Sec. 3] geometric interpretation 21

For all but finitely many ν ∈ Val(K), we have that ν is non-Archimedean and

aij

ν

= 1 for all i and j. This yields

max

i=0,...,N

xi

w

≤ max

i=0,...,N

xi

w

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

max

i=0,...,N

xi

w

≤ Dνw/ν

d

· max

i=0,...,N

xi

w

.

The desired inequality

hnaive

(

ι(x)

)

− hnaive(x) ≤

ν∈Val(K)

log Dν

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

PicC0

(X ).

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

Pn

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 ∈

Γ(Pn

, O(1)).

Let us show two explicit hermitian metrics on O(1) making O(1) into a hermitian

line bundle.