Arakelov divisors and the meaning of real coefficients

Question

I'm learning Arakelov theory on arithmetic surfaces and I have the following general question.

Let $K$ be a number field and consider its ring of integers $O_K$. Moreover let $S:=\operatorname{Spec} O_K$ and consider a regular, projective arithmetic surface $\pi: X\to S$. For every embedding $\sigma:K\to\mathbb C$ we have the fiber at infinity $X_\sigma$ which is a Riemann surface. If we fix a Kalher metric $\Omega_\sigma$ on each $X_\sigma$, then an Arakelov divisor can be written uniquely as: $$\hat D:=D+\sum_{\sigma}\alpha_\sigma X_\sigma$$

where $D$ is a usual divisor of $X$ and $\alpha_\sigma\in\mathbb R$.

I don't understand why the coefficients for the fibers at infinity are in $\mathbb R$ and not in $\mathbb Z$, why do we need this? Foe example consider the real vector $\varepsilon=(\varepsilon_\sigma)_\sigma$ where for every $\sigma$ the real number $\varepsilon_\sigma$ is small enough and construct the Arakelov divisor $$\hat D_\varepsilon:=D+\sum_{\sigma}(\alpha_\sigma+\varepsilon_\sigma) X_\sigma$$ (note that $D$ is fixed)

What are the geometrical differences between $\hat D$ and $\hat D_\varepsilon$?

Answer 1

Changing the coefficient of a divisor by some $\epsilon > 0$ can have a very significant effect, for example it can move you out of the nef cone. Indeed, even in non-Arakelov algebraic geometry it is often very useful to allow real coefficients on divisors. As a specific example, let $A$ be a simple abelian variety with CM by a real quadratic field. Then $\text{NS}(A)\otimes\mathbb Q$ has rank 2, say generated by $D_1$ and $D_2$, but the boundary of the nef cone in $\text{NS}(A)\otimes\mathbb R$ is spanned by divisors of the form $aD_1+bD_2$ with $a/b\notin\mathbb Q$. Another reason one might use real coefficients is that the image of $\text{NS}(X)$ in $\text{NS}(X)\otimes\mathbb R$ is a lattice in a finite dimensional vector space, so now one can use geometry to study $\text{NS}(X)$.