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 $\hat D_\varepsilon$?

  • 1
    $\begingroup$ You can define formal linear combinations with integer coefficients if you want, but already at very early steps in the theory you will find difficulties. For instance, check the definition of principal Arakelov divisors. The coefficients at infinity are not integers (in general). $\endgroup$
    – Pasten
    Jul 14 '16 at 19:25
  • $\begingroup$ Arakelov's arithmetic intersection pairing on arithmetic surfaces has been extended by Deligne and Gillet-Soulé. In these more general approaches, an arithmetic divisor is a pair $(D,g_D)$, where $g_D$ is a Green current at infinity. The choice of a Kähler form allows to normalize the Green current up to scalars (at each place). — Although this does not answer the question, it may help explain the appearance of real coefficients. $\endgroup$
    – ACL
    Jul 15 '16 at 0:23
  • $\begingroup$ You should take a look at Durov's work. Your intuition that $\mathbb{R}$ is "too many" coefficients is correct. I think Durov just uses $Log(\mathbb{Q}^+)$. $\endgroup$ Apr 13 '17 at 12:56

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 $\text{NS}(X)$.


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