Holmstrom 3 Memo notes Arakelov lectures

Incorporate into CT pages.

0: Introduction

1. Infinite descent

A combination of congruence and height arguments. Want a geometry incorporating both (static version of inf descent). This might prove finiteness theorems.

Consider a scheme $X$ over $\mathbb{Z}$. To control height, must look at $X(\mathbb{C})$ (assumed to be smooth) from the point of view of hermitian complex geometry, i.e. we endow holomorphic vector bundles on $X(\mathbb{C})$ with smooth hermitian metrics.

Arakelov geometry combines schemes and hermitian complex geometry. It proved the Mordell conjecture.

No moving lemma for cycles is known over $\mathbb{Z}$. A more dynamic approach than the above would be an adelic variant of Arakelov geometry. This would study a smooth variety over the rationals, and vector bundles on $V$ equipped with metrics at archimedean places, and p-adic analogues of these at finite places. This is yet to be built.

1. The analogy with function fields

Residue formula vs product formula.

Def: An arithmetic variety is a regular scheme, projective and flat over $\mathbb{Z}$. In other words, we consider a system of homogeneous polynomials with integer coeffs. Let $P$ be a point of this projective scheme, i.e. a prime ideal in $S$. Then $f: X \to Spec(\mathbb{Z})$ maps $P$ to $P \cap \mathbb{Z}$. The fiber of $f$ over a finite prime (special fiber) is the variety $f^{-1}(p \mathbb{Z}) = X/p = Proj(S/pS)$ over the finite field $\mathbb{F}_p$. The generic fiber is $f^{-1}((0)) = X_{\mathbb{Q}} = Proj(S \otimes_{\mathbb{Z}} \mathbb{Q})$. We assume that $X$ is regular and that $f$ is flat, i.e. $S$ is torsion free. It follows that $X/p$ is smooth, except for finitely many $p$. At those “bad” primes, it may not even be reduced. We complete the picture by adding the complex points of $X$, and think of them as the “fiber at infinity”. If the fibers have dimension one, then we call $X$ an arithmetic surface, and $X_{\infty}$ will be a Riemann surface. Note that an integral solution of our system of polynomials is a rational point of $X(\mathbb{Z}) = X(\mathbb{Q}) \subset X(\mathbb{C})$, i.e. a section of $f$.

For height arguments, need to study algebraic vb’s $E$ on $X$, endowed with a smooth hermitian metric $h$ on the corresponding holomorphic vector bundle $E_{\infty}$ on $X_{\infty}$. We assume that $h$ is invariant under complex conjugation on $X_{\infty}$. A hermitian vb is a pair $\bar{E} = (E,h)$.

1. The contents of the book

Let $X$, $\bar{E}$ be as above. We shall attach to $\bar{E}$ characteristic classes with values in arithmetic Chow groups. An arithmetic cycle is a pair, consisting of an algebraic cycle on $X$ and a Green current for this cycle. The arithmetic Chow group $\hat{CH}^p(X)$ is the group of such pairs modulo the subgroup generated by pairs $(0, \partial u + \bar{\partial} v)$ and $(div(f), - \log |f|^2)$, where $u$ and $v$ are arbitary currents of the appropriate degree, and $f$ is a nonzero rational function on some irreducible closed subscheme of codimension $p-1$ in $X$.

In chapter 3 we study these groups and show that they have functoriality properties and a graded product structure, at least after tensoring with the rationals. Some difficulties in proving these things: (i) No moving lemma, so must use algebraic K-theory and Adams operations, (ii) Given two arithmetic cycles, need a Green current for their intersection; need to show certain things about log type forms.

Chapter IV: Characteristic classes, for example the Chern character class. This satisfy the usual axiomatic properties for a Chern character. But it does depend on the metric on $E$. Furthermore, it is not additive for arbitrary exact sequences, but it is for orthogonal direct sums. The failure to be additive is given by a secondary characteristic class first introduced by Bott and Chern. Similar results for Chern classes and the Todd class.

Next construction: Direct image map for hermitian vb’s. Let $f:X \to Y$ be a proper flat map between arithmetic varieties, smooth on the generic fiber. Then there is a canonical line bundle $\lambda(E)$ on $Y$ whose fiber at every point $y$ is the determinant of the cohomology of $X_y = f^{-1}(y)$ with coefficients in $E$. We can get a metric on $\lambda(E)$ with ingredients the $L^2$ metric and the Ray-singer analytic torsion, which comes from the zeta function of a Laplacian (details omitted). Can compute some curvature through the Riemann-Roch-Grothendieck formula.

Combining the above with the usual RRG theorem, get a RRG thm for arithmetic Chow groups. Given a proper map $f:X \to Y$ between arithmetic var’s, smooth on the generic fiber, and a hermitian vb $\bar{E}$ on $X$, the thm says that

depends only on the class of $E$ in $K_0(X_{\mathbb{Q}})$.

Application: An existence thm of small sections for powers of ample line bundles. Some of this was input to proof of Mordell, by Vojta. Ref to Faltings for relation to rational pts on abelian var’s.

1: Intersection theory on regular schemes

Will define intersection theory for an arbitrary regular noetherian finite-dimensional scheme. When $X$ is of finite type over a field, can use a Moving lemma for this. No moving lemma is known over a general base, unfortunately. When $X$ is smooth over a Dedekind ring, can use Fulton’s method of the normal cone. But in general, no geometric method is available.

We shall use an isomorphism with a piece of K-theory. All details omitted for now, see Gillet in K-th handbook.

Nice introduction to algebraic K-theory and Chow groups with support. Quite a lot on K-theory, omitted here.

Gersten conjectured that for any regular noetherian finite-dimensional scheme $X$ and for all $p \geq 0$, the group $CH^p(X)$ is isomorphic to the Zariski cohomology group $H^p(X, \mathcal{K}_p)$. In that case, one can define a (bilinear) intersection pairing as the cup product between Zariski cohomology groups, induced by the products on higher K-groups. This definition is valid when $X$ is of finite type over some field.

2. Green currents

Great intro to currents and Green currents. Go through this on paper.

III: Arithmetic Chow groups

Def of Arithmetic Chow groups. Some exact sequences. Intersection pairing. Arithmetic Chow groups are contravariant in $X$ and covariant for maps which are smooth on the generic fiber. Some examples, then assume from now on that $X(\mathbb{C})$ is endowed with a Kahler metric. We can then define a subgroup of our original group(s) by imposiung harmonicity conditions. For $p=1$ and an aritmhetic surface, we recover Arakelov’s original definition. Finally some results on heights of projective var’s.

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