Sec. 3] geometric interpretation 21
For all but finitely many ν ∈ Val(K), we have that ν is non-Archimedean and
= 1 for all i and j. This yields
for every number field L and every w ∈ Val(L) lying above such a valuation.
For every other valuation of K, we find a constant Dν such that
The desired inequality
− hnaive(x) ≤
follows immediately from this.
For the inequality the other way round, one may work with ι−1 instead of ι.
3. Geometric interpretation
3.1. Heights are more closely related to modern algebraic geometry than
this might seem from the definitions given in the sections above. The geometric
interpretation of the concept of a height is the starting point of arithmetic inter-
section theory, a fascinating theory, which we will only touch upon here.
3.2. We return to the assumption that the ground field is K = . This is
done mainly in order to ease notation. The theory would work equally well over an
arbitrary number field.
3.3. Definition. An arithmetic variety is an integral scheme that is projec-
tive and flat over Spec .
3.4. Definition. A hermitian line bundle on an arithmetic variety X is a
pair (L , . ) consisting of an invertible sheaf L ∈ Pic(X) and a continuous hermit-
ian metric on the line bundle L associated with L on the complex space X ( ).
All hermitian line bundles on X form an abelian group, which is denoted by
For the group of all smooth (C∞) hermitian line bundles, one writes Pic(X ).
Pic(X ) is usually called the arithmetic Picard group.
3.5. Example. The projective space
over Spec is an arithmetic vari-
ety. On Pn , there is the tautological invertible sheaf O(1). It is generated by the
global sections X0, . . . , Xn ∈
Let us show two explicit hermitian metrics on O(1) making O(1) into a hermitian