This is my first question on this wonderful site. The following question about Arakelov geometry is gonna be quite long and wide; to be clear one of that kind of questions that are usually ignored. The point is that I haven't found any help in the literature and I feel lost.

Let's fix an **arithmetic surface** $p:X\to \operatorname{Spec }O_K$ ($K$ is a number field). The usual concept of invertible sheaf/line bundle on a complex surface, is substituted by the notion of metrized line bundle. In particular a metrized line bundle $\overline{\mathcal L}$ is a couple $(\mathcal L, \{h_\sigma\})$ where $\{h_\sigma\}$ is a finite collection of "admissible" metrics on the pullback holomorphic line bundles $\mathcal L_\sigma$ on the Riemann surfaces at infinity $X_\sigma$.

This is very cool since we have a reasonable intersection theory of metrized line bundles which employs the Deligne pairing, in paricular to each couple of metrized line bundles $\overline{\mathcal L}$, $\overline{\mathcal M}$ we associate a metrized line bundle $\overline E=\left<\overline{\mathcal L},\overline{\mathcal M}\right>$ on $\operatorname{Spec } O_K$. Again $\overline E$ is a couple made of a projective $O_K$-module of rank $1$ $E$ and a collection of hermitian inner products on $\mathbb C$, one for each embedding of $O_K$ in $\mathbb C$ (up to conjugation). The intersection pairing can be clearly interpreted as pairing on the group of Arakelov divisors which like in the geometric case correspond exactly to metrized line bundles.

**Now let's go to the questions**: In the usual theory of complex surfaces we have crucial "objects" like the complex vector space $H^0(\mathcal L)$ and the Euler-Poincare chacteristic $\chi$.

- Let $\overline {\mathcal L}$ be a metrized line bundle on $X$. What is a reasonable notion of $H^0(\overline{\mathcal L})$? It seems that a good choice should be the subspace of $H^0(\mathcal L)$ made of the global sections $s$ such that $||s_\sigma||\le 1$ for any $\sigma$. These are also called small sections. This definition seems very odd to me, indeed if we interpret this in term of Arakelov divisors and try to create $H^0(\overline D)$ for an Arakelov divisor $\overline D$ it seems that we don't care about what happens at the Archimedean places of $\overline D$ because we only consider effective coefficients at infinity. To be precise: if I modify $\overline D$ only in the archimedian part without changing the sign of the coefficients, I'll get the same set $H^0(\overline D)$. This quite useless! Where is my mistake? What is a better definition of $H^0(\overline{\mathcal L})$ (if there is one)?
- An arithmetic version of $\chi$ can be introduced by means of the determinant of the cohomology. The formula says: $$\chi(\overline{\mathcal L})=\operatorname{deg }\left(\operatorname{det}Rp_\ast{\mathcal L}\right)\,.$$ $ \operatorname{det}R{p_\ast}{\mathcal L}$ is a honest a metrized line bundle on $\operatorname{Spec }O_K$ and a theorem due to Arakelov says that it can be endowed with a canonical metric. At this point we take the (arithmetic) degree and we get our arithmetic $\chi$. Very well, but why do we have such a complicate definition? The usual $\chi$ is a cohomological object but in the arithmetic case we don't have any suitable cohomology of metrized line bundles. Where is the analogy with the geometry? It seems that the only reason is to get a form of the Riemann-Roch theorem which at least formally resembles the classical one.

Thank you for you attention.