X)$ such that
$$dd^cg_Z+\delta_{Z(\mathbb C)}=[\omega_{g_Z}]$$ for a smooth form $\omega_{g_Z}\in A^{(p,p)}(\mathcal X)$.
Note that for any cycle $Z$ there exists a Green's current for $Z$. Consider the      arithmetic variety $X=\mathbb P^d_\mathbb Z= \text{Proj}\mathbb Z[x_0,...,x_d]$ and the cycle
$$Z=\{<a^{(0)},x>=...=<a^{(p-1)},x>=0\}$$ of codimension p,where for $0<i\leq p-1$ and $x=(x_0,..,x_d)$ we set $<a^{(i)},x>=a_0^{(i)}x_0+...+a_d^{(i)}x_d\in \mathbb Z[x_0,...,x_d]$.The Levine form $g_Z$ associated to $Z$ is defined by
$$g_Z(x)=-\log \left(\frac{\sum_{i=0}^{p-1}|<a^{(i)},x>|^2}{\sum_{i=0}^{d}|x_i|^2}\right).\left(\sum_{j=0}^{p-1}\left(dd^c\log\left(\sum_{i=0}^{p-1}|<a^{(i)},x>|^2\right)\right)^j\wedge \omega_{FS}^{p-1-j}\right)$$ where $\omega_{FS}=dd^c\left(|x_0|^2+...+|x_d|^2\right)$ is the Fubini-Study form on $\mathbb P^d_\mathbb C$. The Levine form $g_Z$ associated to $Z$ satises
$dd^cg_Z+\delta_{Z(\mathbb C)}=[\omega_{FS}]$.