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 calculate the $\ast$-product between Green functions. I don't understand the geometry of this product, probably because I don't well understand the geometry behind distribution and currents.

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.

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.

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 calculate the $\ast$-product between Green functions. I don't understand the geometry of this product, probably because I don't well understand the geometry behind distribution and currents.