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Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed points to be $$h_{\bar{L}}(Y) = \log \#\bigl(k(Y)\bigr).$$

If $\dim Y > 0$ we define the height recursively. Let $s$ be a nontrivial rational section of $L$ over $Y$. If $$\text{div}_Y(s)=\sum_\alpha n_\alpha Y_\alpha,$$ then $$h_{\bar{L}}(Y) = \sum_\alpha n_\alpha h_{\bar{L}}(Y_\alpha) - \int_{Y(\Bbb{C})}\log\|s\|c_1(\bar{L})^{\dim Y(\Bbb{C})}.$$

We notice that at the infinite fibre the order of vanishing $-\log\|s\|$ occurs just as in the finite places. Continuing the analogy $c_1(\bar{L})^{\dim Y(\Bbb{C})}$ should be analogous to the height of the infinite fibre.

What is the philosophical reason that $c_1(\bar{L})^{\dim Y(\Bbb{C})}$ shows up as the height of the infinite fibre? In particular, when $Y$ is a curve: Does $c_1(\bar{L})$ in some sense describe the degree of closed points lying over infinity?

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I apologize for answering late.

I think the 1D case has been discussed multiple times in the forum already. The high dimensional case you suggested was first defined by Bost. See page 63 in below:

Théorie de l'intersection et théorème de Riemann-Roch arithmétiques Bost, Jean-Benoît Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Talk no. 731, 46 p.

Actually when I was a graduate student I asked Soule personally on this. His answer was terse - the concept of green currents is a direct analog of what Arakelov did in the original paper.

Translated into this context, the height of $Y$ (with respect to $L$) should be "viewed" as the intersection of $Y$ with an arithmetic variety defined over $\textrm{Spec}(\mathbb{Z})$. The $c_{1}(\bar{L})^{?}$ term in your notation then corresponds to the contribution from infinity from this variety. In Arakelov's original notation we have $D*E=\deg(\epsilon^{*}D)$.

It should be pointed out that proving this definition is sound (unique does not depend on $s$ and the term does exist) carefully is non-trivial (we need to use resolution of singularities by Hironaka, for example) and cannot be done in a MO post. A recently proof can be found in

Arakelov Geometry and Diophantine Applications (Theorem 2.13, page 24)

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  • $\begingroup$ I don't think that this answers my question which is more philosophical in nature. For instance it is clear to me that the $c_{1}(\bar{L})^{?}$ term should correspond to the contribution of the fibre at infinity. It seems to me that $X(\Bbb{C})$ together with $c_{1}(\bar{L})$ takes the role of fibre at infinity. My question is why this is the case. I don't think this is answered anywhere even for arithmetic surfaces. An answer would probably also explain why admissible line bundles are those "living" on the model. $\endgroup$
    – Nuno Hultberg
    Commented Jun 9, 2023 at 10:23
  • $\begingroup$ @NunoHultberg: I do not think this is a valid point. The "philosophical" side of the question is not worth discussing as it is quite well known. See the introduction of Chambert-Loir's article "Arakelov geometry, heights, equidistribution" on arxiv. Maybe you meant something different - I am not sure. For the arithmetic surface case, the original paper should explain this pretty well. $\endgroup$
    – Bombyx mori
    Commented Dec 3, 2023 at 16:44

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