Analogies between classical geometry on complex surfaces and Arakelov geometry

Question

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.

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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 $\chi$.

1. 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)?
2. 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.

Answer 1

These are indeed good questions, and while there is a very good corpus of answers to them, the analogy is not perfect.

0. The non-archimedean analogy

First of all, I would like to go back to the relative situation of a surface $\mathcal X\to B$ fibered over a germ of curve $(B,b)$. Then any local function $f$, resp. local section $s$ of a line bundle $\mathcal L$, may vanish along the special fiber with some multiplicity $m$, and this multiplicity is a valuation: the function $f\mapsto m=v(f)$ satisfies $v(f+g)\geq \min(v(f),v(g))$ and $v(fg)=v(f)+v(g)$. The function $f\mapsto \exp(-v(f))$ behaves as a metric, except that it is non-archimedean.

Now forget the surface $\mathcal X$ and just remember about the generic fiber, viewed as a curve over the complete valued field $F_b$, completion of the field of functions of $B$ with respect to the $b$-adic absolute value.

This furnishes a functor from pairs consisting of a surface $\mathcal X$ over $(B,b)$, line bundles on $\mathcal X$, to pairs consisting of a curve $X$ over $F_b$ and a metrized line bundle on $X$. This extends to vector bundles, in fact.

Under this functor, morphisms of vector bundles go to norm non-increasing morphisms of vector bundles.

This functor is “essentially” fully faithful (one needs some assumptions on $\mathcal X$, say it is normal). Up to blowing-up $\mathcal X$ along closed subschemes of the special fiber, it is “essentially” essentially surjective.

1. What is $H^0(\overline{\mathcal L})$ ?

The answer, which mimicks the above analogy, says that it is the subset of $H^0(\mathcal L)$ of sections of norm $\leq 1$ everywhere.

It is a finite set with essentially no algebraic structure.

There are two propositions for its “dimension”. The earliest one sets $h^0(\mathcal L)=\log \# (H^0(\overline{\mathcal L}))$. More recently, van der Geer, Schoof, Rössler, Bost, etc. have suggested to consider rather the theta-invariant: $$ h^0_\theta(\mathcal L)= \log \left(\sum_{v\in H^0(\mathcal L)} \exp(-\| v\|^2) \right). $$

To answer one concern expressed in your statement of question 1. If you add $m_\sigma$ to the real component at $\sigma$, this multiplies the metric at $\sigma$ by $\exp(-m_\sigma)$. If $m_\sigma>0$, there will be more global sections, in good analogy with the fact that if you add an effective divisor to a divisor, the space of global sections increases.

2. Arithmetic degrees

In any case, rather than this specific $H^0(\overline{\mathcal L})$, it is rather convenient to remember the pair consisting of the free $\mathbf Z$-module $H^0(\mathcal L)$ and of its norm induced by the supremum norm of sections. (One can also introduce useful Euclidean norms by integrating local norms squared agains a fixed volume form; there are comparison results.)

This object is the analogue of a vector bundle over a curve, and can be given an arithmetic degree, satisfying good algebraic properties.

Minkowski's theorem is the analogue of the Riemann-(Roch) theorem giving non-zero sections in $H^0(\overline{\mathcal L})$ provided its arithmetic degree is large enough.

There is an analogue of Serre's duality theorem. Actually, there are two analogues, according to your definition of $H^0$. In the naïve one, it is given by an inequality (Gillet-Soulé, Israel J. Math.). In the theta-version, it comes from the Poisson summation formula and is an exact equality.

3. Arithmetic intersection theory

There I touch your second question. When $\mathcal X$ is a projective surface, there is a nice intersection pairing of line bundles. It can be defined geometrically, intersecting divisors. It can also be defined cohomologically, using Euler-Poincaré functions. (See, for example, the first chapter of Debarre's book on Higher dimensional geometry, for a rapid and clear treatment.)

The same can be done in Arakelov geometry. For arithmetic surfaces (the scope of your question), it is due to Arakelov, Faltings and Deligne; in general, this is due to Gillet-Soulé and Bismut-Gillet-Soulé. These authors define an arithmetic intersection of metrized line bundles, more generally of “arithmetic cycles”, and prove an arithmetic analogue of Grothendieck's Riemann-Roch theorem.

The difficulty with the mere statement of a Riemann-Roch theorem comes from the fact that the naive $H^0$, as we saw, are not algebraic objects, and that there is even no a priori definition of the higher cohomology groups. The idea is then to define the determinant of the cohomology, as a metrized line bundle on the base, whose arithmetic degree would be the analogue of the Euler-Poincaré characteristic.

4. Applications

I cannot resisting mentioning a bunch of applications.

a) This theory fits extremly well, and makes precise, Weil's theory of heights. In Weil's theory, height functions are defined up to bounded ambiguities; here, one has actual functions under the hand.

b) Gillet-Soulé proved an analogue of the Hilbert-Samuel theorem, giving an estimation for the cardinality of $H^0(\overline{\mathcal L}^n)$, where $n\to\infty$. This theorem has been extremly useful, for example in the proof of Bogomolov's conjecture.

c) Zhang proved an analogue of the Nakai-Moishezon theorems, that is, criteria on $\overline{\mathcal L}$ implying that large powers will be generated by sections of norm $<1$. (Basically, the height of every closed subvariety has to be $>0$.)

d) The precise formulas coming out of the arithmetic Riemann-Roch theorem furnish beautiful arithmetic formulas relating heights and special values of $L$-functions. (See the work of Maillot-Rössler, for example.)

e) Bost proved analogues of Lefschetz's hyperplane section theorem on Riemann surfaces, implying for example the triviality of the fundamental groups of some arithmetic surfaces.

f) Bost and myself proved generalizations of the Borel-Dwork criterion, viewed as analogues of the Hironaka-Matsumura theory (algebraization of formal schemes).

g) The formalism is also very useful to formulate in a geometric way the proofs of transcendental number theory (Bost's slope method). It has been used, for example, by Bost-David (explicit version of Masser-Wüstholz isogeny estimates), Bost (generalization of Chudnovsky/André's results about Grothendieck's conjecture to foliations), Gasbarri and Herblot (Schneider-Lang type theorems), etc.