Faltings-Riemann-Roch Theorem

Question

I found the famous Faltings book ``Lectures on arithmetic Riemann-Roch theorem".

In the book, very analytic techniques such as Garding inequality or heat kernel are explained. I have no idea where such analytic tools must come in to prove algebraic theorem.

Because we want to calculate the Euler-Poincare number for the cohomology of vector bundle $E$ that is defined in a purely algebraic manner. So

Question 1: Why do we need to consider such analytic aspects to formulate Riemann-Roch theorem for arithmetic surfaces over Spec $\mathbb{Z}$?

I also found in Lang's book that to formulate Arithmetic Riemann-Roch theorem for arithmetic surface over Spec $\mathbb{Z}$, one might have to take care of Arakelov metric for arithmetic surface over infinite place $R$.

Question 2: What does it mean to ``choose'' a nice metric such as Arakelov-metric for base-changed arithmetic surface over real place? Does it change the structure of the given arithmetic surface? Or is it incomplete to formulate Faltings-Riemann Roch Theorem that we merely consider the algebraic defining equation of arithmetic surface over Spec $\mathbb{Z}$?

Answer 1

I am by no means an expert here, so this is just a "long comment" regarding the question why analytic tools come in. In algebraic geometry intersection theory over complex numbers is a powerful tool to study curves and varieties in general. However, for number theory and arithmetic geometry one needs also other fields, different from the complex numbers. But for other fields, many difficulties arise, and new tools are really necessary - an important example here is Arakelov theory. Arakelov introduced an intersection pairing on arithmetic surfaces. G. Faltings’ arithmetic analogues of Riemann-Roch theorem and adjunction formula from classical intersection theory on surfaces are two outstanding examples for the successes of Arakelov theory. As for archimedian fields, similar things can be done for non-archimedian fields. Harmonic analysis on metrized graphs, for example, is used to study Arakelov-Green functions and related continuous Laplacian operators. Various arithmetic results are obtained after these studies. For example, the proof of "effective Bogomolov conjecture" over function fields of characteristic zero.

In general, to prove an algebraic theorem, often a "new technique" has to be used. There are several examples for this in algebra (and other fields).

Answer 2

I also have to admit I'm a non-expert, and that this is probably more accurately a long comment. I should also add that this just builds off of Dietrich's answer which probably has more direct motivation.

Question 1: Why do we need to consider such analytic aspects to formulate Riemann-Roch theorem for arithmetic surfaces over Spec Z?

In some sense, Arakelov geometry doesn't study $\operatorname{spec}\mathbb{Z}$ directly but studies the 'mythological' proper model of $\mathbb{Q}$, notated as $\widehat{\operatorname{spec}\mathbb{Z}}$ and varieties/schemes over it. The 'valuative criterion for properness' suggests that we need to add points for the missing completions for $\mathbb{Z}$, but this is exactly adding points for the archimedian places. The analytic aspects merely reflect this.

At this point the analogies start stacking up, and the picture is not 100% clear to me. Let me take a shot at the second question then.

Question 2: What does it mean to ``choose'' a nice metric such as Aeakelov-metric for base-changed arithmetic surface over real place?

You should consider a choice of metric on the associated analytic space(or object) as something like fpqc-descent-data for the arithmetic variety. I'm not sure how Faltings does this, if he does at all, but I recently took a course on Arakelov geometry and at least for a short while, to denote an arithmetic surface we'd write $\overline{X}$ for a pair $(X,\mu)$ where $\mu$ is a Kähler form on $X(\mathbb{C})$. My point in bringing up notation, which is typically somewhat boring to discuss, is that one needs to keep in mind that you're studying $\overline{X}$ directly, not necessarily $X$.

[N.B. I say for a short while because we only briefly went over what Arakelov's work, the course was very pointed towards arithmetic intersection theory and basically everything was quasi-projective, giving an obvious choice of a Kähler form]

Does it change the structure of the given arithmetic surface? Or is it incomplete to formulate Faltings-Riemann Roch Theorem that we merely consider the algebraic defining equation of arithmetic surface over Spec Z?

As I said, you should consider it descent-data, and different metrics are something like different compactifications of the same 'affine piece'. We dealt with this issue more with vector bundles, and again we'd write $\overline{E}$ for a pair $(E,h)$ where $E$ was a vector bundle on the scheme and $h$ was a (nice) metric on $E(\mathbb{C})$.

A very concrete example to show how these metrics lead to different structures is via Picard groups. Classically, on $\mathbb{P}^1$ we have denumerably-many non-isomorphic line bundles, but restricted to $\mathbb{A}^1$ they all become isomorphic. Compare this to the Arakelov situation: on $\operatorname{spec}\mathbb{Z}$ every line bundle is trivial but on $\widehat{\operatorname{spec}\mathbb{Z}}$ the line bundles are parametrized by a choice of (conjugation invariant) hermitian metric $h$ on $\mathbb{C}$. Such a metric is determined by the real number $h(1)$, and so we can say $\operatorname{Pic}(\widehat{\operatorname{spec}\mathbb{Z}}) \cong \mathbb{R}$, though we'd normally write this with the hat 'homotopied' leftward: $\widehat{\operatorname{Pic}}(\operatorname{spec}\mathbb{Z})$.

This even looks like the classical situation, where $\operatorname{Pic}(\mathbb{A}^1) \cong 0$ and $\operatorname{Pic}(\mathbb{P^1}) \cong \mathbb{Z}$

The point here is that: yes! It is not enough to formulate the analogue of RR or GRR just considering the algebraic part. We have to consider the algebraic part along with some analytic data to be able to ask the correct question.

Note: There are formalizations which make symbols like $\widehat{\operatorname{spec}\mathbb{Z}}$ mathematically meaningful and well-defined, for example Durov's definitions in his colossal thesis New Approach to Arakelov Geometry (PDF warning!). I'd even go so far as to suggest reading chapter zero of his thesis, it's non-technical and I found it helpful.